Do all Noether theorems have a common mathematical structure?

1. Noether's first and second theorem only apply to classical theories with an action formulation.

2. The quantum analogs are (generalizations of) the Schwinger-Dyson equations and the Ward-Takahashi identities.

The core of the Noether theorem in all contexts where it arises is surprisingly elementary!

From a very general point of view, one considers the following structure.

(i) A set of "states" $xin Omega$,

(ii) A one-parameter group of transformations of the states $phi_u : Omegato Omega$, where $uin mathbb{R}$.

These transformations are requested to satisfy by definition $$phi_tcirc phi_u = phi_{t+u}:, quad phi_{-u}= (phi_u)^{-1}:, quad phi_0 = text{id}tag{0}:.$$

(iii) A preferred special one-parameter group of transformations $$E_t : Omega to Omega $$ representing the time evolution (the dynamics) of the physical system whose states are in $Omega$.

The general physical interpretation is clear. $phi_u$ represents a continuous transformation of the states $xin Omega$ which is additive in the parameter $u$ and is always reversible. Think of the group of rotations of an angle $u$ around a given axis or the set of translations of a length along a given direction.

A continuous dynamical symmetry is a one-parameter group of transformations that commutes with the time evolution, $$E_t circ phi_u = phi_u circ E_t quad forall u,t in mathbb{R}:.tag{1}$$

The meaning of $(1)$ is that if I consider the evolution of a state $$x_t = E_t(x)$$ and I perform a symmetry transformation at each time $$ phi_u(x_t):,$$ then the resulting time-parametrized curve of states is still a possible evolution with respect the said dynamics $$phi_u(x_t) = E_t(phi_u(x)):.$$

These features are shared by the theory of dynamical systems, Lagrangian mechanics, Hamiltonian mechanics, Quantum Mechanics, general Quantum Theory including QFT.

The difference is the mathematical nature of the space $Omega$ and some continuity/differentiability properties of the map $mathbb{R} ni u mapsto phi_u$, whose specific nature depends on the context.

The crucial observation is that, once assumed these quite natural properties, the one-parameter group structure $(0)$ provides a precise meaning of $$X := frac{d}{du}|_{u=0} phi_u$$ and, exactly as for the standard exponential maps which satisfies $(0)$, one has (for us it is just a pictorial notation) $$phi_u = e^{uX}:.$$ $X$ is the generator of the continuous symmetry.

• In quantum theory, $X$ (more precisely $iX$) is a self adjoint operator and hence a quantum observable,

• in dynamical system theory and Lagrangian mechanics $X$ is a vector field,

• in Hamiltonian mechanics $X$ --written as $X_f$ -- is an Hamiltonian vector field associated to some function $f$.

$X$ (or $iX$, or $f$) has another meaning, the one of observable. However, it is worth stressing that this interpretation is delicate and strictly depends on the used formalism and on the mathematical nature of the space $Omega$ (for instance, in real quantum mechanics the said interpretation of $X$ in terms of an associated quantum observable is not possible in general).

Now notice that, for a fixed $tin mathbb{R}$, $$u mapsto E_tcirc e^{uX} circ E^{-1}_t =: phi^{(t)}_u$$ still satisfies $(0)$ as it immediately follows per direct inspection. Therefore it can be written as $$E_tcirc e^{uX} circ E^{-1}_t = e^{uX_t}tag{3}$$ for some time-depending generator $X_t$.

We therefore have a time-parametrized curve of generators $$mathbb{R} ni t mapsto X_t:.$$

The physical meaning of $X_t$ is the observable (associated to) $X$ temporally translated to the time $t$.

That interpretation can be grasped from the equivalent form of $(3)$ $$E_t circ e^{uX} = e^{uX_t} circ E_t tag{4}.$$ The similar curve $$mathbb{R} ni t mapsto X_{-t}$$ has the meaning of the time evolution of the observable (associated to) $X$. One can check that this is in fact the meaning of that curve in the various areas of mathematical physics I introduced above. In quantum mechanics $X_t$ is nothing but the Heisenberg evolution of $X$.

Noether Theorem. ${e^{uX}}_{uin mathbb R}$ is a dynamical symmetry for ${E_t}_{tin mathbb R}$ if and only if $X=X_t$ for all $tin mathbb R$.

PROOF. The symmetry condition $(1)$ for $phi_t = e^{tX}$ can be equivalently rewritten as $E_t circ e^{uX} circ E^{-1}_t = e^{uX}$. That is, according to $(3)$: $e^{uX_t} = e^{uX}$. Taking the $u$-derivative at $u=0$ we have $X_t=X$ for all $tin mathbb R$. Proceeding backwardly $X_t=X$ for all $tin mathbb R$ implies $(1)$ for $phi_t = e^{tX}$. QED

Since $E_t$ commutes with itself, we have an immediate corollary.

Corollary. The generator $H$ of the dynamical evolution $$E_t = e^{tH}$$ is a constant of motion.

That is mathematics. Existence of specific groups of symmetries is matter of physics.

It is usually assumed that the dynamics of an isolated physical system is invariant under a Lie group of transformations.

In classical mechanics (in its various formulations) that group is Galileo's one. In special relativity that group is Poincaré's one. The same happens in the corresponding quantum formulations.

Every Lie group of dimension $n$ admits $n$ one-parameter subgroups. Associated to each of them there is a corresponding conserved quantity when these subgroups act on a physical system according to the above discussion. Time evolution is one of these subgroups.

The two afore-mentioned groups have dimension $10$ and thus there are $10$ (scalar) conserved quantities. Actually $3$ quantities (associated to Galilean boosts and Lorentzian boosts) have a more complex nature and require a bit more sophisticated approach which I will not discuss here; the remaining ones are well known: energy (time evolution), three components of the total momentum (translations along the three axes), three components of the angular momentum (rotations around the three axes).

I won't bother to reproduce the contents of the paper I want to recommend, but I will try to summarize what you'll find in it.

Baez, John C. "Getting to the Bottom of Noether's Theorem." arXiv preprint arXiv:2006.14741 (2020).

The main thrust of the paper is to put applications of Noether's theorem in classical mechanics, quantum mechanics, and statistical mechanics into a common context.

The overall viewpoint taken is largely an algebraic one. The algebraic structures of interest are Lie algebras, Jordan algebras, Poisson algebras, $C^ast$-algebras and some specializations thereof.

The main characters are observables, generators, and a strong connection established between the preceding two things. From there I think the statement of Noether's Theorem boils down to what conservation means in terms of the bracket on the relevant algebraic structure.

Personally, I am fascinated by this paper and am still digesting it. Maybe it doesn't shoehorn all applications of Noether's theorem into a single explanation, but it certainly does a good job of lining up the similarities between parallel applications so that they can be understood in the same way.