Schwinger terms in charge-density commutators

Abstract

The necessity for q-number Schwinger terms in the equal-time commutators between flavor charge densities for chiral fermions in 3 + 1 dimensions is shown. The charge densities here are not coupled to any gauge field. The number of quark species (colors) gives the central charge of an infinite-dimensional Lie algebra. The result is obtained by considering the analogue of the double spectral function in current-current correlation functions.

New Structures in Odd-Odd120I

“…In the neighboring odd-N nuclei [10] , the νh 11/2 band lies closer to the yrast line while the νd 5/2 band and the νd 3/2 band lie closer to the ground state. In the neighboring odd-Z nuclei [6] , the πd 5/2 configuration appears to be the ground state, and bands based on the πh 11/2 , πg 9/2 , πg 7/2 and πd 5/2 configurations gain strong population in heavy-ion induced fusion-evaporation reactions, with the πh 11/2 band being yrast as it originates from the high-j subshell and thus carries the largest alignment. Therefore, it can be expected that the bands built on the πh 11/2 νh 11/2 , πh 11/2 νd 5/2 , πh 11/2 νd 3/2 , πg 9/2 νh 11/2 , πg 7/2 νh 11/2 , and πd 5/2 νh 11/2 configurations in 120 I will be the predominantly populated bands in heavy-ion induced fusion-evaporation reactions.…”

The Gerasimov-Drell-Hearn sum rule and current algebras

“…However, a more careful study carried out recently shows that the commutator actually does have additional terms [10]. Indeed, these terms have quantum numbers that can give rise to modifications of the GDH sum rule.…”

“…The crucial point of ref. [10] is that the vacuum expectation value of the triple commutator of charge densities generated by quark fields cannot vanish, and is proportional to the number of colors carried by these fields. Furthermore, its structure implies that the charge density commutator [V a (x), V b (y)] must have a term symmetric in the a, b indices.…”

“…The minimal (infinite dimensional) Lie algebra that reproduces the correct vacuum expectation value of the triple commutators of the vector and axial-vector charge densities is given by the following relations [10]:…”

“…( 2) share the same quantum numbers as the conventional vector and axial-vector currents. However, except in singular limits [10], they cannot be directly proportional as operators to these currents. The Schwinger terms in the commutators of the charge densities and the spatial current densities are infinite in QCD, while the corresponding quantity in Eq.…”