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

Plain-language counterpart: Counting the equations a gauge theory really has.

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 ${\psi }_i=0$ 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 $\varrho$ of the Lagrange equations are a consequence of the remaining ones.

Read (N16) with that in mind. It is a linear relation among the ${\psi }_i$ and their derivatives, holding identically. If all but $\varrho$ of the ${\psi }_i$ vanish, the relation forces constraints on the rest. The field equations were never $\mu$ independent statements. They were $\mu$ statements with $\varrho$ 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 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:

• $\varrho$ arbitrary functions in the group;
• $\varrho$ identities (N16), holding off-shell;
• $\varrho$ of the field equations following from the others on-shell;
• $\varrho$ 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. 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, $\varrho =n$: the four coordinate functions, and four identities. For electromagnetism, $\varrho =1$: 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 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 $\varrho$ 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 Sources.