The Problem of Constructing the Current Operators in Quantum Field Theory

Abstract

Lorentz invariance of the current operators implies that they satisfy the well-known commutation relations with the representation operators of the Lorentz group. It is shown that if the standard construction of the current operators in quantum field theory is used then the commutation relations are broken by the Schwinger terms. PACS numbers: 03.70+k, 11.10, 11.30-j, 11.40.-q In any relativistic quantum theory the system under consideration is described by some unitary representation of the Poincare group. The electromagnetic or weak current operator Ĵ(x) for this system (where μ = 0, 1, 2, 3 and x is a point in Minkowski space) should satisfy the following necessary conditions. Let Û(a) = exp(ıP̂μa ) be the representation operator corresponding to the displacement of the origin in spacetime translation of Minkowski space by the four-vector a. Here P̂ = (P̂ , P̂) is the operator of the four-momentum, P̂ 0 is the Hamiltonian, and P̂ is the operator of ordinary momentum. Let also Û(l) be the representation operator corresponding to l ∈ SL(2, C). Then Û(a)Ĵ(x)Û(a) = Ĵ(x− a) (1) Û(l)Ĵ(x)Û(l) = L(l)ν Ĵ (L(l)x) (2) where L(l) is the element of the Lorentz group corresponding to l and a sum over repeated indices μ, ν = 0, 1, 2, 3 is assumed.

Cite this paper

@inproceedings{Lev1995ThePO, title={The Problem of Constructing the Current Operators in Quantum Field Theory}, author={Felix M. Lev}, year={1995} }