Particles as Poincaré Representations

Ray composition gives \(U(g)U(h) = \omega(g,h)\,U(gh)\) for a phase \(\omega(g,h)\), since both sides implement \(T_{gh}\) and Theorem 94.3 fixes the implementation up to a phase. In a connected Lie group every element is a finite product of exponentials, \(g = \exp X_1 \cdots \exp X_n\): the image of the exponential map contains a neighbourhood of the identity, and the subgroup generated by any neighbourhood of the identity is open, hence closed — its complement is a union of its own open cosets — hence all of the connected group. Each factor is a square: \(\exp X = \left(\exp \tfrac{X}{2}\right)^{2}\). Hence \(U(g)\) equals, up to a phase, a product of operators \(U(h_i)^{2}\). The square of a unitary operator is unitary, and so is the square of an antiunitary one (two antilinear maps compose to a linear map, and isometry is preserved); phases do not affect (anti)unitarity. A product of unitaries is unitary.

\(x \mapsto \Lambda x + a \mapsto \Lambda'(\Lambda x + a) + a' = (\Lambda'\Lambda)x + (\Lambda' a + a')\); the identity is \((\identity,0)\) and \((\Lambda,a)^{-1} = (\Lambda^{-1}, -\Lambda^{-1}a)\).

Let \(\set{X_a}\) be a basis of the abstract Lie algebra with real structure constants, \(\comm{X_a}{X_b} = f_{ab}{}^{c}X_c\), and let \(\dd U\) denote the induced representation of the algebra, so that \(U(\exp tX_a) = \exp\left(-\tfrac{\ii}{\hbar}t\,T_a\right)\) defines the Hermitian generator \(T_a = \ii\hbar\,\dd U(X_a)\). Since \(\dd U\) preserves brackets,

The structure constants of \(\mathfrak{iso}(3,1)\) were derived in Section 14.3.2 from the realization \(J_{AB} = x_A \pp_B - x_B \pp_A\), \(P_C = \pp_C\) (Equation (14.86)): the brackets are Equations (14.90) and (14.97) together with \(\comm{P_A}{P_B} = 0\), restated for the physical signature in Equations (39.18), (39.19) and (39.20) — granting the identification of that realization with the Lie algebra of \(L^{\uparrow}_{+}\) in the parameterization Equation (94.3), a construction recorded as pending at Equation (14.86) and in Section 39.3.2. Multiplying each by \(\ii\hbar\) and raising indices with \(\eta\) gives Equations (94.5), (94.6) and (94.7).

Direct substitution. As a sample, \(\comm{\hat{K}_1}{\hat{P}_1} = \comm{\hat{J}^{10}}{\hat{P}^{1}} = \ii\hbar\left(\eta^{01}\hat{P}^{1} - \eta^{11}\hat{P}^{0}\right) = \ii\hbar\,\hat{P}^{0} = \tfrac{\ii\hbar}{c}\hat{H}\), and \(\comm{\hat{J}_1}{\hat{J}_2} = \comm{\hat{J}^{23}}{\hat{J}^{31}} = \ii\hbar\,\eta^{33}\hat{J}^{12}\cdot(-1)^{0} = \ii\hbar\,\hat{J}^{12} = \ii\hbar\,\hat{J}_3\), all other \(\eta\) factors vanishing because the index pairs share no common value. The remaining entries are the same computation with permuted indices.

\(AXA^{\dagger}\) is Hermitian when \(X\) is, so \(\Lambda(A)\) is a well-defined real-linear map of \(\R^{4}\); it preserves \(x_{\mu}x^{\mu}\) by Equation (94.10) because \(\det(AXA^{\dagger}) = \abs{\det A}^{2}\det X = \det X\), hence \(\Lambda(A)\) is a Lorentz transformation (Definition 39.1). Homomorphism: \(\Lambda(AB)x \leftrightarrow AB\,X\,B^{\dagger}A^{\dagger} \leftrightarrow \Lambda(A)\Lambda(B)x\). Kernel: if \(AXA^{\dagger} = X\) for all Hermitian \(X\), then \(X = \identity\) gives \(AA^{\dagger} = \identity\), so \(A\) is unitary and commutes with every \(\sigma_i\); a \(2\times2\) matrix commuting with all Pauli matrices is a multiple \(\lambda\identity\) (write it in the basis \(\set{\identity,\sigma_i}\) and use \(\sigma_i\sigma_j = \delta_{ij}\identity + \ii\epsilon_{ijk}\sigma_k\)), and \(\det A = \lambda^{2} = 1\) forces \(\lambda = \pm1\). Connectedness and image: polar decomposition writes every \(A \in \SL(2,\C)\) uniquely as \(A = V\ee^{M}\) with \(V \in \SU(2)\) and \(M\) traceless Hermitian; \(\SU(2)\) is the three-sphere \(\abs{a}^{2}+\abs{b}^{2}=1\) of rows \((a,b),(-\bar b,\bar a)\) and the traceless Hermitian \(M\) form \(\R^{3}\), so \(\SL(2,\C)\) is connected and simply connected (\(S^{3}\times\R^{3}\); the polar factors depend continuously on \(A\), the positive square root and logarithm being continuous on positive matrices, and \(\pi_1(S^{3}) = 0\)). Its continuous image \(\Lambda(\SL(2,\C))\) is then a connected subgroup of \(L\) containing \(\identity\), hence lands in \(L^{\uparrow}_{+}\) (Proposition 39.6: both defining conditions are clopen). That it is all of \(L^{\uparrow}_{+}\) follows from Lemmas 94.11, 94.12 and 94.13: every rotation and every boost is attained, and every \(\Lambda \in L^{\uparrow}_{+}\) is a boost times a rotation. Finally, a surjective continuous homomorphism of Lie groups with discrete kernel is a covering map — a standard fact used here without restatement. In particular \(L^{\uparrow}_{+}\), as the continuous image of the connected \(\SL(2,\C)\), is itself connected: it is the identity component of \(L\), as announced in Section 39.2.4.

Since \((\vect{n}\cdot\vect{\sigma})^{2} = \identity\) (from \(\sigma_i\sigma_j = \delta_{ij}\identity + \ii\epsilon_{ijk}\sigma_k\) and \(\abs{\vect{n}} = 1\)), the exponential series (Equation (7.40)) splits into \(V = \cos\tfrac{\theta}{2}\,\identity - \ii\sin\tfrac{\theta}{2}\,\vect{n}\cdot\vect{\sigma}\). A direct computation with the same product identity gives, for \(X = x^{0}\identity + \vect{x}\cdot\vect{\sigma}\),

which is Rodrigues' rotation formula about \(\vect{n}\) by \(\theta\). Every rotation has an axis and an angle: for \(R \in \SO(3)\), \(\det(R - \identity) = \det\left((R-\identity)\transpose\right) = \det(R^{-1} - \identity) = \det(R^{-1})\det(\identity - R) = -\det(R - \identity)\), so \(\det(R - \identity) = 0\) and \(R\) fixes a unit vector \(\vect{n}\); on the invariant plane orthogonal to \(\vect{n}\), \(R\) is an orientation-preserving linear isometry, i.e. a rotation by some \(\theta\). Hence all of \(\SO(3)\) is attained; at \(\theta = 2\pi\), \(V = \cos\pi\,\identity = -\identity\).

For \(\vect{m} = \vect{e}_3\) we have \(H = \operatorname{diag}\left(\ee^{\eta/2}, \ee^{-\eta/2}\right)\), and in Equation (94.9) the conjugation \(HXH^{\dagger} = HXH\) multiplies the diagonal entries \(x^{0} \pm x^{3}\) of \(X\) by \(\ee^{\pm\eta}\) while leaving the off-diagonal entries \(x^{1} \mp \ii x^{2}\) fixed (\(\ee^{\eta/2}\ee^{-\eta/2} = 1\)): \(x^{0}+x^{3} \mapsto \ee^{\eta}(x^{0}+x^{3})\), \(x^{0}-x^{3} \mapsto \ee^{-\eta}(x^{0}-x^{3})\), which is Equation (94.11). A general axis \(\vect{m}\) is \(R\,\vect{e}_3\) for a rotation \(R\); conjugating \(H\) by the corresponding \(V\) of Lemma 94.11 rotates the boost axis, since \(\Lambda\) is a homomorphism.

Let \(u = \Lambda e_0\), where \(e_0 = (1,0,0,0)\). Then \(u_{\mu}u^{\mu} = 1\) and \(u^{0} = \Lambda^{0}{}_{0} \ge 1\) (Definition 39.5), so \(u = (\cosh\eta,\ \sinh\eta\,\vect{m})\) for some rapidity \(\eta \ge 0\) and unit vector \(\vect{m}\). Let \(B\) be the boost of Lemma 94.12 along \(\vect{m}\) with rapidity \(\eta\); then \(B e_0 = u\), so \(R = B^{-1}\Lambda\) satisfies \(R e_0 = e_0\). A Lorentz transformation fixing \(e_0\) preserves the spatial slice \(x^{0} = 0\) and the Euclidean norm on it (both statements from \(\Lambda\eta\Lambda\transpose = \eta\), Equation (39.2): the first column of \(R\) is \(e_0\), hence so is the first row), so \(R \in \Ogrp(3)\); and \(\det R = \det\Lambda = 1\) places \(R \in \SO(3)\).

The left side is totally antisymmetric in \(\mu\nu\rho\sigma\): swapping any two of them exchanges two terms and flips the sign of the remaining two through the antisymmetry of \(\epsilon\). A totally antisymmetric four-index array in four dimensions is a multiple of \(\epsilon_{\mu\nu\rho\sigma}\); contracting the left side with \(\epsilon^{\mu\nu\rho\sigma}\) shows the multiple is proportional to \(\omega^{\alpha}{}_{\alpha} = \eta^{\alpha\beta}\omega_{\beta\alpha} = 0\).

(i) By Equations (94.6) and (94.7), \(\comm{\hat{J}^{\nu\rho}\hat{P}^{\sigma}}{\hat{P}_{\lambda}} = \comm{\hat{J}^{\nu\rho}}{\hat{P}_{\lambda}}\hat{P}^{\sigma} = \ii\hbar\left(\delta^{\rho}_{\lambda}\hat{P}^{\nu} - \delta^{\nu}_{\lambda}\hat{P}^{\rho}\right)\hat{P}^{\sigma}\); contracted with \(\epsilon_{\mu\nu\rho\sigma}\) each term dies because \(\hat{P}^{\nu}\hat{P}^{\sigma}\) (respectively \(\hat{P}^{\rho}\hat{P}^{\sigma}\)) is symmetric in indices that \(\epsilon\) antisymmetrizes.

(ii) \(\hat{W}_{\mu}\hat{P}^{\mu} = \tfrac12 \epsilon_{\mu\nu\rho\sigma}\hat{J}^{\nu\rho}\hat{P}^{\sigma} \hat{P}^{\mu} = 0\) for the same symmetry reason, using Equation (94.7); the reversed order follows from (i).

(iii) Insert Equations (94.5) and (94.6) into \(\comm{\hat{J}_{\mu\nu}}{\tfrac12\epsilon_{\lambda\alpha\beta\gamma} \hat{J}^{\alpha\beta}\hat{P}^{\gamma}}\) and use the Leibniz rule for commutators; the resulting four \(\epsilon\eta\)-contractions combine, by Lemma 94.16 applied with \(\omega\) the antisymmetrized pair \((\mu\nu)\), into exactly the two terms \(\ii\hbar(\eta_{\nu\lambda}\hat{W}_{\mu} - \eta_{\mu\lambda}\hat{W}_{\nu})\).

\(\comm{\hat{P}^{2}}{\hat{P}_{\lambda}} = 0\) by Equation (94.7), and \(\comm{\hat{P}^{2}}{\hat{J}_{\mu\nu}} = 0\) because Equation (94.6) makes \(\hat{P}\) a four-vector and the square of a four-vector is a scalar: \(\comm{\hat{J}_{\mu\nu}}{\hat{P}_{\lambda}\hat{P}^{\lambda}} = \ii\hbar\left(\eta_{\nu\lambda}\hat{P}_{\mu} - \eta_{\mu\lambda}\hat{P}_{\nu}\right)\hat{P}^{\lambda} + \ii\hbar\,\hat{P}_{\lambda}\left(\delta^{\lambda}_{\nu}\hat{P}_{\mu} - \delta^{\lambda}_{\mu}\hat{P}_{\nu}\right) = \ii\hbar\left(\hat{P}_{\mu}\hat{P}_{\nu} - \hat{P}_{\nu}\hat{P}_{\mu}\right) + \ii\hbar\left(\hat{P}_{\nu}\hat{P}_{\mu} - \hat{P}_{\mu}\hat{P}_{\nu}\right) = 0\) identically. The same computation with Lemma 94.17(iii) kills \(\comm{\hat{J}_{\mu\nu}}{\hat{W}^{2}}\), and \(\comm{\hat{W}^{2}}{\hat{P}_{\lambda}} = 0\) is Lemma 94.17(i). Schur's lemma (Theorem 5.153, with the qualification of Remark 94.1) then makes each a scalar on an irreducible representation; Hermiticity makes the scalar real.

\((\Lambda p)^{0} = \Lambda^{0}{}_{0}p^{0} + \Lambda^{0}{}_{i}p^{i}\). By the Cauchy–Schwarz inequality of Section 6.3 and the row identity \(\left(\Lambda^{0}{}_{0}\right)^{2} = 1 + \sum_i\left(\Lambda^{0}{}_{i}\right)^{2}\) (the \(\mu=\nu=0\) component of Equation (39.2) applied to \(\Lambda^{-1} = \eta\Lambda\transpose\eta\)),

whenever \(\abs{\vect{p}} \le \abs{p^{0}}\) with \(p \ne 0\) (causality) — strictly, since either the causal inequality is strict or \(\sum_i (\Lambda^{0}{}_{i})^{2} < (\Lambda^{0}{}_{0})^{2}\). Hence the first term dominates and carries the sign of \(p^{0}\) (\(\Lambda^{0}{}_{0} \ge 1\), Definition 39.5).

\(p^{2}\) is invariant by definition of \(L\), and \(\operatorname{sgn}(p^{0})\) is invariant on the causal orbits by Lemma 94.20, so the listed classes are unions of orbits. Each class is a single orbit because its standard momentum reaches every member: a rotation (Lemma 94.11) aligns \(\vect{p}\) with \(\vect{e}_3\), and the boost Equation (94.11) with the appropriate rapidity then produces any admissible \(p^{0}\): \(\cosh\eta = p^{0}/(mc)\) in the massive case, \(\ee^{\eta} = p^{0}/\kappa\) in the massless case, \(\sinh\eta = p^{0}/(\mu c)\) in the tachyonic case, the spatial part then matching automatically — which also shows both signs of \(p^{0}\) lie on one tachyonic orbit, \(\sinh\) being unbounded in both directions; the negative-energy rows follow by applying the same transformations to their own standard momenta. The composition of these transformations is the standard boost \(L(p)\), fixed once and for all on each orbit:

with \(R(\hat{\vect{p}})\) a fixed choice of rotation carrying \(\vect{e}_3\) to the unit vector \(\hat{\vect{p}}\) and \(B_{3}\) the boost Equation (94.11); the conventions are \(R(\vect{e}_3) = \identity\) — so that \(L(k) = \identity\) — and a fixed measurable choice of \(R(\hat{\vect{p}})\) elsewhere (no continuous global choice exists); any such choice serves.

\(\Lambda k = k\) says \(\Lambda e_0 = e_0\); the proof of Lemma 94.13 showed exactly that such a \(\Lambda\) is a rotation, and conversely rotations fix \(e_0\). The cover statement is Corollary 94.14.

An algebra element \(\tfrac12\omega_{\mu\nu}\hat{J}^{\mu\nu}\) stabilizes \(k\) iff \(\omega^{\mu}{}_{\nu}k^{\nu} = 0\), i.e.\ \(\omega^{\mu}{}_{0} + \omega^{\mu}{}_{3} = 0\) for every \(\mu\). Writing this out with \(\eta\): for \(\mu = 0\) it gives \(\omega_{03} = 0\); for \(\mu = 1\), \(\omega_{01} = \omega_{13}\); for \(\mu = 2\), \(\omega_{02} = \omega_{23}\); for \(\mu = 3\) it repeats \(\omega_{03} = 0\). Three free parameters remain, multiplying exactly the three combinations Equation (94.17). The brackets follow from Equation (94.5) by direct substitution; as a sample, in \(\comm{\hat{J}^{12}}{\hat{J}^{01}}\) the only surviving term of Equation (94.5) is \(\eta^{\mu\rho}\hat{J}^{\nu\lambda}\) with \((\mu\nu\lambda\rho) = (1,2,0,1)\), giving \(\ii\hbar\,\eta^{11}\hat{J}^{20} = \ii\hbar\,\hat{J}^{02}\), and \(\comm{\hat{J}^{12}}{\hat{J}^{13}} = \ii\hbar\,\eta^{11}\hat{J}^{32} = \ii\hbar\,\hat{J}^{23}\), whose sum is \(\ii\hbar\,\hat{B}\); the other two brackets are the same bookkeeping, with the two disjoint-index commutators \(\comm{\hat{J}^{01}}{\hat{J}^{23}} = \comm{\hat{J}^{13}}{\hat{J}^{02}} = 0\) making \(\comm{\hat{A}}{\hat{B}}\) collapse to \(\comm{\hat{J}^{01}}{\hat{J}^{02}} + \comm{\hat{J}^{13}}{\hat{J}^{23}} = \ii\hbar\left(-\hat{J}^{12} + \hat{J}^{12}\right) = 0\).

Write the integral over the orbit as a four-dimensional one,

which follows from the composition rule for the Dirac delta (Proposition 7.113): at fixed \(\vect{p}\) the argument \(\left(p^{0}\right)^{2} - \vect{p}^{2} - m^{2}c^{2}\) has the single positive root Equation (94.19) with derivative \(2p^{0}\). In the four-dimensional form each factor is invariant under \(p \mapsto \Lambda p\): the delta because \(p^{2}\) is, the step function by Lemma 94.20, and \(\dd^{4}p\) because \(\abs{\det\Lambda} = 1\).

\(W(\Lambda,p)k = L(\Lambda p)^{-1}\Lambda L(p)\,k = L(\Lambda p)^{-1}(\Lambda p) = k\), so \(W(\Lambda,p) \in G_{k}\). Composition: applying Equation (94.22) twice,

so the \(D\) factors compose correctly and \(U(\Lambda')U(\Lambda) = U(\Lambda'\Lambda)\); translations compose by inspection of Equation (94.21), and the mixed products follow from Equation (94.2) applied to \(\Lambda a\). Unitarity: for wave packets \(\ket{\psi} = \sum_{\sigma}\int\dd\mu(p)\, \psi_{\sigma}(p)\ket{p,\sigma}\) the chosen normalization gives \(\norm{\psi}^{2} = \sum_{\sigma}\int\dd\mu(p) \abs{\psi_{\sigma}(p)}^{2}\), and by Equation (94.22) a Lorentz transformation replaces \(\psi_{\sigma}(p)\) by \(\sum_{\sigma'}D_{\sigma\sigma'}(W)\,\psi_{\sigma'}(\Lambda^{-1}p)\): the sum over \(\sigma\) is preserved because \(D\) is unitary, and the integral because the measure is invariant (Lemma 94.26).

Let \(\hat{J}_{\pm} = \hat{J}_1 \pm \ii\hat{J}_2\), so that \(\hat{J}_{\pm}^{\dagger} = \hat{J}_{\mp}\) and, from the algebra,

\(\hat{\vect{J}}^{2}\) commutes with each \(\hat{J}_i\) (same computation as in Theorem 94.18, three-dimensional case), so on an irreducible representation \(\hat{\vect{J}}^{2} = \lambda\hbar^{2} \identity\) with \(\lambda \ge 0\) (it is a sum of squares of Hermitian operators). Diagonalize the Hermitian \(\hat{J}_3\): eigenvalues \(\hbar\sigma\). The first bracket in Equation (94.23) shifts \(\sigma\) by \(\pm1\) under \(\hat{J}_{\pm}\), while

In a finite-dimensional space the ladder must terminate in both directions: there are extremal eigenvalues \(\sigma_{\pm}\) with \(\hat{J}_{+}\ket{\sigma_{+}} = 0\) and \(\hat{J}_{-}\ket{\sigma_{-}} = 0\), so Equation (94.24) forces \(\lambda = \sigma_{+}(\sigma_{+}+1) = \sigma_{-}(\sigma_{-}-1)\), whose only solution with \(\sigma_{+} \ge \sigma_{-}\) is \(\sigma_{-} = -\sigma_{+}\). Writing \(s = \sigma_{+}\): the ladder descends from \(\sigma_{+}\) to \(\sigma_{-}\) in unit steps, so \(2s \in \Z_{\ge0}\) and \(\lambda = s(s+1)\). The states \(\set{\ket{s,\sigma}}_{\sigma=-s}^{s}\) span an invariant subspace (the three generators are linear combinations of \(\hat{J}_3\) and \(\hat{J}_{\pm}\), which act within it), hence by irreducibility the whole space: the dimension is \(2s+1\). Conversely each such \(s\) is realized — Equation (94.24) defines the matrix elements, and the brackets Equation (94.23) are verified directly — so the list is exhaustive. (Infinite-dimensional unitary irreducible representations of a compact group do not occur: every unitary representation of \(\SU(2)\) decomposes into finite-dimensional ones — the Peter–Weyl theorem [Weinberg:1995]; the treatise's own account of \(\SU(2)\), Section 14.2.2, is still pending.)

The classification is Theorem 94.27, Proposition 94.24 and Lemma 94.28, modulo the completeness statement recorded as pending after Theorem 94.27 (irreducibility and exhaustiveness of the induced family). For Equation (94.25) evaluate \(\hat{W}^{2}\) — a Casimir, so any convenient states will do (Theorem 94.18) — on the standard-momentum states \(\ket{k,\sigma}\), \(k = (mc,\vect{0})\). There \(\hat{P}^{\sigma}\) may be replaced by its eigenvalue, in both factors of \(\hat{W}^{2}\): \(\hat{W}\) preserves the momentum eigenspace by Lemma 94.17(i). In Equation (94.12) only \(\hat{P}^{0} = mc\) survives,

the last step because \(\epsilon_{ijk0} = -\epsilon_{0ijk}\) (one cyclic shift of four indices is an odd permutation) and \(\hat{J}_i = \tfrac12\epsilon_{ijk}\hat{J}^{jk}\) Equation (94.4). Hence on these states \(\hat{W}^{2} = \hat{W}_{0}\hat{W}^{0} + \hat{W}_{i}\hat{W}^{i} = -\sum_i\left(mc\,\hat{J}_i\right)^{2} = -m^{2}c^{2}\,\hat{\vect{J}}^{2}\), and Lemma 94.28 supplies the eigenvalue \(\hbar^{2}s(s+1)\): the Pauli–Lubanski square measures the rest-frame angular momentum — the spin.

Derivation. Theorem 94.29: a massive particle species is an irreducible representation \((m,s)\), its polarization space is the \((2s+1)\)-dimensional representation of Lemma 94.28, and any axis may be taken as the \(3\)-axis by Proposition 94.24, so a measurement of the spin projection resolves the \(2s+1\) eigenvalues \(\hbar\sigma\) of \(\hat{J}_3\) (Postulate 77.3). The massless counterpart — helicity \(\pm\hbar\) for the photon — is Proposition 94.32 below. The experimental record is analyzed in Experiment: Stern–Gerlach and [Beth:1936].

Suppose instead \(\hat{A}\) (say) had a nonzero eigenvalue \(a\) on some state. Since \(\comm{\hat{A}}{\hat{B}} = 0\) Equation (94.18), diagonalize both; conjugating by the rotation \(\ee^{-\ii\theta\hat{J}^{3}/\hbar}\) rotates the eigenvalue pair \((a,b)\) by the angle \(\theta\) in the plane (this is Equation (94.18) exponentiated), so the spectrum would contain the whole circle of pairs — a continuous infinity of internal states at fixed momentum, Wigner's “continuous spin” representations [Wigner:1939]. No observed massless particle has more than two polarization states, so observation forces \(a = b = 0\): \(\hat{A} = \hat{B} = 0\) on the physical representation (the same empirical selection as Remark 94.22). What remains of the little group is the one-parameter rotation subgroup; its irreducible unitary representations are one-dimensional because the group is abelian: each \(D_{\theta}\) commutes with the whole representation, so by Schur's lemma (Theorem 5.153, with the qualification of Remark 94.1) it is a scalar, every line is then invariant, and irreducibility forces dimension one (cf. the \(\SO(2)\) analysis of Section 14.2.1). Thus \(D_{\theta} = \ee^{-\ii h\theta}\) with \(\hat{J}^{3}\) acting as the number \(\hbar h\). Quantization of \(h\): in the covering group the rotation subgroup about \(\vect{e}_3\) is \(V(\theta) = \exp\left(-\ii\tfrac{\theta}{2}\sigma_3\right)\) (Lemma 94.11), which closes only at \(\theta = 4\pi\); single-valuedness of \(D\) on the cover demands \(\ee^{-4\pi\ii h} = 1\), i.e. \(2h \in \Z\). (A rotation by \(2\pi\) multiplies the state by \(\ee^{-2\pi\ii h} = (-1)^{2h}\), exactly as for massive spin, Corollary 94.14.)

Evaluate the components of Equation (94.12) on the standard-momentum states, \(p = k = (\kappa,0,0,\kappa)\), replacing \(\hat{P}\) by its eigenvalue (\(k^{0} = k^{3} = \kappa\)):

each line by direct expansion of the \(\epsilon\) sums with \(\epsilon_{0123} = +1\). On the physical representation \(\hat{A} = \hat{B} = 0\) and \(\hat{J}^{3} = \hbar h\) (Proposition 94.31), so \(\hat{W}_{\mu} = \hbar h\,(\kappa,0,0,-\kappa)_{\mu} = \hbar h\,k_{\mu}\) on these states. Both sides of \(\hat{W}_{\mu} = \hbar h \hat{P}_{\mu}\) are four-vectors built on the representation (Lemma 94.17), so equality at the standard momentum propagates to the whole orbit by conjugation with \(U(L(p))\). Contracting with itself, \(\hat{W}^{2} = \hbar^{2}h^{2} \hat{P}^{2} = 0\); taking the spatial part at momentum \(\vect{p}\) gives the stated projection formula (\(\hat{W}^{i}\) reduces to \(p^{0}\hat{J}_i\)-terms along \(\hat{\vect{p}}\)).