Noether's second theorem: two questions

In summary, the conversation discusses Noether's second theorem and its application to different symmetry groups in physics. The first question asks about the possibility of other groups apart from general relativity and if the theorem also applies to quantum field theory. The conversation then delves into the implications of the theorem, including the existence of improper conservation laws and the concept of gauge symmetries. The second question asks about the possibility of movements that do not respect Lagrange's equations, which is answered by explaining the role of global symmetries in physical laws. The conversation ends with a discussion on how Noether's famous paper was ahead of its time and how the general covariance in general relativity is a gauge symmetry.
  • #1
Husserliana97
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A technical subject, well above my level it seems (I'm still learning about quantum physics and special relativity), but one about which I absolutely must get some clear ideas as soon as possible.

From what I 'understand', Noether's second theorem applies to infinite-dimensional symmetry groups. A classic, even historical, example (this is the one that Noether had in mind) is the invariance group of Riemannian spacetimes, i.e. the set of spacetime diffeomorphisms of GR. First question: what other groups are possible, apart from general relativity? Does Noether's second theorem also apply to quantum field theory (= the 'gauge groups')? Or is it confined to general relativity?

This theorem is supposed to show that the invariance of the Lagrangian by the Lie group (infinite in dimension) of certain theories necessarily implies that the field equations proper to these theories satisfy identities, rather than "classical" conservation laws (called "proper conservation laws" by Hilbert), and the quantities thus "improperly" conserved are associated with local symmetries -- and no longer global symmetries, as in Noether's first theorem.

My second question is based on a quotation from an article entitled "On Noether's theorems and gauge theories in Hamiltonian formulation". Here is the extract:

"This brief treatment shows that, when a local group of transformations has a non-trivial global subgroup, it is possible to find some conserved quantities *without requiring Euler- Lagrange equations of motion to be satisfied* and where arbitrary functions are involved. These conservation laws are called improper laws of conservation and we can see the seeds of gauge theories hiding in them."

It's the bit in italics that worries me. Does the second theorem imply the existence, at least as a possibility, of movements that do not respect Lagrange's equations? In other words, violating the principle of stationary action? This is no doubt a naive question (I think I know that you can derive the GR field equation from the action principle), but one that I can't manage to solve on my own.
 
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  • #2
Husserliana97 said:
From what I 'understand'
From where?

Husserliana97 said:
from an article entitled "On Noether's theorems and gauge theories in Hamiltonian formulation"
Please provide a link.
 
  • #3
"From what I understand" ; just a figure of speech. By that, I simply mean that this is what I thought I understood about the second theorem, a very approximate understanding based on Wikipedia and the aforementioned article (which is way too technical for me yet).

As for the article itself :
 
  • #4
Husserliana97 said:
The article
Please provide a link to where this article is available online. We cannot allow attachments.
 
  • #5
Husserliana97 said:
"From what I understand" ; just a figure of speech. By that, I simply mean that this is what I thought I understood about the second theorem, a very approximate understanding based on Wikipedia
Then please give a link to the Wikipedia article.
 
  • #8
Noether's famous paper is really remarkable. It's far more complete than most treatments in modern textbooks and way ahead of its time. In modern lingo, the 1st Noether theorem is about "global symmetries", i.e., true symmetries in physics. Take, e.g., the rotational symmetry of space in Newtonian and special-relativistic physics (the latter for inertial observers). This implies a true symmetry all natural laws, described within these spacetime models, must obey, i.e., symmetry under rotations. This becomes most clear, if formulated in terms of "active transformations", which means that you physically built a new experiment from another one by just rotating the entire setup. This must be a symmetry, i.e., all phenomena must be identical in both experiments. In this sense it's a true symmetry constraint on the physical laws.

The 2nd theorem deals with what we today call "gauge symmetries". They are not "true symmetries" in the sense above, i.e., they do not describe the equivalence of different physical setup of experiments but they describe an redundancy in our description of Nature. E.g., all physical laws must be independent on the choice of arbitrary coordinates, and indeed you can describe any theory such that this is true. E.g., you can describe Newtonian mechanics using arbitrary configuration variables, ##q##, using the Hamilton principle of least action, and the Euler-Lagrange equations look the same, no matter which ##q## you use, but that's not a symmetry constraint on the physics, i.e., it doesn't in any way tell you, how the Lagrange function must look like to be compatible with the "true symmetries of Nature". That's not given by this general diffeomorphism group but by the (global) symmetries of the theory. Here it's the Galilei symmetry of Newtonian spacetime.

In general relativity, which is a field theory, the general covariance ("diffeomorphism invariance") is a gauge symmetry too. It implies that the equations of motion (the Einstein equations of GR) for the pseudometric components ##g_{\mu \nu}(x)## are only unique up to an arbitrary coordinate transformation, but that does not make the theory incomplete since by construction the only physically meaningful quantities are such quantities that don't change under such general coordinate transformations, i.e., scalars, vectors, and tensors.

Another way to derive GR is to make global Poincare symmetry local, i.e., you demand that the physical laws are not only invariant under global Poincare transformations, which is the symmetry of Minkowski space, i.e., special-relativistic spacetime, and which leads to constraints on how the equations in special-relativistic physics should look like, but under local Poincare transformations. Physicswise this means that there are only local inertial frames anymore, and if there is no global frame you end up with a curved spacetime, and this geometry of spacetime describes gravity as "geometrodynamics".

Another example for a gauge theory is classical electrodynamics, when described in terms of the electromagnetic potentials. The equations of motion here are the Maxwell equations, and they determine the electromagnetic potentials only up to a gauge transformation, but that doesn't matter for the physical quantities, which are invariant under such gauge transformations, i.e., if you have found a solution ##\Phi##, ##\vec{A}## for the potentials then the very same situation is described by ##\Phi'=\Phi+\partial_t \chi##, ##\vec{A}'=\vec{A}-\vec{\nabla} \chi## for an arbitrary scalar field ##\chi##. All to make your theory consistent is that the electric charge must be strictly conserved, and this conservation law follows from the Maxwell equations as a constraint, without the need to use the additional equations of motion that describe the motion of the charged matter, although these have to be consistent with charge conservation too. That's an example of Noether's theorem. The charge conservation follows from the equations for the em. fields alone, i.e., impose a consistency constraint to make the theory gauge invariant.

A nice review about the different meaning of global symmetries as true symmetries of the physical laws and gauge symmetries in field theory, see

https://doi.org/10.2172/6129984

A preprint (which should be free to download) can be found here:
https://www.osti.gov/servlets/purl/6129984
 
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