Current matrix elements electromagnetic

Essentially what has enabled us to rewrite the matrix element in this form is the fact that the sources of the electromagnetic potential, that is, the currents of the charged particles obey the continuity equation d j x) — 0. 

An immediate application connects part of the neutron /3-decay matrix element to the electromagnetic form factors of protons and neutrons. Taking for J+ the vector part of the hadronic weak current and using (1.2.11), (1.3.17) yields... 

To begin with, consider the simplest and best known gauge theory— QED. As a result of the gauge invariance there are equations of constraint (the Ward-Takahashi identities mentioned in Section 1.1) that must be satisfied by certain matrix elements. For example, because the photon is coupled to the electromagnetic current in the form J, the amplitude for a photon to cause a transition from some state a) to /3) as shown,... 

This unwelcome discovery is potentially catastrophic for our unified weak and electromagnetic gauge theory. There we have lots of gauge invariance, many conserved currents, both vector and axial-vector, and hence many Ward identities. Moreover the Ward identities play a vital role in proving that the theory is renormalizable. It is the subtle interrelation of matrix elements that allows certain infinities to cancel out and render the theory finite. Thus we cannot tolerate a breakdown of the Ward identities, and we have to ensure that in our theory these triangle anomalies do not appear. 

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