Equal-time commutators and the energy-momentum tensor

Abstract

Starting from a generalization to spinor fields ψ of the equal-time commutator between the time components of the vector and axial vector currents V o i and A o i and the energy density part of the canonical energy-momentum tensor we investigate current-field commutators. Assuming that the equal-time commutators [V o i( X, χ o ), ψ( Y, χ o )] and [A o i( X, χ o ), ψ( Y, χ o )] are proportional to the field itself and have no Schwinger terms we show the absence of Schwinger terms in the equal-time commutators [V k i( X, χ o ), ψ ( Y, χ o )], [A k i( X, χ o ), ψ ( Y, χ o )] and [(ϱ/ϱχ μ)A μ i ( X, χ o , ψ( Y, χ o )] . Thus [(ϱ/ϱχ o , ψ( Y, χ o )] only contains a first-order Schwinger term and [(ϱ/ϱχ o )A o i( X, χ o ), ψ( Y, χ o )] only contains a non-Schwinger term and a first-order Schwinger term. In addition we show that quark model results for [V μ i( X, χ o ), ψ( Y, χ o )] and [A μ i( X, χ o ), ψ( Y, χ o )] follow from assuming the transformation properties of these equal-time commutators to be identical to those of ψγ μγ 4 and ψγ μγ 4γ 5 respectively. The possible deviations from the quark model are found to be given by the equal-time commutators between the spinor current f m ( χ) and the time component of the vector and axial vector currents. Some implications for quark model and algebra of fields current commutators are discussed. Finally we comment on symmetry breaking. We show that if the non-Schwinger part of the equal-time commutator between the time components of conserved currents is the time component of a current, then this current is conserved. Necessary and sufficient conditions are given in order for the non-Schwinger part of the equal-time commutators between the time components of conserved and non-conserved currents or between the time components of non-conserved currents to be the time component of a conserved or non-conserved current.