Poincaré Symmetry and Relativistic Conservation Laws

The generators of spacetime symmetry organize relativistic observables into four translation charges $P^\mu$ and six antisymmetric Lorentz charges $J^{\mu\nu}=-J^{\nu\mu}$. For an isolated Poincaré-invariant system, these are conserved energy-momentum, spatial angular momentum, and boost charges. Their formulas depend on whether the degrees of freedom are particles or fields.

Required background. Calculus and Linear Algebra provides linear maps, antisymmetric tensors, and contractions; Variational Mechanics and Symmetries supplies Noether’s theorem, canonical momentum, and conserved charges.

The Poincaré group

A proper orthochronous Poincaré transformation acts on Minkowski coordinates as

$$x^\mu\longmapsto x'^\mu =\Lambda^\mu{}_\nu x^\nu+a^\mu, \qquad \Lambda\in SO^+(1,3).$$

If $(a_1,\Lambda_1)$ acts first and $(a_2,\Lambda_2)$ second, then

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

The Lorentz transformation rotates or boosts the first translation vector before it is added. Therefore the group is not a direct product but

$$ISO^+(1,3) =\mathbb R^{1,3}\rtimes SO^+(1,3).$$

Translations form a normal Abelian subgroup, and the Lorentz group acts on them. This structure is stated explicitly in Lim 2019, Definition 1.3, printed p. 4 (PDF).

Infinitesimally, $P^\mu$ generates translations and $J^{\mu\nu}$ generates rotations and boosts. The group law already says, without quantum mechanics, that translations commute, momentum transforms as a four-vector under Lorentz transformations, and Lorentz transformations close among themselves. On a classical phase space these generators can act through Poisson brackets. The conserved quantities below follow directly from worldline mechanics or Noether’s theorem, so the classical route does not require an operator representation.

Free point particles

For free spinless point particles with worldlines $x_a^\mu$ and constant momenta $p_a^\mu$, evaluate their positions where the worldlines cross one common inertial spacelike slice. Then

$$P^\mu=\sum_a p_a^\mu, \qquad J^{\mu\nu} =\sum_a\left(x_a^\mu p_a^\nu-x_a^\nu p_a^\mu\right).$$

For particles carrying intrinsic spin, the complete charge instead contains

$$J^{\mu\nu} =\sum_a\left( x_a^\mu p_a^\nu-x_a^\nu p_a^\mu+S_a^{\mu\nu} \right).$$

The spatial components $J^{ij}$ encode orbital angular momentum. The mixed components $J^{0i}$ are boost charges and constrain the center-of-energy motion. Under an active translation $x_a^\mu\mapsto x_a^\mu+b^\mu$,

$$J^{\mu\nu}\mapsto J^{\mu\nu}+b^\mu P^\nu-b^\nu P^\mu,$$

while $P^\mu$ is unchanged. This is an active displacement of the physical system. Under the passive origin change $x_{\rm new}^\mu=x^\mu-b^\mu$, the sign reverses:

$$J_{\rm new}^{\mu\nu} =J^{\mu\nu}-b^\mu P^\nu+b^\nu P^\mu.$$

Angular momentum therefore depends on the chosen origin in a controlled covariant way, provided the active or passive interpretation is stated.

These sums are not automatically complete for interacting particles. If interactions are mediated by a field, the field carries energy, momentum, and angular momentum; discarding it generally spoils conservation.

For a free particle, $J^{0i}=tp^i-x^iE$. Solving for its position gives $x^i(t)=(p^i/E)t-J^{0i}/E$. Thus conservation of the boost charge says that its energy moves uniformly. For an isolated system described by a symmetric stress tensor with vanishing boundary flux, the same argument applies to its center of energy $X^i=E^{-1}\int x^iT^{00}\,d^3x$: $J^{0i}=tP^i-EX^i$ implies $dX^i/dt=P^i/E$. A boost charge has a concrete mechanical meaning even before introducing quantum operators.

Field-theory charges

For fields satisfying their Euler–Lagrange equations, invariance of the action under translations, up to a total derivative, gives a canonical Noether energy-momentum tensor satisfying on shell

$$\partial_\mu T^{\mu\nu}=0.$$

If the fields have suitable falloff and boundary flux vanishes, the four charges

$$P^\nu=\int_{\mathbb R^3}T^{0\nu}\,d^3x$$

are time-independent. Lorentz invariance similarly gives the canonical current

$$\mathcal M^{\lambda\mu\nu} =x^\mu T^{\lambda\nu} -x^\nu T^{\lambda\mu} +S^{\lambda\mu\nu},$$

where $S^{\lambda\mu\nu}$ is the intrinsic-spin current when the fields transform nontrivially under Lorentz transformations. The conserved charge is

$$J^{\mu\nu} =\int_{\mathbb R^3} \mathcal M^{0\mu\nu}\,d^3x.$$

For a scalar field, the spin current vanishes. With a suitable symmetric Belinfante-improved stress-energy tensor, the separate spin term is absorbed into the stress-energy contribution, and the total charge agrees under the stated boundary falloff. Tong derives translation charges and the scalar Lorentz current in Tong, Quantum Field Theory, §§1.3.2–1.3.3, printed pp. 14–17 (PDF); Tong, §4.6, especially Eq. (4.96), printed pp. 98–99 (PDF) displays the intrinsic spin term for a Dirac field.

Optional quantum bridge: the operator algebra

This section may be skipped without affecting the classical conservation laws above or the transfer check below. For this optional bridge. Angular Momentum, Spin, and Rotations introduces quantum generators and commutators.

With $\hbar=1$, let $P^\mu$ act by multiplication by $p^\mu$ on scalar momentum-space wavefunctions and choose the orbital, spin-zero representation

$$J^{\mu\nu} =i\left(p^\nu\frac{\partial}{\partial p_\mu} -p^\mu\frac{\partial}{\partial p_\nu}\right).$$

For wavefunctions with internal spin indices, the full generator is $J^{\mu\nu}_{\rm orb}+S^{\mu\nu}$, where $S^{\mu\nu}$ acts on those indices. The differential operator displayed above is only the orbital part.

The Poincaré algebra is then

$$[P^\mu,P^\nu]=0,$$ $$[J^{\mu\nu},P^\rho] =i\left(\eta^{\mu\rho}P^\nu-\eta^{\nu\rho}P^\mu\right),$$

and the full Lorentz-generator relation is

$$\begin{aligned} [J^{\mu\nu},J^{\rho\sigma}] =i\big(&\eta^{\mu\rho}J^{\nu\sigma} -\eta^{\mu\sigma}J^{\nu\rho}\\ &-\eta^{\nu\rho}J^{\mu\sigma} +\eta^{\nu\sigma}J^{\mu\rho}\big). \end{aligned}$$

Thus, with $J_i=\tfrac12\epsilon_{ijk}J^{jk}$ and $K_i=J^{0i}$,

$$[J_i,J_j]=i\epsilon_{ijk}J_k, \qquad [J_i,K_j]=i\epsilon_{ijk}K_k, \qquad [K_i,K_j]=-i\epsilon_{ijk}J_k.$$

The last relation is the algebraic signature of the rotation generated by noncollinear boosts. To state the representation labels precisely, take $\epsilon^{0123}=-1$ and define the Pauli–Lubanski vector by

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

For a massive positive-energy irreducible representation,

$$P^2=-m^2, \qquad W^2=m^2s(s+1),$$

so the two Casimirs determine the mass $m$ and spin $s$. For an ordinary massless helicity representation, however, $P^2=W^2=0$ for every helicity and $W^\mu$ is parallel to $P^\mu$; helicity is a separate little-group label. Continuous-spin representations form another massless class and lie outside this chapter. Tong, The Standard Model, §1.1.2, printed pp. 9–14 (PDF) develops these massive, helicity, and continuous-spin cases separately.

Transfer check: conservation along a free worldline

For one massive spinless free particle, show directly that $J^{\mu\nu}=x^\mu p^\nu-x^\nu p^\mu$ is conserved, then determine how it changes under an active translation.

Solution

Parametrize the worldline by proper time. Since $p^\mu=m\,dx^\mu/d\tau$ is constant,

$$\frac{dJ^{\mu\nu}}{d\tau} =\frac{p^\mu}{m}p^\nu -\frac{p^\nu}{m}p^\mu +x^\mu\frac{dp^\nu}{d\tau} -x^\nu\frac{dp^\mu}{d\tau} =0.$$

The first two terms cancel by symmetry, and the last two vanish for free motion. Under $x^\mu\mapsto x^\mu+b^\mu$,

$$J'^{\mu\nu} =(x^\mu+b^\mu)p^\nu -(x^\nu+b^\nu)p^\mu =J^{\mu\nu}+b^\mu p^\nu-b^\nu p^\mu.$$

This is the one-particle version of the system transformation law. If an external force acts, $dp^\mu/d\tau\ne0$ and the particle charge alone is not conserved; the missing change must be attributed to the external agent or included fields.

Conservation also requires isolation: flux through a finite boundary changes the charge inside, and an external potential can break translations or Lorentz symmetry. In field theory, canonical and improved stress-energy tensors may differ locally while yielding the same total charges under suitable boundary falloff. These qualifications prevent particle formulas from being silently promoted to general field statements.

The next chapter begins at the operational level: Relativistic Measurement, Particles, and Collisions uses $P^\mu$ to compute observer energies, collision invariants, decays, and thresholds.

References

• Lim, Ian. General Relativity. Notes for Malcolm Perry’s Cambridge Mathematical Tripos Part III course, Michaelmas 2018, last updated January 25, 2019. Definition 1.3, printed p. 4 (PDF).
• Tong, David. Quantum Field Theory. §§1.3.2–1.3.3, printed pp. 14–17 (PDF), and §4.6, especially Eq. (4.96), printed pp. 98–99 (PDF), University of Cambridge, Michaelmas Terms 2006 and 2007. Formulas are translated to $(-+++)$ where needed.
• Tong, David. The Standard Model. §1.1.2, “The Poincaré Group and its Representations,” printed pp. 9–14 (PDF), University of Cambridge Part III Mathematical Tripos lecture notes.