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I know that there are Noether theorems in classical mechanics, electrodynamics, quantum mechanics and even quantum field theory and since this are theories with different underlying formalisms, if was wondering it is possible to find a repeating mathematical pattern. I know that a common "intuitive" explanation is that each symmetry has a corresponding constant quantity - but can we express this in a mathematical way?

In other words: Can all Noether theorems be regarded as special cases of one recipe (in mathematical terms) that works for all formalisms?

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3 Answers 3

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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" $x\in \Omega$,

(ii) A one-parameter group of transformations of the states $\phi_u : \Omega\to \Omega$, where $u\in \mathbb{R}$.

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

(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 $x\in \Omega$ 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 pictorical 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 $t\in \mathbb{R}$, $$u \mapsto E_t\circ e^{uX} \circ E^{-1}_t =: \phi^{(t)}_u$$ still satisfies $(0)$ as it immediately follows per direct inspection. Therefore it can be written as $$E_t\circ e^{uX} \circ E^{-1}_t = e^{uX_t}\tag{3}$$ 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. $\{e^{uX}\}_{u\in \mathbb R}$ is a dynamical symmetry for $\{E_t\}_{t\in \mathbb R}$ if and only if $X=X_t$ for all $t\in \mathbb R$.

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 $t\in \mathbb R$. Proceeding backwardly $X_t=X$ for all $t\in \mathbb R$ 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).

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    $\begingroup$ My question is not a coincidence - I hope this answer gets more attention here, because it is superb :) I had to read it a few times though, because I didn't get what was meant by (0). Then I understood that the number zero $0$ looks like the letter "o" in text-mode. Actually, all numbers look weird in text mode. Is it okay for you, if I put dollar signs around them? $\endgroup$
    – Filippo
    Commented Dec 2, 2020 at 21:20
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    $\begingroup$ Yes, perform all typographic changes you think necessary. Thank you very much for appreciating my answer. $\endgroup$ Commented Dec 2, 2020 at 21:27
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    $\begingroup$ Yes, in fact, Noether theorem refers to a whole system $S$. If there is an external part $S'$ interacting with your system $S$ and this external part evolves, in general, it breaks time homogeneity. In that case the temporal evolution of $S$ is describes by a grupoid $\{F_{t_2,t_1}\}_{t_2\geq t_1}$. Think of the time evolutor of a Schroedinger equation with Hamiltonian which explicitly depends on time. But here we are assuming time homogeneity and the evolutor is a group. Equivalently, the grupoid is trivial $F_{t_2.t_1}= E_{t_2-t_1}$ and there is no preferred time. $\endgroup$ Commented Dec 3, 2020 at 7:47
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    $\begingroup$ it is surprisingly elementary! Spivak wrote, in his Calculus on Manifolds: "Stokes' theorem shares three important attributes with many fully evolved major theorems. 1) it is trivial; 2) it is trivial because the terms appearing in it have been properly defined; 3) it has significant consequences." I feel like modern formulations of Noether's theorem fall into that category too. Expressed in terms of Lie/Poisson brackets, it is algebraically very simple. The relationship described between observables and generators makes it interesting. $\endgroup$
    – rschwieb
    Commented Dec 3, 2020 at 18:52
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    $\begingroup$ The point is that the elementary version I illustrated above takes place in the Hamiltonian formulation, whereas the (more or less equivalent actually weaker) Lagrangian version is much more involved and obscure...And one usually starts with the Lagrangian version to get familiar with the theorem. $\endgroup$ Commented Dec 3, 2020 at 18:56
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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.

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    $\begingroup$ More or less my answer below is the relevant part of Baez' paper you quoted (that also considers the problem of the passage from the Jordan algebra structure to the one of $C^*$-algebra and discusses an old result about it, trying to connect an "obscure" hypothesis with some thermodynamical intepretation. A very nice paper.) $\endgroup$ Commented Dec 2, 2020 at 15:18
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    $\begingroup$ Thank you, I appreciate the recommendation! $\endgroup$
    – Filippo
    Commented Dec 3, 2020 at 9:47
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  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.

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  • $\begingroup$ Thank you for the summary, that's certainly a good starting point for further investigations. $\endgroup$
    – Filippo
    Commented Dec 3, 2020 at 7:30

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