Noether's theorems/History/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 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 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 Sources.