5 · spacetime symmetry and elementary particles

symmetry and projective representations

In quantum mechanics a symmetry is realized by a unitary or anti-unitary operator on Hilbert space (Wigner's theorem). When the symmetry group $G$ is a continuous Lie group, a physical representation is therefore a projective unitary representation $$ U:G\to PU(\mathcal{H})=U(\mathcal{H})/U(1). $$ Projective representations of $G$ are ordinary representations of a central extension of $G$, or equivalently of a covering group when the obstruction lies in $\pi_1(G)$. For the connected Lorentz group $SO^+(1,3)$ the universal cover is $\operatorname{SL}(2,\mathbb{C})$; for the Poincare group the appropriate cover is the inhomogeneous $\operatorname{SL}(2,\mathbb{C})$. All subsequent constructions are understood to be carried out on the appropriate covering group so that true unitary representations may be used.

remark (rays again). the projective character of $U$ is the same $U(1)$ ambiguity already present for pure states in Chapter 1: physical states are rays, so only $PU(\mathcal{H})$ is forced a priori by Wigner symmetry.

semidirect products and systems of imprimitivity

The Poincare group is a semidirect product $$ G=N\rtimes H,\qquad N=\mathbb{R}^{1,3},\quad H=SO^+(1,3). $$ More generally, let $N$ be an abelian normal subgroup of a Lie group $G=N\rtimes H$. The unitary dual $\hat N$ carries a natural action of $H$. Mackey's theory of induced representations classifies the irreducible unitary representations of $G$ in terms of this action.

definition (system of imprimitivity). a system of imprimitivity based on a $G$-space $X$ is a projection-valued measure $E$ on $X$ together with a unitary representation $U$ of $G$ satisfying $$ U(g)E(\Delta)U(g)^{-1}=E(g\cdot\Delta) $$ for Borel sets $\Delta\subset X$.

When $X$ is the dual group $\hat N$ and $E$ is the spectral measure of the restriction of $U$ to $N$, the system of imprimitivity encodes the joint spectral theory of the abelian normal subgroup. Mackey's imprimitivity theorem asserts that every irreducible system of imprimitivity (under standard regularity hypotheses) is equivalent to one induced from a closed subgroup.

example (translations as a spectral measure). for the Poincare group, $N$ is spacetime translations and $\hat N$ is momentum space. the spectral measure of the four translation generators is supported on a Lorentz orbit; covariance under $H$ forces that measure to be quasi-invariant under Lorentz transformations.

induced representations

Let $K\subset G$ be a closed subgroup and let $\sigma$ be a unitary representation of $K$ on a Hilbert space $\mathcal{H}_\sigma$. The induced representation $\operatorname{Ind}_K^G\sigma$ is realized on a space of square-integrable sections of the homogeneous vector bundle $$ G\times_K\mathcal{H}_\sigma\to G/K. $$ Equivalently, it may be realized on functions $f:G\to\mathcal{H}_\sigma$ satisfying the covariance condition $$ f(gk)=\sigma(k)^{-1}f(g) $$ and square-integrable with respect to a quasi-invariant measure on $G/K$. The inducing construction is functorial and respects direct integrals; it is the principal tool for building representations of semidirect products from representations of stabilizers.

example (induction from a little group). once a momentum $p$ is fixed and $K$ is its stabilizer in the Lorentz cover, inducing a spin or helicity representation of $K$ produces the full one-particle Hilbert space as $L^2$ sections over the orbit $G\cdot p\cong G/K$.

orbit-stabilizer analysis for the poincare group

Restrict attention to the translation subgroup $N=\mathbb{R}^{1,3}$. Its unitary dual is again $\mathbb{R}^{1,3}$, identified with momentum space via $$ p\mapsto\bigl(a\mapsto e^{ip\cdot a/\hbar}\bigr). $$ The Lorentz group $H=SO^+(1,3)$ acts on momentum space by the natural dual action. The orbits under this action fall into distinct classes according to the value of the invariant $p^2=m^2$ (with $c=1$) and the sign of the energy:

• massive orbits: $p^2=m^2>0$, $p^0>0$ (and the negative-energy counterpart);
• massless orbits: $p^2=0$, $p^0>0$ (and the negative-energy counterpart);
• tachyonic orbits: $p^2<0$;
• the zero orbit $\{0\}$.

Physical particles correspond to the forward massive and massless orbits. The stabilizer (little group) of a representative momentum on each orbit determines the remaining quantum numbers.

remark (units). restoring $c$, the mass shell reads $p^2=m^2c^2$. hereafter $c=1$ as in the classical relativistic notes.

massive particles and spin

Choose a rest-frame momentum $p=(m,0,0,0)$ with $m>0$. Its little group is the rotation group $SO(3)$ (or its cover $\operatorname{SU}(2)$). Irreducible unitary representations of $\operatorname{SU}(2)$ are labelled by a spin $s=0,\tfrac12,1,\tfrac32,\dots$ and act on a $(2s+1)$-dimensional Hilbert space. The corresponding induced representation of the Poincare group is irreducible and is labelled by the pair $(m,s)$.

States within the representation are further distinguished by three-momentum $\mathbf{p}\in\mathbb{R}^3$ and spin projection $m_s=-s,\dots,s$ along a chosen axis. These are the massive elementary particles of definite mass and spin.

example (electron). the Dirac electron of mass $m_e$ realizes $(m_e,\tfrac12)$. the $(2s+1)=2$ little-group degrees of freedom are the two spin states; the continuous momentum label parametrizes the forward mass shell.

massless particles and helicity

Choose a lightlike momentum $p=(E,0,0,E)$ with $E>0$. Its little group is the Euclidean group $\operatorname{ISO}(2)=SO(2)\ltimes\mathbb{R}^2$ (double-covered by a corresponding extension of $U(1)$). In any finite-dimensional unitary representation relevant to standard particle physics, the translational $\mathbb{R}^2$ factor must act trivially; the remaining $SO(2)$ (or $U(1)$) representations are labelled by a helicity $\lambda\in\tfrac12\mathbb{Z}$.

The induced representation of the Poincare group is irreducible and is labelled by the pair $(0,\lambda)$. Because a continuous orthochronous Lorentz transformation cannot reverse the sign of helicity, a massless particle of helicity $\lambda$ is distinct from one of helicity $-\lambda$ unless additional discrete symmetries are imposed. Photons ($\lambda=\pm 1$) and gravitons ($\lambda=\pm 2$) furnish the classic examples; neutrinos were long assigned $\lambda=-\tfrac12$ in the two-component description.

remark (continuous-spin representations). nontrivial unitary representations of the $\mathbb{R}^2$ little-group translations exist and yield continuous-spin massless particles. they are excluded from the standard particle catalog by locality and finite helicity content, but they appear in the complete Mackey classification.

projective representations and discrete quantum numbers

The full spacetime symmetry group may be enlarged by parity, time-reversal, and charge conjugation. These discrete operations are represented by unitary or anti-unitary operators that intertwine the irreducible representations of the proper orthochronous Poincare group. Their presence or absence, together with the possibility of central extensions, accounts for additional discrete labels (intrinsic parity, and so on) and for the distinction between particles and antiparticles.

The CPT theorem, proved later in the quantum-field setting, guarantees that the combined operation is always a symmetry of a local relativistic theory.

example (photon helicities). under parity a helicity $\lambda$ state maps to $-\lambda$. electromagnetism therefore packages $\lambda=+1$ and $\lambda=-1$ into a single particle species once parity is imposed as a symmetry of the free theory's state space organization.

from geometry to particle species

The geometry of Minkowski space determines the isometry group $\mathcal{P}$. The dual action of $\mathcal{P}$ on momentum space $\widehat{\mathbb{R}^{1,3}}$ partitions momenta into orbits $\mathcal{O}_p\cong\mathcal{P}/K_p$, where $K_p$ is the little group of a representative $p$. Mackey induction $\operatorname{Ind}_{K_p}^{\mathcal{P}}\sigma$ from an irreducible unitary representation $\sigma$ of $K_p$ then produces an irreducible unitary representation of $\mathcal{P}$. The orbit invariant $p^2=m^2$ is the mass; the label of $\sigma$ is the spin $s$ or helicity $\lambda$.

An elementary particle is identified with one such irreducible representation $\pi$: its state space is the representation space $\mathcal{H}_\pi$, its energy-momentum spectrum is the support of the spectral measure of the translation generators (the orbit $\mathcal{O}_p$), and its internal angular-momentum degrees of freedom are the representation space of $\sigma$.

summary

Elementary particles are the irreducible unitary representations of spacetime symmetry. For the Poincare group the classification proceeds by Mackey induction from the little groups of the momentum orbits. Massive particles are labelled by mass $m>0$ and spin $s\in\tfrac12\mathbb{N}_0$; massless particles are labelled by helicity $\lambda\in\tfrac12\mathbb{Z}$.

Systems of imprimitivity encode the spectral theory of the translation subgroup, while induced representations build the full state space from the little-group data. The construction supplies the precise quantum numbers that later chapters will promote to relativistic wave equations, Fock spaces, and local quantum fields.