Ward identity

This result is known as renormalization group equation for the (L, M)-cumu-lant. It is clearly just a Ward identity of the dilatation group (see Sect. 10.2.1). 

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

Eq. (2) presents the basis for the covariant renormalization approach. The explicit expressions are known for E Ten(E), X u 6 in momentum space. For obtaining these expressions the standard Feynman approach [11,12] or dimensional regularization [13] can be used. They are free from ultraviolet divergencies but acquire infrared divergencies after the renormalization. However, these infrared divergencies, contained in X 1) and cancel due to the Ward identity X -1) = —A1 1 and the use of the Dirac equation for the atomic electron in the reference state a) ... 

The transformation properties of Q lj are often given in terms of Ward identities which are found by taMng the derivative of Eq. (10,32) ... 

There is no general relation between the vertex r (piP2fe) and the propagators QpJ k). Still such a relation exists in an important particular case, when Pi = P2- This relation is called the Ward identity . It is convenient to introduce the short vertex ... 

See, for instance, the Ward identities in perturbation theory discussed in P. Noziere, Le problems a N corps, Dunod Cie, Paris, 1963, p. 243ff. 

Basko, D.M. (2009) Calculation of the Raman G peak intensity in monolayer graphene role of Ward identities. New. Phys., 11(9), 095011. 

Eq.(2.27) together with the Dyson equation (2.1 ) (with t defines so-called "0-derivable" approximations. 0 derivability guarantees then a self-consistent approximation in many-body perturbation theory which, for a given order of approximation, satisfies various conservation laws. For example the particle-hole attraction in eq.(2.9) is given by (Ward identity) 6 0/6g6g. 

This irreducible e-h interaction, via a Ward identity, also determines the particle-hole interaction in the Bethe-Salpeter equation (2.7) for the two-particle propagator. Ideally, we should... 

This relationship, like = 0, gives rise to constraints amongst certain matrix elements. We shall refer to these as the axial Ward identities. For example, if we consider the diagrams in Fig. 9.3 in which two photons couple to the axial current and the pseudo-scalar current respectively then we ought to find, because of (9.5.13), that the scalar product of ki + k2)fji with the first amplitude should equal the second amplitude. 

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. 

It turns out, similar to the above, that regularization of the triangle diagrams leads to correct results for the vector current Ward identities. The analogue of (9.5.13) is that the divergence of the axial-vector current vanishes, so the RHS of (9.5.15) should be zero, and this fails to hold. We must therefore construct our theory so that the coefficient of 75 in the expression for (fci - - A 2) A i/p is zero for algebraic reasons. 

---