Ward-Takahashi identity

The Ward-Takahashi identity , named after the British physicist John Clive Ward and the Japanese physicist Yasushi Takahashi , is a relation between correlation functions in quantum electrodynamics . They received their general form in 1957 by Takahashi; the special case of Ward identity had already been established by Ward in 1950.

The case of non-Abelian gauge theories is more complicated, here you have the Slavnov-Taylor identities .

General

The Ward-Takahasi identity can be derived from the Dyson-Schwinger equations and reads:

${\displaystyle {\begin{aligned}k_{\mu }M^{\mu }\left(k;p_{1},\dots ,p_{n};q_{1},\dots ,q_{n}\right)=&\ e\sum M_{0}\left(p_{1},\dots ,p_{n};q_{1},\dots ,q_{i}-k,\dots ,q_{n}\right)\\&-e\sum M_{0}\left(p_{1},\dots ,p_{i}+k,\dots ,p_{n};q_{1},\dots ,q_{n}\right)\end{aligned}}}$

It is

• ${\displaystyle M^{\mu }}$the Fourier transform of a correlation function that contains the Dirac current:

${\displaystyle M^{\mu }(k;p;q)=\int \mathrm {d} ^{4}z\prod _{i,j=1}^{n}\mathrm {d} ^{4}x_{i}\mathrm {d} ^{4}y_{j}e^{-\mathrm {i} (zk+\sum y_{j}p_{j}-x_{i}q_{i})}\left\langle \Omega \left|Tj^{\mu }(z)\prod \Psi (x_{i}){\bar {\Psi }}(y_{j})\right|\Omega \right\rangle }$

• ${\displaystyle M_{0}}$the correlation functions in which the momentum of the Dirac current was added / subtracted to the momentum of an incoming or outgoing fermion :

${\displaystyle M_{0}(p;q)=\int \prod _{i,j=1}^{n}\mathrm {d} ^{4}x_{i}\mathrm {d} ^{4}y_{j}e^{-\mathrm {i} (\sum y_{j}p_{j}-x_{i}q_{i})}\left\langle \Omega \left|T\prod \Psi (x_{i}){\bar {\Psi }}(y_{j})\right|\Omega \right\rangle }$

Ward identity

The special case of Ward identity can be derived from the Ward-Takahashi identity if

• ${\displaystyle M^{\mu }}$is a matrix element of the scattering matrix and
• on the right side all incoming and outgoing fermions are on shell , i.e. real, observable and obeying the energy-momentum relation .

Then the two terms on the right side of the identity cancel each other out with the aid of the LSZ reduction formula, and only the left side remains:

${\displaystyle k_{\mu }M^{\mu }=0}$

The Ward identity makes an important contribution to the renormalization of quantum electrodynamics, as it reduces the degree of divergence of photon loops as a symmetry-maintaining property . As a result, there is no hierarchy problem in quantum electrodynamics .

Individual evidence

1. ↑ Yasushi Takahashi, Nuovo Cimento , Ser 10, 6 (1957) 370.
2. ↑ JC Ward, Phys. Rev. 78, (1950) 182
3. ↑ Andrei Slavnov: Slavoňov Taylor identities, Scholarpedia