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{{DISPLAYTITLE:The recovery in field theory}}

By the 1970s "Noether's theorem" was standard in every graduate course in field
theory. This page traces how it got there, and notes what the route cost.

== The waypoints ==

The recovery was not a single event and no one paper did it. The following are
the points usually named, and they are named here as the commonly cited route,
not as a verified account.

'''Belinfante and Rosenfeld, 1939 and 1940.''' The stress-energy tensor that the
first theorem produces is the canonical one, which is neither unique nor
generally symmetric. Their improvement adds a divergence term that symmetrises
it without changing the conserved charges. This put the theorem's output into
the form general relativity needs, and made the non-uniqueness a subject rather
than an embarrassment.

'''Bergmann and the constrained-Hamiltonian programme, from 1949.''' Work on the
Hamiltonian formulation of general relativity ran directly into the fact that
the second theorem describes: a theory with local symmetry has fewer independent
equations than fields, and its Hamiltonian formulation therefore carries
constraints. This is the second theorem rediscovered from the inside, by people
solving a concrete problem.

'''Hill, 1951.''' A review article in ''Reviews of Modern Physics'' on
Hamilton's principle and the conservation theorems. This is the paper most often
credited with putting the theorem in front of physicists in a form they could
use, and it is also, on the usual account, where the restriction to the first
theorem in its field-theory special case became the standard presentation.

'''Yang and Mills, 1954, and the gauge theories that followed.''' Utiyama in
1956 and Kibble in 1961 extended the gauge idea to gravitation. Once physics had
several theories with local symmetry rather than one, the second theorem stopped
being a remark about general relativity and became the general structural fact
about gauge theories.

'''Trautman, from 1962.''' Conservation laws in general relativity, treated
properly as an instance of the theorem rather than as a special problem.

'''Ward and Takahashi, 1950 and 1957.''' The quantum statement. A symmetry of
the action constrains the correlation functions of the quantised theory, and the
Ward-Takahashi identity is what survives of the classical conservation law. This
is also where the boundary appears: a classical symmetry can fail to survive
quantisation, and the anomaly is the failure.

== What the route cost ==

The theorem physics recovered is not the theorem Noether proved, and the
differences are traceable to the route.

'''One theorem, not two.''' What entered the textbooks was the first theorem.
The second entered separately and under other names — constraints, Noether
identities, Bianchi identities — and its connection to the same 1918 paper was
often not made.

'''One direction, not two.''' The converses were dropped almost everywhere. A
textbook statement of the theorem usually proves that a symmetry gives a
conservation law and stops.

'''A special case, not the general statement.''' Physics needed one independent
variable or four, first derivatives, and transformations that do not mix the
dependent and independent variables. That is what got stated. Noether's actual
statement allows arbitrary derivative order and full mixing.

'''An addition credited backwards.''' Divergence invariance is
[[Noether's theorems/History/Bessel-Hagen and the first uses|Bessel-Hagen's]],
and is now routinely presented as part of Noether's theorem.

None of these losses is a mistake in the physics. A special case correctly
stated is still correct. They are losses in what the theorem is understood to
say, and they are why
[[Noether's theorems/History/What Noether's theorem now means|the name now means
different things]] in different rooms.

''Provenance: this page contains no verbatim quotation and no page reference was
verified against the papers named. The route above is the one the secondary
literature commonly gives; the attributions are standard but were not checked
against the primary papers. See [[Noether's theorems/Sources|Sources]].''

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