Relativistic Invariance in Quantum Field Theory and Commutation Relations for Densities

Abstract

The conditions for the relativistic invariance in quantum field theory are investigated on a fairly general assumption. Relations are obtained in the form of the equal-time com­ mutator of energy and momentum densities. These basic relations must be satisfied by any fields having intrinsic spin and local interactions, and they generalize Schwinger's results. When we quantize relativistic field equations involving higher spin particles in accordance with the canonical quantization method, we encounter serious difficulties on account of the presence of the constraint equations on these field. In this paper we investigate the condition for relativistic invariance in quantized field theory under a fairly general assumption, and formulate it in a form independent of detailed structure of the theory, namely, independent of the type of local interactions, magnitude o£ particle. spins, and statistics. We assume the unique existence of energy and momentum densities of interacting matter fields which satisfy the conservation law. From these densi­ ties one can construct the ten infinitesimal operators of the inhomogeneous Lorentz group by three-dimensional surface integrals on an arbitrary space-like surface. These ten generators of the inhomogeneous Lorentz group*) are characteristic of the dynamical system and satisfy the fundamental commutation relations of the IHLG. The conservation laws express the time variation of the densities in terms of their space derivatives on the t =constant hyperplane. Hence the conditions for relativistic invariance in field theory reduce to the sets of equal time commutation relations of the energy and momentum density of the dynami­ cal system. We seek for the general solution to these commutation relations. One can construct the energy-momentum tensor of free field having intrinsic spin larger than two. 1 ) As is well known in these cases it is not easy to con­ struct invariant Lagrange functions leading simultaneously to the equation of motion and the constraint equations. The fundamental commutation relation for the energy density of the dy­ namical system was first obtained by Schwinger in the problem of establishing the relativistic invariance of non-Abelian gauge fields.:l) Schwinger 3 ) discussed