Energy-Momentum Conservation Implies Translation Invariance: Some Didactic Remarks

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Abstract
We discuss from a rigorous viewpoint two more-or-less familiar cases where energy-momentum conservation implies invariance under space-time translations. First, if a closed linear operator on a Hilbert space has a domain that is invariant under spectral projections belonging to the four-momentum operators, and if it ``conserves energy-momentum,'' it necessarily commutes with the appropriate representation of the translations. (Bounded operators, such as the S matrix, are a special case.) At least for separable spaces, the domain restriction characterizes the closed operators for which the theorem is true. Second, if a bounded bilinear form between momentum states of m and n particles in a Fock space (or more generally, a bounded multilinear form) conserves energy momentum, the corresponding tempered distribution has a conservation delta function at points where the mass shell is a C∞ manifold; but no derivatives of delta functions can occur. In this connection, we are led to a result that seems to be new: the cluster parameters (``connected amplitudes'') of a family of bounded bilinear forms, labeled by (m, n), are also bounded bilinear forms. The two systems, of course, mutually conserve energy momentum.