## December 13, 2010

### The Three-Fold Way (Part 3)

#### Posted by John Baez

Last time we classified irreducible unitary group representations into three kinds: real, complex and quaternionic. But what does this mean for physics?

Well, since elementary particles are often described using representations like this, particles must come in three kinds: real, complex and quaternionic!

Of course the details depend not just on the particle itself, but on the group of symmetries we consider. But still, it sounds pretty far-out. What sort of particle is quaternionic?

This time we’ll look at the simplest example: an electron, regarded as a representation of $SU(2)$. People usually describe its state with a pair of complex numbers. But in fact, it makes a lot of sense to use a single quaternion!

We’ll see why in a while. But first, in case you fell asleep last time, let me remind you what we proved — we need it now. There are three choices for an irreducible unitary representation of a group $G$ on a complex Hilbert space $H$:

1. Our representation may not be isomorphic to its dual, in which case we call it truly complex.

2. It may be isomorphic to its dual thanks to an invariant antiunitary operator $J: H \to H$ with $J^2 = 1.$ In this case we call it real, because it’s the complexification of a representation on a real Hilbert space. And in this case there’s an invariant nondegenerate bilinear form $g : H \times H \to \mathbb{C}$ with $g(v,w) = g(w,v) ,$ also known as an orthogonal structure on $H$.

3. It may be isomorphic to its dual thanks to an invariant antiunitary operator $J: H \to H$ with $J^2 = -1 .$ In this case we call it quaternionic, because it comes from a representation on a quaternionic Hilbert space. In this case there’s an invariant nondegenerate bilinear form $g : H \times H \to \mathbb{C}$ with $g(v,w) = -g(w,v) ,$ also known as a symplectic structure on $H$.

This is the three-fold way.

Let’s consider an example: $G = SU(2)$. In physics this group is important because its representations describe the ways a particle can transform under rotations. There is one irreducible representation for each ‘spin’ $s = 0, \frac{1}{2}, 1, \dots$ When $s$ is an integer, this representation describes the angular momentum states of a boson; when $s$ is a half-integer (meaning an integer plus $\frac{1}{2}$) it describes the angular momentum states of a fermion.

For example, take any spin-$\frac{1}{2}$ particle, and consider only its rotational symmetries. Then we can describe this particle using the obvious unitary representation of $SU(2)$ on $\mathbb{C}^2$. This is called the spin-$\frac{1}{2}$ representation, and it’s quaternionic.

Why? Because we can think of a pair of complex numbers as a single quaternion, and $SU(2)$ as the group of unit quaternions. The obvious representation of $\SU(2)$ on $\mathbb{C}^2$ then turns out to be the action of unit quaternions on $\mathbb{H}$ via left multiplication. And since this commutes with right multiplication by quaternions, we’ve got a quaternionic representation!

In slogan form: qubits are not just quantum—they are also quaternionic!

More generally, all particles of half-integer spin are quaternionic, while particles of integer spin are real — as long as we consider them only as representations of $SU(2)$.

We shall see why this is true shortly, but I can’t resist making a little table that summarizes the pattern:

IRREDUCIBLE REPRESENTATIONS OF $SU(2)$

\begin{aligned} integer \; spin & bosonic & \quad & real & J^2 = 1 \\ half-integer \; spin & fermionic & & quaternionic & J^2 = -1 \end{aligned}

Why does it work this way?

We’ve seen it’s true for the spin-$\frac{1}{2}$ representation. But we can build all the irreducible unitary representations of $SU(2)$ as symmetrized tensor powers of the spin-$\frac{1}{2}$ representation. The $n$th symmetrized tensor power, $S^n(\mathbb{C}^2)$, is the spin-$s$ representation with $s = n/2$. At this point a well-known general result will help:

Theorem: Suppose $G$ is a Lie group and $H, H' \in Rep(G)$. Then:

• $H$ and $H'$ are real $\implies$ $H \otimes H'$ is real.
• $H$ is real and $H'$ is quaternionic $\implies$ $H \otimes H'$ is quaternionic.
• $H'$ is quaternionic and $H'$ is real $\implies$ $H \otimes H'$ is quaternionic.
• $H$ and $H'$ are quaternionic $\implies$ $H \otimes H'$ is real.

To see this, just notice that $H$ is real (resp. quaternionic) if and only if it can be equipped with an invariant antiunitary operator $J: H \to H$ with $J^2 = 1$ (resp. $J^2 = -1$). So, pick such an antiunitary $J$ for $H$ and also one $J'$ for $H'$. Then $J \otimes J'$ is an invariant antiunitary operator on $H \otimes H'$, and $(J \otimes J')^2 = J^2 \otimes {J'}^2 .$ This makes the result obvious.

In short, the ‘multiplication table’ for tensoring real and quaternionic representations is just like the multiplication table for the numbers $1$ and $-1$… and also, not coincidentally, just like the usual rule for combining bosons and fermions!

Next, note that any subrepresentation of a real representation is real, and any subrepresentation of a quaternionic representation is quaternionic. It follows that since the spin-$\frac{1}{2}$ representation of $\SU(2)$ is quaternionic, its $n$th tensor power is real or quaternionic depending on whether $n$ is even or odd, and similarly for the subrepresentation $S^n(\mathbb{C}^2)$. This is the spin-$s$ representation for $s = n/2$. So, the spin-$s$ representation is real or quaternionic depending on whether $s$ is an integer or half-integer.

But what is the physical meaning of the antiunitary operator $J$ on the spin-$s$ representation? In fact, it describes time reversal.

To see this, let’s call our representation $\rho : \SU(2) \to \U(H)$, where $H$ is the Hilbert space $S^n(\mathbb{C}^2)$. Choose any element $X$ in the Lie algebra $su(2)$ and let $S = d\rho(X)$. Then $J \, \exp(t S) = \exp(t S) \, J$ for all $t \in \mathbb{R}$, so differentiating gives $J S = S J$

But the operator $S$ is skew-adjoint. As discussed earlier, in quantum mechanics observables are self adjoint. To turn $S$ into an observable, say $A$, we should write $S = i A$. This observable $A$ tells us the particle’s angular momentum along some axis.

Since $J$ anticommutes with $i$, we have $J A = -A J$ So, for every state of our spin-$s$ particle, say $v \in H$, we have a new state $J v$ where the expected value of the observable $A$ is exactly opposite: $\langle J v, A J v \rangle = - \langle J V, J A v \rangle = - \langle A v, v \rangle = - \langle v, A v \rangle$ So, the antiunitary operator $J$ reverses angular momentum! Since the time-reversed version of a spinning particle is a particle spinning the opposite way, physicists call $J$ time reversal.

It seems natural that time reversal should obey $J^2 = 1$, as it does in the bosonic case; it may seem strange to have $J^2 = -1$, as we do for fermions. Shouldn’t applying time reversal twice get us back where we started? It actually does, in a crucial sense: the expectation values of all observables are unchanged when we multiply a unit vector $v \in H$ by $-1$. Still, this minus sign should remind you of the equally curious minus sign that a fermion picks up when you rotate it a full turn. Are these signs related? Is there a nice way to see the connection?

It seems so, and we’ll talk about that next time…

Posted at December 13, 2010 5:05 AM UTC

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### Re: The Three-Fold Way (Part 3)

Again a little advertising for the nLab: One of the most famous “theorems” in quantum field theory (QFT) says that nature is invariant under the simultaneous change of time (T), all charges (C), and parity (P), it is often called PCT or CPT theorem.

Since there are several different frameworks, with different axioms, for quantum field theories, it would be more accurate to talk about PCT theorems instead of the PCT theorem. And sometimes physicists use the word “theorem” for a statement that is not a theorem in the strict sense of mathematics, but:

There are fully rigorous approaches to QFT where the PCT theorem is actually a theorem with a proof, more details and references can be found on the nLab:

PCT theorem

Posted by: Tim van Beek on December 14, 2010 1:21 PM | Permalink | Reply to this

### Re: The Three-Fold Way (Part 3)

more details and references can be found on the nLab: PCT theorem.

Tim, I have edited the formatting of the statement of the definition of the PCT operator slightly. Check if you are still happy with it.

Hopefully somebody will find the energy to reproduce more of the story in Borchers-Ingvason article at the entry.

Posted by: Urs Schreiber on December 14, 2010 3:39 PM | Permalink | Reply to this

### Re: The Three-Fold Way (Part 3)

I’ll have a look :-)

Maybe others will, too…

Posted by: Tim van Beek on December 14, 2010 6:56 PM | Permalink | Reply to this

### Re: The Three-Fold Way (Part 3)

Interesting. In the Wightman approach, one must have, it says,

Covariance of the theory under the (connected part of the) Poincaré group.

I suppose it’s only natural, then, that PCT has to be proved a different way in curved spacetimes, when we have to forsake Poincaré symmetry. Neat!

Posted by: Blake Stacey on December 14, 2010 5:13 PM | Permalink | Reply to this

### Re: The Three-Fold Way (Part 3)

Robert Wald tells a nice story about the replacement of the Poincaré group on curved spacetimes in the review article that Blake linked to:

It is worth mentioning that there was an interesting historical interplay between microlocal analysis and quantum ﬁeld theory in curved spacetime. In the late 1960’s Hormander visited the Institute for Advanced Study in Princeton and interacted with Wightman. Wightman explained to Hormander what the “Feynman propagator” is in Minkowski spacetime, and a characterization of “Feynman parametrices” in a general curved spacetime in terms of wavefront set properties can be found in the classic paper of Duistermaat and Hormander [14]. Conversely, Wightman realized that the methods of microlocal analysis could potentially be useful in the formulation of quantum ﬁeld theory in curved spacetime. For example, in de Sitter spacetime, there is no globally timelike Killing ﬁeld and therefore no global notion of energy that is positive.

Remark: “Energy needs to be positive” means that there has to be a lower bound on the energy that a state can have, lest e.g. the vacuum becomes instable and starts to decay to states with lower and lower energy. This condition is sometimes called the “spectrum condition”, e.g. on the page Haag-Kastler vacuum representation, and is formulated using the (representation of the) Poincarè group for QFT on the Minkowski spacetime.

Therefore, it does not appear that one could impose a global spectral condition on a quantum ﬁeld analogous to the requirement of positivity of energy in the Minkowski case. However, one might be able to impose a “microlocal spectral condition” on the local quantum ﬁeld observables.

Shortly after his interactions with Hormander, Wightman had a student, S. Fulling, who was interested in quantum ﬁeld theory in curved spacetime, and he suggested to Fulling that he investigate the possible application of microlocal analysis to quantum ﬁeld theory in curved spacetime. However, after spending some effort in studying microlocal analysis, Fulling decided that his efforts would be better spent on other projects. Among the other projects that Fulling then investigated in his thesis was the inequivalence of different quantization schemes. In particular, he showed that quantization in the “Rindler wedge” of Minkowski spacetime using a Lorentz boost Killing ﬁeld to deﬁne a notion of “time translations” gave rise to a different notion of “vacuum state” than the restriction of the usual Minkowski vacuum to this region. This work provided the mathematical basis for Unruh’s subsequent analysis discussed above [9].

However, Wightman had to wait another twenty years before he had another student interested in quantum ﬁeld theory in curved spacetime. When Radzikowski began to apply the methods of microlocal analysis to analyze Kay’s conjecture, Wightman was well prepared to provide plenty of encouragement.

What Kay’s conjecture was, as well as a big part of the rest of the story of QFT on curved spacetimes, can be found here:

• Robert Wald: “The History and Present Status of Quantum Field Theory in Curved Spacetime”, arXiv.
Posted by: Tim van Beek on December 14, 2010 7:06 PM | Permalink | Reply to this

### Re: The Three-Fold Way (Part 3)

Part of Hollands and Wald is already explicitly categorical: starting from a background ${\bf M}$, they define a QFT by writing the coefficients $\mathcal{C}({\mathbf M})$ of the operator product expansion. The “OPE-coefficient systems” $\mathcal{C}({\mathbf M})$ are the objects in a category whose morphisms are maps which preserve certain characteristics of the field theory; the algebra of observables $\mathcal{A}({\mathbf M})$ and the state space $\mathcal{S}({\mathbf M})$ are objects in categories on the other end of functors from the OPE-coefficient-system category.

OK, off to learn more about operator product expansions so I can make sense of this (particularly what happens when you add interactions to the free theory).

Posted by: Blake Stacey on December 14, 2010 9:49 PM | Permalink | Reply to this

### Re: The Three-Fold Way (Part 3)

Part of Hollands and Wald is already explicitly categorical:

There is a formulation of perturbative quantum field theory by Kevin Costello and Owen Gwilliam that is vaguely reminiscent of the kind of approach that Hollands is taking. The difference is probably that here the basic idea of a vertex operator algebra is interpreted in higher category theory: formalized as a factorization algebra.

They have, remarkably, a wiki on which the upcoming publication on this is being produced: Factorization algebras in perturbative quantum field theory.

Posted by: Urs Schreiber on December 14, 2010 10:41 PM | Permalink | Reply to this

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