## March 22, 2009

### Unitary Representations of the Poincaré Group

#### Posted by John Baez

John Huerta’s paper on grand unified theories made some people eager to discuss aspects of group representation theory that this paper deliberately avoided.

For example, in relativistic quantum mechanics, we classify particles as unitary irreducible representations (or ‘irreps’) of a group like this:

$G \times P$

Here $G$ is a compact Lie group depending on the theory of physics we happen to be studying, called the ‘internal symmetry group’. For example, the Standard Model has $G =$ U(1) $\times$ SU(2) $\times$ SU(3).

$P$, on the other hand, is the Poincaré group! This is, roughly speaking, the group of symmetries of Minkowski spacetime. So, this is the same regardless of our theory, unless we posit extra dimensions or something funky like that.

A unitary irrep of $G \times P$ is always built by tensoring a unitary irrep of $G$ with one of $P$. So, the project of classifying particles splits into two parts: one depending on $G$, one depending on $P$.

In this thread let’s talk about the second part! If we do, we’ll learn why physicists classify particles according to their mass and spin (or more precisely, helicity). So, when we hear them mutter something about a ‘massless left-handed spin-1/2 particle’, we’ll know which representation of the Poincaré group they’re talking about.

Just to kick off the discussion, let me say exactly what group I’m calling the Poincaré group! It’s really

$P = SL(2,\mathbb{C}) \ltimes \mathbb{R}^4$

If you’re feeling pedantic you might call this ‘the universal cover of the connected component of the isometry group of Minkowski spacetime’ — but let’s just call it the Poincaré group, since this is the group that matters in the classification of particles!

In particular, the isometry group of Minkowski spacetime is the semidirect product

$O(3,1) \ltimes \mathbb{R}^4$

where $\mathbb{R}^4$ is our friend the translation group and $O(3,1)$ is the Lorentz group, the group of all linear transformations of $\mathbb{R}^4$ that preserve the Minkowski metric.

$O(3,1)$ has four connected components: the component containing the identity, the component containing time reversal ($T$), the component containing parity (also called $P$), and the component containing $P T$. The identity component is usually called $SO_0(3,1)$. This group is not simply connected — but of course its universal cover is, and this universal cover is isomorphic to $SL(2,\mathbb{C})$: the group of $2 \times 2$ complex matrices with determinant 1.

Since $SO_0(3,1)$ acts on $\mathbb{R}^4$, so does its double cover $SL(2,\mathbb{C})$. So, we can form the semidirect product

$P = SL(2,\mathbb{C}) \ltimes \mathbb{R}^4$

and this is the group that governs our universe when we combine special relativity and quantum mechanics.

Okay, so that’s what I mean by the Poincaré group.

Next: in a famous paper, Wigner made a good stab at classifying all the unitary representations of this group! You can see his paper here:

• E. P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals of Mathematics 40 (1939), 149–204.

I think he completely classified the ‘positive energy’ representations, which are the ones of greatest importance in physics. This leaves out exotic, unobserved representations like negative-energy particles, tachyons and other freaks too scary to mention here. But to be honest, I’m not sure exactly how far he got. What really matters to me are the positive energy reps.

I hope that we can use this blog entry as a forum to find out what Wigner proved, and understand its implications for particle physics. If we all team up, we’ll all learn a lot of stuff.

If you don’t know how to start, try the short, dull Wikipedia articles about representation theory of the Poincaré group and Wigner’s classification theorem, and ask a question about them. We could make them a lot better, that’s for sure!

Posted at March 22, 2009 11:08 PM UTC

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### Re: Unitary Representations of the Poincaré Group

Heh – thanks, John! (I was actually pestering John with some email questions about this today. But he thought it would be better to have this out in the open.)

Personally, what I want is a “nice” (meaning written in clean modern mathematical language and notation) statement and proof of what Wigner did. I’ve seen it sketched in a physics book or two, and it could be made a lot nicer I think.

Just to try to get the ball rolling, let me comment on John’s slightly cryptic mention of $SL_2(\mathbb{C})$, and how it’s supposed to act on Minkowski spacetime $\mathbb{R}^4$.

The trick is to identify $\mathbb{R}^4$ with the space of Hermitian matrices. Recall that a complex matrix $X$,

$\left( \array{ a & b \\ c & d } \right)$

is Hermitian if its equals its conjugate transpose $X^*$: concretely, if $a = \bar{a}$, $b = \bar{c}$, $d = \bar{d}$. Apparently, then, every Hermitian matrix can be written in the form

$\left( \array{ t - z & x + i y \\ x - i y & t + z } \right)$

for a unique choice of real $t, x, y, z$. Notice the nice fact that the determinant of this matrix is precisely

$t^2 - x^2 - y^2 - z^2,$

the Minkowski norm of the spacetime vector $(t, x, y, z)$.

Now, let $SL_2(\mathbb{C})$ act on Hermitian matrices $X$ by

$SL_2(\mathbb{C}) \times Herm \to Herm$ $(P, X) \mapsto P X P^*$

Since matrices $P$ in $SL_2(\mathbb{C})$ have determinant 1, it is clear that

$det(X) = det(P X P^*)$

which, under the identification $Herm \cong \mathbb{R}^4$, just means that the induced action of $SL_2(\mathbb{C})$ on $\mathbb{R}^4$ preserves the Minkowski norm, or the metric on spacetime used in relativity theory.

It’s not hard to see that the topological group $SL_2(\mathbb{C})$ is connected. Thus, if $SO_0(3, 1)$ is the connected component of the Lorentz group (the group of automorphisms of $\mathbb{R}^4$ which preserve the Minkowski metric), we have produced a map

$SL_2(\mathbb{C}) \to SO_0(3, 1)$

It turns out this map is onto, and that the kernel is $\{I, -I\}$. This quotient map is in fact the universal cover; it restricts to another famous universal cover:

$SU(2) \to SO(3)$

There are a number of details which ought to be written up, but this gives the basic idea. In studying continuous group representations, it’s usually convenient to pass to the universal cover (which I’m sure is why John introduced it). All of this is to be related furthermore to spinors, and to the distinction between bosons and fermions.

Posted by: Todd Trimble on March 23, 2009 1:37 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I think S. Sternberg’s Group Theory and Physics contains a proof. It also contains a nice discussion of the various covers of full disconnected Lorentz group as I recall.

Posted by: Aaron Bergman on March 23, 2009 1:48 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Thank you Aaron – I’ll see if I can get hold of Sternberg’s book.

What I’m hoping is to see a complete and mathematically precise proof that fits in just a few pages. I’m happy to assume some basic functional analysis as background (e.g., spectral theory), even though my own background may stop a bit short of what is actually needed. For example, it seems from what I’ve read that momentum operators, which are unbounded and densely defined, will inevitably come into play, but I haven’t carefully studied spectral theory for unbounded self-adjoint operators.

What I don’t want is what physicists call a mathematical proof. I should immediately temper this by saying that I am in awe of what physicists have accomplished, but my experiences in trying to read some of the best tell me to take what they call “mathematics” with a big rock of salt. (I’d be willing to listen to heuristic explanations, of course, but in the end I want an honest proof!)

Posted by: Todd Trimble on March 23, 2009 2:03 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I should also say that it’s not just convenient to pass to the universal cover; from the point of view of physics, it is necessary. The reason why is that the relevant things for physics are not the representations of the group but rather the projective representations. The theorem which I can never remember the precise statement of says that a projective rep lifts to an honest rep of the universal cover. However, with the disconnected parts, you can pick up some phases leading to the various covers I referred to in the previous comment.

Posted by: Aaron Bergman on March 23, 2009 1:51 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Thanks for giving $SL(2,\mathbb{C})$ a proper introduction, Todd! Let me add a few comments.

If we take the Poincaré group and mod out by the ‘translation subgroup’ $\mathbb{R}^4$, we get $SL(2,\mathbb{C})$ as a quotient group. So, every unitary irrep of $SL(2,\mathbb{C})$ gives a unitary irrep of the Poincaré group for which translations act trivially.

This means that a full classification of unitary irreps of the Poincaré group must include a classification of unitary irreps of $SL(2,\mathbb{C})$. That’s a fascinating subject in itself. But, these ‘translation-trivial’ irreps of the Poincaré are not generally considered physical interesting, because they describe ‘particles’ that don’t change at all when you move them! They’re really not like particles at all, more like diffuse omnipresent entities. They could be important in physics, but nobody seems to believe that yet.

These are among the scary freaks I alluded to. We can rule them out by demanding that we want ‘positive-energy’ unitary irreps of the Poincaré group — i.e., those for which time translation acts as

$exp(-i H t)$

where $H$ is a nonnegative, nonzero selfadjoint operator.

(In physics, $H$ is called the ‘Hamiltonian’ or ‘energy’.)

Interestingly, many physically interesting irreps of the Poincaré group are closely related to finite-dimensional, nonunitary irreducible representations of $SL(2,\mathbb{C})$, using Mackey’s technology of ‘induced representations’. I suspect that any modern clean proof of Wigner’s theorem will get into this technology.

Posted by: John Baez on March 23, 2009 2:05 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Two things: here’s a link to Wigner’s orginal paper that I found on a Wikipedia page. It is old fashioned, but still a good read.
And it seems a little harsh to call it Mackey’s technology, since as far as I know, Wigner’s paper was the first to use induced representations for Lie groups. Section 6 is called ‘Reduction of the representations of the inhomogeneous Lorentz group to representations of a “little group”’.
(I realise that later on Mackey did a lot of work and made it a respectable method)

Posted by: Simon on March 23, 2009 3:37 AM | Permalink | Reply to this

### Representations of a little group

When I was making my way back into physics thanks to Henk van Dam and especially Tom Kephart, the language problem was a challenge. When Tom referred to the little group’, it took me a while to realize it meant an isotropy subgroup. Even more challenging, the arrow went from the big group to the little group! because it was really denoting the restriction of irreps.
Hopefully more of us on both sides of the linguistic divide are becoming bilingual.

Posted by: jim stasheff on March 23, 2009 1:24 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

So for these positive-energy irreps,

We see thus that there corresponds to every invariant quantum mechanical system of equations such a representation of the inhomogeneous Lorentz group. This representation, on the other hand, though not sufficient to replace the quantum mechanical equations entirely, can replace them to a large extent… the representation can replace the equation of motion, it cannot replace, however, connections holding between operators at one instant of time.

I went on to ask, last time I quoted this, whether there will be a similar story to tell about similar representations of 2-groups.

Posted by: David Corfield on March 23, 2009 10:11 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Speaking of how $SL(2,\mathbb{C})$ is a double cover of $SO_0(3,1)$, one can see this very vividly by pondering how Möbius transformations

$z \mapsto \frac{a z + b}{c z + d}$

give conformal transformations of the heavenly sphere. Each point in the heavenly sphere (or mathematically, the Riemann sphere) is a light ray going through your eye, so Lorentz transformations act on this sphere. But, the cool part is that they are precisely the Möbius transformations.

And the really beautiful thing is that this fact has analogues where we replace $\mathbb{C}$ by any normed division algebra! The groups $SL(2,\mathbb{R})$, $SL(2,\mathbb{C})$, $SL(2,\mathbb{H})$ and $SL(2,\mathbb{O})$ are the double covers of $SO_0(2,1)$, $SO_0(3,1)$, $SO_0(5,1)$, and $SO_0(9,1)$, respectively. I explain this here.

Posted by: John Baez on March 23, 2009 2:50 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

You can also take the split real forms of $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ and construct the double covers of SO(2,2), SO(3,3) and SO(5,5). This is not relevant to physics.

Posted by: Bruce Westbury on March 23, 2009 7:45 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Unless you’re a dirty two-timing rat, $SO(2,2)$ is not relevant to physics. Still, it’s interesting.

Posted by: John Baez on March 23, 2009 9:26 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Unless you’re a dirty two-timing rat, SO(2,2) is not relevant to physics.

Given the revival in the last five years or so of twistor techniques and other ways of analytically continuing momenta to compute scattering amplitudes, a lot of physicists are now two-timing rats.

Posted by: Matt Reece on March 23, 2009 9:30 PM | Permalink | Reply to this

### Triangular fish symbol?

What is that triangular alpha or triangular fish symbol? :)

Posted by: Eric on March 23, 2009 2:07 AM | Permalink | Reply to this

### Re: Triangular fish symbol?

Nevermind. I’m half asleep. Don’t mind me.

Posted by: Eric on March 23, 2009 2:09 AM | Permalink | Reply to this

### Semidirect Product

In case anyone else doesn’t recognize that symbol, it denotes a semidirect product of two groups where one acts as automorphisms of the other. The little triangle points to the guy being acted on. The guy being acted on winds up becoming a normal subgroup of the semidirect product.

In our example, the Lorentz transformations are acting on $\mathbb{R}^4$.

Posted by: John Baez on March 23, 2009 2:24 AM | Permalink | Reply to this

### Re: Semidirect Product

Thanks. I panicked when I saw the unfamiliar symbol and asked before seeing that right before you wrote the symbol you said it was the semi-direct product :B

Your additional explanation helped though. I don’t remember seeing semi-direct products back when I learned this stuff the first time.

By the way, the splitting into an internal symmetry group and the Poincare group makes me think of Kaluza-Klein theories. Is there a neat way to formulate things in one big group where the distinction between spacetime and the internal gauge group disappears?

Posted by: Eric on March 23, 2009 8:46 AM | Permalink | Reply to this

### Kaluza-Klein mechanism

By the way, the splitting into an internal symmetry group and the Poincare group makes me think of Kaluza-Klein theories. Is there a neat way to formulate things in one big group where the distinction between spacetime and the internal gauge group disappears?

Yes, indeed, as you say yourself, this is the idea of Kaluza-Klein theory: the internal group is really the diffeomorphisms of the second factor $Y$ of a higher dimensional pseudo-Riemannian spacetime which locally looks like $M^4 \times Y$ with $Y$ a compact Riemannian manifold of very small Riemannian volume.

The basic idea is simple and works well, except for one little subtlety: in the full context of general relativity (as opposed to the special relativistic setup considered in the discussion here so far) the volume of the “internal space” $Y$ is itself dynamical, and hence apart from the desired Poincaré and internal group, the Kaluza-Klein mechanism always yields one more phenomenon: so-called “scalar moduli” which parametrize these scale degrees of freedom of the internal manifold.

As these moduli fields are not being observed, taken at face value this rules out KK-theories. On the other hand, since apart from the the KK-mechanism is so nice, one may start thinking about extending the idea slightly so as to take care of this problem. There are plenty of ideas on this and it becomes a long story. I’ll just mention this aspect:

In the string-theoretic context, after many years of being an open problem, the moduli were finally shown to be “fixed” i.e. achieve a constant dynamics which would, generally speaking, be consistent with these fields not being observed, by assuming that certain higher degree differential form fields (higher gauge theory fields) called “fluxes” in this context, are in a classical condensate state. So higher gauge theory comes to the rescue of the old KK-mechanism!

See also maybe the table at the very end of the entry Connes on spectral geometry of the standard model which lists which aspects of particle physics are encoded by which aspects of a Kaluza-Klein geometry.

Posted by: Urs Schreiber on March 23, 2009 12:50 PM | Permalink | Reply to this

### Diagonalization (was Re: Semidirect Product)

John’s explanation of semi-direct product back here is almost a good jumping-off point for asking about one part of the Wigner classification (as described by physicists) that has me puzzled.

So, we have an exact sequence of topological groups

$0 \to \mathbb{R}^4 \to SL_2(\mathbb{C}) |\times \mathbb{R}^4 \to SL_2(\mathbb{C}) \to 1$

[hey, how do you make that semidirect fish symbol?], where the thing in the middle is the Poincaré group, in which $\mathbb{R}^4$ appears as a normal abelian subgroup.

A unitary (Hilbert space) representation $H$ of the Poincaré group thus restricts to a unitary representation of $\mathbb{R}^4$, taking $a \in \mathbb{R}^4$ to a unitary operator $U_a: H \to H$. Since all of these $U_a$ commute, we should (at least heuristically) be able to “simultaneously diagonalize” them. Following the physicist notation, this would mean we have orthonormal basis elements which they write as $|p\rangle$, so that we have

$U_a|p\rangle = e^{-i p \cdot a}|p\rangle$

where $p \cdot a$ denotes the dot product of a (momentum) 4-vector $p$ with the position 4-vector $a$.

What I want to know is: how rigorous can this be made, or what else do we have to say to make just this bit rigorous? My problem is this: that it looks like these “state vectors” $|p\rangle$ are interpreted physically as states of definite momentum $p$. But such critters do not belong to the Hilbert space $H$ (I mean they’re not $L^2$ functions in any honest way AFAIK) – it’s just some loose physicist way of talking, as best I can make out right now.

So how would a mathematician say it precisely, to make everything right?

Posted by: Todd Trimble on March 23, 2009 3:26 PM | Permalink | Reply to this

### Re: Diagonalization (was Re: Semidirect Product)

[hey, how do you make that semidirect fish symbol?]

\ltimes gives $\ltimes$, \rtimes gives $\rtimes$.

(Thanks, internet!)

Posted by: Tim Silverman on March 23, 2009 6:07 PM | Permalink | Reply to this

### Re: Diagonalization (was Re: Semidirect Product)

Thanks, Tim!

Posted by: Todd Trimble on March 23, 2009 6:11 PM | Permalink | Reply to this

### Re: Diagonalization (was Re: Semidirect Product)

So how would a mathematician say it precisely, to make everything right?

I believe the way this is usually interpreted is with a rigged Hilbert space. The basic idea, as I understand it, is as follows. Given an unbounded normal operator $T$ on a separable Hilbert space $H$, it is possible to find a topological vector space $\Phi$ and a continuous embedding of $\Phi$ into $H$ such that the image of $\Phi$ is dense and contained in the domain of $T$, and such that the dual space $\Phi'$ contains a complete set of eigenvectors for $T$. In other words, there is a measure $\mu$ on the spectrum $\sigma(T)$ such that, for almost every $\lambda \in \sigma(T)$, there exists $\phi_{\lambda} \in \Phi'$ such that $\phi_{\lambda}(T x) = \lambda\phi_{\lambda}(x)$ for any $x \in \Phi$, and furthermore,

(1)$\Vert x\Vert^2 = \int_{\sigma(T)} \vert \phi_\lambda(x)\vert^2 d\mu(\lambda).$

I believe this statement is more or less equivalent to the usual spectral theorem for unbounded normal operators.

As a simple example, we can recover the Fourier transform from the spectral theory of $\frac{d}{d x}$. We embed the Schwarz space $\mathcal{S}$ into $L^2(\mathbb{R})$; its dual is the space $\mathcal{S}'$ of tempered distributions. The eigenvectors of $\frac{d}{d x}$ lying in $\mathcal{S}'$ are the functions $e^{i k x}$ for $k \in \mathbb{R}$. (Note that none of these lie in $L^2(\mathbb{R})$ itself.) If we take $\mu$ to be a properly normalized Lebesgue measure on $\mathbb{R}$, then the completeness equation above is just a statement of Plancherel’s theorem.

Posted by: Evan Jenkins on March 23, 2009 8:27 PM | Permalink | Reply to this

### Re: Diagonalization (was Re: Semidirect Product)

Thanks very much, Evan. I’d heard of these rigged Hilbert spaces, but never sat down and studied them. I appreciate your taking the time to provide precise statements.

I’m going to guess that one can rigorously state and even prove the content of Wigner’s theorem without recourse to rigged Hilbert spaces, but that they can provide a technical convenience along the way. It’s nice that some of the heuristic arguments of physicists can be made completely rigorous in this fashion.

Well, I may as well ask: would (or do) slicked-up modern-day proofs of Wigner’s theorem actually pass through the technology of rigged Hilbert spaces, or are there other preferred methods?

(Aspects of this discussion to be conducted in other comments…)

Posted by: Todd Trimble on March 23, 2009 10:25 PM | Permalink | Reply to this

### Re: Diagonalization

Todd wrote:

I’m going to guess that one can rigorously state and even prove the content of Wigner’s theorem without recourse to rigged Hilbert spaces, but that they can provide a technical convenience along the way.

Exactly right! It’s not hard to derive the theory of rigged Hilbert spaces from the spectral theorem, which I introduced elsewhere in this discussion. Then, even in situations where a self-adjoint operator doesn’t actually have eigenvectors lying in the Hilbert space we’re studying, we can think of those eigenvectors as vectors in a larger space.

Even more than a technical convenience, this is a conceptual convenience. Deep down in our hearts, we all want self-adjoint operators to have eigenvectors. The physicists achieve this by brazenly asserting their existence; the theory of rigged Hilbert spaces makes it true.

Alternatively, we can use the theory of direct integrals to get the job done. As we pass from algebra to analysis, we need to go beyond addition and think about integrals. Similarly, we need to go beyond direct sums and think about direct integrals.

Why? For a large class of Lie groups, can write any strongly continuous unitary representation as a direct integral of irreps. Direct sums aren’t enough.

This class of Lie groups includes all abelian Lie groups, compact Lie groups, and connected real algebraic Lie groups. In particular, it includes $\mathbb{R}$.

Each strongly continuous unitary irrep of $\mathbb{R}$, is 1-dimensional, given by the formula

$U(t) |p\rangle = e^{-i t p} |p \rangle$

for some real number $p$.

For example, the translation action of $\mathbb{R}$ on $L^2(\mathbb{R})$ decomposes as a direct integral over $p \in \mathbb{R}$ of all the 1-dimensional irreps shown above. But because it’s a direct integral instead of a direct sum, the vectors $|p \rangle$ don’t actually lie in our Hilbert space $L^2(\mathbb{R})$. Only ‘linear combinations’ of the form

$\psi = \int \widehat{\psi}(p) |p \rangle dp$

with $\widehat{\psi} \in L^2(\mathbb{R})$ are vectors in our Hilbert space!

And so, as Evan pointed out, the theory of the Fourier transform is a special case of what we’re talking about: unitary representations of $\mathbb{R}$.

Well, I may as well ask: would (or do) slicked-up modern-day proofs of Wigner’s theorem actually pass through the technology of rigged Hilbert spaces, or are there other preferred methods?

Some people love rigged Hilbert spaces; others shun them. It’s really just a matter of taste. What we need, either way, is the technology that underlies the theory of rigged Hilbert spaces: the spectral theorem!

Posted by: John Baez on March 23, 2009 11:57 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

If we move on to the physics. Is a good place to start the Dirac wave equation? I used to think the Hilbert space was the $L^2$-sections of the spinor bundle on spacetime. However I was thinking of $SL(2,\mathbb{C})$ and not the Poincare group. As we have the electron and positron then it also seems that once I have the correct representation of the Poincare group then I should tensor it by a two dimensional vector space.

Posted by: Bruce Westbury on March 23, 2009 7:53 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

In some private email with John, I was groping towards trying to understand the nuts and bolts underlying just the basic QED interaction “electron emits a photon”, interpreted as a certain intertwining operator between Hilbert space representations of Poincaré $\times U(1)$, which in an earlier thread I denoted as

$[electron] \to [electron] \otimes [photon]$

The first thing I had wanted to do was understand just the representations $[electron]$, $[photon]$.

John wrote back and said

If it’s relativistic quantum mechanics you’re trying to understand, what you really need is the classification of unitary representations of the Poincare group, which are most beautifully described as spaces of solutions of Poincare-invariant linear PDE.

He also outlined a possible pedagogical game plan for me:

You might start by pondering massive spin-0 field (Klein-Gordon equation) before going on to spin-1/2 (Dirac equation) or spin-1 (Maxwell’s equations), which are trickier.

So: it’s not just Wigner’s classification we should be after (which would be very nice, of course) – we should also try to see how it plays out in specific cases.

What I mean, for example, is this: say we define a “Klein-Gordon particle” as an irreducible unitary representation of the Poincaré group with “mass” $m$ > 0 and “spin” 0. (All these terms will have to be defined as we go along, but never mind this for now.) If I understand what Wigner’s classification is trying to say, it should be possible, just on the basis of this very bare-bones description, not only to assert that this not only characterizes up to isomorphism which irrep we mean, but in principle to be able to extract from this description the Klein-Gordon equation itself, and then we could go on to describe its space of solutions and get some real feel for this representation (and here I take it that there are classical solutions, and quantized solutions – but all of this is a bit murky to me as of this writing).

Is this asking too much of Wigner? :-)

(Incidentally, I also wonder if there are known Klein-Gordon particles in quantum field theory? A quick Google search suggested to me that there’s the hypothetical Higgs boson which has positive mass and spin 0, and there are hypothetical supersymmetric counterparts to some known particles which would have these characteristics, but any known particles?)

Posted by: Todd Trimble on March 23, 2009 2:42 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Incidentally, I also wonder if there are known Klein-Gordon particles in quantum field theory? A quick Google search suggested to me that there’s the hypothetical Higgs boson which has positive mass and spin 0, and there are hypothetical supersymmetric counterparts to some known particles which would have these characteristics, but any known particles?

Right: no. Fundamental spin 0-particles play a crucial role in theory but have as yet not been observed.

It’s not just the Higgs, which, while unobserved, is generally taken to be a central ingredient of the standard model of particle physics.

It’s also the inflaton, which, while unobserved, is generally taken to be a central ingredient in the standard model of cosmology.

These are two spin-0 fields regarded as necessary to make our current best picture of the word work. On top of that, there is a plethora of unobserved spin-0 particles which are predicted by various attempts to refine the standard model to something more encompassing. Explaining why all these spin-0 particles are there in principle but not observable in practice has been occupying modern high energy physics to a great deal. Not the least, the infamous l***sc**e discussion is a result of trying to understanding this.

On the other hand, the Higgs may simply show up in LCH in a year or two and there’ll be no mystery left about that.

Posted by: Urs Schreiber on March 23, 2009 6:05 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Todd writes:

Incidentally, I also wonder if there are known Klein–Gordon particles in quantum field theory?

Let’s pretend we’re physicists and say ‘spin-0 particle’ to mean ‘unitary irrep of the Poincaré group given by the space of solutions of the Klein-Gordon equation.’

As Urs points out, the only spin-0 particle in the Standard Model is the mysterious ‘Higgs boson’ — the one they’re currently trying to snare with that enormous Large Hadron Collider near Geneva!

The inflationary cosmology also wants an ‘inflaton’, which is not part of the Standard Model, and quite mysterious indeed.

But let’s remember our history! Just as today’s elementary particles may no longer be ‘elementary’ come tomorrow, yesterday’s elementary particles may not be considered elementary today — but they still exist. Restricting attention to particles that happen to be considered elementary today is like only listening to the current hits on the radio, and ignoring the Beatles and Billie Holiday. That would be very sad!

The Beatles of spin-zero particles are the pions: $\pi^+$, $\pi^0$ and $\pi^-$. (This group has no Ringo.) We should not forget them!

Yukawa predicted that some particle like these should exist, in order to carry the force that attracts neutrons and protons and protons and binds them into nuclei. It was later found… see the bottom of page 6 in John Huerta’s paper for the beginning of this wonderful story.

So, if you look at older quantum field theory books, like the excellent one by Bjorken and Drell, you’ll see a nice theory where there are two particles: a spin-1/2 particle called the ‘nucleon’ that comes in two ‘isospin states’ (proton and neutron), and a spin-0 particle called the ‘pion’ that comes in three isospin states ( $\pi^+$, $\pi^0$ and $\pi^-$). This theory ultimately turned out to be wrong… but it also led Yang and Mills to invent gauge theory, so it’s very much worth pondering!

The pions are just a few of the many spin-0 mesons we see in nature…

Posted by: John Baez on March 23, 2009 6:56 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

The Beatles of spin-zero particles…

Yes, therefore I wiseay said fundamental spin 0-particles have not been observed. But of course what counts as fundamental might change.

As for non-fundamental spin-0 particles: there is not just those bound states once considered fundamental, but pretty much every bound state of spin 0 which one might wish to think of.

Which is the reason why spin-0 particles play a central role in so many computations. Say cosmologists are computing primordial density fluctuations: in first approximations this is always done in terms of effective Klein-Gordon particles.

Posted by: Urs Schreiber on March 23, 2009 7:59 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Since it may not have been obvious, let me say I didn’t mean my question about Klein-Gordon particles to signify that I wouldn’t be interested in them if they didn’t correspond to something ‘known’ and ‘fundamental’. I can appreciate that there may be reasons both mathematical and physical to study them, even if that is the case – my question was more ‘just out of curiosity…’: something for my general education.

I’m glad John and Urs went beyond a simple ‘yes’ or ‘no’ answer; it’s all very interesting!

Posted by: Todd Trimble on March 24, 2009 12:19 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

As long as it’s a learning experience, I’ll throw in something about fundamental scalar fields which hasn’t been mentioned at all: they shouldn’t happen. Their existence violates certain “naturalness” considerations, which basically boil down to saying that our theories shouldn’t need exact cancellations between very large numbers to maintain small ones — unless, that is, such a cancellation is enforced at some “deeper” level by a symmetry. So actually the pions are very nice as scalar fields, because they are Goldstone bosons, so escape this naturalness problem. Gauge bosons are also protected, by having the gauge symmetry in the first place.

Now for those who care a bit more:

When we write down a field theory, we are really encoding how spatially separate bits of the field interact with each other. We usually demand that the theory is “local”, which is to say that they only interact in their immediate (spacetime) neighbourhood. In theory, this is the mathematically ideal — infinitely small, infinitely close together. In practice however, our instruments only measure over an extended volume (in both space and time). So we usually incorporate this into theory as saying that the field has an ultraviolet cutoff — that is, fourier modes above some limit simply don’t exist. Concretely, we set integrals over momenta to have an upper limit (there are other ways, but they are technical improvements on the same idea).

But what if we have deliberately crude instruments (or if you live in a technologically inferior civilisation that can’t even build machines to probe the Planck scale)? Well, if we don’t excite too much the modes with length scale smaller than the instrument size, then we ought to be able to fudge in their effects by some statistical averaging. Concretely, you would take your Feynman sum over paths and perform the sums which are over the frequencies you now don’t care about. The actual effect of this is that you end up changing the dimensionless constants of your Lagrangian, even changing some of them from zero to non-zero. This coarse graining strictly loses you information, but that’s not a bad thing, as long as you end up focusing on the stuff you want. In fact, for physics, the ideal theory just talks about what you care about, and the rest somehow end up not mattering, or are just absorbed into some constants. So actually, there are good reasons to look at what happens in the limit of your instruments being macroscopically sized. This flowing of the constants is usually called renormalisation.

Now for some magic: most of the numbers will actually go downwards, and only a finite number will go up or stay the same. We name these irrelevant, relevant and marginal, respectively. Relevant terms which grow to much greater than unity are telling you that you should change your basic set of fields — because they are no longer weakly coupled together, and you will no longer be able to treat their interactions perturbatively. For instance, QCD goes from quarks and gluons to being confined, and it then makes more sense to talk about hadrons and mesons; in superconductors, you stop talking about electrons, and talk about Cooper pairs instead. Now we can make precise what I said earlier about naturalness: irrelevant terms will go to zero, relevant terms will necessitate a different set of fields to continue the renormalisation, and only the marginal ones should remain at the largest scales.

The mass term of a scalar field is relevant. This means that as we look at lower and lower energy phenomenon, the scalar field should actually drop out entirely of the set of fields we wish to consider, since it’s so massive that no process will actually create it. The only way for it to stay, is if somehow the various other numbers conspired to keep it small yet non-zero — but this essentially requires very large numbers to cancel off against each other, almost yet not quite. It will require, as a consequence, that we measure some non-zero, dimensionless numbers to absurd precision to be able to have any predictive power.

So what about the Higgs? The currently accepted method of introducing it is essentially chosen because of simplicity — it’s the simplest way to break electroweak symmetry and end up with things as we see it. However, because of the naturalness problem I’ve (tried, but probably failed) to explain, there are good reasons to believe that the picture fails just above the electroweak scale. This is essentially why we bothered to spend the billions to build the LHC — we’ve got actual theoretical reasons to believe that there are new, experimentally accessible phenomenon right above when we’re supposed to see the Higgs.

Posted by: Gen Zhang on March 26, 2009 12:07 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

By the way, when I seem to get overheated and start scolding someone — say, for forgetting about the pions, or only caring about particles that are currently considered fundamental — it’s often because I’m preaching to the world at large, trying to pound home points that ‘everyone needs to learn’. The people who’re actually daring to speak up in this conversation are just a tiny fraction of the people listening in.

In this case, it’s not that I actually believe Urs forgot about pions, or Todd is only interested in particles that are currently considered fundamental. I just want everyone in the world to understand that quantum field theories can be useful, important and interesting even if they’re ‘just approximations’ to the best theory we have now. The mathematical beauty of physics is not just the beauty of the ‘final theory’ — which after all we don’t even have. It’s the beauty of many theories fitting together in a complicated web.

Posted by: John Baez on March 24, 2009 1:28 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

By the way, when I seem to get overheated and start scolding someone — say, for forgetting about the pions, or only caring about particles that are currently considered fundamental — it’s often because I’m preaching to the world at large […]

I think I was just trying to say: oh, if you are not restricting yourself to particles which are fundamental, well then there is a large supply of spin-0 particles: just take any bound state with total spin vanishing. For instance I am a spin-0 particle to good approximation, for that matter!

Model builders therefore compute with compound spin-0 particles all the time. Not just with pions. For instance most computations of quantum fluctuations on cosmological backgrounds are done this way, for instance.

On top of that, it might be that even the spin-0 particles which are considered fundamental today, notably the Higgs, are secretly actually composite particles. There are various theories postulating this. None of them considered overly likely today, but certainly possible.

Posted by: Urs Schreiber on March 24, 2009 9:59 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

The people who’re actually daring to speak up in this conversation are just a tiny fraction of the people listening in.

True that.

Posted by: Theo on March 26, 2009 4:22 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Todd wrote:

What I mean, for example, is this: say we define a “Klein-Gordon particle” as an irreducible unitary representation of the Poincaré group with “mass” m > 0 and “spin” 0. (All these terms will have to be defined as we go along, but never mind this for now.) If I understand what Wigner’s classification is trying to say, it should be possible, just on the basis of this very bare-bones description, not only to assert that this not only characterizes up to isomorphism which irrep we mean, but in principle to be able to extract from this description the Klein-Gordon equation itself, and then we could go on to describe its space of solutions and get some real feel for this representation…

You’re right. Any reasonable description of the mass-$m$ spin-0 representation of the Poincaré group will quickly point us towards the Klein-Gordon equation

$( \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial}{\partial z^2} + m^2 ) \phi = 0$

There is a nice Hilbert space of solutions of this equation on which the Poincaré group acts as unitary transformations — and this gives us our representation!

(and here I take it that there are classical solutions, and quantized solutions –– but all of this is a bit murky to me as of this writing).

Fear not: the Hilbert space I’m talking about is a space of ordinary real-valued functions

$\phi : \mathbb{R}^4 \to \mathbb{R}$

satisfying above equation.

I suspect the confusion embodied in your parenthetical remark arises from various things you’ve read in quantum field theory books. The functions I’ve described above are called classical solutions of the Klein–Gordon equations, or classical fields, but in relativistic quantum mechanics they represent quantum states of a single mass-$m$ spin-$0$ particle. When we treat arbitrary collections of such particles, through a process called ‘second quantization’, we have moved from doing relativistic quantum mechanics to quantum field theory. At this stage we meet a more sophisticated entity called a ‘quantum field’, which obeys an equation of the exact same appearance.

As you can see, this is sort of confusing! And most books explain this stuff in a truly wretched style, guaranteed to cause endless puzzlement.

Luckily, we are not talking about quantum field theory on this thread! We are talking about unitary representations of the Poincaré group, and now you’ve met your first one: the Hilbert space of suffficiently nice functions

$\phi : \mathbb{R}^4 \to \mathbb{R}$

satisfying

$( \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial}{\partial z^2} + m^2 ) \phi = 0$

Of course this meeting has been quite informal, since I haven’t told you what ‘sufficiently nice’ means, or why sufficiently nice real-valued solutions form a complex Hilbert space, or why the Poincaré group acts as unitary operators on this space!

But we’re just getting started… I hope you and other folks look at the exercise here, and ask questions if it’s too hard. That will soon lead us to the Klein–Gordon equation, if we persist.

Posted by: John Baez on March 25, 2009 12:02 AM | Permalink | Reply to this

### Less than zero energy to discuss this; Re: Unitary Representations of the Poincaré Group

“Luckily, we are not talking about quantum field theory on this thread!”

Otherwise we WOULD have to discuss negative energy states.

Posted by: Jonathan Vos Post on March 25, 2009 6:12 AM | Permalink | Reply to this

### Review Article

There’s a very nice recent review article about exactly this topic, explaining the mathematical (and historical) context:

http://arxiv.org/abs/0809.4942

Norbert Straumann

Unitary Representations of the inhomogeneous Lorentz Group and their Significance in Quantum Physics

Note these representations give the space of “single-particle wavefunctions”. To construct the actual space of states for a relativistic theory of free particles you need to build a Fock space out of these spaces (i.e. use the symmetric or anti-symmetric part of the tensor algebra built out of this space).
You also need to deal with the fact that these spaces are polarized, i.e. describe both positive and negative energy states, and take this into account in how you build the Fock space.

Posted by: Peter Woit on March 23, 2009 8:11 PM | Permalink | Reply to this

### Unitary Representations of the Real Line

Todd wrote:

A unitary (Hilbert space) representation $H$ of the Poincaré group thus restricts to a unitary representation of $\mathbb{R}^4$, taking $a \in \mathbb{R}^4$ to a unitary operator $U_a: H \to H$. Since all of these $U_a$ commute, we should (at least heuristically) be able to “simultaneously diagonalize” them. Following the physicist notation, this would mean we have orthonormal basis elements which they write as $|p\rangle$, so that we have

$U_a|p\rangle = e^{-i p \cdot a}|p\rangle$

where $p \cdot a$ denotes the dot product of a (momentum) 4-vector $p$ with the position 4-vector $a$.

What I want to know is: how rigorous can this be made, or what else do we have to say to make just this bit rigorous?

All of this can be made as rigorous as we want. But first let me set your question in its proper context.

We’re trying to understand unitary representations of the Poincaré group, but since the translation group $\mathbb{R}^4$ is a subgroup of this, we soon realize it’s a prerequisite to understand unitary representations of $\mathbb{R}^4$.

But $\mathbb{R}^4$ has a lot of subgroups that look just like $\mathbb{R}$! So, before tackling $\mathbb{R}^4$, it’s a prerequisite to understand unitary representations of $\mathbb{R}$.

So, we cry: HELP! WHAT ARE UNITARY REPRESENTATIONS OF THE REAL LINE LIKE?

And now the two trusty warhorses of mathematical physics trot to our rescue: Stone’s theorem, and the spectral theorem.

Stone’s theorem says:

Any strongly continuous unitary representation of $\mathbb{R}$ on a Hilbert space is of the form $exp(-i t A)$ for a unique (possibly unbounded) self-adjoint operator $A$ on this Hilbert space. Conversely, any such operator gives a strongly continuous unitary representation of $\mathbb{R}$ by this formula.

The spectral theorem finishes the job by saying what self-adjoint operators are like. There are many ways to state this theorem, but here’s a quick one:

Suppose $A$ is a (possibly unbounded) self-adjoint operator on a Hilbert space. Then this Hilbert space is isomorphic to $L^2(X)$ for some measure space $X$, and making use of this isomorphism, $A$ becomes a multiplication operator:

$(A \psi)(x) = a(x) \psi(x)$

where

$a : X \to \mathbb{R}$

is some measurable function. And in this situation,

$(exp(- i t A) \psi)(x) = e^{-i t a(x)} \psi(x)$

If you want to see proofs of both these theorems, and more details, try my lecture notes on quantum theory and analysis. Unfortunately I don’t prove the version of the spectral theorem stated above.

Now, it may not be obvious how these theorems relate to what you wrote! But, stick them together and see what happens.

Posted by: John Baez on March 23, 2009 10:13 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

By the way, if you stick these theorems together, you should see more precisely what I meant when I said that every strongly continuous unitary rep of $\mathbb{R}$ is a direct integral of 1-dimensional reps of the form:

$U(t) |p\rangle = e^{-i t p} |p \rangle$

where $p$ is some real number… and $|p \rangle$ is just a cute name for any basis vector for our 1d Hilbert space.

You can see this even if you don’t really know what a direct integral is!

Posted by: John Baez on March 24, 2009 12:41 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Regarding Stone’s theorem, here is one thing that bothers me a little. It gives a way of understanding strongly continuous unitary representations of $\mathbb{R}$ (and of $\mathbb{R}^4$, etc.).

I understand this “strong” topology (on the group of unitary operators $U(H)$ where $H$ is a Hilbert space) to be the weakest (i.e., smallest) topology such that all the evaluation maps $ev_h: U(H) \to H$ are continuous. It’s weaker than the topology I’m more used to thinking about, given by the usual norm on bounded linear operators and then restricting to the subspace of unitary operators.

So at the outset, what do we even mean by a unitary Hilbert space representation of the Poincaré group $P$? Should we understand in fact a continuous group homomorphism

$P \to U(H)$

where $U(H)$ is equipped specifically with this so-called “strong” topology?

(And if that’s the case, why don’t more people come out and say this is in fact what we’re doing? (-: )

Posted by: Todd Trimble on July 10, 2011 12:37 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Hi, Todd!

So at the outset, what do we even mean by a unitary Hilbert space representation of the Poincaré group $P$? Should we understand in fact a continuous group homomorphism

$P \to U(H)$

where $U(H)$ is equipped specifically with this so-called “strong” topology?

Yes, that’s what we mean. When people study unitary representations of Lie groups on Hilbert spaces, they almost always restrict attention to strongly continuous representations. One reason, as you note, is that Stone’s theorem sets up a 1-1 correspondence between strongly continuous unitary representations of $\mathbb{R}$ on a Hilbert space and self-adjoint operators on that Hilbert space.

Of course, there’s also a ‘baby Stone’s theorem’ that gives a 1-1 correspondence between norm continuous unitary representations of $\mathbb{R}$ on a Hilbert space and bounded self-adjoint operators on that Hilbert space. But restricting attention to norm continuous unitary representations would be crippling, since lots of interesting self-adjoint operators are unbounded.

(And if that’s the case, why don’t more people come out and say this is in fact what we’re doing? (-: )

Umm… because they’re physicists and don’t know it? Or else they’re mathematicians so used to this stuff that they don’t bother mentioning it? Either way, the effect is the same.

Posted by: John Baez on July 10, 2011 1:02 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Thank you, John! That’s very good to know.

The fact that physicists don’t come out and say this is not very surprising; it just doesn’t seem to be part of the prevailing culture to get involved with such details. Maybe the mathematicians do come out and say this, but I’ve not seen it (admittedly, I’m not all that well-read in the literature on the mathematical side). It’s the type of thing that I get hung up on, and I’ll bet there are hundreds or even thousands like me in this respect.

Posted by: Todd Trimble on July 10, 2011 3:00 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Just in case anyone is following this here and wondering: there is more discussion of this here on the nForum – a little bit right now and hopefully more in the near future.

I have just had one of the most insanely busy weeks of my life, with an article I am co-authoring, a book that I am co-editing and a bachelor thesis that I am advising by cosmic coincidence all wanting to finish right now… and I must go on vacation tomorrow morning.

But once I have a minute, I hope to join Todd on his quest to write out these things more clearly and more publicly on the $n$Lab…

Posted by: Urs Schreiber on July 10, 2011 11:49 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Have a great (and well-deserved) vacation, Urs! I am looking forward to working on this when you are ready.

In the meantime, maybe I’ll have some questions for John. I am looking over this thread and thinking about the various elements here, and I would love to get this material settled in my mind once and for all, and write it up in the Lab.

Posted by: Todd Trimble on July 11, 2011 1:25 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

While I have not followed the discussion on the nForum (sorry), I tried to explain the choice of the “strong topology” over at the nLab on the page topological group a while ago.

One important fact to note - also on that page - is that the group of unitary operators is a complete metrizable topological group in the strong topology, which is nice :-)

Posted by: Tim van Beek on July 11, 2011 8:46 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Just a sidenote: everybody interested in operator theory (linear operators on Hilbert spaces) should know about the following topologies on operators:

• norm topology,

• strong topology,

• ultraweak topology,

• weak topology

ordered from stronger to weaker. I mention this just in case anybody is wondering why the strong topology is called strong topology while it is weaker than the norm topology. (And yes, the ultraweak topology could also be renamed as infra strong topology.)

Posted by: Tim van Beek on July 11, 2011 8:54 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Wow. I wouldn’t have guessed that ultraweak was less weak than weak.

Posted by: Tom Leinster on July 11, 2011 6:48 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Thanks, Tim – that’s a nice easy example you gave at the nLab page on topological groups, for why the strong topology is important.

Posted by: Todd Trimble on July 11, 2011 7:42 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Tom wrote:

Wow. I wouldn’t have guessed that ultraweak was less weak than weak.

Yeah, it’s a famously dumb name. There’s also the ultrastrong topology, which while stronger than the strong topology is still weaker than the norm topology.

Who made up these names? Von Neumann?

The ultraweak and ultrastrong topologies have similar names because they can be defined in a similar way. Let $L(H)$ denote the space of bounded linear operators on a Hilbert space $H$. Tensoring with the identity on a countably infinite-dimensional Hilbert space $K$ gives an inclusion

$L(H) \hookrightarrow L(H \otimes K)$

The weak topology on $L(H \otimes K)$ then restricts to the ultraweak topology on $L(H)$, and the strong topology on $L(H \otimes K)$ restricts to the ultrastrong topology on $L(H)$.

So, ‘ultra’ here refers to tensoring with a Hilbert space of countably infinite dimension. Given this, it would probably make more sense to call these topologies ‘$\sigma$-weak’ and ‘$\sigma$-strong’. Some people do.

However, these topologies are only useful in fairly specialized contexts, so beginners should ignore them at first. The really important topologies on $L(H)$ are the norm, strong and weak topologies, so these are the ones that it pays to get a good intuition for.

Find a sequence of bounded operators that converges to zero weakly but not strongly. Find one that converges to zero strongly but not in norm. Show that a sequence of unitary operators converges to a unitary operator strongly iff it converges to that operator weakly, so strongly continuous 1-parameter unitary groups are the same as weakly continuous ones. And so on.

All this is very nicely explained in:

• Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, volume 1: Functional Analysis, Academic Press, 1980.

I really love this book, and also their volume on Fourier Analysis and Self-Adjointness.

Posted by: John Baez on July 13, 2011 3:57 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

My first (probably naive) thought: does this generalise to an ultra^n-weak topology?

Posted by: Tom Ellis on July 13, 2011 7:04 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

I don’t see any useful way to define ‘ultran-weak’ or ‘ultranstrong’ topologies, since tensoring with two Hilbert spaces of countably infinite dimension is the same as tensoring with one.

Posted by: John Baez on July 13, 2011 7:27 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Or because they are physicists talking to a restricted community who KNOW what’s going on or are presumed to?

Posted by: jim stasheff on July 11, 2011 12:40 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Could be, Jim. I’d like to know what would be the typical reaction to a graduate student who asks such questions in class. (I have a shelf full of quantum mechanics texts written by physicists, and I’ve never seen these points raised; have you? This leads me to believe that it’s not an issue that physicists think is worth bringing up.)

Posted by: Todd Trimble on July 11, 2011 1:21 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

I went into math largely because physicists weren’t interested in questions like this — not because I didn’t want to do physics.

If you ask questions like this in a physics class, it’s likely the teacher will make fun you. They’re regarded as signs of an overly delicate sensibility.

If you want to learn a lot of rigorous stuff about unitary group representations in physics, this book is good:

• George Mackey, Unitary Group Representations in Physics, Probability, and Number Theory, Addison-Wesley, Redwood City, California, 1989.
Posted by: John Baez on July 11, 2011 2:51 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

I am feeling that we – the scientific community – need to get over this way of talking “the physicists do this” and “the mathematicians do that”. I hear this so often in talks, and it is a waste of everybody’s time, I think. Recently I was in an otherwise excellent QFT talk over in the physics department that spent about 10 minutes complaining about “those crazy math guys”, literally. It was 10 lost minutes, but was actually no worse than what you hear in math departments the other way round all the time.

(I have a shelf full of quantum mechanics texts written by physicists, and I’ve never seen these points raised; have you?

Yes, I have.

And I once pointed out which texts a person like you should read when reading about quantum mechanics.

The standard QM textbook that does live up to what you are looking for is

Rudolf Haag, Quantum physics – Fields, particles, algebras Springer (1992)

Plenty of discussion of the various topologies on operator algebras in there. It starts discussing four different ones, their uses and non-uses, and then eventually concentrates on the strong topology.

(If you have trouble getting hold of a copy, let me know and I’ll help out.)

Also, any textbook following this one will do. For instance Hans Halvorson’s very nice book, which is available on the arXiv.

It is of course true that lots of QM courses don’t go much into mathematical details. Often for a good reason: there is so much else to do, somebody has to build cell-phones and lasers and satellites, too.

But for instance if you learn your QM in Hamburg, say with Prof. Fredenhagen, you learn these things. In general, if you want to see anything about quantum field theories discussed in a way at least very close to what you, Todd, would hope to see, look for articles by Fredenhagen at al. See also the references in the introduction in a famous book on the current status of the mathematical foundations of QFT.

Posted by: Urs Schreiber on July 11, 2011 5:22 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Okay, now I’m feeling guilty about criticizing my old physics teachers for making fun of me when I was a student and asked questions about mathematical issues. Luckily, that didn’t stop me from studying and working on both math and physics. I love them both and am equally likely to criticize either ‘side’.

We could hope that someday there won’t be ‘sides’, but I think there will always be a rivalry here. As long as it’s a friendly rivalry, I don’t think it so bad. Of course it gets boring at times when people recycle clichés.

Posted by: John Baez on July 11, 2011 5:52 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

I meant no disrespect to the physicists (and I actually wanted to write to avoid giving an impression of dissing). I had in mind something like what you said:

It is of course true that lots of QM courses don’t go much into mathematical details. Often for a good reason: there is so much else to do, somebody has to build cell-phones and lasers and satellites, too.

but maybe I should have come out and said something similar. I fully respect the fact that theoretical physics is a full-time intellectual discipline, even if you don’t get into mathematical details.

More importantly for our purposes here: thanks for the tips, Urs! I might have trouble getting a copy of Haag. I think you did put me in touch once before with Halvorson, and I got stuck near page 1, in the sense that I had little idea what was going on. Maybe I should try again?

Posted by: Todd Trimble on July 11, 2011 6:17 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Urs wrote:

It is of course true that lots of QM courses don’t go much into mathematical details. Often for a good reason: there is so much else to do, somebody has to build cell-phones and lasers and satellites, too.

Right, usually physicists who attend QM classes don’t know much about functional analysis or Hilbert spaces, and you can’t cover that, too, in such a class. I attended the QM class of Detlev Buchholz in Göttingen, another leading figure of AQFT, and he never told us about operator algebras :-)

The standard QM textbook that does live up to what you are looking for is

• Rudolf Haag, Quantum Physics – Fields, Particles, Algebras, Springer, 1992.

That’s about QFT - and QFT is way more complicated than QM. For QM I would recommend for example

• F. Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics.

Strocchi follows an axiomatic approach with observables as elements of $C^*$ algebras, but in the much simpler setting of QM.

Haag explains a lot of connections to physics, which leads to a nonlinear and somewhat repetitive exposition (necessarily), which more mathematically oriented books like

• Hellmut Baumgärtel, Operator Algebraic Methods in Quantum Field Theory: a Series of Lectures, Akademie Verlag, 1995.

do not do. The advantage of the latter book is that it is very concise and concentrated on a strict math style (definition, theorem, proof). But, as I said, you won’t find any connections to the motivation on the physical side in it.

Posted by: Tim van Beek on July 12, 2011 5:56 AM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

@ Urs: I am feeling that we, the scientific community , need to get over this way of talking :the physicists do this and the mathematicians do that. I hear this so often in talks, and it is a waste of everybody’s time, I think. Recently I was in an otherwise excellent QFT talk over in the physics department that spent about 10 minutes complaining aboutthose crazy math guys’, literally.

Sounds like s/he was a bit defensive.

I don’t think I’ve ever heard a similar 10 minute !! rant in a math talk - at worst remarks like:
at a physical level of rigor
or
the physicists have given us a homework assignment
or…

In fact, if it weren’t for the physicists inadvertently suggesting interesting MATH problems,
what would some of us be doing now???

Posted by: jim stasheff on July 12, 2011 12:20 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Sounds like s/he was a bit defensive.

He was all the opposite of being defensive.

I don’t think I’ve ever heard a similar 10 minute !! rant in a math talk

It was probably even longer. The reason he kept coming back to it was that the point of his talk was demonstrating that he correctly dealt with a subtle issue of QFT on a de Sitter background. To amplify his achievement, he kept pointing out that the mathematical physicists that he talked to (of course he didn’t say who, so maybe he was just talking to the wrong people) continuously did the wrong thing where he did the right thing, even after he tried to point it out to them.

He felt genuinely superior to “these math guys” who probably did something in an exact way, but did the wrong thing in an exact way.

When formulating a theory of reality, one has to do the right thing in the right way. Mathematicians tend to do it right, but not necessarily do the right thing right. Physicists tend to not do it right, but they tend to know what is the right thing to do.

My point is: it’s not much use to complain about the shortcomings of the other team. One could argue that it’s not the physicist’s job to give precise formulations of their theories. It’s the mathematician’s job.

Look at historical examples: today we think about special and general relativity not the way Einstein found them, but the way the mathematicians Poincaré and Hilbert formalized them. Einstein already did a herculean job in figuring out what the right thing to do is. The mathematician’s job was to see how to do that right thing in the right way.

Posted by: Urs Schreiber on July 13, 2011 5:19 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Urs wrote

When formulating a theory of reality, one has to do the right thing in the right way. Mathematicians tend to do it right, but not necessarily do the right thing right. Physicists tend to not do it right, but they tend to know what is the right thing to do.

That’s very beautifully (and wittily) put, and I sense there’s a lot of rightness about it. :-) I like how it gives each group its due, and how they complement one another.

Posted by: Todd Trimble on July 13, 2011 8:31 PM | Permalink | Reply to this

### Re: Unitary Representations of the Real Line

Amen to that! It’s what I meant - paraphrasing Bott - about physicists giving us homework assignments.

Posted by: jim stasheff on July 14, 2011 1:53 PM | Permalink | Reply to this

### Alternatives to Poincare Group?

When I look at the Poincare group

$P = SL(2,\mathbb{C}) \ltimes \mathbb{R}^4,$

what I see is an explicit assumption about cosmology. It says the universe is topologically $\mathbb{R}^4$.

What if the universe is not globally $\mathbb{R}^4$? How might the representation theory be modified if the universe was globally hyperbolic or some kind of torus? I can imagine how global topology could easily impact the family of particles in the theory.

Posted by: Eric on March 23, 2009 10:55 PM | Permalink | Reply to this

### Alternatives to Poincare Group? Forget ‘em!

Eric wrote:

It says the universe is topologically $\mathbb{R}^4$.

Actually it says more: it says the universe is geometrically Minkowski spacetime.

What if the universe is not globally $\mathbb{R}^4$?

Different geometries of spacetime have different symmetry groups, and we need to classify the representations of those symmetry groups to understand particles on those spacetimes.

For lumpy, bumpy irregular spacetimes with very small symmetry groups, this approach doesn’t work. Then we’re doing ‘quantum field theory on curved spacetime’ in full generality.

But right now, right here, we’re trying to understand unitary representations of the Poincaré group. So far, we’ve just gotten about $1/50$th of the way there. So, it’s not time to start talking about generalizations! We have a long road to travel, and I’ll try to resist tempting byways.

Posted by: John Baez on March 24, 2009 12:22 AM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

Hmph! You beat me to the comment box by 4 minutes on this one. Let’s just say, in general spacetimes, the whole concept of “particle” gets kind of shaky.

Posted by: Tim Silverman on March 24, 2009 1:29 AM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

Even if the background is fixed?

I vaguely remember in grad school, we would often assume “periodic boundary conditions”, which is equivalent (I think) to assuming space was a 3-torus. Then we let the size of the torus go to infinity.

The only reason I bring it up at all is that I seem to recall (from one of John’s lectures ages ago) that some of the mathematical difficulties with representations of the Poincare group have to do with the fact that $\mathbb{R}^4$ is infinite (or something). I was wondering if $\mathbb{R}^4$ might not be the simplest starting point. Is there a simpler group that becomes the Poincare group in some limit, e.g. a 4-torus in the limit as the radius in all 4 dimensions foes to infinity? I didn’t intend to introduce additional complexity and was actually hoping to start simpler.

Posted by: Eric on March 24, 2009 2:01 AM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

Eric wrote:

… I seem to recall (from one of John’s lectures ages ago) some of the mathematical difficulties with representations of the Poincaré group have to do with the fact that $\mathbb{R}^4$ is infinite (or something).

Yes, the fact that $\mathbb{R}^4$ is noncompact makes life a bit difficult — but maybe even more important is the fact that the Poincaré group itself is noncompact. The representation theory of compact Lie groups is a piece of cake, by comparison, because all the irreps are finite-dimensional.

Is there a simpler group that becomes the Poincare group in some limit, e.g. a 4-torus in the limit as the radius in all 4 dimensions foes to infinity?

Unfortunately not. The Poincaré group is 10-dimensional: 4 translations, 3 rotations, 3 boosts. The isometry group of a 4-torus is just 4-dimensional: only translations.

The Poincare group is 10-dimensional, and 10 = $4(4+1)/2$. In general the maximum dimension of the symmetry group of an $n$-dimensional spacetime is $n(n+1)/2$, and the cases where the maximum is reached are known.

In dimension 4, if we’re talking about Lorentzian manifolds — meaning 1 time dimension plus 3 space — the options are these: Minkowski spacetime, DeSitter spacetime, and anti-DeSitter spacetime. For all of these, the isometry groups are noncompact and the relevant irreps are infinite-dimensional. The representation theory is probably easiest, or at least most familiar, in the Minkowski case. So, that’s what we’re talking about.

If you’re desperate to get a compact Lie group, to get a bunch of finite-dimensional reps, you can switch to Riemannian manifolds — where all 4 dimensions are spacelike. Then the maximally symmetric 4d ones are 4d Euclidean space, 4d hyperbolic space, and the 4-sphere. The isometries of the 4-sphere form the group $SO(5)$ — check that this is indeed 10-dimensional — and only in this case do we get a group that’s compact.

People know everything there is to know about the irreps of SO(5), and Todd probably knows plenty of this stuff, since he and Jim spent ages discussing Dynkin diagrams. There is a link between representations of SO(5) and representations of the Poincaré group, but it’s so indirect that without knowing the Poincaré reps ahead of time, it would be a major feat to see how this works! Starting from the 4-sphere, you have to take the limit as the radius goes to infinity and replace one dimension of space by one dimension of time, to reach Minkowski spacetime. Each procedure is separately fascinating, but I don’t have the energy to lead us through both of them.

So, I really urge that we dive in and directly attack Wigner’s classification of irreps of the Poincaré group. It’s incredibly beautiful math and incredibly important physics.

Posted by: John Baez on March 24, 2009 2:35 AM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

Eric wrote:

I vaguely remember in grad school, we would often assume “periodic boundary conditions”, which is equivalent (I think) to assuming space was a 3-torus. Then we let the size of the torus go to infinity.

[…]

Is there a simpler group that becomes the Poincare group in some limit, e.g. a 4-torus in the limit as the radius in all 4 dimensions goes to infinity?

As mentioned, the answer to this question is no. But to be fair, I should say that it is quite interesting to ponder particles on spacetimes like $\mathbb{R} \times T^3$ and $\mathbb{R} \times S^3$.

When space is the 3-torus $T^3$, the isometry group is just 4-dimensional: space and time translations. This means we get concepts of momentum and energy, but no angular momentum – bummer! On on the other hand $T^3$ is a compact abelian group so we can take full advantage of Fourier series: that’s why people like ‘periodic boundary conditions’.

When space is the 3-sphere $S^3$, the isometry group is 7-dimensional: 6 dimensions for the rotational symmetries of the 3-sphere, plus time translation. So, we get 6 conserved quantities reminiscent of momentum and angular momentum, together with energy. Note that momentum and angular momentum are ‘unified’, since there’s no real difference between a ‘translation’ of the 3-sphere and a ‘rotation’. But, we still don’t get anything like Lorentz boosts.

My whole thesis was actually about quantum field theory on $\mathbb{R} \times S^3$!

Posted by: John Baez on March 24, 2009 3:29 AM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

Would it be instructive at all to reduce the dimension by one? Do the same issues of compactness appear in (2+1)-dimensional Minkowski space? I suspect so. Is a Lorentzian $\mathbb{R}\times S^2$ compact?

In the spirit of Flatland and Sphereland, would the particle families observed on Flatland be the same as the particle families observed on Sphereland? I’ve probably read this somewhere before, but I keep wondering if the particles that you observe can tell you about the topology of the universe.

If there is any merit to that idea, are there any compact spaces that would give rise to “some” of the particles we observe? How certain are we that the universe we live in is noncompact? Is there a no-go theorem that tells us there is no way we could observe the particles we see if our universe was compact?

Believe me, I’m trying to get excited about Wigner’s theorem and representations of noncompact Lie groups, but I want to be reasonably sure I’m learning about our universe and not simply mathematics for mathematics sake (which I have nothing against, its just not for me).

Posted by: Eric on March 24, 2009 6:09 PM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

How certain are we that the universe we live in is noncompact?

A while ago there was a little hype about some people having found alleged evidence in the cosmic microwace background that the universe is spatially compact. That evidence turned out to be an illusion. See maybe this.

Beyond that, nobody has much of a clue.

But for our discussion the important point is: it doesn’t matter really for all practical purposes.

Even if the universe were a doughnut on large scales, the particles observed in a localized (on planet earth, say) accelerator would behave for all practical purposes as particles on $\mathbb{R}^3 \times \mathbb{R}$.

The large scale geometry of the universe is relevant for questions of particle theory only in as far as you want to understand questions of matter creation etc. in an expanding universe or the like. This is not what one looks at in the context of GUTs!

That’s why people working on ambitious programs to do something like a GUT and unify gravity with it work on two different ends of the same bone: on the one hand studying particles on flat Minkwoski space, on the other end study cosmological questions. The former is a “local” question for most purposes.

Notice that this is different if and when one considers small compact dimensions, as in the context of Kaluza-Klein theory mentioned above: if a factor of spacetime is so small that particles can entirely probe it while zipping through your local accelerator, then the full non-flat geometry of that compact space is relevant.

Posted by: Urs Schreiber on March 24, 2009 6:33 PM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

Thanks Urs.

The first part of your comment:

Even if the universe were a doughnut on large scales, the particles observed in a localized (on planet earth, say) accelerator would behave for all practical purposes as particles on $\mathbb{R}^3\times\mathbb{R}$.

told me that my suspicion about the global geometry affecting the family of particles observes in a local accelerator was incorrect. I was about to go on happy with that knowledge, but then your second comment confused me again:

Notice that this is different if and when one considers small compact dimensions, as in the context of Kaluza-Klein theory mentioned above: if a factor of spacetime is so small that particles can entirely probe it while zipping through your local accelerator, then the full non-flat geometry of that compact space is relevant.

When you say “small compact dimensions”, I imagine a spacetime of the form

$(Big Spacetime)\times(Small Compact Space)$

Would it still be the case that the particles observed would not depend on the Big Spacetime as long as it locally it looks like Minkowski space in the neighborhood of the accelerator? The geometric dependence you were referring to was the geometry of the Small Space only?

Ok. I think I get it. Are you saying the particles observed in a local accelerator depend only on the local geometry and do not depend in any way on the global geometry? The small compact dimensions are part of the “local geometry” so will determine the family of particles observed in a local accelerator?

That seems a little disappointing, but it wouldn’t be the first time one of the long-held beliefs was dispelled :)

Posted by: Eric on March 24, 2009 7:14 PM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

Are you saying the particles observed in a local accelerator depend only on the local geometry and do not depend in any way on the global geometry? The small compact dimensions are part of the “local geometry” so will determine the family of particles observed in a local accelerator?

Yes, sure. If you use the term “local geometry” in this sense, namely loosely in the sense “local as measured by distance with respect to the given (pseudo)-Riemannian metric”.

When you say “small compact dimensions”, I imagine a spacetime of the form

$(Big Spacetime) \times (SmallCompactSpace)$

Essentially, yes. As we said before in the discussion about Kaluza-Klein theories.

Posted by: Urs Schreiber on March 24, 2009 8:20 PM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

Eric wrote:

Would it be instructive at all to reduce the dimension by one?

Yes, if you’re interested in subtleties like anyons, which can arise only when space is 2-dimensional. In our universe, the classification of irreps of the Poincaré group will reveal particles of spin $0,1,2,\dots$ called bosons, and particles of spin $1/2, 3/2, 5/2...$ called fermions. In flatland we can have particles with any spin we like, say spin 3.509.

Posted by: John Baez on March 24, 2009 7:12 PM | Permalink | Reply to this

### Re: Alternatives to Poincare Group? Forget ‘em!

Thank you. This is great stuff :)

I started asking a bunch of basic questions, but found a good article. I’ll have a look there first, but thought I would share it in case anyone is interested.

Quantum numbers for particles in de Sitter space

The de Sitter groups are of interest in elementary particle physics for several reasons. First of all, by definition, an elementary physical system in a de Sitter world would be a system described by a wavefunction transforming according to an irreducible unitary representation of the de Sitter group. Complete sets of commuting operators in the enveloping algebra of the de Sitter algebras (i.e. Casimir operators of the entire group plus, e.g., Casimir operators of a certain chain of subgroups) will then provide the quantum number of such particles in definite states. A large literature exists on elementary particle theory in de Sitter space, in particular dealing with problems of localization, the positivity of energy (or lack thereof), generalizations of the Dirac equation and other invariant equations, etc. (seem e.g., Refs. 4-11 and many others). A large body of work also exists on the representation theory of the de Sitter group (see, e.g., the classical papers 12-14).

Aside from the aspect of considering particle or field theory in curved space and this incorporating some aspects of gravitational interactions, the de Sitter world may be of interest in that it provides a possible way of avoiding the O’Raifeartaigh theorem.15 Indeed, while it is not possible to combine the Poincare group and an internal symmetry group, like $SU(3)$ into a larger group, providing a discrete mass spectrum, such a unification is possible if one of the de Sitter groups is taken as the space-time group. 10,16”

The last sentence seems particularly intriguing.

Posted by: Eric on March 24, 2009 11:59 PM | Permalink | Reply to this

### Not So Fast! (was: Alternatives to Poincare Group? Forget em!)

However, it may be possible to combine the centrally extended Poincare’ group with internal symmetries. The proof of the Raifeartaigh theorem breaks down precisely at that point where the 11th generator is involved.

This situation is relatively easy to understand, in retrospect and it contains an important object lesson (“trivial central charges matter too!”).

The Galilei group is missing an important piece which makes it impossible to provide representations for systems of non-zero mass. When the mass (M) is included alongside the momentum (P) and kinetic energy (H), this leads to two invariants: P^2 - 2MH and M, itself.

After this inclusion is made, the Galilei group becomes the Bargmann group, which is equivalently described (BTW) as the restricted form of the 4+1 de Sitter group that leaves a null direction invariant.

In non-relativistic theory, non-trivial interactions can be incorporated by adding a potential (U) to H. With this inclusion, for a system with mass M = m, the invariant takes on the value -2mU.

In relativity, the roles played by M and U are reversed. M is present, but not U. The missing piece that needs to be added is now U, rather than M.

The central extension of the Poincare’ group is equivalently described as the restriction of the 4+1 de Sitter group that leaves a space-like direction invariant, and it has a continuously deformation to the Bargmann group.

The absence of U from the Poincare’ group is best seen as follows. The Poincare’ group has the total energy (E), and momentum (P). Splitting E into the kinetic part H and a boost-invariant part (mu): E = H + mu c^2 and replacing E by the “relativistic mass” M = E/c^2, the mass shell invariant can be written: P^2 - E^2/c^2 = P^2 - 2MH + H^2/c^2.

It is here that you can best see the correspondence with the Bargmann group. The Bargmann linear invariant is mu = M, while the Poincare’ linear invariant is mu = M - H/c^2.

The quadratic invariant goes from P^2 - 2MH to P^2 - 2MH + H^2/c^2. For systems with rest mass mu = m, this invariant is 0. There is no relativistic version of -2 mu U.

With the centrally extended Poincare’ group, however, the Lie bracket for the momentum P and moment K assumes the form [P, K] = H/c^2 + mu, with mu being the 11th generator added to the Lie algebra.

Now there is sufficient room to put in a relativistic version of U, by adopting the relation mu = m - U/c^2. As a result, the quadratic invariant becomes -2 mu U - U^2/c^2.

And it is at this point where not only the Raifeartaigh theorem breaks down, but also the Leutwyler Theorem, and probably also the the Haag Theorem and the Coleman-Mandula Theorem (this is what I’m presenting verifying).

The fact that the Poincare’ group has only “trivial” central extensions probably obscured this alternative, which has been hiding in plain sight since at least the time of Bakamjian and Thomas’s symplectic construction of non-trivial N-body relativistic dynamics in the 1950’s.

The object lesson to this is: trivial central extensions matter too! And therein lies the answer to the question originally posed: the alternative to the Poincare’ group is its central extension.

Posted by: Mark Hopkins on March 15, 2013 10:58 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Do we need Wigner’s classification? I agree this is an important piece of work that we should all study. However I anticipate that for the Standard Model and the three GUTs we are nominally studying that there are three representations of the Poincare group we need to construct. One for the fermions, one for the gauge bosons, one for the Higgs boson(s?). What I really mean here is that we need a representation of the Poincare group for each irrep of the gauge group.

Posted by: Bruce Westbury on March 24, 2009 8:17 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Bruce wrote:

Do we need Wigner’s classification? I agree this is an important piece of work th at we should all study. However I anticipate that for the Standard Model and the three GUTs we are nominally studying that there are three representations of the Poincare group we need to construct.

Right: the Standard Model and the GUTs we are studying only involves the massive spin-0 rep (Higgs), the massless spin-1/2 rep (fermions), and the massless spin-1 rep (gauge bosons).

To be pedantic: the massless reps each come in two versions, ‘left-handed’ and ‘right-handed’. The theories we are studying use both. But the left-handed and right-handed reps differ only by a mirror reflection, so the math works very much the same way.

To be even more pedantic: each massive rep is actually a continuum of different reps, parametrized by a ‘mass’ $m \gt 0$.

If I ever actually get around to discussing Wigner’s classification, I’ll start by describing these specific reps — since I don’t believe in talking about general results without looking at examples first.

Posted by: John Baez on March 24, 2009 11:21 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Since everyone else is being coy or shy I’ll stick my neck out.

Starting with finite groups where this theory is known as Clifford theory. Let $N$ be a normal subgroup of $G$. Then conjugation gives a homomorphism from $G$ to the automorphism group of $N$. Hence $G$ acts on the set of isomorphism classes of representations of $N$. Let $V$ be an (irreducible) representation of $N$. Define $I$ to be the stabiliser of the isomorphism class of $V$, a subgroup of $G$. This is known as the inertia group. Then $N$ is a subgroup of $I$. Also $V$ can be regarded as a representation of $I$. Then induce this representation of $I$ to a representation of $G$.

For Lie groups the theory is known as Mackey theory. The outline is similar but we are now working with Hilbert spaces so someone needs to explain what we mean by induction.

For the Poincare group the normal subgroup is Minkowski space $M$. The irreducible representations are one dimensional and correspond to the dual vector space. We identify $M$ with the dual vector space so we have a one dimensional representation for each point of $M$. Next we have to look at the orbits. The norm is invariant so we look at points with fixed norm. If the norm is positive then we have a single orbit. The stabiliser of a point is (the double cover of) $SO(1,2)$. If the norm is negative then we have a single orbit and the stabiliser of a point is $SO(3)$. If the norm is zero then there are three orbits, namely the future light cone, the zero vector, the past light cone. For the future and past light cones the stabiliser of a point is (the double cover of) the isometry group of the Euclidean plane. For the origin the stabliser is (the double cover of) the Lorentz group $SL(2,\mathbb{C})$.

Now the Poincare group is the semi-direct product of the Minkowski space and the Lorentz group. Then the inertia group is the semi-direct product of the Minkowski space and the stabiliser of the point. This means that inducing from the stabliser to the Lorentz group gives the same Hilbert space as inducing from the inertia group to the Poincare group.

I have not checked but I am guessing that the stabiliser is known as the little group.

It appears that the norm corresponds to the parameter JB refers to as mass. Negative mass is rejected at this stage as unphysical but there is a comment on this thread that we will come to them if we get as far as quantum field theory.

The next stage is to decompose these induced representations.

Posted by: Bruce Westbury on March 25, 2009 9:19 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Bruce wrote:

Since everyone else is being coy or shy I’ll stick my neck out.

Yay! Besides you, the only other people who have actually expressed an interest in working through the proof of Wigner’s theorem are Todd and I. So, we may move rather slowly, with lots of onlookers… but that’s fine.

Almost everything you write looks great; just a few minor comments for now.

Starting with finite groups where this theory is known as Clifford theory…

I hadn’t seen that term. It’s certainly good to understand the finite group case of ‘Mackey theory’, where the analysis and geometry go away and we’re left with algebra.

For the Poincaré group the normal subgroup is Minkowski space $M$. The irreducible representations are one dimensional and correspond to the dual vector space. We identify $M$ with the dual vector space so we have a one dimensional representation for each point of $M$. Next we have to look at the orbits. The norm is invariant so we look at points with fixed norm.

Let me introduce a bit of physics notation for what you’re saying, so we can bridge the two cultures.

The usual coordinates on $M$ are $t,x,y,z$: we call $t$ time and $x,y,z$ the three coordinates of space. A point with coordinates $(t,x,y,z)$ is called a spacetime point. The norm you’re talking about is called

$x \cdot x = x^2 + y^2 + z^2 - t^2$

although we should be very aware that most particle physicists use the opposite convention:

$x \cdot x = t^2 - x^2 - y^2 - z^2$

so many of their signs will be the opposite of yours.

The dual coordinates on the dual vector space $M^*$ are called $E,p_x,p_y,p_z$: we call $E$ energy and $p_x,p_y,p_z$ the three components of momentum. A point with coordinates $(E,p_x,p_y,p_z)$ is called an energy-momentum vector.

Following the particle physics sign convention, the norm on $M^*$ is called mass squared:

$m^2 = E^2 - p_x^2 - p_y^2 - p_z^2$

Now let me translate a few of your remarks into physics lingo and make a few comments.

Case 1.

If the norm is positive then we have a single orbit. The stabiliser of a point is (the double cover of) SO(1,2).

Right:

If the mass squared is negative then we have a single orbit, the one-sheeted hyperboloid

$E^2 - p_x^2 - p_y^2 - p_z^2 = m^2 \lt 0$

and the stabilizer of any point is the double cover of $SO_0(1,2)$.

The irreps we get from this case are called tachyons, because in physics they describe particles that go faster than light. These are considered unphysical for various reasons. In particular, the hyperboloid includes points where the energy $E$ is negative, so we will not get positive-energy irreps from this case.

Case 2.

If the norm is negative then we have a single orbit and the stabiliser of a point is [the double cover of] SO(3).

If the mass squared is positive then we actually get two orbits, the sheets of this two-sheeted hyperboloid:

$E^2 - p_x^2 - p_y^2 - p_z^2 = m^2 \gt 0$

and the stabilizer of any point is the double cover of $SO(3)$.

The irreps we get from this case are called tardyons, because in physics they describe particles that move slower than light.

The two orbits give two subcases: the positive-energy irreps coming from the sheet $E \gt 0$, and the negative-energy irreps coming from the sheet $E \lt 0$.

The positive-energy tardyons are of great importance in physics, and they’re usually just called massive particles.

The negative-energy tardyons are considered unphysical, though Dirac’s research on antimatter made the mistake of assuming there was some relation between antiparticles and negative energy, and this mistake lingers on in the public mind.

Case 3.

If the norm is zero then there are three orbits, namely the future light cone, the zero vector, the past light cone. For the future and past light cones the stabiliser of a point is (the double cover of) the isometry group of the Euclidean plane. For the origin the stabiliser is (the double cover of) the Lorentz group, SL(2,$\mathbb{C}$).

Right:

If the mass squared is zero we get three orbits, namely the subsets of the light cone

$E^2 - p_x^2 - p_y^2 - p_z^2 = m^2 = 0$

for which $E \gt 0$, $E = 0$ and $E \lt 0$, respectively. The stabilizers are as you’ve described them.

The irreps we get in this case are called luxons, since for the most part they describe particle that move at the speed of light.

The three orbits give three subcases:

The subcase of physical interest is, as usual, the case where $E \gt 0$, since this case gives positive-energy irreps. In the classification scheme I’m describing here, these are called positive-energy luxons — but physicists usually just call them massless particles.

The subcase $E = 0$ gives irreps of the Poincaré group that factor through the double cover of the Lorentz group. These irreps are just irreps of $SL(2,\mathbb{C})$, so we can skip this subcase if we want. Translations act trivially on these irreps, so in physics they don’t correspond to localized objects deserving to be called particles. Particle physicists generally ignore this subcase.

The subcase $E \lt 0$ corresponds to negative-energy luxons, which are again considered unphysical.

So, as you see, everything you wrote sounds much more impressive and mysterious when transcribed into physics terminology!

Posted by: John Baez on March 25, 2009 6:25 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

OK, since everybody else has gone quiet, I’ll step up. I’ve been cheating and looking at Shlomo Sternberg’s delightful Group Theory and Physics, so I think I can more or less explain the derivation of the Klein-Gordon equation.

As Bruce explains, the group $G = SL_2(\mathbb{C}) \ltimes \mathbb{R}^4$ acts on the space $\widehat{N}$ of irreducible representations of $N = \mathbb{R}^4$. The representation of $N$ corresponding to the point $p = (E, p_x, p_y, p_z)$ will be the unitary character

(1)$\chi_p: N \to \mathbb{T}, X \mapsto e^{-i g(p, X)},$

where $g(p, X)$ is shorthand for $E t - p_x x - p_y y - p_z z$, where $X = (t, x, y, z)$.

Next, we pick a point $p$ with $g(p, p) = m^2 \gt 0$ and $E \gt 0$. As John points out, such a point corresponds to a massive particle. The stabilizer of the point $p$ is isomorphic to $H = SU(2) \ltimes \mathbb{R}^4$. Since we are considering “spin-zero” representations, we want $SU(2)$ to act trivially. So the point $p$ corresponds to a representation of $H$ where $SU(2)$ acts trivially, and the $\mathbb{R}^4$ part acts like $\chi_p$. We’ll abuse notation and continue to call this representation $\chi_p$.

Now, we want to turn this representation of $H$ into a representation of $G$. We do this by constructing a certain vector bundle over $G / H$ and taking its $L^2$ sections (where integrability is taking with respect to an invariant measure). Specifically, we are looking at the vector bundle $E = G \times \mathbb{C} / H$, where $H$ acts on $G \times \mathbb{C}$ by $h \cdot (g, s) \mapsto (g h, \chi_p(h^{-1})s)$. $G$ acts on this vector bundle by left-multiplication on the left coordinate, and this action gives us our representation of $G$ on global sections $\Gamma(E)$.

I claim that this bundle is actually trivial. Define a function $S: G \to \mathbb{C}$ by $S(A X) = \chi_p(-X)$, where $A \in SL_2(\mathbb{C})$, $X \in N$. This function is equivariant with respect to the right action of $H$, so it defines a function $\sigma: G / H \to E$ by $\sigma(g H) = (g, S(g))$. We then get a bundle isomorphism from $E$ to the trivial bundle by sending a section $\gamma \in \Gamma(E)$ to the function $\gamma / \sigma$.

We have thus constructed a unitary representation of $G$ on $L^2(G / H)$. But $H$ was the stabilizer of the point $p$ in $\widehat{N}$, so what we have is a representation of $G$ on $L^2$ functions on the $E \gt 0$ branch of the hyperboloid $g(p, p) = m^2$ in $\widehat{N}$. We can then think of a function $f \in L^2(G / H)$ as a certain distribution on $\widehat{N}$ supported on this hyperboloid, i.e., a distribution of the form

(2)$u = f\delta(g(p, p) - m^2),$

where we assume that $f$ vanishes along the negative energy branch of the hyperboloid and is square integrable with respect to the invariant measure on the positive energy branch.

A distribution will be of the form (2) if and only if $(g(p, p) - m^2)u = 0$. Taking the inverse Fourier transform of this equation turns the multiplication operators $E^2$, $p_x^2$, $p_y^2$, and $p_z^2$ into the differentiation operators $-\frac{\partial^2}{\partial t^2}$, $-\frac{\partial^2}{\partial x^2}$, $-\frac{\partial^2}{\partial y^2}$, and$-\frac{\partial^2}{\partial z^2}$, respectively, so we recover the Klein-Gordon equation

(3)$\left(\frac{\partial^2}{\partial t^2} -\frac{\partial^2}{\partial x^2} -\frac{\partial^2}{\partial y^2} -\frac{\partial^2}{\partial z^2} + m^2 \right)(\mathcal{F}^{-1} u) = 0.$

We have thus realized our representation of $G$ on the space of solutions of the Klein-Gordon equation whose Fourier transforms satisfy certain constraints (namely, positive energy and integrability).

Anyway, I may have gotten a few details wrong, so please feel free to correct me as needed. A similar story holds for representations with nonzero spin, but with the additional complication that our vector bundle is nontrivial. But we should probably work the kinks out of the “easy” spin-zero case before moving on.

Posted by: Evan Jenkins on March 27, 2009 7:42 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Evan, thanks for this. This path to the Klein-Gordon equation is quite helpful.

I do have a question (for Evan or whoever is reading this), having to do with various uses of the word “trivial”.

There is on the one hand the trivial 1-dim representation of a group like $SU(2)$. This corresponds to a irrep of spin $0$.

There is on the other hand a “trivial bundle”, which is usually understood to mean a product bundle.

For the current discussion, let’s fix a mass $m \gt 0$. It seems to me that for any spin, the induced bundle, which lives over the hyperboloid sheet consisting of 4-momenta $p = (E, p_x, p_y, p_z)$ satisfying

$E^2 - p_{x}^2 - p_{y}^2 - p_{z}^2 = m^2, \qquad E \gt 0,$

will automatically be trivial in this second sense, because this hyperboloid sheet is contractible.

This seems to be in conflict with the sentence that includes the phrase “with the additional complication that our vector bundle is nontrivial”, unless the word ‘trivial’ there was actually meant in the first sense.

Just checking.

Posted by: Todd Trimble on July 14, 2011 7:08 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Todd wrote:

I do have a question (for Evan or whoever is reading this), having to do with various uses of the word “trivial”.

There is on the one hand the trivial 1-dim representation of a group like SU(2). This corresponds to a irrep of spin 0.

There is on the other hand a “trivial bundle”, which is usually understood to mean a product bundle.

I don’t know what Evan meant by “trivial”, but here’s a guess. He was talking about certain bundles $p: E \to B$ where a group $G$ acts on $E$ and $B$ and the map $p$ is equivariant. I guess these are called ‘equivariant bundles’ or something.

There’s probably a standard notion of ‘trivial equivariant bundle’ which is stronger than the notion of ‘trivial bundle’ you’re referring to. Maybe that’s what Jenkins was talking about.

Actually I can think of two candidates for such a notion. One is that $E \cong B \times V$ in the category of $G$-spaces, and $p: E \to B$ is the obvious projection. Another adds the condition that $V$ be a vector space on which $G$ acts trivially.

If you examine Jenkins’ argument, maybe you can see if he’s using ‘trivial bundle’ in one of these senses.

Posted by: John Baez on July 15, 2011 6:35 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

John B wrote:

Actually I can think of two candidates for such a notion. One is that $E\simeq B\times V$ in the category of $G$-spaces, and $p:E\to B$ is the obvious projection. Another adds the condition that $V$ be a vector space on which $G$ acts trivially.

I tend to use “equivariantly trivial” in the first sense: equivariant trivialization. I haven’t seen the second condition (triviality of action of $G$ on $V$) being imposed.

Posted by: Eugene Lerman on July 15, 2011 3:37 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Evan Jenkins wrote:

OK, since everybody else has gone quiet, I’ll step up. I’ve been cheating and looking at Shlomo Sternberg’s delightful Group Theory and Physics, so I think I can more or less explain the derivation of the Klein-Gordon equation.

Great! Todd has said he isn’t giving up, but he’s busy just now. So thanks, Evan, for jumping in and pushing the ball forward!

Everything you say looks right to me. So, all I can offer is a little bit of material that connects what you’re saying to some of what I wrote. A while back, I wrote something like this:

Stone’s theorem says:

Any strongly continuous unitary representation of $\mathbb{R}$ on a Hilbert space is of the form $exp(-i t A)$ for a unique (possibly unbounded) self-adjoint operator $A$ on this Hilbert space. Conversely, any such operator gives a strongly continuous unitary representation of $\mathbb{R}$ by this formula.

The spectral theorem says:

Suppose $A$ is a (possibly unbounded) self-adjoint operator on a Hilbert space. Then this Hilbert space is isomorphic to $L^2(X)$ for some measure space $X$, and making use of this isomorphism, $A$ becomes a multiplication operator:

$(A \psi)(x) = a(x) \psi(x)$

where

$a : X \to \mathbb{R}$

is some measurable function. And in this situation,

$(exp(- i t A) \psi)(x) = e^{-i t a(x)} \psi(x)$

Now, stick these theorems together — perhaps with a little glue — and see what happens!

If we do this, we get:

Let $X$ be a measure space and let

$a : X \to \mathbb{R}$

a measurable function. Then there is a strongly continuous unitary representation $U$ of $\mathbb{R}$ on $L^2(X)$ given as follows:

$(U(t)\psi)(x) = e^{-i t a(x)} \psi(x)$

Moreover, every strongly continuous unitary representation of $\mathbb{R}$ is unitarily equivalent to one of this form!

So, this is a nice concrete description of all the strongly continuous unitary representations of the very first Lie group to be born at the beginning of time: the real line.

Let’s look at a baby example. Let’s take $X$ to be single point! Then the function $a : X \to \mathbb{R}$ is really just a real number, and

$L^2(X) \cong \mathbb{C}$

So, we’re getting a 1-dimensional representation of $\mathbb{R}$, given by

$U_a(t) \psi = e^{- i t a} \psi$

Moreover, our theorem assures that every strongly continuous unitary representation of $\mathbb{R}$ is of this form!

If we leave out that annoying adjective ‘strongly continuous’, there are many more — at least if you believe in the Axiom of Choice. But let’s not go there. From now, in this thread I’ll use rep to mean ‘strongly continuous unitary representation’.

Okay, now let’s return to the case of a general measure space $X$. If we pick a measurable function $a : X \to \mathbb{R}$, we get a rep $U$ of the real line thanks to our theorem. But each point of $X$ is giving us a one-dimensional rep of the real line. And, I hope you see there’s some sense in which our rep $U$ is ‘built’ from all these 1-dimensional reps.

What is this sense, exactly? If $X$ is a finite set, our rep $U$ is a direct sum of these 1-dimensional reps $U_{a(x)}$, one for each point $x \in X$. But in general, $U$ is a direct integral of these 1-dimensional reps.

You may not know what a ‘direct integral’ is, but the point is that $L^2(X)$ is built from lots of copies of $\mathbb{C}$, one for each point of $x$, in a way that involves integrals. And the same is true of our rep $U$ of the real line on $L^2(X)$! It’s fun to work out exactly what’s going on here… and then direct integrals will lose their terror, because they’re all pretty much like this.

Now, maybe someone can step up the plate and tackle some of these:

1. Take the rep of the real line on $L^2(\mathbb{R})$ that acts by translating functions:

$(V(t) \psi)(x) = \psi(x - t)$

Hit it with our classification of reps of the real line and see what you get!

2. Guess which 1-dimensional reps $V$ is a direct integral of. What famous thing are we secretly talking about here?
3. Generalize Stone’s theorem and the spectral theorem from $\mathbb{R}$ to $\mathbb{R}^n$. What’s the classification of reps of the additive group $\mathbb{R}^n$? What do the one-dimensional reps look like?
4. Generalize problem 1 to $\mathbb{R}^n$. Take this rep of $\mathbb{R}^n$ on $L^2(\mathbb{R}^n)$:

$(V(y) \psi)(x) = \psi(x - y)$

where now $x,y \in \mathbb{R}^n$. Hit it with your classification of reps of $\mathbb{R}^n$ and see what it looks like.

5. Generalize problem 2 to $\mathbb{R}^n$. What famous thing are we talking about now?
6. Why is all this incredibly closely linked to what Evan was just talking about? How does the hyperboloid

$E^2−p_x^2−p_y^2−p_z^2=m^2$

get into the game? How does the Klein–Gordon equation get into the game? Can you see why $L^2(\mathbb{R}^4)$ is a direct integral of Hilbert spaces of the form $L^2(hyperboloid)$?

Posted by: John Baez on March 30, 2009 12:09 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Just a quick note to say: yes, you’re absolutely right, I haven’t given up, far from it! I’ve actually been doing some thinking and reading as time allows; I also had a longish response to the very useful comments you and Evan have been making, but I accidentally erased it (after having the window open for a few days, gestating – very annoying).

And yes, I’m very distracted and busy with other non-mathematical things, I’m sad to say. I really want to get back to this conversation and carve out the blocks of time to do so. But I’m still hanging around, listening in. Please continue!

Posted by: Todd Trimble on March 30, 2009 3:00 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Glad you’re still around! Personally I’d prefer seeing a short, sketchy post from you soon to having you ponder a post for a week and then accidentally delete it.

What I mean is: I think of this as a conversation, and I fully intend to make lots of dumb mistakes — so I’d be upset if everyone else waited until their comments were perfectly polished before saying anything.

Posted by: John Baez on March 30, 2009 7:08 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Just popping my head round the door to say that I’m still interested in this thread even if I don’t feel able to contribute (twin problems of time and timidity).

And of course the glib answer to Q1-5 (though it doesn’t give the details, which I guess is what you’re getting at) is either “Fourier transform” or “C*(Rn) = C0(Rn)”

Posted by: Yemon Choi on March 30, 2009 5:29 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

The glib answer is indeed ‘Fourier transform’. If the details are obvious to everyone then we don’t need to go into them, but otherwise we probably should. Trying to understand the representation theory of the Poincaré group without full control of the Fourier transform is not a good idea, since the Fourier transform is precisely the representation theory of the ‘easy’ part of the Poincaré group: the translation group.

Posted by: John Baez on March 30, 2009 7:12 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

1. Applying the two theorems gives: There exists a measurable space X, and an isomorphism F:L_2(R)->L_2(X) such that V(t)\psi = F^{-1}(exp(-ita)F\psi) for some measurable function a on X.

Posted by: Gantumur on March 30, 2009 8:12 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Gantumur wrote:

Applying the two theorems gives: There exists a measurable space $X$, and an isomorphism $F:L^2(\mathbb{R}) \to L^2(X)$ such that

$V(t)\psi = F^{-1}(exp(-i t a)F\psi)$

for some measurable function $a$ on $X$.

Right!

In fact, $X = \mathbb{R}$ and $a(x) = x$. $F$ is a version of the Fourier transform.

And now we see a flaw in how I stated the spectral theorem. It tells us that the space $X$ and the function $a$ exist, but it doesn’t say what they are.

We could state a stronger version of the spectral theorem that would remedy this defect… but for now maybe we should skip this step and believe that $F$ is the Fourier transform.

Posted by: John Baez on April 4, 2009 2:36 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

The $X$ here is the spectrum of $A$? Which can be identified with a subspace inclusion $i: X \hookrightarrow \mathbb{R}$ (with the measurable function $a: X \to \mathbb{R}$ being this very inclusion)?

Posted by: Todd Trimble on July 11, 2011 9:14 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Todd wrote:

The $X$ here is the spectrum of $A$? Which can be identified with a subspace inclusion $i: X \hookrightarrow \mathbb{R}$ (with the measurable function $a: X \to \mathbb{R}$ being this very inclusion)?

Wow, you’re asking a question about something I said two years ago! Let me try to remember what we were talking about…

Initially, I said:

The spectral theorem says:

Suppose $A$ is a (possibly unbounded) self-adjoint operator on a Hilbert space. Then this Hilbert space is isomorphic to $L^2(X)$ for some measure space $X$, and making use of this isomorphism, $A$ becomes a multiplication operator:

$(A \psi)(x) = a(x) \psi(x)$

where

$a : X \to \mathbb{R}$

is some measurable function. And in this situation,

$(exp(- i t A) \psi)(x) = e^{-i t a(x)} \psi(x)$

Here $X$ is not the spectrum of $A$, except when the ‘multiplicity’ of each point in the spectrum is 1.

To see what I mean, let $A$ be an $n \times n$ self-adjoint matrix. If each eigenvalue of $A$ has multiplicity 1, then we can take $X$ to be the spectrum of $A$. Otherwise, we can’t. Consider for example:

$A = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$

The spectrum of $A$ contains one point, but we need $X$ to contain two points.

This is why I later said:

And now we see a flaw in how I stated the spectral theorem. It tells us that the space $X$ and the function $a$ exist, but it doesn’t say what they are.

You can fix this flaw, but it’s not as easy as simply taking what I said and adding the remark that $X$ is the spectrum of $A$ and $a: X \to \mathbb{R}$ is the inclusion. That’s why I was, and am still, too lazy to fix the flaw. But I could be persuaded to fix it if I thought someone really needed to know how. In general $X$ is some kind of ‘bundle’ of measurable spaces over the spectrum of $A$, where the cardinality of the fiber, the ‘multiplicity’, can vary from point to point.

Posted by: John Baez on July 13, 2011 4:59 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Oh, okay – thanks. I’m working right now on a long post which tries to bring together and summarize some thoughts of mine and others in this long, long thread, and it will vaguely touch upon this point. (But more should be said.)

Posted by: Todd Trimble on July 13, 2011 3:23 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Two standard references for spectral multiplicity theory are

• Paul R. Halmos: “Introduction to Hilbert Space and the Theory of Spectral Multiplicity”

and

• Dunford and Schwartz: “Linear Operators Part II: Spectral Theory”

which also covers unbounded self adjoint operators.

Posted by: Tim van Beek on July 13, 2011 7:15 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

By the way, one reason I was being so reluctant to describe the space $X$ in my statement of the spectral theorem here is that I don’t know a functorial way to build it. But yesterday I came up with a way that might work. It’s nonfunctorial but that may be unavoidable. Take the operator $A$ and choose a maximal abelian algebra of bounded operators commuting with $A$. Since $A$ is self-adjoint this algebra is a commutative C*-algebra; take $X$ to be its Gelfand spectrum.

Posted by: John Baez on July 14, 2011 3:07 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Is there not something you could do with the groupoid of maximal abelian algebras of bounded operators commuting with $A$?

Posted by: David Corfield on July 14, 2011 8:54 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Perhaps. Or the category of abelian subalgebras containing $A$. And that starts reminding me of Chris Heunen and Bas Spitter’s paper, as explained in week257.

Posted by: John Baez on July 14, 2011 9:05 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Or the category of abelian subalgebras containing A. And that starts reminding me of Chris Heunen and Bas Spitter’s paper…

…on Bohrification.

By the way, a student of mine is currently looking into the generalization of this from quantum mechanics to quantum field theory. There a possibly interesting interplay between causal locality in spacetime on the on hand and Bohr-locality by classical contexts on the other appears: a net of observables of a quantum field theory is causally local precisely if the presheaf of ringed toposes assigned to it by Bohrification satisfies spatial descent by local geometric morphisms. The details are here.

Posted by: Urs Schreiber on July 15, 2011 8:01 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Jim wrote:

Is there something on this blog or the nLab addressing the ‘no go’ theorem asserting that there can not be particles of spin > 2 - which assumes Lie algebra reps hiding in the formulation.

For starters, everything in this blog entry is about a single noninteracting relativistic quantum particle. This context is called ‘relativistic quantum mechanics’. And in this context, particles of any spin $0, \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, \dots$ are allowed.

That famous ‘no-go theorem’ says that there’s no way to get particles of spin > 2 to interact with other particles in a renormalizable way when spacetime has dimension 4. This is a result about quantum field theory, rather relativistic quantum mechanics. And this result tends to get passed along as a ‘folk theorem’: I’ve never actually seen a precise statement together with a proof. I would love to see one.

I’m sure someone has written about this result carefully somewhere, but I think it would hard to exclude all loopholes: I think you’d need to rule out ‘unexpected cancellations of divergences’. People are still finding unexpected cancellations of divergences in supergravity, where you have particles of spin 2. So, I’d be a bit surprised if there were a fully rigorous proof of this ‘no-go theorem’ at present!

Posted by: John Baez on July 15, 2011 7:03 AM | Permalink | Reply to this

### Higher Unitary Representations of the Poincaré Group

This may not be the right place for this comment so transfer if need be. One of the posts referred to spin reps. Is there something on this blog or the nLab addressing the ‘no go’ theorem asserting that there can not be particles of spin > 2 - which assumes Lie algebra reps hiding in the formulation.

Posted by: jim stasheff on July 14, 2011 2:02 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I wonder if anyone would like to say a bit about the representation for non-zero spins. In Weinberg’s The Quantum Theory of Fields, Section 2.5, this is cooked up in a workable but slightly unsatisfying way: as well as picking a representative momentum vector in each orbit, for every other momentum vector in the orbit you have to nominate a “standard” Lorentz transformation that takes the representative momentum to the specified one. Then you use this to map the representation of the isotropy subgroup to different fibres.

Sternberg’s treatment seems more elegant … but I can’t quite figure out what he’s doing! On p138 he lists all the steps, but in step (2) he gives a representation of the semidirect product of the isotropy subgroup and the Abelian normal subgroup, then step (3) just says “construct the corresponding vector bundle and let $G$ act on its sections”. But I can’t see where the action of the full group $G$ on the whole bundle comes from; for every $g\in G$, the action ought to give you a linear map between fibres, but … where exactly do those maps come from?

Posted by: Greg Egan on April 15, 2009 1:11 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I wrote:

the action ought to give you a linear map between fibres, but … where exactly do those maps come from?

D’oh! Sternberg explains all of this back on p105.

Posted by: Greg Egan on April 15, 2009 3:25 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

For the benefit of anyone without their own copy of Sternberg’s Group Theory and Physics (helpfully catalogued by my local bookstore as Group Therapy and Physics), here’s some of the detail.

First, I guess it’s worth sketching in general how induced representations work. Given a group $G$ with a subgroup $H$, and a representation $s$ of $H$ on a vector space $V$, we define a left action of $H$ on $G\times V$ by $h\cdot (g, v) = (g h^{-1}, s(h)v)$. We write $[(g,v)]$ for the orbit, or equivalence class, that contains $(g,v)$.

We then define $E = (G\times V)/H$ as the set of orbits of that action of $H$, $M = G/H$ as the set of right cosets of $H$, and the projection $\pi: E\to M$ by $\pi ([(g,v)]) = g H$, where of course it makes no difference if we re-describe the orbit $[(g,v)]$ as $[(g h^{-1}, s(h)v]$ for any $h\in H$ because $(g h^{-1}) H = g H$.

For each $x\in M$, choose $g$ to be any element of $G$ such that $x = g H$. Define $E_x = \pi^{-1}(x)$, and $\phi_g:V\to E_x$, $\phi_g(v) = [(g,v)]$.

The map $\phi_g$ is onto: for any $[(k,w)]\in E_{(g H)} = \pi^{-1}(g H)$, we have $k=g h_1^{-1}$ for some $h_1\in H$, so $k^{-1} g\in H$, $(k^{-1} g)\cdot (g, s(g^{-1} k)w) = (k,w)$, so $\phi_g(s(g^{-1} k)w) = [(g, s(g^{-1} k)w)] = [(k,w)]$.

The map $\phi_g$ is one-to-one: if $\phi_g(v) = \phi_g(w)$, then $[(g,v)]=[(g,w)]$, so for some $h_1\in H$, we have $h_1\cdot (g,v) = (g,w)$, or $(g h_1^{-1}, s(h_1)v) = (g,w)$; equating the first coordinates requires $h_1=e$, and $s$ is a representation so $s(e)=1_V$, and $v=w$.

Since $\phi_g$ is a bijection between $E_x$ and the vector space $V$, we can make $E_x$ into a vector space by defining $\alpha p + \beta q \equiv \phi_g(\alpha \phi_g^{-1}(p) + \beta \phi_g^{-1}(q))$, for all $\alpha, \beta \in \mathbb{R}, p, q \in E_x$. But is this independent of our choice of $g$? If we chose $g h$ instead of $g$, we’d have $\phi_{g h}(v) = [(g h,v)] = [(g, s(h)v)] = \phi_g(s(h)v)$, so $\phi_{g h}=\phi_g\circ s(h)$, and $\phi_{g h}^{-1}=s(h^{-1})\circ \phi_g^{-1}$. Then:

$\phi_{g h}(\alpha \phi_{g h}^{-1}(p) + \beta \phi_{g h}^{-1}(q)) = (\phi_g\circ s(h))(\alpha (s(h^{-1})\circ \phi_g^{-1})(p) + \beta (s(h^{-1})\circ \phi_g^{-1})(q)) = \phi_g(\alpha \phi_g^{-1}(p) + \beta \phi_g^{-1}(q))$

in agreement with our original definition.

We define the action of $G$ on $E$ by $g_1\cdot [(g,v)] = [(g_1 g,v)]$, or in other words $g_1\cdot \phi_g(v) = \phi_{g_1 g}(v)$. We then have:

$\pi(g_1\cdot [(g,v)]) = \pi[(g_1 g,v)] = (g_1 g) H = g_1\cdot (g H) = g_1\cdot \pi([(g,v)])$

That is, $\pi$ is a $G$-morphism. This also means that the action maps fibres to fibres, $g_1:E_{(g H)}\to E_{g_1\cdot (g H)}$. What’s more, the action of $g_1$ restricted to the fibre $E_{(g H)}$ is $\phi_{g_1 g}\circ \phi_g^{-1}$, passing from $E_{(g H)}\to V \to E_{g_1\cdot (g H)}$, and this is linear simply by virtue of the way we’ve defined the vector space operations on the $E_x$.

We get a representation $r$ of $G$ on the vector space $\Gamma(E)$ of sections of the bundle $E$ by:

$(r(g_1)f)(x) = g_1\cdot f(g_1^{-1}\cdot x)$

Now, finally getting back to the Poincaré group, we’ve chosen an orbit in the space of 4-momentum vectors, and a representative vector $p$. Each 4-momentum $p$ gives us a unitary character (or 1-dimensional irrep) $\chi_p$ of the translation group $\mathbb{R}^4$. We define $L_p$ to be the subgroup of $SL(2,\mathbb{C})$ that fixes $p$, then we take $H = L_p \ltimes \mathbb{R}^4$ as the subgroup of $G = SL(2,\mathbb{C}) \ltimes \mathbb{R}^4$ in the previous construction. Given a representation $\rho$ of $L_p$ on some vector space $V$, we get the representation $s$ of $H$ by defining $s(k x)v=\chi_p(x)\rho(k)v$, for $k\in L_p, x\in \mathbb{R}^4$.

I’ve run out of energy, but if I’ve grasped all this correctly the underlying fibre bundle is independent of the choices of coset representatives needed in the construction, and any difference in choices amounts to a gauge transformation.

Posted by: Greg Egan on April 16, 2009 8:28 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Greg wrote:

I wonder if anyone would like to say a bit about the representation for non-zero spins…

Hi! I’d been a bit annoyed at how this thread had gone dead. Now, looking back at it, I see there was plenty left to say about Evan Jenkin’s post — I’d just forgotten to do it. And now you’ve joined in! So, there’s plenty for me to talk about… but my laptop died a while back, and this computer in the engineering building at Glasgow isn’t loaded with the necessary fonts, so it’s tough to read what people have written. So, I’ll just say a little bit now.

The ‘induced bundle’ trick is indeed crucial to understanding representations of the Poincaré group. In generality, we can start with a Lie group $G$ acting smoothly and transitively on a smooth manifold $M$. The stabilizer of your favorite point $x \in M$ will be a Lie subgroup $H \subseteq G$, and we have

$M \cong G/H$

Starting from this, there’s a process that takes any representation $s$ of $H$ on a vector space $V$ and turns it into a vector bundle $E$ over $M$ — the so-called ‘induced bundle’. Even better, the group $G$ acts on this bundle, and the projection

$\pi : E \to M$

gets along with the action of $G$:

$\pi(g e) = g \pi(e) .$

So, we say the $E$ is a $G$-equivariant vector bundles over $M$.

You’ve described how this works, so let me just gild the lily a bit.

First, the ‘process’ I described is actually a functor.

There’s a category

$Rep(H)$

of linear representations of $H$, and a category

$Vect(M,G)$

of $G$-equivariant vector bundles over $M$. The induced bundle construction gives a functor

$L: Rep(H) \to Vect(M,G)$

But, if you think about it, you’ll notice there’s also a functor going back the other way:

$R: Vect(M,G) \to Rep(H)$

If you give me a $G$-equivariant vector bundle $E$ over $M$, I can take its fiber over your favorite point $x$, and I get a vector space — and this becomes a representation of the stabilizer group $H$, thanks to how $G$ acts on $E$.

This functor is simpler than the induced bundle construction!

Whenever we have functors going both ways between two categories, we should suspect that they’re adjoints. The simpler functor often amounts to ‘forgetting’ something. This forgetful functor is usually the right adjoint. It’s partner going the other way, the left adjoint, usually involves ‘constructing’ something instead of ‘forgetting’ something.

And indeed, that’s what’s happening here! Technically, what I’m saying is that

$hom(L V, F) \cong hom(V, R F)$

Here $V$ is a representation of $H$ — I hope you forgive me for calling it $V$, which is the name for the vector space on which $G$ acts, instead of the more pedantic full name for a representation, which is something like $s: G \to GL(V)$.

Similarly, $F$ is a $G$-equivariant vector bundle over $M$ — and again I hope you forgive me for calling it this instead of something like $\pi : F \to M$, or something even more long-winded that gives a name to how $G$ acts on $F$ and $M$.

$L V$ is the induced bundle corresponding to $V$.

$R F$ is the fiber of $F$ over your favorite point $x$, which becomes a representation of $G$.

And this:

$hom(L V, F) \cong hom(V, R F)$

says that $G$-equivariant vector bundle maps from $L V$ to $F$ are in natural 1-1 correspondence with intertwining operators from $V$ to $R F$.

This might be fun to check.

But my real point is this: whenever you see any sort of ‘forgetful’ process, you should wonder if it has a left adjoint, a construction which in some loose sense is the ‘reverse’ of forgetting. Why? Because these left adjoints tend to be important.

Endowed with this heuristic, as soon as you see there’s a rather obvious ‘forgetful’ process that takes a $G$-equivariant vector bundle over $M$ and gives a representation of $H$ on the fiber over $x \in M$, you will seek the ‘reverse’ process — and then you’ll rediscover the induced bundle construction!

And why is this so great? Well, there’s also a process that takes any representation of $G$ and restricts it to a representation of $H$:

$R': Rep(G) \to Rep(H)$

And this too, has a left adjoint:

$L' : Rep(H) \to Rep(G)$

which is called the induced representation trick.

Given all this, it shouldn’t be a shock that induced representations are very closely related to induced bundles!

And given that, it shouldn’t be a shock that when $G$ is the Poincaré, we can use induced bundles to get representations of the Poincaré starting from representations of a smaller group $H$.

This is how I like to think about what you’ve just explained. Of course there’s a lot more to say. For example, we should take the functors $L,R,L',R'$ and try to fit them all into some commutative diagram that summarizes what we’re doing. But maybe someone else can take a crack at that.

Posted by: John Baez on April 16, 2009 1:19 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

It’s great that Greg and John go through the trouble of typing all these detailed technical explanations into blog comments.

Might it be a good idea to type the explanations directly into an $n$Lab entry? There it would be archived in a form that increases the chance that some time from now people will still find and profit from the result of this effort.

Posted by: Urs Schreiber on April 16, 2009 2:17 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Hmm, I think of this as more of a conversation or ‘class’ than an ‘entry’. But perhaps I should conduct future ‘classes’ in the $n$Lab, rather than here. That would probably decrease participation (bad), but make it easier for the end result to be a nicely written explanation of the subject (good).

Posted by: John Baez on April 16, 2009 2:23 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I mean, Greg for instance effetcively wrote an entry, above “What is an induced representation?”

One could just copy-and-past this into an entry [[induced representation]] over on the nLab. It seems to be precisely the kind of material from blog discussion that we would want to archive on the nLab.

And this doesn’t have to move the discussion from here to the $n$Lab and hence to kill it here. We could just supply here a link to the $n$Lab entry.

In two years, it is unlikely that people will be able to systematically find informaiton buried in the long threads here. The $n$Lab would make it easier to find and navigate the information which we do accumulate. And the $n$Lab entry can itself still link back to the discussion here. We just did it that way for [[co-Yoneda lemma]], for instance, for which the discussion tok place almost entirely on the blog, but the result it now archived in a coherent form on the $n$Lab, which in this form cannot be found on the blog.

Posted by: Urs Schreiber on April 16, 2009 2:35 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

About 6 months ago, Urs wrote with regard to a detailed and helpful comment of Greg Egan:

Might it be a good idea to type the explanations directly into an nLab entry? There it would be archived in a form that increases the chance that some time from now people will still find and profit from the result of this effort.

Urs already took the step of pasting some of this in the nLab (under induced representation). My own interest in getting back to unitary irreps of the Poincaré group has been reawakened a bit recently (thanks to a kind soul who emailed me recently), and so I’d like to try to get started on that in the nLab.

I’ve made a start on this by having a go at writing up Poincaré group in the nLab. I’m on something of a learning curve here, so experts should feel free to pop by.

Posted by: Todd Trimble on October 19, 2009 2:52 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I have already gotten an email from a prospective grad student saying this thread is the best modern online treatment of unitary irreps of the Poincaré group. I wonder if that was the same kind soul who emailed you…

Anyway, if this disorganized thread is the best thing online so far, even a halfway decent $n$Lab entry is bound to be a really valuable resource!

Posted by: John Baez on October 19, 2009 8:49 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Anyway, if this disorganized thread is the best thing online so far

Posted by: Todd Trimble on October 19, 2009 9:19 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Based on Greg’s comment above I have started an $n$Lab entry

$n$Lab: induced representation

Posted by: Urs Schreiber on April 16, 2009 3:05 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I’ve pasted in John’s explanation before Greg’s account.

It would be very useful sifting through old posts for good exposition. Perhaps we could have a place to record discoveries of good exposition to be extracted to nLab. For the three of us with access to the itex, it would be easier for us to do this extraction.

Posted by: David Corfield on April 17, 2009 12:56 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I meant to mention this before, but your comment is along similar lines. It would be GREAT if there were an “extract itex” feature or something. On occasions, I’ve copied the rendered equations and pasted them into either nLab or comments here. The results sometimes contain itex commands along with a lot of gibberish.

An “extract itex” feature would assist anyone in transferring material from here to nLab. Not to mention, assist in quoting material here in comments.

Just a thought.

Posted by: Eric on April 17, 2009 4:22 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Thanks, David, for adding further material to the entry!

Eric writes:

It would be GREAT if there were an “extract itex” feature or something. On occasions, I’ve copied the rendered equations and pasted them into either nLab or comments here. The results sometimes contain itex commands along with a lot of gibberish.

Sure, good point. I can’t implement this in the software, but I do offer to extract the source code of any comment or entry that you ask me for.

Just send me a request by email with the link and/or the precise date on which it was posted.

In the long run, what would be best were if everyone who feels like posting an exposition or explanation of anything to the blog would (create if necessary and then) fill it into the respective $n$Lab entry. Then one can either include in the blog comment a link to that or just copy-and-paste the same into the comment here. But the point is that the version on the $n$Lab will live on and improve over time, whereas the version here on the blog is likely to be simply forgotten after a while.

Posted by: Urs Schreiber on April 17, 2009 4:38 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Urs wrote:

In the long run, what would be best were if everyone who feels like posting an exposition or explanation of anything to the blog would (create if necessary and then) fill it into the respective nLab entry.

I’ll try to do that in the future, though I must admit there’s a certain psychological barrier for an amateur like me: posting something in a blog discussion means “this is me desperately hoping I know what I’m talking about”, whereas inserting something in an nLab entry seems to be making a stronger claim to be correct and authoritative. I guess I’ll just have to try to find the right path to steer between wasting exposition on blog comments that will never be polished or expanded or referred to again … and the risk of spamming the nLab with low-quality doodling.

Posted by: Greg Egan on April 18, 2009 4:23 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I must admit there’s a certain psychological barrier for an amateur like me:

Thanks for saying this, that’s good to know. I must be that other $n$-Café regulars have similar reservations, and I am hoping that we can eventually removed these barriers.

posting something in a blog discussion means “this is me desperately hoping I know what I’m talking about”, whereas inserting something in an nLab entry seems to be making a stronger claim to be correct and authoritative.

Yes, i understand. And in the end we are hoping the $n$-Lab to be a source for reliable information. Among other things: we also want the $n$Lab to be a place where we can jointly develop ideas, expositions and whatever else the content of an entry may. This necessitates to allow for imperfect intermediate stages. The gain is that eventually and incrementally what once was imperfect becomse more refined and may give rise to further useful material.

I don’t know if you followed the activity we had on the $n$Lab in the last months, as logged at latest changes. But if you do that, you’ll see that we regularly re-examined entries and improved and corrected them if necessary – and it is often necessary.

Another thing we do is that we drop a remark on how robust we think the material is. Like “I think this is true, but please check.”

If you have such reservations about material you enter, or else if you are worried that existing material you come across may not be fully correct, a good thing to do is to leave a “query box” as described on the HowTo page.

You can find an example for a recent disucssion about whether or not something is right at semi-abelian cartegory. But there are many more entries which contain in green query boxes disucssions about what’s really going on.

The big advantage of all this is that in the end it increases the chance that there actually is a net progress. This is something I see lack of in our blog activity: the net progress.

For instance you might enjoy noticing that a new contributor, Richard Body, kindly implemented a small correction to your exposition of induced representation on the $n$Lab: see the red/green colored pieces here.

This is good, I’d say. On the blog such a small correction would probably be not considered worth an extra comment, and even if it would be hard for later readers to piece together a comment and its erratas.

So, please don’t be shy with contributing to the $n$Lab. There more of us do, the better it will become.

(That reminds me: there is still an entry light mill waiting to be expanded…)

Posted by: Urs Schreiber on April 18, 2009 3:20 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

helpfully catalogued by my local bookstore as Group Therapy and Physics

You're being ironic, but that's a feature, not a bug. By thus misspelling the title in the catalogue, the publisher dramatically increases the chances that bookstores will choose to pick it up. <g>

Posted by: Toby Bartels on April 17, 2009 12:03 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

M x G-equivariant vector bundle F, π1: FM Representation R(F) of H on π1–1(x) h π1 h π1 h π1 h G / H G-equivariant vector bundle L(V), π2: (G×V) / HG / H Representation of H on V h π2 h π2 h π2 h R Restrict to action of H on a single fibre π1–1(x) L Construct induced bundle h ⋅ (g, v) = (g h–1, h v) π2: (G×V) / HG / H π2([(g, v)]) = g H g1[(g, v)] = [(g1g, v)] Intertwiner i i : π1–1(x) → V i h = h i Vector bundle morphism (f, m) f : FL(V) m : MG / H m π1 = π2 f f g = g f

NB: I’m holding on to the SVG and itex files for this comment, so if I get everything clear I’ll add it to the nLab entry.

John wrote:

$G$-equivariant vector bundle maps from $L V$ to $F$ are in natural 1-1 correspondence with intertwining operators from $V$ to $R F$. This might be fun to check.

The Lie group $G$ acts smoothly and transitively on the smooth manifold $M$. We have chosen a point $x\in M$, and $H\subseteq G$ is the subgroup that stabilises $x$.

(1) Given an intertwiner $i:\pi_1^{-1}(x)\to V$ in the above diagram, how do we construct a vector bundle morphism from $F$ to $L(V)$?

Since $G$ acts transitively on $M$, we can always define a function $k : M\to G$ such that $k(y) x = y$. [I guess we can also make $k$ smooth given that the action of $G$ is smooth, though it’s not immediately obvious to me how to do that.] Note that for any $g\in G$, $k(g y)^{-1} g k(y) x = x$, so $k(g y)^{-1} g k(y)$ will lie in $H$.

We define the map $m : M \to G/H$ by $m(y) = k(y) H$, for each $y\in M$. Note that since $k(x)\in H$, we have $m(x) = e H$, where $e$ is the identity of $G$.

We define the map $f : F \to L(V)$ by $f(w) = [(k(\pi_1(w)), i(k(\pi_1(w))^{-1} w) )]$ for each $w\in F$. Because $k(\pi_1(w)) x = \pi_1(w)$, $k(\pi_1(w))^{-1}$ will map the entire fibre to which $w$ belongs to $\pi_1^{-1}(x)$, the domain of the intertwiner $i$. And we have:

$\pi_2(f(w)) = k(\pi_1(w)) H = m(\pi_1(w))$

The map $f$ is a linear map between the fibres $\pi_1^{-1}(y)$ and $\pi_2^{-1}(m(y))$, because, along with the linearity of $i$, the vector space structure on the fibres of $L(V)$ is defined so all maps of the form $v\to [(g,v)]$ are linear. So, $m$ and $f$ together give us a vector bundle morphism from $F$ to $L(V)$. In order to be a morphism in the category of $G$-equivariant vector bundles, $f$ should also commute with the action of $G$. We have:

$f(g w) = [(k(\pi_1(g w)), i(k(\pi_1(g w))^{-1} g w) )] = [(k(g \pi_1(w)), i(k(g \pi_1(w))^{-1} g w) )]$

Let’s abbreviate $\pi_1(w)$ as $y$ and define $h=k(g y)^{-1} g k(y)$, which as we’ve noted above must lie in $H$. Then we have:

$f(g w) = [(k(g y), i(h k(y)^{-1} w) )] = [(k(g y), h i(k(y)^{-1} w) )] = [(k(g y) h, i(k(y)^{-1} w) )]$

$= [(g k(y) , i(k(y)^{-1} w) )] = g\cdot [(k(y) , i(k(y)^{-1} w) )] = g\cdot f(w)$

(2) Given a vector bundle morphism from $F$ to $L(V)$, consisting of maps $m : M \to G/H$ and $f : F \to L(V)$, how do we construct an intertwiner $i:\pi_1^{-1}(x)\to V$?

Suppose, initially, that $m(x)=e H$. If we restrict $f$ to the fibre $\pi_1^{-1}(x)$, it will give us a linear map to the fibre $\pi_2^{-1}(e H)$ of $L(V)$. There is a linear bijection $\phi_e:V\to \pi_2^{-1}(e H)$, defined by $\phi_e(v)=[(e,v)]$, and we can put $i = \phi_e^{-1}\circ f$. Then $i$ will be linear, and for any $w\in \pi_1^{-1}(x)$ and any $h\in H$, we have $i(h w) = \phi_e^{-1}(f(h w)) = \phi_e^{-1}(h f(w))$. Now if $f(w)$ is the conjugacy class $[(e,v)]$ then $h f(w) = [(h, v)] = [(e, h v)]$. So $i(h w) = \phi_e^{-1}(h f(w)) = h v = h i(w)$.

But I’m stuck on the case $m(x)\ne e H$. You’d think it would be easy enough to construct an intertwiner between $\pi_1^{-1}(x)$ and the fibre $\pi_1^{-1}(m^{-1}(e H))$ that we could compose with our previous construction, but the group action on $F$ gives linear maps between fibres that are not intertwiners unless $G$ is Abelian. Any suggestions?

Posted by: Greg Egan on April 20, 2009 12:24 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Greg,

amazingly nice diagram! it induces a new vision of what the $n$Lab might look like one fine day.

Posted by: Urs Schreiber on April 20, 2009 8:19 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

The point is to not wait until everything is perfect before including it. Doing that keeps others from being able to help. If there is an error on anything that gets on the nLab, an expert can surely fix it. No need to be perfect.

PS: Very cool diagrams! I tried without much success making diagrams in SVG, so having this on nLab would also serve as a tutorial for SVG, i.e. I can copy it and modify it.

Posted by: Eric on April 20, 2009 3:44 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I did say I’d put it on nLab soon; I’ve got the source files, I don’t need other people to do this. But at this precise moment I think it’s worth waiting for some further discussion on this thread.

One point I’d welcome any comments on is this slightly mysterious business of the map I call $k: M\to G$, with $k(y)x=y$. Because the action of $G$ on $M$ is transitive it’s obvious that there’s such a function, and it generally won’t be unique. But it’s not clear to me whether there’s some nice way to construct a specific $k$, say one that would be guaranteed to be smooth if the action of $G$ is smooth. Weinberg has a map $L$ with a similar purpose in his discussion of the Poincaré irreps, and offers some mystifying advice and prescriptions about it as if it ought to be obvious what he’s doing and why … but it’d be great to hear from anyone who can illuminate this whole business in a more general context.

Posted by: Greg Egan on April 20, 2009 4:16 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Because the action of $G$ on $M$ is transitive it’s obvious that there’s such a function, and it generally won’t be unique.

It'll be unique if the action of $G$ on $M$ is also free. I haven't been following completely, but perhaps this is an assumption that you really want but forgot to state?

Posted by: Toby Bartels on April 20, 2009 5:09 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

It’ll be unique if the action of $G$ on $M$ is also free. I haven’t been following completely, but perhaps this is an assumption that you really want but forgot to state?

No, we don’t want to assume that, because then the subgroup $H$ of elements that leave $x$ fixed would necessarily be trivial. That would make the mathematics much less interesting, but worse, we know for a fact that it’s not true in the physical applications we’re interested in: the action of the Poincaré group on momentum 4-vectors. We get to make the action transitive simply by declaring $M$ to be a single orbit of a given 4-vector under that action, but $H$ will be a group like $SO(3)$.

Posted by: Greg Egan on April 20, 2009 11:32 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Ah right, $H$ is a stabiliser.

But in that case, you might still be ok. Regardless of how well-defined $k$ is, as long as the results that really matter are independent of your choices, then you should have well-defined (and even smooth) maps. Certainly that's the case for $m$, although I got lost tracing $f$.

But it's certainly not true that, given any group $G$ acting transitively on any manifold $M$ and any point $x: M$ that there exists a smooth map $k: M \to G$ such that $k(y) x = y$ for all $y: M$. For example, let $M$ be the sphere and let $G$ be its isometry group; then any smooth map $M \to G$ must be constant (because any $\pi_2(SO(3))$ is trivial), which won't work here.

Posted by: Toby Bartels on April 21, 2009 12:06 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I'm not sure why I thought that any map homotopic to a constant map must itself be constant just now. But, anyway … maybe somebody can do something with all that.

Posted by: Toby Bartels on April 21, 2009 2:15 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Greg wrote:

One point I’d welcome any comments on is this slightly mysterious business of the map I call $k : M \to G$, with $k(y) x = y$. Because the action of $G$ on $M$ is transitive it’s obvious that there’s such a function, and it generally won’t be unique. But it’s not clear to me whether there’s some nice way to construct a specific $k$, say one that would be guaranteed to be smooth if the action of $G$ is smooth.

I get the feeling you’ve figured this out by now, but anyway: you don’t need this $k$ to be everywhere smooth, which is good because usually you cannot choose it to be everywhere smooth.

Toby gave a good example, though not the right proof. There’s no way to smoothly choose a rotation $k(y)$ in $SO(3)$ as a function of $y \in S^2$ such that $k(y)$ maps the north pole to $y$.

Why? Well, if such a thing existed, we’d have a map $k: S^2 \to SO(3)$ and a map $j: SO(3) \to S^2$ such that

$j k = 1$

Namely, let $j$ send $g \in SO(3)$ to $g x$, where $x \in S^2$ is the north pole.

But we can’t have this, for topological reasons. (I leave this as an exercise for budding topologists.)

Luckily we can always make a locally smooth choice of $k$ near any given point, and that should be enough.

Posted by: John Baez on April 21, 2009 4:58 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Ah, I managed to forget that the representation of $H$ on a generic fibre of $F$ is not the same as the group action on the bundle as a whole. On the fibre $\pi_1^{-1}(y)$, we can get a representation of $H$ from $\rho_y(h,w)= (k(y) h k(y)^{-1})\cdot w$, where $k : M\to G$ is our map that gives $k(y)\cdot x = y$.

Given that representation of $H$ on each fibre, for any $y, z\in M$ we can define an intertwiner $i_{y,z} : \pi_1^{-1}(y)\to \pi_1^{-1}(z)$ by $i_{y,z}(w)=(k(z) k(y)^{-1})\cdot w$. Then:

$i_{y,z}(\rho_y(h,w)) = (k(z) h k(y)^{-1})\cdot w = \rho_z(h,i_{y,z}(w))$

So when we’re given our $G$-equivariant vector bundle morphism from $F$ to $L(V)$ and it maps some point $y\in M$ to $e H\in G/H$, we just use $i_{x,y} : \pi_1^{-1}(x)\to \pi_1^{-1}(y)$ first, followed by the construction I gave previously, to get an intertwiner $i : \pi_1^{-1}(x)\to V$.

Posted by: Greg Egan on April 20, 2009 3:03 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I thought I could rescue the case where the vector bundle morphisms don’t preserve the point $x$, but what I wrote about this here doesn’t actually work. I think it’s necessary to restrict to a subcategory of $Vect(M,G)$ in which the morphisms preserve $x$.

On that basis, I’ve now written up bijections for morphisms going in both directions, in the nLab entry. I’d be happy if someone can prove me wrong and make this work for the more general case!

Posted by: Greg Egan on April 21, 2009 12:59 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Greg wrote:

I think it’s necessary to restrict to a subcategory of $Vect(M,G)$ in which the morphisms preserve $x$.

I was a bit vague when describing $Vect(M,G)$, but I very definitely wanted the morphisms to act as the identity on the base space $M$. So, the ‘subcategory’ you’re talking about is just what I meant by $Vect(M,G)$. All is well.

It’s worth noting that when people talk about ‘the category of vector bundles over $M$’, they usually — say, 85.7% of the time — assume that the morphisms act as the identity on the base space $M$. That’s what I was doing. But sometimes it’s important to consider a bigger category where we allow arbitrary diffeomorphisms of the base space. This is especially true in gauge fields coupled to gravity, where $M$ is spacetime and a typical symmetry is a combination of a gauge transformation and a diffeomorphism of $M$.

Here of course we are considering $G$-equivariant vector bundles and morphisms that get along with the $G$ action. That vastly restricts what the morphisms can do to the base space, especially when $G$ acts transitively on the base space. But, we still have the option of either demanding that these morphisms act as the identity on the base space, or not.

Posted by: John Baez on April 22, 2009 5:52 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

John wrote:

I very definitely wanted the morphisms to act as the identity on the base space $M$.

Ah, good! That makes everything quite easy – especially the bijection for the morphisms going in the directions you originally suggested.

When I get some more energy I should show that this is a natural isomorphism, though, which I haven’t actually done in the nLab entry yet.

Posted by: Greg Egan on April 22, 2009 6:17 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

And after all that work, I just realised I gave a bijection between the morphisms that go in the opposite direction to those JB described …

Posted by: Greg Egan on April 20, 2009 3:16 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Sorry to have fallen quiet just as this thread is heating up, but my laptop is dead — I’m using my wife’s now — and right now I’m dead tired, having just flown back from Wisconsin. It may take me a few days to catch up with life and get back in the swing of things.

And after all that work, I just realised I gave a bijection between the morphisms that go in the opposite direction to those JB described…

Maybe that’s not such a big deal. In the category of finite-dimensional vector spaces we have a natural isomorphism

$hom(V,W) \cong hom(W,V)^*$

which makes sense because the hom-sets are actually vector spaces. This is also true in the category of finite-dimensional representations of $G$, and maybe it’s also true in the category of $G$-equivariant vector bundles over your transitive $G$-space $M$. If so, your isomorphism

$hom(F, L V) \cong hom( R F, V)$

would yield an isomorphism

$hom(L V, F)^* \cong hom(V, R F)^*$

and thus

$hom(L V, F) \cong hom(V, R F)$

Perhaps I’m deluding myself, but this might explain what’s going on. I’ll have to think about this again when I’m more awake!

Posted by: John Baez on April 21, 2009 4:34 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I am reading Schweber’s “An Introduction to Relativistic Quantum Field Theory” (just going into chapter 4), and so far, it has offered many physical insights that I have missed previously in other treatments. And I have just found this thread! I’ll be reading the discussions here in detail, and since I have just read Schweber’s account on the inhomogeneous Lorentz group, it is a very timely post for me.

So I ask you, the experts, what do you think of Schweber’s exposition? I know his book dates from 1961, so I suppose the subject today is introduced under a different perspective than the one given there. But so far, I think Schweber’s book is well organized and instructive, specially with the physical point of view in mind, even though I may be missing a modern perspective. Any advice about using this book?

Another book I have been skimming through is “Group Theory in Physics” by Wu-Ki Tung, which appears to be a good reference, more mathematically inclined. Chapter 10 of this book deals with the Poincaré group. In particular, he promisses to illustrate “the complementary roles played by the finite dimensional (non-unitary) representations of the Lorentz group and by the (infinite dimensional) unitary representations of the Poincaré group in physical applications”. But I have not reached to that point yet. He uses Wigner’s induced rep. method in his exposition.

Any comments on these references are welcomed, and in case they help the discussions here, I’ll be following them with interest.

Thanks,
Christine

Posted by: Christine Dantas on April 18, 2009 2:56 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Also, I am intrigued by this reference from 1956:

Phys. Rev. 102, 568 - 581 (1956)
Synthesis of Covariant Particle Equations
Leslie L. Foldy

which suggests “the existence of several distinct relativistic theories for particles of any given spin”. And that “Conjectures are made as to the physical significance of these different possibilities when the equations are second-quantized. It is shown that each of the conventional theories employs only one of the available possibilities for these transformations, the choices being different for integral and half-integral spin theories.”

Was there any further developments on the nature of those “conjectures”?

Posted by: Christine Dantas on April 18, 2009 3:38 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

The article described below is a complete redo of Wigner’s sections 1-4 and much of 5-7. It was originally intended for this forum, but is too long. Even the summary to follow is.

Plus, everything is in PDF or in HTML making use of HTML’s math and layout facilities that are not supported in n-Category Cafe.

Instead, the entire thread, as well as my reply, have been translated into HTML and placed under the web link attached to my name on the blog header. For reference, it is:

The Wigner Classification for Galilei Poincare and Euclid

and its supplements:

Poincare Representations - n-Category Cafe (the thread)

The Wigner Classification for General 4-Space Signatures (the reply)

A. P. Balachandra, G. Marmo, B.-S. Skagersam, A. Stern,
Gauge Symmetries and Fibre Bundles, Applications in Particle Dynamics, Lecture Notes in Physics 188;

whose coordinatization of Non-Helical Sector 1L (described below) goes beyond what I describe; and

N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, 1998 Springer (particularly sections III.1 and I.2.5).

Related links that tie into the article are listed at the end of the reply and include:

The Jordan Decomposition in the Unified Group

the detailed analysis that subsumes Wigner’s section 4B.

Towards a General Theory of Signature and Signature Change,

Dimension and Signature

both of these lie at the root of the analysis carried out.

Poisson Algebras, Poisson Manifolds, Symplectic Manifolds, Poisson Bracket

these cover the foundation of the analysis used to subsume Wigner. The articles significantly expand on (and clean up) the Wikipedia originals, with running examples involving the Heisenberg and Spin algebras.

Unification of Galilei, Poincare and Euclidean Symmetry

(UI-Chicago, 2008 October 7)

An on-line version of a talk given at a seminar at UIC, hosted by Kauffman; this includes a vastly expanded account of NOVA’s two Einstein bio series, including not just the respective web sites but also transcribed and annotated copies of several of the original papers by Einstein and Lorentz.

The Newton-Wigner Position Operator - derived from T. F. Jordan

The coordinatization of the class 1L (and the no-go result for carrying out the same for non-Helical class 2L).

The Missing Heisenberg Relation

What happened to the 4th Heisenberg relation?

Definition of Mass

Mass defined from first principles; from the 11th generator of the Unified Group

This is a summary of the sections.

0. Unravelling and Subsuming Wigner

1. Wigner’s Section 2: Linear Representation Theory vs. Symplectic Reduction

Symplectic reduction and the Poisson-Lie manifold over the Lie Group.

2. If Irreps and Particles are Synonymous, then where is the Vacuum Particle?

The concept of irrep as particle has obvious gaps (as indicated by the title of the section). A more comprehensive account of what an irrep actually is - one that includes the Wigner class 3 (and class 4) sectors - starts from an entirely different viewpoint: irreps corresponding to elementary systems that are to be thought of as media, not particles; e.g., an isotropic medium is one which has the rotation generator J as an invariant; a quasi-vacuum, one which has the boost generator K as an invariant. An isotropic quasi-vacuum is a vacuum. A vacuon (which is Wigner’s class 3) is a translation invariant medium.

3. Wigner’s Section 3: The Wigner Sectors for General Signatures

This signficantly expands Wigner’s discussion in Section 3.

The Wigner/von Neumann classes include the following: the Tardion (classes 1/4E, 1L, 1G) which are media with a Staton state; Statons (class 1/2A), which are Archimedia at Absolute Rest, Luxons (class 2L), which are the null media, Lorentzian Tachyons (class 4L), which are Lorentzian media with a Synchron frame, Archimedean Tachyons (class 4A), which are Archimedia in Absolute Motion and Synchrons (class 2/4G) which are Galilean media which support instantaneous action at a distance transfers of impulse across space.

4. Wigner’s Section 4A: Symmetry and Signature

This is an expansion of Wigner’s section 4A which starts out from a general theory of (possibly-degenerate) signatures, replacing the orthogonal group with the bi-orthogonal groups. In addition, treatment by linear representation theory is expanded (and yet, simplified) by its generalization to the non-linear representation theory of Poisson-Lie manifolds and symplectic reduction.

5. Wigner Section 5: Central Extensions

This covers much of what Wigner’s section 5 is dealing with, with a surprising twist, that puts to the lie the notion that the 10 generators of Poincare are enough even when retricting focus to the Lorentzian case.

6. Wigner’s Section 4B-D: The Jordan Decomposition

Wigner botched this part of the proof, along with the rest of section 4 by working with the group, itself, rather than just its Lie algebra. Initially, I thought he did this because he was treating global issues. But a close reading of section 4 shows that he’s only working with the connected subgroup - which completely defeats the point of his analysis!

The correct way to the analysis is thus is with the Lie algebra, not the Lie group! As a bonus, the simplicity of this approach allows one to run through the general cases: all signatures, in one fell swoop; and to even treat the inhomogeneous group (which Wigner does not do, either).

The subsections include

6.1. The Characteristic Equations of the Infinitesimal Generators.

The Jordan classes of the transformations are easily determined on the Lie algebra. Jordan decomposition remains invariant under exponentiation, therefore this applies to the Lie group as well.

6.2. Wigner Section 4B: The Jordan Decomposition

The classes are [(1111)], the identity transformation (all signatures); [(211)] the Galilean boost (Galilean) and Poincare shift (Archimedean); [(31)] the Null Boost (Lorentzian); [(11)zz*] Rotations (all signatures); [zz*zz*] general Euclidean transformations; [(11)zz*] general Lorentz transformations; [(11)11] general Galilean transformation and [(11)11] general Archimedean transformations.

Historically speaking…

All of the Archimedean members of the family are rooted all the way back in the Hellenistic Era, except the Poincare shift. The Galilean boost is, historically, the first bona fide space-time symmetry and is rooted in early modern Europe; while the Poincare shift (and its relativization to the Lorentz boost) are from the late 19th century, dating from Poincare’ study in global time synchronization. (He was one of those involved in the standardization into our present-day time zones).

6.3. Wigner’s Section 4C: Uniqueness of the Boost

This is trivially handled if doing this with the Lie algebra, rather than the Lie group.

The analysis is done here.

6.4. Wigner’s Section 4D: Simplicity of the Lorentz Group

Similarly, this is much more easily dealt with in the Lie algebra, instead of the Lie group. Here, the result generalizes: the Unified Group is simple for all signatures, except the Euclidean. The decomposition in the Euclidean sector is well-known.

This analysis is also done here.

7. Wigner’s Section 6-7: Symplectic Decomposition

No analysis in terms of linear spaces. Instead: a far more general analysis in terms of non-linear representation. Irrep is replaced by symplectic leaf.

7.1. The Vacuon Sectors

Covers the 3 classes of translation-invariant media: the generic vacuon, the quasi-vacuum and the vacuum.

7.2. The Archimedean Tachyon Sectors

Covers a special subclass of translation non-invariant media: those specific to the Archimedean signature, corresponding to systems in motion.

7.3. Wigner’s Section 6: Translation Invariants

The translation-invariant sectors are the vacuons; while the translation non-invariant sectors subdivide into the Archimeden Tachyon, the Helical and Non-Helical sectors.

Part of Wigner’s section 6 deals with the translation-invariants. Here, the analysis is done more simply. Out of it naturally emerge the Pauli-Lubanski 4-vector.

7.4. The Non-Helical Sectors

This cross-classifies with the von Neumann–Wigner classification to yield the non-helical Tardions, Tachyons, Luxons, Synchrons and Statons.

7.5. The Helical Sectors

Neither this, nor the following sections have yet been fully written up here, but a link to a parallel analysis for the non-Archimdean cases is provided.

The helical subdivision, like the non-helical subdivision, also cross-categories with respect to the Tardion, Tachyon, Luxon/Synchron/Staton classification.

8. Mass-Energy-Momentum vs. Velocity for the Translation Non-Invariant Sectors

9. Coordinatization of the Translation Non-Invariant Sectors

Once you get a coordinatization (which is an application of the Darboux Theorem), you then have the canonical 1-form and 2-forms, as well as the basis for defining the invariant measures on both the phase space and the configuration space. The general non-helical translation non-invariant sector has 4 Darboux pairs. In the tardion and helical cases, this decomposes into a Heisenberg triple plus a complementary pair for spin.

It is the Darboux coordinates that you then proceed to quantize by whatever favorite method you have at your disposal. In turn, it is this which governs the linear space representations.

This is the part of the analysis from Wigner which has not (yet) been included. But for the most part, it is fairly routine – as opposed to the much more complex problem of trying to quantize the whole Lie group!

10. Schroedinger Equation for Arbitrary Signatures

The Schroedinger equation is just an expression of the quadratic mass-shell invariant specialized to the Galilean sector … combined with the linear invariant that comes from the 11th generator. Both specialize across all the signature types. In the Lorentzian sector, the resulting equation is equivalent to the Klein-Gordon equation, up to a Foldy-Woutheusen transformation.

11. The Dirac Equation for Arbitrary Signatures

Similarly, the Dirac equation can be done across the board to all signature types and all sectors (including the tachyons and synchrons).

12. Position and Time Operators

This carries out the analysis of all the possibilities, devising a system of differential equations for a prospective position operator and time operator, built on the Poisson-Lie manifold that contains the symmetry group. Time operators exist for the Synchrons and (apparently) one of the Archimedean Tachyon sectors; while position operators exist for the non-helical tardion and helical sectors.

13. Many Body States and the No-Interaction Theorem

If I write this up, my intent is to combine BOTH the Leutweiler and Haag theorems into one; the former being the classical case of the general result, the latter the quantized version.

In addition, this can be generalized to arbitrary signatures (i.e. non-relativistic forms of the Haag and Leutweiler theorems).

Posted by: Mark Hopkins on June 30, 2009 12:11 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Zounds! Great stuff!

I especially like the idea of irreps as ‘media’. I’d thought about the ‘vacuum’ myself, and how its translation-invariance means it’s completely unlocalized. But I hadn’t thought about all the other ‘vacua’.

I also love the idea of treating all signatures, including degenerate ones, in a unified way.

At one point I’d been mourning the fact that this thread didn’t go as far as I’d wanted. Now it’s gone much further! Not only you, but also Greg Egan, have taken it in directions that go way beyond what I’d imagined.

… everything is in PDF or in HTML making use of HTML’s math and layout facilities that are not supported in n-Category Cafe.

Of course the nCafé has TeX abilities that allow for math that goes way beyond HTML. It’s a bit of work to master, especially since it’s slightly different than ordinary TeX in a few ways (as discussed here). But for long articles or research papers it’s better to use the nLab than the nCafé.

Posted by: John Baez on June 30, 2009 8:57 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Thanks for fixing up my reply. I didn’t know you were in Wisconsin in April until I saw the thread. If it was in Milwaukee, I’m almost certain you’d have been surprised at just how much everything’s changed: the architecture is totally different than the last time you were here, and it’s not just downtown; but everywhere.

The translated version in HTML, by the way, has everything except the SVG diagram and it also fixed up the replies that were not originally in MathML.

Much of what’s in the SVG (2009 April 20 12:24) and the earlier reply (2009 April 16 13:19) I’ve already (re-)written up a short article on a while back

Induced Representations
http://federation.g3z.com/Physics/index.htm#InducedRep

(Actually, I’ve also gone through line-by-line the first few chapters of Mackey’s monograph, copying, editing, annotating everything, reworking the analysis, or trying to; but I’ve never put this on-line yet).

The blog’s language, itself, may be big (and TeX); but the subset of the language actually used by people in the blog (including your articles) is readily convertible to HTML.

Since I translated everything by hand, then I now know enough to go in both directions. But I don’t know if the blog allows direct insertion of MathML source.

You’re only seeing the end of my replies, not the beginning; I have a strict deadline of July 1 (local time) on everything, before a month-long moratorium. So, there are a few things that popped up in the thread I wanted to reply to all at once.

The 2009 March 25 18:25 reference to “case 1” negative mass-squared can better be understood as positive impulse-squared. This generalizes action at a distance to a relativistic setting, where “instantaneous” now means “instantaneous in somebody’s frame of reference”.

If you think of these as particles, the interpretation is complicated, but Sudarshan has made a business out of doing this. On the other hand, if you lump these in the same category as the vacuua, then you have a completely different interpretation: you’d be no more seeking out “particles of tachyon” than you would “particles of vacuum”. The interpretation is now of a type of medium that supports the Coulomb part of interactions. Only, in the Lorentzian case (unlike the Galilei case of Synchrons), the “instantaneous” nature is askew in time.

This is the gap that probably lies directly behind the Leutweiler theorem.

The March 25 06:12, March 24 02:35, March 24 23:59 replies: the projective representation of the Galilei group is in de Sitter space. Tying all the signatures together in such a way as to contain the centrally extended Galilei group means there’s a projective rep for the Poincare group too (and an 11th generator). The Galilei, Lorentzian and Euclidean signatures are all different projections in a common 5-D de Sitter geometry. Hence, the reference in my HTML reply to Steven King’s Langoliers (travellers who get stuck in an anomalous 4-D projection of 5-D space in which ordinary time is frozen).

(It’s also the second time I’ve publicized the scandalous secret that Steven King is actually the disguised alter ego of Steven Hawking).

The 11th generator takes the bottom out of the energy scale, putting everything back into the same place it was in non-relativistic theory – a scale without an absolute 0.

In the 5-d projective representation, the Klein-Gordon mass is a 5-luxon. So, the m-squared part of the equation corresponds to the partial derivative with respect to a 5th coordinate (u). The state is an eigenvector with respect to the partial derivative operator on u. Thus, it has a phase factor. So, in a quantum setting, u is interpreted as the phase.

This is the underpinning of the Foldy-Woutheusen (sp?) transformation.

Foldy and Woutheusen were originally intending to redo all of the foundation of physics by excising the particle concept and rebuilding everything from scratch within the transformation theory of the spacetime symmetry group. The reference mentioned in the reply from April 18 14:56 is from that period; but the F-W transformation is about the only present-day survival left from that time.

Posted by: Mark Hopkins on July 1, 2009 3:29 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

There are a couple extra comments I should make (partly in relation to section 7B of Wigner) before I run off and try to get a paper I’m working on (due late July) completed.

The explicit tie-in between irreps and symplectic leaves is contained in the Landsman reference, Propositions I.2.6.8-9 and Corollary I.2.6.10. What you’re about to see, however, is that you’re going to get a LOT clearer picture of what’s going on by staying at the classical level with the Poisson manifold structure. I’m referring in particular to the funny business going on with E(2) and spin (as well as SO(2,1) and spin).

Let x denote the axis parallel to the momentum 3-vector P. Then, Jx, the x component of the angular momentum 3-vector J is the helicity. The Pauli-Lubanski 3-vector W decomposes as W = (M Jx, Wy, Wz), where M is the “relativistic mass”; and the 4th component W0 as Px Jx = |P| Jx.

This part of the Poisson-manifold structure yields a function subspace whose Poisson bracket closes. In particular, you have {Jx, Wy} = Wz, {Jx, Wz} = -Wy, and {Wy, Wz} = (M^2 - alpha beta P^2) Jx.

This is the “spin algebra”.

This expression is generic to all signatures (with alpha and beta being the signature parameters, as described in my original reply) and all sectors except the translation-invariant sector (the “vacuons”).

Landsman, in Section IV.3.1 replicates the Wigner types 1-4 classes in equation (3.3), giving in order the Vacuons (L/L), the Tardions (L/SO(3)), the Luxons (L/E(2)) and the Tachyons (L/SO(1,2)); the quotients giving the reduction of the Lorentz group L.

He replicates the standard treatments of the various sectors without adding anything new. In particular, the Poincare group is the semidirect product of L with Minkowski space M. Though this is not quite accurate: this M is not the coordinate space, but the space of translation generators – momentum space. So, his “coadjoint” action is on the position space: the dual of M. On the other hand, the “coadjoint” actions on the spin algebras are on the double-dual (putting us back in the Poisson manifold). So, there’s a fragementation of approaches even here.

Everything becomes much clearer when staying back at the classical level and even when keeping the signature dependency intact. The 3 translation non-invariant classes are based on the sign of the bracket for {Wy,Wz}: the sign of the invariant (M^2 - alpha beta P^2), where M is the “relativistic mass”. If positive, this yields the tardions; if negative, tachyons; and if 0, it yields the luxon/synchron/staton classes. The synchrons occur for alpha = 0, M = 0; the statons for beta = 0, M = 0. The beta = 0 class (the Archimedean signature) only has statons and tachyons for its translation non-invariant sectors.

For the positive M^2 - alpha beta P^2, one can proceed to define the “rest mass” m as the square root of the invariant, and then define the spin by Jx = Sx, Wy = m Sy, Wz = m Sz. This results in the usual spin algebra for the tardions. In turn, it also results in a position vector r (the Newton-Wigner vector), which gives you the correct expression for the spin-orbit decomposition of J.

This ties in with section 7A of Wigner.

The situation is fairly well-known and standard. For zero spin, the Poisson bracket reduces to 0 and the symplectic leaf is 0-dimensional. For positive spin, one can use spherical coordinates (Sx,Sy,Sz) = S (cos T, sin T cos F, sin T sin F) and one finds that S is an invariant, while Sx = S cos T is the complement of the angular coordinate F: {F, Sx} = 1.

The symplectic leaves are either the point or 2-sphere.

On the other hand; the zero sign gives you the algebra generated by the brackets: {Jx, Wy} = Wz, {Jx, Wz} = -Wy, {Wy, Wz} = 0. This is E(2).

This ties in with section 7B of Wigner, as well as Landsman preview of his section IV.3 (he has a 36 page summary previewing the 396 pages in his book that followed the summary).

The Poisson manifold structure is best seen by converting to polar coordinates: (Wy, Wz) = H (cos T, sin T) with either H = 0 or H > 0.

The result is that H turns out to be an invariant.

If H > 0, the brackets reduce to the symplectic form given by {T, Jx} = 1. These are the Darboux coordinates.

The Helicity is the complement of the angular coordinate T. The symplectic leaves are cylinders of constant H.

This is the “continuous spin” sector of Luxons.

The non-compact Darboux pair effectively captures a piece of what been 3-space position-momentum conjugacy in the tardion sector. One way of interpreting this is as the pair cooresponding to the 3rd affine coordinate for a null trajectory. But I haven’t seen this worked out and don’t know exactly what pieces will fit where in this puzzle.

The case H = 0 yields Jx as an invariant, with the Poisson-bracket reducing to 0. The symplectic leaves are single points: (Jx,Wy,Wz) = (Jx,0,0).

This is the one used to model Luxons in contemporary theories. Then the designation of Jx as helicity dovetails with the usual terminology.

For tachyons, the brackets become {Jx,Wy} = Wz, {Jx,Wz} = -Wy, {Wy,Wz} = (M^2 - alpha beta P^2) Jx.

Tachyons give us a relativistic version of the synchron: an instantaneous action at a distance transfer of impulse, except that “instantaneous” is now only with respect to a designated set of frames (the tachyons’s “synchron” frames). This is analogous to the rest state of tardions.

The factor in front of Jx is negative: the negative square of the impulse Pi associated with the tachyon. So, the 3rd bracket reads {Wy,Wz} = -(Pi/c)^2 Jx.

This can be normalized by taking Sx = Jx, Hy = c Wy/Pi, Hz = c Wz/Pi. The result is {Sx,Hy} = Hz, {Sx,Hz} = -Hy, {Hy,Hz} = -Sx.

As before, one can write (Hy,Hz) = (H cos T, H sin T). Then it remains the case once again that {T,Sx} = 1, with these two coordinates becoming complementary pairs. The only difference from the Luxon case, is that H is not an invariant; but rather H^2 - Sx^2 is. So, there are 3 subclasses of tachyon depending on the sign of this invariant.

In effect, what had originally been one of the 3-space Heisenberg pairs has moved over into the “spin algebra”.

It’s also for this reason that neither the Luxon nor Tachyon sectors have the usual 3-space Heisenberg pair for position and momentum, when the spin algebra is non-trivial. It’s only the case where Wy = 0 = Wz that you can move forward on that. Those were the translation non-invariant sectors I dubbed “Helical”; while the ones with non-zero Wy and Wz I dubbed “Non-Helical”.

The Helical sectors apparently all have position vectors, because once you have the 3-vector W being parallel to the 3-vector P, their cross product drops out. But it’s the cross-product of W and P which leads to a deviation from the non-relativistic form of the position vector, in the first place. So, this means you can continue to use the non-relativistic form of the position vector. The usual spin-orbit decomposition for J applies, but the spin part is now collinear with P (that is: the helicity Sx may be non-zero, but the components Wy, Wz are both zero).

All the translation non-invariant sectors have 8-dimensional symplectic leaves as their main sectors; plus specialized cases consisting of symplectic leaves of lesser dimension. The 4 Darboux pairs for the Tardions are the 3 (non-compact) position-momentum pairs; and the 1 pair involving Sx and the angle T. (When quantized, Sx corresponds to what’s usually called m). It’s the distribution of the compact vs. non-compact Darboux pairs that seems to mix up when going outside the Tardion sector to the “continuous spin” Luxons and Tachyons. But I still don’t know how to get this to seamlessly connect to the Tardion sector’s coordinatization.

Posted by: Mark Hopkins on July 4, 2009 4:45 AM | Permalink | Reply to this

### Contemporary Treatment of the Representation Problem

The listing I give for the Jordan decomposition should read [(1111)] identity, [(211)] Galilean boost and Poincare shift, [(31)] null boost, [(11)11] Lorentz boost, [(11)zz*] circular rotation, [2zz*] generic Archimedean and Galilean transformations, [11zz*] generic Lorentz transformations, [zz*zz*] generic Euclidean transformations.

The reference to a non-relativistic version of Haag is still somewhat of a conjecture. When the tidal wave of No Go crashes ashore the coastline of the Galilean domain, it may turn out to be nothing more than a small ripple. But, then the question naturally arises: can what works in the Galilean sector be pulled back to the Lorentzian sector? (Notice the tacit reference to adjunction here? Exercise: what is the adjunction that connects non-relativistic theory – with all its particulars, like absolute time, non-trivial interactions, etc. – to relativistic theory?)

On the question of Symplectic reduction vs. linear reps, where I point out that you can take the Darboux coordinates and then proceed to quantize by whatever your favorite method is: in fact, this is essentially what Landsman does in the remainder of section III.1, and it comprises parts 6-11.

In part I, Landsman also talks about the quantization of the Poisson-Lie and Poisson manifold structures. The quantized structures are referred to as Poisson spaces and are discussed in I.2.6.

If all you really want to do was to review Wigner’s result and to try to get a more contemporary understanding of it; following up on the earlier comment “we’re trying to understand unitary representations of the Poincare group… So, it’s not time to start talking about generalizations!” (which is not a good idea, since the larger context is where all the murky issues get answered), then there’s really no reason to go to Wigner, at all.

As I mentioned in the HTML reply: Wigner didn’t cover Tachyons or Vacuons (other than the Vacuum), so you’re no better off going to the original than you are going to more recently-published references.

The relevant material is covered in a more contemporary fashion in depth in Landsman, section IV.3, parts 1-3. This goes well beyond this application, even; and is part of a comprehensive treatment on reduction and induction (the former is the classical version of the latter, the latter is the quantized version of the former).

In particular, parts 4-9 also gets into discussing U(1) representations, capping off with the discussion of the vacuum angle in section 9.

To his credit Landsman does not come out and assert that the theta angles and theta vacuua are a purely quantum feature of gauge theory (since they’re not). In fact, they’re not only rooted in classical theory (the axial permittivy coefficients in electromagnetism and their gauge-theoretic generalization), but ARE essentially classical, even in quantum theory. That’s what lies behind the comment gauge-invariant theory admits inequivalent quantizations, classified by theta (i.e. the state spaces for different thetas do not coherently superpose with one another, hence theta is an essentially classical unquantizable variable).

The importance of the theta exercise is that it brings out a case-in-point showing the hazard of adopting the theoretical bias that everything is purely quantum. Rather, the world must be treated in the larger setting of a classico-quantum theory. This is probably a large part of why Landsman said his treatise was originally going to be titled Tractatus Classico-Quantummechanicus.

In turn, this is why it’s important to cast as much of Wigner as possible in a paradigm-universal form: that is, in classical language, with all the unnecessary entanglement with quantum theory removed. Quantizing is something you can do after the classical version is taken care of.

I’m still rummaging through section 5 of Wigner to extract out the part that pertains to central charges vs. the part that pertains to the global structure. I’m not quite sure why the two are lumped together: is the issue of universal covering space the discrete version of central extension or something? (e.g. a discrete form of central extension to cover Z2(P) x Z2(T)).

But anyway: it’s only with a proper understanding of what’s going on in Landsman IV.1.9, that one can properly appreciate the parallel discussion by Sardanashvily and his people, regarding the essentially classical nature of the metric. (Here: the metric plays an analogous role to the theta variable).

Section IV is about the only part of my copy of Landsman whose pages are not yet discolored with overuse. It goes WAY beyond even what I discussed here, providing a classico-quantum account of both reduction and induction. In particular, symplectic reduction is done in IV.1. IV.2.8 (covariant quantization) and IV.2.10 (quantization of singular reduction) also seem to be particularly relevant to my previous discussion.

Actually, now that I look at it, it’s not entirely unused. I got these 2 pictures in the margin at the start of section IV in my copy of Landsman dated at the end of 2004. Those two pictures, alone, cover the equivalent of most of section IV.1.

But the interesting story they tell is another topic for another time (and it would take a long time to explain how constraint manifolds and first class vs. second class constraints relate to widowed spouses, because that’s basically what my pictures show).

Posted by: Mark Hopkins on June 30, 2009 10:41 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

A few additional comments are in order regarding the nature of the discrete symmetry group: just as there is a central extension to the symmetry group (which, as alluded to above, gets really interesting when expanding the survey to cover the different 4-dimensional signatures), there is one for the discrete subgroup spanned by parity P and time translation T.

The same analysis done in my expanded reply applies here, but now at the group level rather than the Lie algebra level. The result is that there are 8 possible projective extensions with real coefficients – 2 with complex coefficients. The complex cases are given by PT = TP with P^2 = I = T^2; and the non-Abelian case in which P, T generate a quaternion algebra, with PT = -TP and P^2 = -1 = T^2. I don’t know if these have names, but one might think of them as Spin(+) and Spin(-) respectively.

One has to be careful in handling the fermions correctly. The actual extension you want is what is known as Sin(-), which is Spin(-) without the P. In it, T^2 = -1 and the fourth power is required to get back to 1. The other is Sin(+).

This is discussed in CQG 12 (1995) 2231-2241.

The importance of expanding the analysis to general signatures becomes immediately apparent the moment you start to think of path integrals. There is no Wick rotation for curved backgrounds. The best you can do is wing it by restricting to a well-behaved family of curved spaces and trying to link them one-by-one to Euclidean variants. This is possible with Schwarzschild and is the original inspiration behind Hawking’s black hole/thermodynamics relation.

But backing up into the wild unexplored territory of arbitrary Lorenztian background, you’re out in the cold. So, one alternative is to try and replace the Wick rotation idea with something more physically meaningful and intuitive – an actual continuous deformation from Lorentzian to Euclidean signature.

If one insists on being able to recover isotropic backgrounds in each member of the continuous family of spaces, this requires staying within one of the 4 signatures where 3 of the dimensions are treated the same. Continuity means you have to pass through a degenerate boundary case when making the transition over from Lorentzian to Euclidean. The only two possibilities are where light speed goes to infinity (the Galilean signature) or where light speed goes to 0 (the Archimedean signature).

Thus, the problem of trying to transform Lorentzian to Euclidean background is intimately tied up with the problem of trying to form a continuous connection not just between the different signatures, but the representations and field theories themselves. That means, for instance, formulating a well-defined “Galilean limit” for electromagnetism and gauge theory … and an “Archimedean limit”, as well.

I’m surprise no mention was made of Ashtekar when I brought up Landsman and the notion of falling back to Poisson manifolds and to (the equivalent) of coadjoint orbits. In fact, Ashtekar showed back in 1997 that the “Poisson bracket” and Poisson algebra structure can be recovered in exact form at the quantum level, with the need to take classical limits. He accomplishes this trick by “unhatting” operators (i.e. the GNS transform). The state space becomes a bona fide Poisson manifold, and the unhatting of -i times the commutator of two observables A and B is the Poisson bracket of the unhatted versions of A and B.

So, it is also in this way that it makes sense to fall back to the non-linear representation theory on Poisson manifolds, rather than just sticking with the linear representation theory on Hilbert spaces.

Posted by: Mark Hopkins on October 20, 2009 11:04 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I printed out a copy of Wigner’s paper to take a closer look at it. It appears that his section 7B does cover both the “helical” and “cylindrical” luxons, after all. The photons lie in the former, the “continuous spin” sector is the latter. He divides the treatment of the little group E(2) into case a (cylindrical) and case b (helical). His treatment matches the one I gave in an earlier reply – but in the math and notation of linear Hilbert space representations.

The REAL question I wanted to get to here is what’s up with the other sectors? In particular, the position and time operators.

I mentioned in an earlier reply that all the cases where the Pauli-Lubanski vector W are parallel to the momentum P yield position vectors. This is wrong. Only the case W = 0 does. The other case is the helical luxons, which are well-known not to have a position vector.

There are, in fact, a whole slew of no go results governing position and time operators. They go along the lines: tardions have no time operator, luxons have neither time nor position operators, tachyons have no position operators. An exception is made for spin 0 (W = 0).

All of these deficits have a common root, and there is a remedy that addresses them all. The cause of the problem is the conflict with the mass shall invariant. Since each irreducible sector (that is, each symplectic leaf) admits only up to 4 conjugate pairs of coordinates, and one of them is taken up by the Pauli-Lubanski algebra, that leaves only 3 left to handle prospective position and time operators.

This means that one of the coordinates must be “classicalized”. This offsets of the imposition of the mass-shell invariant.

For tardions, one “classicalizes” the t coordinate. As a result, the equation of motion dA/dt has a term involving the Poisson bracket {A,H}, as well as an extra term involving the partial of A with respect to t. This second term comes about because after t is classicalized it’s removed from the set of canonical coordinates, so the extra term compensates.

For the “synchron” sector, the situation is somewhat the opposite: here, it’s the time operator (t) that’s one of the canonical coordinates, while the longitudinal coordinate (z) becomes “classicalized”.

So, the general approach one may adopt is that before trying to devise a position and time operator, one should first reduce the canonical brackets for the position and momentum. This is done in the following steps.

First, rewrite the coordinates as u = A’z + B’t, v = C’z + D’t, and conjugate momenta as P_u = AP + BE, P_v = CP + DE. We then classicalize the u coordinate by requiring {u,P_u} = 0 = {u,P_v}; or {u,P} = 0 = {u,E}.

Conjugacy implies that AA’ + CC’ = 1, AB’ + CD’ = 0, BA’ + DC’ = 0 and BB’ + DD’ = -1.

(This transformation, one may notice, bears a strong similarity to what’s known as the Bogoliubov transform).

Second, we continue to treat v and P_v as conjugate, imposing the relation {v,P_v} = 1. But now because of the mass-shell constraint, one has a non-zero bracket for {v,P_u}. Thus, for instance, in the tardion sector, v = z, and P_u = -E, and the non-zero bracket is just the Heisenberg equation for the z coordinate.

Remarkably, it turns out that the resulting relations are independent of A and B, depending only on C and D; with the dependence only on the combinations c = CP/(CP + DE) and s = DE/(CP + DE). Thus, one is led to the notion of reduced relations controlled by a “chronicity parameter” c. The case (c,s) = (1,0) will allow you to move forward and derive the Newton-Wigner operator for tardions. The case (c,s) = (0,1) will work with tachyons, while the case (c,s) = (1/2, 1/2) works with luxons.

The reduced relations all have the form
{z,P} = c
{z,E} = c P/E
{t,P} = -s PE/|P|^2
{t,E} = -s

So, if you start out with these relations, it should be possible to move forward with the construction of position and time operators for each sector; thus both generalizing the construction of the Newton-Wigner operator AND by-passing (or explaining away) the whole slew of no go theorems.

Posted by: Mark Hopkins on December 1, 2009 2:15 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

The Wigner Classification for Galilei Poincare and Euclid

seems broken - federation.g3z.com cannot be found…

Posted by: Arnold Neumaier on December 1, 2009 3:05 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

This thread has been for me a very useful and helpful discussion. After a fair amount of time spent going back and forth over it, I think the fog in my brain is at last beginning to clear, after (sadly) years of confusion about what is, after all, very basic quantum physics.

I’d like to try to summarize what I’ve so far understood. (Looking back over it, I acknowledge that it’s pretty long-winded, and that it repeats some points already made by Greg Egan, Bruce Westbury, Evan Jenkins, and – of course – John Baez. I owe a debt of gratitude to them all; the fact I haven’t linked back to their comments is due to sheer laziness on my part.)

We are trying to understand unitary irreducible representations of the Poincaré group $Poin = SL_2(\mathbb{C}) \ltimes \mathbb{R}^4$, aka “elementary particles” in quantum mechanics. A unitary representation consists of a Hilbert space $H$ and a continuous group homomorphism $Poin \to U(H)$ to its unitary group with the strong operator topology, and “irreducible” has the usual meaning.

(At this first pass, we won’t worry about technical details like “strong operator topology”. There are various spots in the outline below, particularly every place where the words “direct integral” appear, that implicitly require as formal background some form of spectral theory and Stone’s theorem, and there details of operator topologies will certainly be involved.)

At the outset, John told us that each unitary irrep is uniquely determined by two parameters:

• A continuous parameter $m$ called the mass. This has something to do with how the irrep $Poin \to U(H)$ restricts to a representation of the normal subgroup of translations $\mathbb{R}^4 \hookrightarrow Poin$, a noncompact group.
• A discrete parameter $s$ called the helicity (or “spin”). This specifies a unitary irrep of a compact subgroup of $SL_2(\mathbb{C})$ called a “little group”. If the mass is positive, this little group will be a copy of $SU(2)$ [whose unitary irreps are parametrized by half-integers, called spin numbers]. Other possibilities for the little group arise if the mass is zero.

By using the method of induced bundles, we can reconstruct the unitary irrep of $Poin$ from its mass and helicity. I’d like to try to outline how I think the whole thing works (in some cases minor twists on things others have already said).

Let’s start by discussing “mass”. This has to do with how the restricted representation

$\mathbb{R}^4 \hookrightarrow Poin \to U(H)$

decomposes into a “direct sum” of irreducible unitary representations of $\mathbb{R}^4$. Except that in the representation theory of a noncompact Lie group like $\mathbb{R}^4$, we won’t use a direct sum exactly – we use a slightly more elaborate construction called a direct integral of representations. We’ll come to this later; for now let’s start with irreps of $\mathbb{R}^4$.

We know what unitary irreps of $\mathbb{R}^4$ look like. Because $\mathbb{R}^4$ is a locally compact commutative group, its unitary irreps are 1-dimensional and are specified by characters

$\chi: \mathbb{R}^4 \to U(\mathbb{C}) = S^1$

or in other words by elements of the Pontryagin dual of $\mathbb{R}^4$, which is isomorphic to $\mathbb{R}^4$. This isomorphism takes an element $p \in \mathbb{R}^4$, which physicists call a 4-momentum, to the character $\chi_p$ defined by

$\chi_p: \mathbb{R}^4 \to U(\mathbb{C}))$

$\chi_p(x) = e^{-i p \cdot x}$

Here the $\cdot$ in the exponent refers to a bilinear form based on the Minkowski form of signature $(1, -1, -1, -1)$. (If I were paying more mind to the physical dimensions, I guess I ought to have a factor $\frac1{\hbar}$ in the exponent as well, referring here to Planck’s constant.)

The mass (or should we say mass squared?) of a momentum $p$ is defined by $m^2 = p \cdot p$; in other words, if we write $p = (E, p_x, p_y, p_z)$, then

$m^2 = E^2 - p_{x}^2 - p_{y}^2 - p_{z}^2.$

Mass is a relativistic invariant; let’s see what that implies. Consider the action of $Poin$ on the character group, by pulling back the action of $Poin$ on the translation group $\mathbb{R}^4$:

$\mathbb{R}^4 \stackrel{g}{\to} \mathbb{R}^4 \stackrel{\chi}{\to} U(H)$

(abusing language here: $g$ denotes both an element of $Poin$ and the linear transformation it induces on $\mathbb{R}^4$). By the Pontryagin isomorphism, this gives an action on momenta via the definition

$\chi_{p \cdot g} \coloneqq \chi_p \circ g$

The relativistic invariance of mass means that this right action carries a 4-momentum $p$ to another 4-momentum $p \cdot g$ of the same mass.

Let us return now to a unitary irrep of $Poin$. We restrict the irrep $Poin \to U(H)$ to a rep of $\mathbb{R}^4$. This decomposes into a direct sum (or direct integral) of a whole bunch of little 1-dimensional reps of $\mathbb{R}^4$ indexed by 4-momenta:

$H = \int_{p \in M} V_p$

where $M$ indexes the 4-momenta which occur as irreducible components, and the $V_p$ is the eigenspace attached to a given $p$ (i.e., the sum of the 1-dim components whose character is $\chi_p$).

Now I think an important point is that if $H$ is irreducible as a representation over $Poin$, then the 4-momenta that occur in $M$ all have the same mass. (That is, if $p$ and $p'$ occur in the direct integral but have different masses, then they would have to belong to different irreducible components of the original representation of $Poin$.) So whatever the domain of integration $M$ is, it ought to be contained in the locus of momentum space given by the equation $p \cdot p = m^2$.

That mass is what we mean by the mass of the unitary irrep of $P$.

Even better, under the assumption of irreducibility, the domain $M$ should be exactly the orbit of any one of these $p$ under the Poincaré group action. (If $M$ has more than one orbit, this would contradict irreducibility.) For example, in the relatively simple case of massive particles ($m \gt 0$), I believe $Poin$ acts transitively on that component of the locus $p \cdot p = m^2$ contained in the “forward light cone” – one of the two sheets of the hyperboloid – and this will be our $M$.

The transitive action of $Poin$ on $M$ defines a connected groupoid whose objects are points of $M$ and where morphisms $p \to p'$ are elements $g \in Poin$ such that $p \cdot g = p'$. The mapping $p \mapsto V_p$ defines a linear representation of this groupoid, and the action of $Poin$ on $H$ is completely determined from this groupoid representation (we retrieve the action by taking a direct integral – a little bit about this below). Since the groupoid is connected, this representation is determined by its restriction to any one of the automorphism groups $Aut(p)$ in the groupoid; in fact we have an identification of $M$ as a homogeneous space

$M = Poin/Aut(p)$

where $Aut(p)$ is the stabilizer of a point $p \in M$. The subgroup $\mathbb{R}^4 \hookrightarrow Aut(p)$ acts trivially on $V_p$, so if we form the quotient $L_p$ in the exact sequence

$0 \to \mathbb{R}^4 \to Aut(p) \to L_p \to 1$

then we are really interested in the action of $L_p$ on $V_p$. The group $Aut(p)$ is then the semidirect product $L_p \ltimes \mathbb{R}^4$.

The quotient $L_p$ is the so-called “little group” at $p$. It is a compact subgroup of the spin cover $SL_2(\mathbb{C})$ of the Lorentz group; by definition it is the stabilizer subgroup of $p$ in $SL_2(\mathbb{C})$. When $m \gt 0$, we may choose $p = (m, 0, 0, 0)$ as a representative momentum, and see that $p$ is fixed by any element in (the spin cover of) the spatial rotation group $SO(3)$. In other words, the little group here would be the spin cover $SU(2)$. Any other little group $L_{p'}$ would be a subgroup conjugate to $SU(2)$.

Summarizing then: if we know how a chosen little group $L_p$ acts on $V_p$, then there is a unique extension of this action to a groupoid representation as above, and from there we retrieve the action of $Poin$ on $H$ by a direct integral construction on the groupoid representation. Both of these processes are additive functors. Therefore, if the action of $L_p$ on $V_p$ were reducible, so must then be the action of $P$ on $H$.

Hence a necessary condition for the unitary representation $P \to U(H)$ to be irreducible is that any (and therefore every) accompanying little group representation $L_p \times V_p \to V_p$ also be irreducible. But we know about the irreducible representations in a case like $L_p = SU(2)$; as we mentioned, they are parametrized by a discrete parameter called spin (usually presented as a half-integer: $0, \frac1{2}, 1, \frac{3}{2}, \ldots$). For massless particles, other types of little groups can arise (e.g., $SO(1, 2)$), and ‘helicity’ is the general term for a discrete parameter which parametrizes their unitary irreps.

So, this gives the way in which each unitary irrep of $P$ gives rise to a mass and a helicity, and we have some idea of how to go the other way, putting a mass and helicity together to form a unitary irrep. Let’s go into some more detail on this.

Above I said that we form a representation of a groupoid. This groupoid is really a topological groupoid. Its underlying span is of the form

$\array{ & & Poin \times M & & \\ & \swarrow \mathrlap{\pi_2} & & \searrow \mathrlap{\alpha} & \\ M & & & & M }$

where $M$ is the coset space $Poin/Aut(p)$ and $\alpha$ is the canonical action on cosets. The representation of the groupoid is exactly what Greg Egan was telling us about before, the bundle induced from the representation of $V_p$ over $L_p$. I would write this induced bundle as

$\array{ Poin \otimes_{Aut(p)} V_p \\ \downarrow \mathrlap{\pi} \\ M \mathrlap{= Poin/Aut(p)} }$

where the tensor product refers to modding out by a relation of the form $(g h, v) \sim (g, h \cdot v)$. Hopefully that helps make manifest the way the groupoid is supposed to act: it boils down to the fact that $Poin$ acts naturally on the bundle.

Then, when I referred to a direct integral construction being performed on the groupoid representation, I am referring to something equivalent to taking the $L^2$ sections of the induced bundle. This should give back the Hilbert space $H$ that underlies the original unitary irrep, if I understand things correctly.

I have some concerns about what I have written, but I’ll pause for now, and think about it some more, and maybe ask some questions later.

Posted by: Todd Trimble on July 13, 2011 7:24 PM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

Hi, Todd! This looks great. The use of groupoids could be an interesting advance over the standard treatment.

You might someday want, or at least need, to get into the theory of ‘measurable fields of Hilbert spaces’, which lies behind the theory of direct integrals. This is explained really well here:

• J. Dixmier, Von Neumann Algebras, North-Holland, Amsterdam, 1981.

and also in a more friendly, less detailed way here:

• G. W. Mackey, Unitary Group Representations in Physics, Probability and Number Theory, Benjamin–Cummings, New York, 1978.

But if you have trouble getting ahold of these books, this quick summary might come in handy.

Posted by: John Baez on July 15, 2011 6:59 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I really liked chapter 14 (direct integrals and decompositions) of

* Kadison, Ringrose: “Fundamentals of the Theory of Operator Algebras. Advanced Theory”

This is the last chapter of a +1000 page book, so I should add that you don’t have to master all the material that is in there before you read chapter 14. I certainly don’t. And it has - like the whole book - a very user friendly style, which is especially true for the proofs.

(The link to the summary does not work for me, is it broken?)

Posted by: Tim van Beek on July 15, 2011 9:59 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

I fixed the link. If your browser does not take you to page 35 of the PDF file, please go there! The paper is long, but the summary of measurable fields of Hilbert spaces is short.

Posted by: John Baez on July 15, 2011 10:12 AM | Permalink | Reply to this

### Re: Unitary Representations of the Poincaré Group

There’s a coincidence which bothers me for quite a while:

1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So for photons, gluons, … we have 2 transversal polarizations i.e. 2 physical d.o.f.

2) For gauge theories like QED and QCD it is well known that the 4-vector carries two unphysical d.o.f. which can be eliminated by gauge fixing (a la Dirac, Gupta-Bleuler, BRST, …). An obvious way to see this is to
i) use the temporal gauge A° = 0 to eliminate one unphysical d.o.f.
ii) keep the corresponding Gauss law constraint G ~ 0 to define the physical Hilbert space as its kernel; this eliminates the second unphysical d.o.f.
So gauge fixing results in 4-2 = 2 physical d.o.f., too.

The method 2) gives us exactly the two helicity states described in 1) But 1) is using Poincare invariance whereas 2) is using gauge invariance w/o ever looking at Poincare invariance. So it seems that it’s sheer coincidence that 1) and 2) arrive at the same results.

Is there some deeper relation?

Posted by: Tom on October 17, 2012 9:42 AM | Permalink | Reply to this

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