Ward-Takahashi identity

The number of Group V diagrams is reduced from 518 to 47 using the Ward-Takahashi identity and time-reversal symmetry of diagrams. 

Due to this form invariance of the Lagrangian under the renormalization prescription the stracture of the theory, which e.g. expresses itself in Dyson s equations and Ward-Takahashi identities, remains completely unchanged. The renormalized Greens functions, i.e. the vacuum expectation values of f and A, are thus simply obtained from... 

Eqn (1.1.4) is an example of a Ward-Takahashi identity , a relation that must be satisfied by the matrix elements of any operator that possesses... 

It should be stressed that gauge invariance plays a vital role in proving that the theory is renormalizable. Technically, as mentioned in Section 1.1, it gives rise to the Ward-Takahashi identities amongst matrix elements (Ward, 1950 Takahashi, 1957). 

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,... 

Takahashi, Y. (1957). On the generalized Ward identity, Nuovo Cimento 6, 371-375. 

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