Axial and parity anomalies and vacuum charge: A direct approach

We study,the axial and,parity,anomalies,in abelian,gauge,theories,using,the direct,yet,intuitive,approach,of counting,the,relative,number,of states,of one chirality with respect to the other. A fundamental,gauge invariant quantity, the determinantal ratio, is introduced to this purpose. We find that the number of states,is conserved,and,that,the,gauge,fields differentially,phase,shift states,of opposite,chirality at infinite energies. This implies,a relative change,of the density of states at infinite energies which must be compensated,by a rearrangement,of the,density,of states,at finite energies.,We then,derive,a sum,rule which,yields two,alternative,formulae,for the,index,of a Dirac operator.,One,expresses,the index in terms of its high energy behavior, and the other in terms of the low energy,properties;,these,are the,“zero modes”,of definite chirality. Two,examples are worked,out,in detail to clarify our,general,result. The physics,of the axial anomaly,is shown,to translate,into that,of the parity anomaly in (2 + 1) dimensions, in which parity and chirality have interchanged roles. We also analyze,the,vacuum,charge,in regard,to its high,and,low,energy origin. The,possibility,of spectral,flow,is formulated,and,briefly,discussed.,In short, we provide a physical interpretation of certain mathematical indices, relate them to an extended version of Levinson’s theorem of potential scattering, and simplify,their evaluation. 2 1. Axial Anomaly, density of states and indices