BFP Rauch: 1 revision imported

2026-09-10T13:29:56Z

1 revision imported

← Older revision	Revision as of 13:29, 10 September 2026
(No difference)

NT>NoetherVolumeBuild: Noether theorems volume, initial import (S-05)

2026-09-10T00:00:00Z

Noether theorems volume, initial import (S-05)

New page

{{DISPLAYTITLE:Fewer independent field equations than fields}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Counting the equations a gauge theory really has|Counting the equations a gauge theory really has]].''

[[File:NT-F-22.png|thumb|right|360px|alt=Three columns of stacked bars. Eight field variables on the left, eight field equations in the middle with the last two shaded to mark them as dependent, and six independent facts on the right. An arrow labelled minus two joins the middle column to the right. The caption explains that two free functions in the symmetry give two identities and two equations that follow from the others.|Fields against independent field equations]]

The second theorem's consequence, stated as a count.

== What setting the Lagrange expressions to zero does ==

For the first theorem, imposing <math>\psi_{i}=0</math> turns divergence
relations into conservation laws. For the second it does something else
entirely, and Noether states it in one sentence in her first section: Theorem II
says that <math>\varrho</math> of the Lagrange equations are a consequence of the
remaining ones.

Read (N16) with that in mind. It is a linear relation among the
<math>\psi_{i}</math> and their derivatives, holding identically. If all but
<math>\varrho</math> of the <math>\psi_{i}</math> vanish, the relation forces
constraints on the rest. The field equations were never <math>\mu</math>
independent statements. They were <math>\mu</math> statements with
<math>\varrho</math> relations among them.

'''No conservation law appears at any point in this paragraph.''' That is the
whole difference from the first theorem, and it is why
[[Noether's theorems/First theorem/On-shell and off-shell|the on-shell page]]
insisted the two passages be kept apart.

== The count ==

One arbitrary function in, one identity out, one field equation lost as an
independent statement. The bookkeeping is exact and it runs:

* <math>\varrho</math> arbitrary functions in the group;
* <math>\varrho</math> identities (N16), holding off-shell;
* <math>\varrho</math> of the field equations following from the others on-shell;
* <math>\varrho</math> functions' worth of freedom in any solution, carrying no physical content.

The last line is the one physics feels. A solution is determined only up to the
action of the group, so initial data does not fix the future — not because the
theory is indeterministic, but because part of what looks like a difference
between solutions is a relabelling.

Hilbert had reached this from the other end in 1915, stating for his own
invariant that four of his fundamental equations follow from the rest, and had
spelled out the consequence for prediction in his second communication.
[[Noether's theorems/History/Hilbert and Klein|The history track]] gives that
sequence. What was missing, and what the second theorem supplies, is that the
count is a theorem about the group and not a feature of one Lagrangian.

== Why gauge theories are built this way anyway ==

The redundancy is not a cost that gauge theories pay for some other benefit. It
is what having a local symmetry means, and any theory with one has it.

For general relativity, <math>\varrho=n</math>: the four coordinate functions,
and four identities. For electromagnetism, <math>\varrho=1</math>: the single
gauge function, and one identity, which is why the potentials are not determined
by the field equations. Each case is on its own page, in
[[Noether's theorems/Applications/Yang-Mills theory|the applications track]].

== What this does not say ==

'''It does not say the theory has too few equations.''' The count of equations
matches the count of physically distinct quantities exactly. It is the count of
'''field variables''' that exceeds both.

'''It does not say the redundant equations are useless.''' They hold, they are
true, and they are used. They are simply not independent.

''Provenance: the statement that <math>\varrho</math> of the Lagrange equations
are a consequence of the rest is Noether's, from her first section, paraphrased.
The reading of Hilbert's Theorem I and his remark on prediction is summarised
from an OCR-damaged copy of his second communication and is not quoted. The count
above is this volume's own arrangement of her result. See
[[Noether's theorems/Sources|Sources]].''

Noether's theorems/Second theorem/Fewer independent field equations than fields - Revision history

BFP Rauch: 1 revision imported

NT>NoetherVolumeBuild: Noether theorems volume, initial import (S-05)