Poincaré Group

The Poincaré group combines spacetime translations with Lorentz transformations. It is the symmetry group of Minkowski spacetime with no preferred origin. Its quantum representations explain why mass and spin label free relativistic particles, and why a finite-component covariant wavefunction is different from the infinite-dimensional Hilbert space of its momentum states. This page introduces the group and its invariants without attempting the full classification of representations.

Required background. Lorentz Transformations supplies boosts and rotations; the Energy–Momentum Relation supplies the mass shell. We also assume familiarity with Hermitian generators of unitary symmetries.

Translations and Lorentz transformations form a semidirect product

A Poincaré transformation acts on coordinates as

$$x' = \Lambda x+a.$$

Apply $(\Lambda_1,a_1)$ first and $(\Lambda_2,a_2)$ second. Substitution gives

$$(\Lambda_2,a_2)(\Lambda_1,a_1) =(\Lambda_2\Lambda_1,\,a_2+\Lambda_2a_1).$$

The inverse is $(\Lambda^{-1},-\Lambda^{-1}a)$. Translations form an abelian normal subgroup, but they do not commute with general Lorentz transformations: the latter rotate or boost the translation vector. This is the meaning of the semidirect product $\mathbb R^{1,3}\rtimes SO^+(1,3)$ for the connected group. There are four translation parameters and six Lorentz parameters.

As a concrete example, first translate a point by $a=(0,L,0,0)$ and then apply an $x$ boost. The effective translation in the boosted coordinates is

$$\Lambda a=(-\gamma\beta L,\gamma L,0,0).$$

Reversing the order would leave the translation vector $(0,L,0,0)$. The group multiplication therefore cannot be replaced by independent addition of the two translation vectors.

Generators and their algebra

Continuous spacetime symmetries act projectively on quantum rays; passing to the covering group permits a unitary representation on the Hilbert space, including half-integer spin. Work on a common invariant domain where the following commutators are defined. Let $P^\mu=(H/c,\mathbf P)$ denote translation generators, $\mathbf J$ angular momentum, and $\mathbf K$ boost generators with units of action. Choose signs so that $J_i=\epsilon_{ijk}M^{jk}/2$ and $K_i=M^{0i}$ for the antisymmetric Lorentz generators, with

$$[M^{\mu\nu},P^\rho] =i\hbar(\eta^{\nu\rho}P^\mu-\eta^{\mu\rho}P^\nu).$$

In this convention the spatial algebra is

$$\begin{aligned} [J_i,J_j]&=i\hbar\epsilon_{ijk}J_k,& [J_i,K_j]&=i\hbar\epsilon_{ijk}K_k,\\ [K_i,K_j]&=-i\hbar\epsilon_{ijk}J_k,& [P^\mu,P^\nu]&=0, \end{aligned}$$

and the energy–momentum relations are

$$\begin{aligned} [J_i,P_j]&=i\hbar\epsilon_{ijk}P_k,&[J_i,H]&=0,\\ [K_i,P_j]&=-i\hbar\delta_{ij}H/c,&[K_i,H]&=-i\hbar cP_i. \end{aligned}$$

Here $P_i$ in the three-vector equations means a Euclidean Cartesian component, not the lowered spacetime component $P_\mu$. With this choice, a passive boost is represented by $U(B(\boldsymbol\xi))=\exp(+i\boldsymbol\xi\cdot\mathbf K/\hbar)$, while a right-handed rotation uses $U(R)=\exp(-i\boldsymbol\theta\cdot\mathbf J/\hbar)$. Reversing the definition of $\mathbf K$ reverses both its parameter sign and the displayed mixed commutators.

The negative sign in $[K_i,K_j]$ distinguishes boosts from rotations and encodes the rotation generated by non-collinear boosts. The commutator $[K_i,H]$ says that a boost generally changes the energy of a state. Poincaré invariance does not mean that the Hamiltonian commutes with boosts.

Mass and spin as invariants

The first Casimir operator is

$$P_\mu P^\mu=H^2/c^2-\mathbf P^2.$$

It commutes with all generators. On a massive irreducible one-particle representation its value is $m^2c^2I$. The second invariant is built from the Pauli–Lubanski vector,

$$W^\mu=\frac12\epsilon^{\mu\nu\rho\sigma} P_\nu M_{\rho\sigma}.$$

The explicit components in this convention are $W^0=-\mathbf P\cdot\mathbf J$ and $\mathbf W=-(H/c)\mathbf J+\mathbf P\times\mathbf K$. For a massive particle in its rest frame $W^0=0$ and $W_\mu W^\mu=-m^2c^2\mathbf J^2$. Consequently

$$W_\mu W^\mu=-m^2c^2\hbar^2s(s+1)I.$$

The nonnegative integer or half-integer $s$ is the spin. In the rest frame, the subgroup preserving the standard momentum $(mc,\mathbf0)$ is the rotation group; its double cover supplies the familiar $2s+1$ spin states.

For massless finite-helicity representations, $P^2=0$ and $W^2=0$. These two numbers alone do not distinguish different helicities. One needs the action of the massless little group, the subgroup preserving a reference null momentum. Continuous-spin representations also exist mathematically; excluding them requires a restriction beyond merely writing $P^2=0$. This is why the full Wigner classification is more than the dispersion relation plus a chosen number of wavefunction components.

Covariant components and unitary particle states

A Dirac amplitude has four components at each spacetime point. A one-particle state, however, carries a continuous momentum label and a spin label. A normalizable state is a superposition over momenta, so its Hilbert space is infinite dimensional even when the spin space is finite dimensional.

Lorentz boosts on the covariant spinor components are not unitary in the ordinary Euclidean component norm. This does not contradict unitarity on the particle Hilbert space: momentum arguments, integration measure, and spin transformations participate together. The invariant mass-shell measure is one part of that construction. The covariant Dirac equation displays the component intertwining relation.

The group algebra is a kinematic constraint, not a complete interaction theory. A prescribed external field can break translation or boost symmetry, so the external-field Hamiltonian need not realize every free-particle symmetry. Lorentz covariance of equations with transformed backgrounds is different from invariance of one fixed background.

Exercises

1. Using the displayed commutators, verify that $[K_i,H^2/c^2-\mathbf P^2]=0$.

Solution

Since $[H,P_i]=0$, the first term gives $-2i\hbar HP_i/c$. The second gives $\sum_j([K_i,P_j]P_j+P_j[K_i,P_j])=-2i\hbar HP_i/c$. They cancel, as required for the invariant mass.

2. For $s=1/2$, compute the massive $W^2$ eigenvalue. Can the same formula at $m=0$ distinguish a scalar from a massless spinor?

Solution

$W^2=-3m^2c^2\hbar^2/4$. Substituting $m=0$ sets it to zero for both finite-helicity examples and loses the helicity information. Massless representations have no rest frame; their little-group analysis is distinct.

3. Explain why an electrostatic potential centered at one spatial point can preserve rotations but break translations, even though the Dirac equation is written covariantly.

Solution

A central potential depends only on distance from its center and is unchanged by rotations about that center. Translating the state while leaving the potential fixed changes its potential energy. Transforming both the state and the source gives a covariant description of a different background; it is not a symmetry of the original fixed background.

References

• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, chapter 2 — the Poincaré algebra and one-particle states.
• E. P. Wigner, “On Unitary Representations of the Inhomogeneous Lorentz Group,” Annals of Mathematics 40, 149–204, 1939, doi:10.2307/1968551 — classification by momentum orbits and little groups.