Poincaré generators in field theory

For a real scalar, the translation current gives a symmetric stress tensor built from derivatives and the Lagrangian. Its zero-zero component is energy density and its zero-spatial components are momentum density with raised spacetime indices. Spatially integrating them gives the total four-momentum when surface flux vanishes. The metric matters: raising a spatial derivative changes its sign, so a three-vector gradient cannot be substituted blindly for every indexed derivative.

In the free quantum theory, local stress components are composite distributions defined through a vacuum subtraction or equivalent Wick prescription. Their spatial integrals act on a suitable dense domain, with limits and boundary terms controlled. The integrated translation generators have commuting components in the ordinary Poincaré representation. Their commutators with a scalar field give spacetime derivatives. This relation specifies the active translation convention and checks that the Hamiltonian agrees with the previously derived Heisenberg evolution.

Rotations use the spatial moment of momentum density. A scalar has no independent intrinsic spin term in its canonical Lorentz current; spinor and vector fields require the corresponding spin current or an improved stress tensor. Improvements can rearrange local densities without changing total charges when their surface contributions vanish. Splitting angular momentum into orbital and spin pieces can depend on gauge and convention, while the total physical rotation generator is the relevant conserved quantity.

Define the boost generator as the spatial moment of energy density minus time times total momentum. This chooses the sign opposite to a frequently used upper-index zero-spatial Lorentz generator. Its conservation states that the center of energy moves with velocity total momentum divided by total energy for an isolated state. The center of energy is a weighted moment of a field expectation, not a new claim that every relativistic field has a simple covariant single-particle position operator.

Rotations make momentum transform as a vector. Boosts mix energy and momentum, and two boosts along different directions differ by a rotation. Their commutation relations are therefore not independent optional checks. A cutoff that singles out a frame may obscure or break these relations before a symmetry-restoring continuum prescription is applied. Agreement of one mode's dispersion with the mass shell does not alone establish the full generator algebra.

For free particles, total four-momentum adds across occupied modes, including antiparticles with positive energy. The invariant mass of a multiparticle state depends on relative motion and can exceed the sum of rest masses. In an interacting theory, boosts generally contain interaction contributions as well as the Hamiltonian, while total asymptotic momenta still label scattering states under suitable assumptions. Writing classical Noether integrals is consequently a formal starting point; constructing self-adjoint quantum generators and a unitary representation is an additional mathematical task.