Noether's theorems/Mathematics/The Ward-Takahashi identity

Plain-language counterpart: The theorem after quantum mechanics.

This page derives nothing. It is a summary, and it says so here rather than in a provenance line at the bottom.

Deriving the Ward-Takahashi identity requires either the path integral or the operator formalism of quantum field theory, neither of which this volume introduces. What follows states the shape of the result and marks the boundary of what has been checked.

The question

Both of Noether's theorems are classical. They concern an integral, its stationary points, and the identities that invariance forces. Quantum mechanics does not have stationary points; it has amplitudes summed over all histories, including the ones the classical theory rejects.

So the classical conservation law cannot survive unchanged. The question is what replaces it.

The answer, stated

UNVERIFIED, all of the following.

A symmetry of the classical action constrains the correlation functions of the quantised theory. The constraint takes the form of a relation among those functions, called the Ward-Takahashi identity, which reduces to the classical conservation law when the quantum corrections are dropped.

The identity is not a statement that a quantity is conserved. It is a statement that certain combinations of correlation functions are equal, and the conserved current appears inside them rather than as the subject of the sentence.

For a theory with a local symmetry, the corresponding identities are what guarantee that the unphysical parts of the description — the slack identified on the counting page — do not contaminate physical predictions. That is the second theorem's shadow in the quantum theory, and it is the reason the identities are load-bearing rather than decorative.

What must not be said

The Ward-Takahashi identity is not Noether's theorem restated, and the claim is common enough to be worth refusing here as the moment-map claim is refused on its page.

The two are related by a limit, not by a translation. Noether's theorems are statements about a variational integral and hold identically in all arguments. The Ward-Takahashi identity is a statement about expectation values in a particular quantised theory, and depends on how that quantisation was carried out — which is exactly what makes anomalies possible.

The honest position of this page

A reader who needs this result should go to a text on quantum field theory. This page exists to mark that the classical theorems have a quantum counterpart, to name it, and to say plainly that the counterpart has not been derived here and that nothing elsewhere in this volume depends on it.

That is a smaller page than the charter's candidate list implies. It is the page that can be written honestly at this depth.

Provenance: no source was reached for this page and nothing on it is derived. Every substantive statement is tagged UNVERIFIED above. See Sources.