As the proof of the theorem goes, because the dynamics is determined by the stationarity of the action. Noether's theorem is restricted to continuous transformations which leaves the action invariant (i.e., symmetries of the action).

Strangely, in classical physics the sole purpose of the action is to be stationary. This changes in the Feynman path integral formulation of quantum mechanics, where also a new investigation of Noether's theorem is required.

Kåre Olaussen: Do you think that the "second theorem" describes right the case of the Feynman formulation in "k" paths. i.e. where each parameter (1,...,k) in Noether's theorem corresponds to a possible path of the quantum leap? Might the Feynman interpretation serve as a visualization of the way for generalizing from the "first" to the "second theorem"? Is the action again statuionary, but in all paths rather than a single one as in clasical mechanics?

I can't see that this is a good way to look at the quantum problem. In my view the situation is better illustrated by the following simple example: Define

Z(a, b) = ∫∫ dx dy exp(-S(x2 + y2)) exp(ax+by),

where the integration runs over the full xy-plane, with exp(-S) is rapidly vanishing as x2 + y2 → ∞. Show that the (i) invariance of S under rotations, and (ii) invariance of the integration measure dx dy under rotations, implies that Z depends on the combination a2+ b2 only.

Note that the requirement on the integration measure is new, and implies that there may be classical Noether conservation laws which don't survive quantisation.

An equation of motion can be formulated as an optimisation problem. That's how action is built. Actually, building action for a new equation of motion is fun.

The theorems of Emmy Noether simplify the derivation of General Relativity, so the answer seems to yes for describing physical quantities.

Jerry Decker: Emmy Noether's theorems might be formulated abstractly and mathematically in terms of differential equations, manifolds, homeomorphisms, symmetries, etc. Action would turn out to be a physically dimensionless mathematical variable defined by the functional on a certain set of Lagrangians also physically dimensionless. Anyway, Emmy Noether, a mathematician with huge experience and contributions rather than a physicist, has preferred to formulate them physically.

It seems to me that the only difference between the mathematical and physical formulation would consists in the concept of time in each case: it is reversible as any mathematical variable after a mathematical interpretation, but irreversible as a unique physical quantity. In fact classical, mechanics and general relativity utilize reversible mathematical models unlike thermodynamics, in which the irreversibility of time is postulate implicitly in its principles.

Do you think that the theorems would be true after a proprely mathematical reformulation? Does the irreversibility of time assist the generation of physical dimensionality, and if yes, how?

Igor Goliney: Indeed, the principle of least action means right that motion is an optimization, and action is that quantity able to equate them. However, it refers to the smooth motion of classical mechanics. What about the discrete motion of quantum mechanics (in "all paths" in Feynman's interpretation)? Is a quantum leap an optimization, too? And an optimization of what if yes? It seems to me that the "second theorem" is the key to the quantum case. Really?

Kåre Olaussen: Maybe I am wrong, but it seems to me that the ultimate sense of your example is: what is stationary in quantum mechanics is probability for the integral of probability distribution is a unit identically. Of course, it is true, and any description of quantum dynamics is always to that stationarity of probability. (Your function Z(a,b) corresponds to a (real) functional of the wave functions.) This does not contradict to the "second theorem" for a manifold can be reversely restored to a set of wave functions and ultimately, to a set of probability distributions. Of course, my question was about the visualization and maybe your vizualization is better and clearer, indeed.

The quantum mechanical evolution is even smoother than the classical one. But the conservation has a different meaning. The wave function is not obliged to be an eigenfunction of any operator. The quantities are conserved only in average.

I see no difficulty in formulating Schrödinger equation as optimization problem. In fact, such approach is common to the quantum field theories. They start with the Lagrangian which implies action, of course.

As for the Feynman's approach, I am just familiar with it. I was not taught the art of integrating in the function space in my math course. But as far as I know, Feynman uses the classical action. Anyway, it is fully equivalent to the simple Schrödinger or Heisenberg picture.

Igor Goliney: I mean "The wave function is not obliged to be an eigenfunction of any operator". It is very intersting to me. What do you mean? Entanglement? Or something else? How should "any operator" be interpreted? What is the link between that kind of wave functions and "motion as optimization"? And also: It seems to me that the state corresponding to that wave function should be immeasurable in principle. Or not? If not, how should its measurable values be defined (if not as the eigenvalue of some operator, to which the wave function is an eigenfunction)? Sorry if I have not understood something.

No, this is not entanglement. This is called superposition. It is in the heart of quantum mechanics as it allows interference effects. it states that if the wave function psi_1 is possible and the wave function psi_2 is possible then the wave function a*psi_1 + b* psi_2 is also possible. a and b here are any complex numbers. Even if funstions 1 and 2 are eigenfunctions of an operator, the sum is not (except for the case of equal eigenvalues).

Operators in quantum mechanics correspond to observables. At a measurement the wavefunction collapses to an eigenfunction of the operator. This is the most murky point in quantum mechanics and raises a flood of discussions. Up to the measurement the evolution is smooth as silk. So, the mentioned state is measurable. The wave function would collapse. But prior to the measurement it is free to interfere.

Noether's theorem was formulated before the quantum mechanics of course.

The complex separable Hilbert space usually utilized in quantum mechanics is a vector space and the superposition of any two vectors interpretable as wave functions is necessary a vector of the same space according the definition of vector space. Any vector can serve as a basis, i.e. eigenfunction of some Hermitian operator and implies a measurable eigenvalue without referring to the "collapse". The collapse is necessary to explain only the random result of a single measurement.

Oh, you know. So, if you measure symmetry you get a random result. That is how the conservation laws in quantum mechanics are peculiar. But do work in a different sense.

In the medieval style of puzzles and misleadings (if it is necessary), what does the following mean: a.b → ∫ a(b)db? ;)

As far as I can read it, this is a definition of the point symbol in a.b. But I am disappointed.

Yes, but the interesting is the interpretation of the generalization of "." to that integral in the present context. So, this is extremely unsufficient without any link to the context. A possible link might be the following extended hint (but again only a hint ;) ):

Why does entropy increase? It doesn't increase actually, it increases only to future, but it decreases in the past exactly as much as it has increased in the present. So, it is constant, it conserves, but is divided into two parts, in the past and in the present. equal in module, but opposite in sign.

Its counterpart is the physical quantity of action, which changes or conserves correspondingly.

It seems to be the most fundamental physical law underlying and thus explaining even the course of time. It means: Information (resp. entropy or action) is what conserves most fundamentally. Energy conserves only in the present, but integrally it is always zero for as much energy is conserved in the present as energy has vanished in the past.

Thus, the theorems of Emmy Noether (1919) are the most fundamental physical theorems ever have been created.

The physical meaning of the first one is: the present moment as a point divides disjunctively and thus orthogonally action into two always constant parts though each one changing in the course of time. The one is a symmetry referable to the past, and the other one is a conservation referable to the present. The conservation is directly constant. The symmetry is constant in a generalized sense: it means that all elements of the symmetry are equivalent to each other as to the symmetry. So all elements are equal to each element and thus the mapping of the symmetry into itself can be considered as a set theory generalization of any constant function.

Then what does the course of time do? It transforms an element of symmetry into a value of a constant function. In other words, any moment of time measures for it transforms a thing which is an element of symmetry into a number, which is a value of constant function.

Thus the measurement of the present moment choices randomly an element of the symmetry, however any choice is indistinguishable from any other right because of the symmetry.

The second theorem meaning right the case of quantum mechanics generalizes the first theorem if the present moment is interpreted as a compact area rather that as a point, and thus the past and future are not disjunctive to each other or orthogonal to each other. They overlap each other partly therefore defining the present right by that overlapping.

Action is again conserved in the particular area of the present, but in a generalized form described in the conditions of the second theorem:

It conserves as a manifold defined by "k" differential equations corresponding to "k" parameters (interpretable as "k" paths in Feynman's interpretation) instead of the condition "k=1" in the first theorem.

One can add that the manifold is equivalent to a subset (possibly subspace) of the separable complex Hilbert space and ultimately to a set (possibly a topological subspace or neighborhood) of probability distributions including correspondingly "k" wave functions or "k" probability distributions.

If one uses the "language of manifold", one describes the present in terms of general relativity as a space-time continuum.

If one uses the "language of wave functions or probability distributions" one describes the present in terms of quantum mechanics as a quantum system and its states.

Anyway, those are only two languages for the description of the same, namely the present. Then the problem of quantum gravity should be interpreted as how to be proved the equivalence of both descriptions in the two different languages above. The second theorem of Emmy Noether gives the key for the solution.

An equation of motion can be formulated as an optimisation problem. Building action for a new equation of motion is fun.

You can use the documents of Neuenschwander, Dwight E.

See for example:

Neuenschwander, Dwight E. (2010). Emmy Noether's Wonderful Theorem. Johns Hopkins University Press. ISBN 978-0-8018-9694-1.