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Noether theorems volume, initial import (M-07)

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{{DISPLAYTITLE:The Ward-Takahashi identity}}

''Plain-language counterpart: [[Noether's theorems/Explainers/The theorem after quantum mechanics|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
[[Noether's theorems/Second theorem/Fewer independent field equations than fields|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
[[Noether's theorems/Mathematics/The symplectic form of the theorem and the moment map|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
[[Noether's theorems/Mathematics/Anomalies|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
[[Noether's theorems/Sources|Sources]].''

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