Converse of Noether's theorem?
David Feder
In a recent lecture on QED Noether's theorem was discussed. This
theorem, as far as I could tell, states that if the Lagrangian is
invariant under certain transformations, then something is conserved
(i.e. invariance of the Lagrangian under a spatial rotation implies
conservation of angular momentum etc.) It was pointed out that the
converse has not been proven, though there is no physical evidence to
disprove it either. I was wondering if there have been any attempts to
prove the converse of Noether's theorem, in general, or whether there is
some reason why this would be impossible. It strikes me as odd that it
wouldn't work both ways!
David Feder (fe...@physun.physics.mcmaster.ca)
Daniel E. Platt
It never occured to me that it wouldn't work both ways. The reason I
thought it did work both ways is that if you wanted to express conservation
of some quantity, how would you write it in terms of a Lagrangian, except
through Noether's theorm? Also, when people wish to conserve something,
they tend to construct a field with an internal symmetry whose Lagrangian
conserves the quantity -- essentially using Noether's theorem backwards
as a rule of construction. Is this incorrect?
Dan
--
-------------------------------------------------------------------------------
Daniel E. Platt pl...@watson.ibm.com
The views expressed here do not necessarily reflect those of my employer.
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Sean Merritt
At a more fundemental level if you consider the Action Integral and
look to see if it is(isn't) invariant as Noether did, that may be better
place to start.
Let .
A = Int[t - t']dt { L (q,q,t) }
perform the variation on this.
The problem is that if L is explicilty L(t) there may not be
a kinosthenic variable, a coordinate q that has a momenta
p in the Lagrangian but does not appear itself
(ie d/dq(L) = 0, the d's are partials), as is the case with
the conservation of angular momentum.
If you could show that absence of kinosthenic variables
precludes conservation laws, that would be suggestive
although not definitive.
-sjm
--
Sean J. Merritt | "Every revolt is a cry of innocence
Dept of Physics Boston University| and an appeal to the essence of being."
mer...@macro.bu.edu | Albert Camus, The Rebel
John C. Baez
>
The converse to Noether's theorem has indeed been studied. It's easier
to formulate the precise converse in terms of the Hamiltonian approach
than in the Lagrangian approach. Noether's theorem in the Hamiltonian
approach states that a quantity F is conserved under the time evolution
generated by the Hamiltonian H if F generates a one-parameter group of
symmetries preserving the Hamiltonian H. In fact the "if" is an "if and
only if" so one has a converse. This converse is easy to prove, since
one simply switches the letter F and H in the sentence:
The quantity F is conserved under the one-parameter group
generated by the quantity H if F generates a one-parameter group of
symmetries preserving H
to get the converse! I.e., the Hamiltonian formulation of Noether's
theorem reveals that this theorem IS ITS OWN CONVERSE! Pretty cool, I'd
say.
John C. Baez
>In article <1993Jan15.1...@mcshub.dcss.mcmaster.ca>, fe...@physun.physics.mcmaster.ca (David Feder) writes:
>|>
>|> In a recent lecture on QED Noether's theorem was discussed. This
>|> theorem, as far as I could tell, states that if the Lagrangian is
>|> invariant under certain transformations, then something is conserved
>|> (i.e. invariance of the Lagrangian under a spatial rotation implies
>|> conservation of angular momentum etc.) It was pointed out that the
>|> converse has not been proven, though there is no physical evidence to
>|> disprove it either.
>It never occured to me that it wouldn't work both ways. The reason I
>thought it did work both ways is that if you wanted to express conservation
>of some quantity, how would you write it in terms of a Lagrangian, except
>through Noether's theorm?
Of course it's true that the nicest way to find conserved quantities is
via Noether's theorem, but that's no proof that it's the *only* way. To
supplement my previous post I guess I had better add that there ARE
conserved quantities that are not dervied from Noether's theorem. These
include things like parity (from discrete symmeties in quantum theory!)
and things like topological charges (in both quantum and classical field
theory). Topological charges are simply conserved quantities that at the
classical level are constant on each component of phase space. That's
how they manage to make an end run around the "converse of Noether's
theorem" that I stated in my previous post - without actually
contradicting it!
The theme of symmetries and conserved quantities is very important in
theoretical physics, and people have found some interesting extra twists
in the last couple of decades: spontaneous symmetry breaking,
topological charges, and anomalies, for example. So this stuff, while
very basic in a sense, still has life to it.
David Feder
I would have agreed with you completely, but I was recently reading
Golstein (1980), p. 594, where he finds (at least) one exception to the
statement that the converse is true - he cites as an example the
sine-Gordon and Korteweg-deVries equations, the latter describing fields
with soliton solutions.
Since in this case there are an infinite number of conserved quantities,
does this then imply an infinite number of symmetries in the Lagrangian?
Furthermore, both of the above-mentioned equations are non-linear; what
could be import of this to the standard interpretation of Noether's
theorem?
As he mentions himself, research in solitons (as of 1980) was in its
early stages. I should further mention that whereas Noether's theorem
makes intuitive sense in both the 'forward' and 'reverse' directions I
know of no formal proof.
David Feder (fe...@physun.physics.mcmaster.ca)
Daniel E. Platt
I have vague memories of Goldstone bosons, symmetry breaking in phi^4 vector
models, etc... Its been a decade since I've read through my Amit, and its
tended to run together in my memory. My recollection is that, if you have
a vector field with a phi^4 kind of term, there is a transition where
the average field value moves from 0 to a finite value (a la Landau's
phase transition model). If you expand the path integral terms about
that extremum, you get a contribution to the Green function that looks
like a mass where there was none before. Since there is now a non-zero
value for the expectation value of the field, the field must have some
direction, and symmetry is broken. I'm not sure I recall what happens
to the critical exponents (been too long). I have the uneasy feeling
I have when several marginally related pieces are floating around...
Its been longer since I've looked at Bjorken and Drell, which is my primary
reference to Noether's theorem. It seemed to me that they DID handle charge
conservation through Noether's theorem in that text. Is that also a
mis-remembered piece?
The other part of my post was that it seemed to me that people used
Noether's theorem to construct fields with symmetries that would
satisfy conservation laws. Is that also mistaken?
John C. Baez
>Its been longer since I've looked at Bjorken and Drell, which is my
>primary reference to Noether's theorem. It seemed to me that they DID handle
>charge conservation through Noether's theorem in that text. Is that also a
>mis-remembered piece?
No, you're right, ordinary electric charge conservation follows from
Noether's theorem. The kind of conservation law that does not follow
from Noether's theorem is conservation of *topological* charges, of the
sort that topological solitons carry. This is a post-Bjorken&Drell
development.
>The other part of my post was that it seemed to me that people used
>Noether's theorem to construct fields with symmetries that would
>satisfy conservation laws. Is that also mistaken?
No, again you're right on track. People know what they want to be
conserved and cook up Lagrangians with the appropriate symmetries to get
the conservation laws they want.