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 KleinGordon 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)
selfadjoint 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) selfadjoint 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 1dimensional 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 onedimensional rep of the real line. And, I hope you see there’s some sense in which our rep $U$ is ‘built’ from all these 1dimensional reps.
What is this sense, exactly? If $X$ is a finite set, our rep $U$ is a direct sum of these 1dimensional reps $U_{a(x)}$, one for each point $x \in X$. But in general, $U$ is a direct integral of these 1dimensional 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:

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!

Guess which 1dimensional reps $V$ is a direct integral of. What famous thing are we secretly talking about here?

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 onedimensional reps look like?

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.

Generalize problem 2 to $\mathbb{R}^n$. What famous thing are we talking about now?

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 $L^2(hyperboloid)$?
Re: Unitary Representations of the Poincaré Group
The article described below is a complete redo of Wigner’s sections 14 and much of 57. 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 nCategory 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  nCategory Cafe (the thread)
The Wigner Classification for General 4Space Signatures (the reply)
Related reading includes:
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 NonHelical 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
(UIChicago, 2008 October 7)
An online 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 NewtonWigner Position Operator  derived from T. F. Jordan
The coordinatization of the class 1L (and the nogo result for carrying out the same for nonHelical 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 PoissonLie 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 quasivacuum, one which has the boost generator K as an invariant. An isotropic quasivacuum 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 (possiblydegenerate) signatures, replacing the orthogonal group with the biorthogonal groups. In addition, treatment by linear representation theory is expanded (and yet, simplified) by its generalization to the nonlinear representation theory of PoissonLie 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 4BD: 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 spacetime 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 presentday 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 wellknown.
This analysis is also done here.
7. Wigner’s Section 67: Symplectic Decomposition
No analysis in terms of linear spaces. Instead: a far more general analysis in terms of nonlinear representation. Irrep is replaced by symplectic leaf.
7.1. The Vacuon Sectors
Covers the 3 classes of translationinvariant media: the generic vacuon, the quasivacuum and the vacuum.
7.2. The Archimedean Tachyon Sectors
Covers a special subclass of translation noninvariant media: those specific to the Archimedean signature, corresponding to systems in motion.
7.3. Wigner’s Section 6: Translation Invariants
The translationinvariant sectors are the vacuons; while the translation noninvariant sectors subdivide into the Archimeden Tachyon, the Helical and NonHelical sectors.
Part of Wigner’s section 6 deals with the translationinvariants. Here, the analysis is done more simply. Out of it naturally emerge the PauliLubanski 4vector.
7.4. The NonHelical Sectors
This crossclassifies with the von Neumann–Wigner classification to yield the nonhelical 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 nonArchimdean cases is provided.
The helical subdivision, like the nonhelical subdivision, also crosscategories with respect to the Tardion, Tachyon, Luxon/Synchron/Staton classification.
8. MassEnergyMomentum vs. Velocity for the Translation NonInvariant Sectors
9. Coordinatization of the Translation NonInvariant Sectors
Once you get a coordinatization (which is an application of the Darboux Theorem), you then have the canonical 1form and 2forms, as well as the basis for defining the invariant measures on both the phase space and the configuration space. The general nonhelical translation noninvariant 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 massshell 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 KleinGordon equation, up to a FoldyWoutheusen 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 PoissonLie 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 nonhelical tardion and helical sectors.
13. Many Body States and the NoInteraction 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. nonrelativistic forms of the Haag and Leutweiler theorems).
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 longwinded, 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 halfintegers, 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 1dimensional 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 4momentum, 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 4momentum $p$ to another 4momentum $p \cdot g$ 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 1dimensional reps of $\mathbb{R}^4$ indexed by 4momenta:
$H = \int_{p \in M} V_p$
where $M$ indexes the 4momenta which occur as irreducible components, and the $V_p$ is the eigenspace attached to a given $p$ (i.e., the sum of the 1dim 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 4momenta 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 socalled “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 halfinteger: $0, \frac1{2}, 1, \frac{3}{2}, \ldots$). 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.
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.