From the Little Algebra to the Labels of a Massive Representation

Contents

Conventions, and what is assumed
Statement
The geometry of the mass shell
The induced representation
The classification, and the count
The observed case

This appendix supplies the step left owed after Lemma 18.68 of Lie Groups, Lie Algebras, and Fibre Bundles: the passage from the algebra that fixes a timelike momentum to the count of labels of a massive irreducible representation of the inhomogeneous algebra, and with it the arithmetic of Equation (18.107),

\begin{equation}\tag{A36.1} 1 + \left\lfloor\frac{D-1}{2}\right\rfloor = \left\lceil\frac{D}{2}\right\rceil\ep \end{equation}

What Lemma 18.68 establishes is that the subalgebra annihilating a timelike $p$ is $\mathfrak{so}(D-1)$, of rank $\lfloor(D-1)/2\rfloor$. What is owed is the representation theory that converts a rank into a count: that a massive irreducible representation is determined by the mass together with an irreducible representation of that subalgebra, and that no further invariant of the enveloping algebra escapes those labels.

The construction is Wigner's method of induced representations [Wigner:1939]. Particles as Poincaré Representations carries it out in the observed four dimensions — Theorem 99.27 is the construction and Theorem 99.29 the result — and this appendix does the same in general $D$, with two differences of emphasis: the geometric ingredients (transitivity of the Lorentz group on the mass shell, the standard boost, the invariant measure) are proved here rather than quoted, and the arithmetic of the label count is separated cleanly from the classification theorem it rests on. That classification theorem is the one genuine import, and Remark A36.11 says exactly what it is.

Under rule 7 of this treatise the general-$D$ statement is the natural mathematical one and is stated as such; every worked instantiation below is $3+1$.

Conventions, and what is assumed

Indices $A,B,\ldots$ run from $1$ to $D$ and $\eta_{AB} = \diag(+1,\ldots,+1,-1)$ is the flat metric of signature $(D-1,1)$ in the ordering of Notation 18.1, so that the time direction is the last one and $x^{D} = ct$, as in Remark 18.88. A vector $p$ is timelike when $\eta_{AB}p^{A}p^{B} < 0$, which is the convention of Lemma 18.68.

The symmetry group is the connected inhomogeneous group

\begin{equation}\tag{A36.2} G = \R^{D} \rtimes \SO(D-1,1)^{\uparrow}\ec \qquad (\Lambda',a')(\Lambda,a) = (\Lambda'\Lambda,\ a' + \Lambda' a)\ec \end{equation}

whose Lie algebra is $\mathfrak{iso}(D-1,1)$ of Equation (18.99), with brackets Equations (18.84), (18.90) and (18.97); the translation $a$ carries the SI unit $\mathrm{m}$. A quantum symmetry is a projective unitary representation, and by Theorem 99.3 — proved in Wigner's Theorem on Quantum Symmetries — each symmetry is implemented by a unitary or antiunitary operator unique up to a phase, while Corollary 99.4 rules out the antiunitary branch for a connected group. Bargmann's analysis then replaces the projective representations of $G$ by the ordinary unitary representations of its universal cover $\tilde{G}$ [Bargmann:1954], whose Lorentz factor is $\Spin(D-1,1)$. Everything below is written for $G$; for the half-integer labels every $\Lambda$ is read as its lift to $\Spin(D-1,1)$ and every little-group element as an element of $\Spin(D-1)$, exactly as Theorem 99.27 does in four dimensions.

The generators are normalized as in Remark 18.66: with $P_{A} = \pp_{A}$ from Equation (18.82) the momentum operator is $\hat{P}_{A} = -\ii\hbar P_{A}$, of SI unit $\mathrm{kg}\,\mathrm{m}/\mathrm{s}$, and with $J_{AB}$ from Equation (18.86) the angular-momentum operator is $\hat{J}_{AB} = -\ii\hbar J_{AB}$, of unit $\mathrm{J}\,\mathrm{s}$.

Remark A36.1 (A sign of signature).

In the mostly-plus ordering fixed above, a timelike momentum satisfies $\eta_{AB}p^{A}p^{B} = -m^{2}c^{2}$ and therefore $\hat{P}^{2} = -m^{2}c^{2}$ on a momentum eigenstate. Equation (18.105) records the same content with the opposite overall sign, being written in the mostly-minus convention customary in Particles as Poincaré Representations; the two differ by $\eta \mapsto -\eta$ and nothing else. Only $\abs{\hat{P}^{2}} = m^{2}c^{2}$ is used below, and the ratio Equation (18.106) is unaffected by the choice, both invariants changing sign together.

Statement

Definition A36.2 (The mass shell and the standard momentum).

For $m > 0$ put

\begin{equation}\tag{A36.3} \mathcal{M}_{m} = \set{p \in \R^{D} \mid \eta_{AB}p^{A}p^{B} = -m^{2}c^{2},\ p^{D} > 0}\ec \qquad k^{A} = mc\,\delta^{A}_{D}\ec \end{equation}

so that $k \in \mathcal{M}_{m}$ is the rest momentum, of components $\vect{p} = 0$ and $p^{D} = mc$. The little group is the stabilizer $G_{k} = \set{\Lambda \in \SO(D-1,1)^{\uparrow} \mid \Lambda k = k}$. Rests on Notation 18.1 and Lemma 18.68.

Theorem A36.3 (Labels of a massive representation).

Let $D \ge 4$ and $m > 0$. The unitary equivalence classes of irreducible, strongly continuous, positive-energy unitary representations $U$ of $\tilde{G}$ with $\hat{P}^{2} = -m^{2}c^{2}\,\identity$ are in bijection with the unitary equivalence classes of irreducible unitary representations $\sigma$ of $\Spin(D-1)$, the bijection being $U \cong U^{(m,\sigma)}$ with $U^{(m,\sigma)}$ the induced representation Equation (A36.8) below. Consequently the class of $U$ is fixed by the mass $m$ together with the $\lfloor(D-1)/2\rfloor$ invariant labels of $\sigma$ furnished by Theorem 18.21, and by no fewer numbers; a massive irreducible representation therefore carries exactly $\lceil D/2\rceil$ labels, which is Equation (A36.1). Every central element of the enveloping algebra acts on $U$ as a scalar that is a function of those labels. Rests on Lemma 18.68, Theorem 18.21 and Definition A36.2.

The bijection itself — the classification — is the imported ingredient; the geometry that makes it meaningful, the construction of $U^{(m,\sigma)}$, and the arithmetic of the count are proved here.

The geometry of the mass shell

Lemma A36.4 (The little group).

$G_{k} = \SO(D-1)$, acting on the first $D-1$ coordinates, and its Lie algebra is the $\mathfrak{so}(D-1)$ of Lemma 18.68. It is compact. Rests on Definition A36.2, Equation (18.23) and Lemma 18.68.

Proof.

Derives Lemma A36.4. Let $\Lambda k = k$, that is $\Lambda e_{D} = e_{D}$ where $e_{D}$ is the unit vector in the time direction. Since $\Lambda$ preserves $\eta$, it preserves the orthogonal complement $e_{D}^{\perp} = \R^{D-1}$, on which $\eta$ restricts to the Euclidean metric; so the restriction lies in $\Ogrp(D-1)$. Its determinant equals $\det\Lambda = 1$ because $\Lambda$ acts as the identity on the remaining direction, whence $\Lambda|_{\R^{D-1}} \in \SO(D-1)$; conversely every such rotation, extended by $e_{D} \mapsto e_{D}$, preserves $\eta$, has determinant $1$ and is orthochronous. Compactness is that of a closed bounded subset of $\R^{(D-1)\times(D-1)}$: the rows of an orthogonal matrix are unit vectors, and orthogonality is a closed condition. Differentiating a curve through the identity gives exactly the subalgebra of Lemma 18.68, namely the elements of $\mathfrak{so}(D-1,1)$ annihilating $k$.

∎

Lemma A36.5 (Standard boost and transitivity).

For $p \in \mathcal{M}_{m}$ write $\vect{p} = (p^{1},\ldots,p^{D-1})$ and $\abs{\vect{p}}$ for its Euclidean length, so that $p^{D} = \sqrt{\abs{\vect{p}}^{2} + m^{2}c^{2}}$. The matrix

\begin{equation}\tag{A36.4} L(p)^{i}{}_{j} = \delta^{i}{}_{j} + \frac{p^{i}p^{j}}{\abs{\vect{p}}^{2}} \left(\frac{p^{D}}{mc} - 1\right)\ec\qquad L(p)^{i}{}_{D} = L(p)^{D}{}_{i} = \frac{p^{i}}{mc}\ec\qquad L(p)^{D}{}_{D} = \frac{p^{D}}{mc}\ec \end{equation}

with $i,j = 1,\ldots,D-1$, is a proper orthochronous Lorentz transformation, depends continuously on $p$ with $L(k) = \identity$, and satisfies $L(p)k = p$. In particular $\SO(D-1,1)^{\uparrow}$ acts transitively on $\mathcal{M}_{m}$, and $\mathcal{M}_{m}$ is a single orbit. Rests on Definition A36.2 and Equation (18.23).

Proof.

Derives Lemma A36.5. Put $\hat{n}^{i} = p^{i}/\abs{\vect{p}}$, $\cosh\zeta = p^{D}/(mc)$ and $\sinh\zeta = \abs{\vect{p}}/(mc)$; these are consistent because $\cosh^{2}\zeta - \sinh^{2}\zeta = \left((p^{D})^{2} - \abs{\vect{p}}^{2}\right)/(mc)^{2} = 1$ by Equation (A36.3), and $\zeta \ge 0$ is determined. In these terms Equation (A36.4) reads

\begin{equation}\tag{A36.5} L(p) = \identity + \left(\cosh\zeta - 1\right) \left(\hat{n}\hat{n}\transpose + e_{D}\,e_{D}\transpose\right) + \sinh\zeta\left(\hat{n}\,e_{D}\transpose + e_{D}\,\hat{n}\transpose\right)\ec \end{equation}

$\hat{n}$ being regarded as a vector in $\R^{D}$ with vanishing last component. Reading off the entries confirms that this is Equation (A36.4): the $\hat{n}\hat{n}\transpose$ term gives $L^{i}{}_{j}$, the two mixed terms give $L^{i}{}_{D} = L^{D}{}_{i} = \sinh\zeta\,\hat{n}^{i} = p^{i}/(mc)$, and the $e_{D}e_{D}\transpose$ term raises $L^{D}{}_{D}$ from $1$ to $\cosh\zeta = p^{D}/(mc)$. It is the identity on the $(D-2)$-dimensional subspace orthogonal to both $\hat{n}$ and $e_{D}$, and on the plane they span it is $\begin{pmatrix}\cosh\zeta & \sinh\zeta\\ \sinh\zeta & \cosh\zeta\end{pmatrix}$ in the basis $(\hat{n},e_{D})$. Restricted to that plane $\eta$ is $\diag(+1,-1)$, and $\cosh^{2}\zeta - \sinh^{2}\zeta = 1$ is exactly the statement that the matrix preserves it; on the complement $L(p)$ is the identity and the two blocks are $\eta$-orthogonal. So Equation (18.23) holds at the group level and $L(p) \in \Ogrp(D-1,1)$. Its determinant is $\cosh^{2}\zeta - \sinh^{2}\zeta = 1$ and its $(D,D)$ entry is $\cosh\zeta \ge 1 > 0$, so it is proper and orthochronous.

It maps $k$ to $p$. From Equation (A36.4), $\left(L(p)k\right)^{i} = L(p)^{i}{}_{D}\,mc = p^{i}$ and $\left(L(p)k\right)^{D} = L(p)^{D}{}_{D}\,mc = p^{D}$.

Continuity at $\vect{p} = 0$. The only term in Equation (A36.4) whose form is singular there is $p^{i}p^{j}\left(p^{D}/(mc) - 1\right)/\abs{\vect{p}}^{2}$, and

\begin{equation*} \frac{p^{D}}{mc} - 1 = \frac{\sqrt{\abs{\vect{p}}^{2}+m^{2}c^{2}} - mc}{mc} = \frac{\abs{\vect{p}}^{2}} {mc\left(\sqrt{\abs{\vect{p}}^{2}+m^{2}c^{2}} + mc\right)}\ec \end{equation*}

so the quotient by $\abs{\vect{p}}^{2}$ is bounded and tends to $0$; the whole term vanishes with $\vect{p}$, leaving $L(k) = \identity$. The remaining entries are manifestly continuous.

Transitivity. For $p,q \in \mathcal{M}_{m}$ the element $L(q)L(p)^{-1}$ is in $\SO(D-1,1)^{\uparrow}$ and carries $p$ to $q$. Conversely a Lorentz transformation preserves $\eta_{AB}p^{A}p^{B}$, and an orthochronous one preserves the sign of $p^{D}$ on timelike vectors, so it maps $\mathcal{M}_{m}$ into itself.

∎

Lemma A36.6 (The invariant measure).

The measure

\begin{equation}\tag{A36.6} \dd\mu(p) = \frac{\dd^{D-1}\vect{p}}{p^{D}}\ec \qquad p^{D} = \sqrt{\abs{\vect{p}}^{2}+m^{2}c^{2}}\ec \end{equation}

on $\mathcal{M}_{m}$ is invariant under $\SO(D-1,1)^{\uparrow}$. Rests on Definition A36.2 and Lemma A36.5.

Proof.

Derives Lemma A36.6. Consider on $\R^{D}$ the measure

\begin{equation*} \dd\nu(p) = \dd^{D}p\; \delta\!\left(\eta_{AB}p^{A}p^{B} + m^{2}c^{2}\right) \theta\!\left(p^{D}\right)\ec \end{equation*}

with $\theta$ the unit step. Every factor is invariant: $\dd^{D}p$ because $\abs{\det\Lambda} = 1$, the delta because its argument is a Lorentz scalar, and the step because $\Lambda$ is orthochronous. Now carry out the $p^{D}$ integration. Writing $f(p^{D}) = \abs{\vect{p}}^{2} - (p^{D})^{2} + m^{2}c^{2}$, the only zero with $p^{D} > 0$ is $p^{D} = \sqrt{\abs{\vect{p}}^{2}+m^{2}c^{2}}$, at which $\abs{f'} = 2p^{D}$; so $\delta(f)\,\theta(p^{D})\,\dd p^{D}$ integrates to $1/\left(2p^{D}\right)$ and $\dd\nu = \dd^{D-1}\vect{p}/(2p^{D}) = \tfrac{1}{2}\dd\mu$. A constant multiple of an invariant measure is invariant.

∎

Remark A36.7 (The SI dimension of the measure).

Each $p^{i}$ carries the unit $\mathrm{kg}\,\mathrm{m}/\mathrm{s}$, so $\dd\mu$ of Equation (A36.6) carries $\left(\mathrm{kg}\,\mathrm{m}/\mathrm{s}\right)^{D-2}$; in the observed $D = 4$ that is $\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}$. A wave function on the mass shell therefore carries the reciprocal of the square root of that unit, so that $\int\dd\mu(p)\,\abs{\psi(p)}^{2}$ is a pure number, as a probability must be. One may of course divide $\dd\mu$ by $(mc)^{D-2}$ to obtain a dimensionless measure; nothing below depends on the choice, and the unnormalized form is kept because it is the one in which Lemma A36.6 is transparent.

The induced representation

Proposition A36.8 (Construction).

Let $\sigma$ be a unitary representation of $G_{k} = \SO(D-1)$ — or of $\Spin(D-1)$, for the covering group — on a finite-dimensional Hilbert space $V_{\sigma}$. On

\begin{equation}\tag{A36.7} \mathcal{H}_{\sigma} = L^{2}\!\left(\mathcal{M}_{m},\dd\mu;V_{\sigma}\right) \end{equation}

define, for $a \in \R^{D}$ and $\Lambda \in \SO(D-1,1)^{\uparrow}$,

\begin{equation}\tag{A36.8} \left(U^{(m,\sigma)}(\identity,a)\psi\right)(p) = \ee^{-\ii\,\eta_{AB}p^{A}a^{B}/\hbar}\,\psi(p)\ec\qquad \left(U^{(m,\sigma)}(\Lambda,0)\psi\right)(p) = \sigma\!\left(W(\Lambda,\Lambda^{-1}p)\right) \psi\!\left(\Lambda^{-1}p\right)\ec \end{equation}

with the Wigner rotation

\begin{equation}\tag{A36.9} W(\Lambda,q) = L(\Lambda q)^{-1}\,\Lambda\,L(q)\ec \qquad q \in \mathcal{M}_{m}\ec \end{equation}

and $U^{(m,\sigma)}(\Lambda,a) = U^{(m,\sigma)}(\identity,a)\,U^{(m,\sigma)}(\Lambda,0)$. Then $W(\Lambda,q) \in G_{k}$, and $U^{(m,\sigma)}$ is a strongly continuous unitary representation of $G$ with

\begin{equation}\tag{A36.10} \hat{P}_{A}\psi(p) = p_{A}\,\psi(p)\ec\qquad \hat{P}^{2} = -m^{2}c^{2}\,\identity\ec \end{equation}

and positive energy, $p^{D} > 0$ throughout the spectrum. Rests on Lemmas A36.4, A36.5 and A36.6.

Proof.

Derives Proposition A36.8. The Wigner rotation lies in the little group. Using $L(q)k = q$ from Lemma A36.5,

\begin{equation*} W(\Lambda,q)\,k = L(\Lambda q)^{-1}\Lambda L(q)k = L(\Lambda q)^{-1}\left(\Lambda q\right) = k\ec \end{equation*}

so $W(\Lambda,q) \in G_{k}$ by Definition A36.2, and it depends continuously on $(\Lambda,q)$ because $L$ does.

The composition law. Put $q = \Lambda^{-1}\Lambda'^{-1}p$, so that $\Lambda q = \Lambda'^{-1}p$ and $\Lambda'\Lambda q = p$. Then

\begin{align} W(\Lambda',\Lambda q)\,W(\Lambda,q) &= \left[L(\Lambda'\Lambda q)^{-1}\Lambda' L(\Lambda q)\right] \left[L(\Lambda q)^{-1}\Lambda L(q)\right] \nn\\ &= L\!\left(\Lambda'\Lambda q\right)^{-1}\Lambda'\Lambda L(q) = W\!\left(\Lambda'\Lambda, q\right)\ec \tag{A36.11} \end{align}

the two middle factors cancelling. Applying Equation (A36.8) twice and using Equation (A36.11) together with the homomorphism property of $\sigma$,

\begin{equation*} \left(U(\Lambda',0)U(\Lambda,0)\psi\right)(p) = \sigma\!\left(W(\Lambda',\Lambda q)\right) \sigma\!\left(W(\Lambda,q)\right)\psi(q) = \sigma\!\left(W(\Lambda'\Lambda,q)\right)\psi(q)\ec \end{equation*}

which is $\left(U(\Lambda'\Lambda,0)\psi\right)(p)$. The translations compose by inspection, $\ee^{-\ii p\cdot a/\hbar} \ee^{-\ii p\cdot a'/\hbar} = \ee^{-\ii p\cdot(a+a')/\hbar}$, and the mixed relation required by Equation (A36.2) is

\begin{equation*} \left(U(\Lambda,0)U(\identity,a)\psi\right)(p) = \ee^{-\ii\,(\Lambda^{-1}p)\cdot a/\hbar} \sigma(W)\psi(\Lambda^{-1}p) = \ee^{-\ii\,p\cdot(\Lambda a)/\hbar} \left(U(\Lambda,0)\psi\right)(p)\ec \end{equation*}

because $\eta$ is $\Lambda$-invariant: $(\Lambda^{-1}p)\cdot a = p\cdot(\Lambda a)$. That is $U(\Lambda,0)U(\identity,a) = U(\identity,\Lambda a)U(\Lambda,0)$, the semidirect product law.

Unitarity. For the translations the factor is a phase. For the Lorentz part,

\begin{equation*} \norm{U(\Lambda,0)\psi}^{2} = \int_{\mathcal{M}_{m}}\dd\mu(p)\, \norm{\sigma\!\left(W(\Lambda,\Lambda^{-1}p)\right) \psi(\Lambda^{-1}p)}_{V_{\sigma}}^{2} = \int_{\mathcal{M}_{m}}\dd\mu(p)\, \norm{\psi(\Lambda^{-1}p)}_{V_{\sigma}}^{2}\ec \end{equation*}

since $\sigma$ is unitary on $V_{\sigma}$, and the last integral equals $\norm{\psi}^{2}$ by the invariance of $\dd\mu$ (Lemma A36.6). Surjectivity is $U(\Lambda,0)^{-1} = U(\Lambda^{-1},0)$. Strong continuity follows from the continuity of $L$, $W$ and $\sigma$ together with dominated convergence on the dense set of compactly supported continuous $\psi$.

The momentum spectrum. The one-parameter groups of translations are $U(\identity,a) = \exp\left(-\ii\,\eta_{AB}\hat{P}^{A}a^{B} /\hbar\right)$, which is the normalisation of Remark 18.66; comparing with the first line of Equation (A36.8), whose phase is $-\ii\,\eta_{AB}p^{A}a^{B}/\hbar$ for every $a$, the generator $\hat{P}^{A}$ acts as multiplication by the number $p^{A}$, which is Equation (A36.10); then $\hat{P}^{2} = \eta^{AB}p_{A}p_{B} = -m^{2}c^{2}$ everywhere on $\mathcal{M}_{m}$ by Equation (A36.3), and $p^{D}>0$ there by construction.

∎

Proposition A36.9 (Reducible little-group data give reducible representations).

If $V_{\sigma}$ has a proper nonzero $\sigma$-invariant subspace $V'$, then $L^{2}(\mathcal{M}_{m},\dd\mu;V')$ is a proper nonzero $U^{(m,\sigma)}$-invariant subspace of $\mathcal{H}_{\sigma}$. Hence irreducibility of $U^{(m,\sigma)}$ requires irreducibility of $\sigma$. Rests on Proposition A36.8.

Proof.

Derives Proposition A36.9. Both operations in Equation (A36.8) act on the value $\psi(q) \in V_{\sigma}$ either by a scalar phase or by $\sigma(W)$ with $W \in G_{k}$, and both preserve $V'$; the argument $q$ is merely relabelled. So the subspace of functions with values in $V'$ is invariant, and it is proper and nonzero because $V'$ is.

∎

The classification, and the count

The converse of Proposition A36.9, and the statement that the list Equation (A36.8) is exhaustive, is the imported ingredient.

Theorem A36.10 (Wigner–Mackey classification; quoted).

Let $D \ge 4$, $m > 0$, and let $U$ be a strongly continuous unitary representation of $\tilde{G}$ that is irreducible, has positive energy and satisfies $\hat{P}^{2} = -m^{2}c^{2}\identity$. Then $U$ is unitarily equivalent to $U^{(m,\sigma)}$ of Equation (A36.8) for some irreducible unitary representation $\sigma$ of $\Spin(D-1)$; conversely $U^{(m,\sigma)}$ is irreducible whenever $\sigma$ is; and $U^{(m,\sigma)}$ and $U^{(m,\sigma')}$ are equivalent if and only if $\sigma$ and $\sigma'$ are [Wigner:1939] [Weinberg:1995]. Rests on Proposition A36.8 and Definition A36.2.

Remark A36.11 (What is quoted here).

Theorem A36.10 is the one statement of this appendix that is not proved in this treatise, and it is worth being exact about what it costs.

Its content is Mackey's imprimitivity theorem [Mackey:1952], specialized by Wigner [Wigner:1939] to the inhomogeneous Lorentz group. Three analytic inputs go into it, none of which this treatise develops: the spectral theorem for a strongly continuous unitary representation of the translation group $\R^{D}$ (the Stone–Naimark–Ambrose–Godement theorem, which supplies the projection-valued measure whose support is the momentum spectrum — Stone's one-parameter case is [Stone:1932] and the general statement is standard functional analysis [Reed:1972]); the identification of the commutant of that measure's multiplication algebra with the decomposable operators, which is what turns an operator commuting with all translations into a measurable field $p \mapsto A(p)$ of matrices; and the disintegration of the Hilbert space over the orbit, which converts the field into a representation of the stabilizer. Given those three, the classification is the argument whose geometric half is proved above: the spectrum is a $\Lambda$-invariant subset of the hyperboloid Equation (A36.3), which is one orbit by Lemma A36.5, so the spectrum is the whole mass shell; the explicit continuous section $L$ of Equation (A36.4) then trivializes the bundle of fibres, and the residual freedom is exactly a representation of $G_{k}$.

Two things follow from that accounting. First, the imported theorem is functional analysis, not group theory: the Lie-theoretic content — which subalgebra fixes a timelike momentum, what its rank is, and how many labels its representations carry — is proved, in Lemma 18.68 and in Theorem 18.21. Second, the existence half of the classification, which is what a physical application actually uses, is proved here without any import: Proposition A36.8 constructs $U^{(m,\sigma)}$ and computes its invariants, and Proposition A36.9 supplies one direction of the irreducibility criterion. What is quoted is the exhaustiveness of the list, together with the converse irreducibility statement.

One further standard fact is used silently and is named here: a unitary representation of a compact group decomposes into finite-dimensional irreducible ones (the Peter–Weyl theorem [Weinberg:1995]), which is why $V_{\sigma}$ may be taken finite-dimensional in Proposition A36.8. Lemma 99.28 records the same point for $\SU(2)$ in four dimensions.

Proof of Theorem A36.3. Derives Theorem A36.3. The bijection between equivalence classes is Theorem A36.10. It remains to count.

The little algebra and its rank. By Lemma A36.4 the little group is $\SO(D-1)$, whose Lie algebra is the $\mathfrak{so}(D-1)$ of Lemma 18.68; and since $D \ge 4$, that algebra has $D-1 \ge 3$ and is semisimple, of rank $\lfloor(D-1)/2\rfloor$ by Proposition 18.22. Passing to $\Spin(D-1)$ changes neither the algebra nor its rank.

How many labels $\sigma$ carries. Theorem 18.21 applied to $\mathfrak{so}(D-1)$ says that the centre of its enveloping algebra is a polynomial algebra in $\lfloor(D-1)/2\rfloor$ algebraically independent Casimir elements, and that a complete set of invariant labels for its finite-dimensional irreducible representations consists of that many numbers and of no fewer. The representation $\sigma$ is finite-dimensional by the Peter–Weyl theorem (Remark A36.11), so exactly $\lfloor(D-1)/2\rfloor$ numbers fix its class.

Adding the mass. The number $m$ is an invariant of $U$ independent of those: it is read off $\hat{P}^{2}$, which by Equation (A36.10) is $-m^{2}c^{2}$ and involves the translation generators alone, whereas the little-group Casimirs are built from $\hat{J}$. Distinct masses therefore give inequivalent representations for every $\sigma$, and distinct $\sigma$ give inequivalent representations for every mass, by Theorem A36.10. The complete label set is thus $m$ together with the $\lfloor(D-1)/2\rfloor$ labels of $\sigma$, and by Equation (18.108) of Lemma 18.68 their number is

\begin{equation*} 1 + \left\lfloor\frac{D-1}{2}\right\rfloor = \left\lceil\frac{D}{2}\right\rceil\ec \end{equation*}

which is Equation (A36.1) and Equation (18.107).

No further invariant survives. Let $C$ be any central element of the enveloping algebra $U\!\left(\mathfrak{iso}(D-1,1)\right)$. On the irreducible $U$ it acts as a scalar by Corollary 18.18, and that scalar depends only on the unitary equivalence class of $U$ — an intertwiner conjugates $C$ to itself. By the bijection the class is the pair $(m,[\sigma])$, so the scalar is a function of $m$ and of the $\lfloor(D-1)/2\rfloor$ labels of $\sigma$: no invariant of the enveloping algebra can separate two representations that those numbers already identify.

∎

The observed case

Example A36.12 ($D = 4$).

Take $D = 4$, the observed spacetime. Then $\lfloor(D-1)/2\rfloor = \lfloor 3/2 \rfloor = 1$ and $\lceil D/2\rceil = 2$: a massive particle carries exactly two labels.

The little group is $\SO(3)$ by Lemma A36.4, its cover is $\SU(2)$, and its algebra has rank one (Proposition 18.22 with $D-1 = 3$), so Theorem 18.21 allows a single Casimir — the $\vect{J}^{2}$ of Definition 18.40. Its irreducible representations are labelled by $s$ with $2s \in \N\cup\set{0}$ (Theorem 18.43), of dimension $2s+1$, and the Casimir takes the value

\begin{equation}\tag{A36.12} \hat{\vect{\mathcal{J}}}^{2} = \hbar^{2}\,s(s+1)\,\identity\ec \qquad \text{of SI unit } \mathrm{J}^{2}\,\mathrm{s}^{2}\ec \end{equation}

$s$ itself being dimensionless. The two labels of Theorem A36.3 are therefore $m$, of unit $\mathrm{kg}$, and $s$.

In four dimensions — and, of the cases treated in Lie Groups, Lie Algebras, and Fibre Bundles, only there — both labels are realized by genuine Casimir elements of the inhomogeneous algebra itself: $\hat{P}^{2}$ and the Pauli–Lubanski square $\hat{W}^{2}$ of Theorem 18.65, whose values are Equation (18.105). That $\hat{W}^{2}$ measures the rest-frame angular momentum, so that $\hat{W}^{2} = -m^{2}c^{2}\hat{\vect{\mathcal{J}}}^{2}$ on the fibre over $k$, is the computation carried out in Remark 18.66 and again in Theorem 99.29; combined with Equation (A36.12) it gives the second line of Equation (18.105), and the ratio Equation (18.106) isolates the $\hbar^{2}s(s+1)$ that survives. Theorem 99.27 is Proposition A36.8 written out for this case, with $\dd\mu$ of Equation (A36.6) replaced by the equivalent normalization used there. Rests on Theorems 18.43, 18.65 and A36.3.

Remark A36.13 (Why the higher invariants are not constructed here).

Remark 18.67 states that in general $D$ the invariants beyond $\hat{P}^{2}$ are built from antisymmetrized products of $\hat{J}$ with $\hat{P}$, the Pauli–Lubanski construction Equation (18.103) being available only at $D = 4$ because it contracts a $D$-index Levi-Civita symbol with one $J$ and one $P$. That construction is not carried out in this appendix, and it is not needed for the theorem: the count Equation (A36.1) follows from the classification and from Theorem 18.21 applied to the little algebra, neither of which requires the invariants to be exhibited as elements of the enveloping algebra. What has been proved is that a massive irreducible representation is specified by $\lceil D/2\rceil$ numbers and by no fewer, and that any central element of the enveloping algebra is a function of them. The explicit general-$D$ construction of the higher invariants would say which polynomial in $\hat{J}$ and $\hat{P}$ realizes each label; only the $D = 4$ case is used anywhere in this book, and it is Theorem 18.65.

Remark A36.14.

Theorem A36.3 discharges the obligation recorded after Lemma 18.68: the lemma supplies the little algebra and its rank, and the theorem converts that rank into the label count Equation (18.107) quoted in Remark 18.67 — two labels at $D = 4$ and three at both $D = 5$ and $D = 6$, the ceiling and not the floor. The $3+1$ instantiation, Example A36.12, is the mass and the spin, and it is the only case the physical parts of this treatise use; its detailed development, including the massless representations that this appendix does not treat, is Particles as Poincaré Representations.