Lorentz transformation properties of nonconserved charges

Abstract

If a current jμ( x) is conserved, then the charge Q defined as the three-space integral of j0( x) is a Lorentz scalar. In this paper we investigate the Lorentz transformation properties of Q assuming that jμ( x) is not conserved; that is, ∂μ jμ( x)≠0. We find that Q, which now depends on time, transforms as the infinite direct sum of the spin-0 parts of the (0,2j+1) finite-dimensional tensor representations of the Lorentz group plus the spin-0 component of an infinite-dimensional indecomposable representation of the Lorentz group, [(0,1)→(1,0)]:[(0,1)→(1,0)] ⊕∑∞j=0⊕ (0,2j+1), where we are using the Gel’fand, Milnos, and Shapiro notation. More simply, if T μ1μ2⋅⋅⋅μn(x) is an n-index traceless, symmetric tensor density and S(x) is a density transforming as the spin-0 component of the infinite-dimensional representation, we find that Q transforms as the infinite sum S(x)+T(x)+T00(x)+T0000(x) +T000000(x)+⋅⋅⋅ .