2018 brings with it the centenary of a major milestone in mathematical physics: the publication of Amalie (“Emmy”) Noether’s theorems relating symmetry and physical quantities, which continue to be a font of inspiration for “symmetry arguments” in physics, and for the interpretation of symmetry within philosophy.

In order to celebrate Noether’s legacy, the University of Notre Dame and the LSE Centre for Philosophy of Natural and Social Sciences are co-organizing a conference that will bring together leading mathematicians, physicists, and philosophers of physics in order to discuss the enduring impact of Noether’s work.

There’s a registration fee, which you can see on the conference website, along with a map showing the conference location, a schedule of the talks, and other useful stuff.

I’m looking forward to analyzing the basic assumptions behind various generalizations of Noether’s first theorem, the one that shows symmetries of a Lagrangian give conserved quantities. Having generalized it to Markov processes, I know there’s a lot more to what’s going on here than just the wonders of Lagrangian mechanics:

I’ve been trying to get to the bottom of it ever since.

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4 Responses to The Philosophy and Physics of Noether’s Theorems

It should be interesting to deepen the geometrical structures of this fabulous Noether Theorem,

Jean-Marie Souriau gave a first tentative of geometrization of Noether Theorem with the invention of the “Moment map” in his book published in 1969 “Structure of Dynamical Systems”: https://link.springer.com/book/10.1007/978-1-4612-0281-3
Souriau gave it the name of “Symplectic Noether Theorem”.
More details are given in Professor Yvette Kosmann-Schwarzbach Book: “Les théorèmes de Noether: Invariances et lois de conservations au XXième siècle.

The components of the Moment map are the invariants of the Noether Theorem. The Souriau moment map gives a geometrical meaning to Noether Invariants, replacing pure algebraic approach by a cohomological one.

Another issue is the link with Cohomology, and the non-equivariant case and the action of the group on the moment map and dual Lie Algebra, with the notion of (Souriau) cocycle. This case is also related to Souriau “Lie Groups Themodynamics” providing a covariant Gibbs density.

As explained by Thomas Delzant at 2010 CIRM conference “Action Hamiltoniennes: invariants et classification”, organized with Michel Brion:
[1] Delzant T., Wacheux C., Actions Hamiltonniennes, vol.1 n°1, pp.23-31, Rencontres du CIRM « Action Hamiltoniennes : invariants et classification, organisé par Michel Brion et Thomas Delzant, CIRM, Luminy, 2010
We can read in this paper“The definition of the moment map is due to Jean-Marie Souriau…. In the book of Souriau, we find a proof of the proposition: the map J is equivariant for an affine action of G on g* whose linear part is Ad *…. In Souriau’s book, we can also find a study of the non-equivariant case and its applications to classical and quantum mechanics. In the case of the Galileo group operating in the phase space of space-time, obstruction to equivariance (a class of cohomology) is interpreted as the inert mass of the object under study”. We can uniquely define the moment map up to an additive constant of integration, that can always be chosen to make the moment map equivariant (a moment map is G-equivariant, when G acts on g∗ via the coadjoint action) if the group is compact or semi-simple. In 1969, Souriau has considered the non-equivariant case where the coadjoint action must be modified to make the map equivariant by a 1-cocycle on the group with values in dual Lie algebra g∗.

This approach has also been developed by André Lichnerowicz in the paper:
[2] Lichnerowicz A., Medina A., On Lie groups with left-invariant symplectic or Kählerian structures, Letters in Mathematical Physics, Volume 16, Issue 3, pp 225–235, October 1988

or in the Books of Jean-Louis Koszul:
[3] Koszul, J.L., Introduction to Symplectic Geometry, Science Press, Beijing, 1986 (Chinese), translated by SPRINGER in 2018 with foreword by Charles-Michel Marle

and the Book of Paulette Libermann and Charles-Michel Marle:
[4] Paulette Libermann and Charles-Michel Marle, Géométrie symplectique. Bases théoriques de la mécanique (Symplectic Geometry and Analytic Mechanics. Reidel, Boston 1987).

I know about the moment map, since I took symplectic geometry in grad school from Victor Guillemin, who was very excited about the moment map. And I’ve also been fascinated by how mass is a 2-cocycle on the Galilei group: that story was retold in Guillemin and Sternberg’s book Symplectic Techniques in Mathematical Physics.

But I would like to better understand the history of geometric quantization… and there’s always more math to learn, too! So your links will be useful.

(I know many of the Americans involved in geometric quantization: my thesis advisor Irving Segal at MIT who helped get this theory started, his student Bertram Kostant who was also teaching at MIT when I was a student there, Victor Guillemin at MIT, Shlomo Sternberg at Harvard, and Alan Weinstein at Berkeley. I never met Marsden, alas. And I’m much less familiar with the French side of the story: of course I know about Souriau and Lichnerowicz and Koszul, but I never met any of them. I’ve met Yvette Kosmann-Schwarzbach a couple of times in Paris, and I’ll see her again at this conference!)

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It is too late to submit posters. Deadline: 1 August.

Whoops. Fixed.

It should be interesting to deepen the geometrical structures of this fabulous Noether Theorem,

Jean-Marie Souriau gave a first tentative of geometrization of Noether Theorem with the invention of the “Moment map” in his book published in 1969 “Structure of Dynamical Systems”:

https://link.springer.com/book/10.1007/978-1-4612-0281-3

Souriau gave it the name of “Symplectic Noether Theorem”.

More details are given in Professor Yvette Kosmann-Schwarzbach Book: “Les théorèmes de Noether: Invariances et lois de conservations au XXième siècle.

The components of the Moment map are the invariants of the Noether Theorem. The Souriau moment map gives a geometrical meaning to Noether Invariants, replacing pure algebraic approach by a cohomological one.

Another issue is the link with Cohomology, and the non-equivariant case and the action of the group on the moment map and dual Lie Algebra, with the notion of (Souriau) cocycle. This case is also related to Souriau “Lie Groups Themodynamics” providing a covariant Gibbs density.

As explained by Thomas Delzant at 2010 CIRM conference “Action Hamiltoniennes: invariants et classification”, organized with Michel Brion:

[1] Delzant T., Wacheux C., Actions Hamiltonniennes, vol.1 n°1, pp.23-31, Rencontres du CIRM « Action Hamiltoniennes : invariants et classification, organisé par Michel Brion et Thomas Delzant, CIRM, Luminy, 2010

We can read in this paper“The definition of the moment map is due to Jean-Marie Souriau…. In the book of Souriau, we find a proof of the proposition: the map J is equivariant for an affine action of G on g* whose linear part is Ad *…. In Souriau’s book, we can also find a study of the non-equivariant case and its applications to classical and quantum mechanics. In the case of the Galileo group operating in the phase space of space-time, obstruction to equivariance (a class of cohomology) is interpreted as the inert mass of the object under study”. We can uniquely define the moment map up to an additive constant of integration, that can always be chosen to make the moment map equivariant (a moment map is G-equivariant, when G acts on g∗ via the coadjoint action) if the group is compact or semi-simple. In 1969, Souriau has considered the non-equivariant case where the coadjoint action must be modified to make the map equivariant by a 1-cocycle on the group with values in dual Lie algebra g∗.

This approach has also been developed by André Lichnerowicz in the paper:

[2] Lichnerowicz A., Medina A., On Lie groups with left-invariant symplectic or Kählerian structures, Letters in Mathematical Physics, Volume 16, Issue 3, pp 225–235, October 1988

or in the Books of Jean-Louis Koszul:

[3] Koszul, J.L., Introduction to Symplectic Geometry, Science Press, Beijing, 1986 (Chinese), translated by SPRINGER in 2018 with foreword by Charles-Michel Marle

and the Book of Paulette Libermann and Charles-Michel Marle:

[4] Paulette Libermann and Charles-Michel Marle, Géométrie symplectique. Bases théoriques de la mécanique (Symplectic Geometry and Analytic Mechanics. Reidel, Boston 1987).

Thanks for all those nice links!

I know about the moment map, since I took symplectic geometry in grad school from Victor Guillemin, who was very excited about the moment map. And I’ve also been fascinated by how mass is a 2-cocycle on the Galilei group: that story was retold in Guillemin and Sternberg’s book

Symplectic Techniques in Mathematical Physics.But I would like to better understand the history of geometric quantization… and there’s always more math to learn, too! So your links will be useful.

(I know many of the Americans involved in geometric quantization: my thesis advisor Irving Segal at MIT who helped get this theory started, his student Bertram Kostant who was also teaching at MIT when I was a student there, Victor Guillemin at MIT, Shlomo Sternberg at Harvard, and Alan Weinstein at Berkeley. I never met Marsden, alas. And I’m much less familiar with the French side of the story: of course I

know aboutSouriau and Lichnerowicz and Koszul, but I never met any of them. I’ve met Yvette Kosmann-Schwarzbach a couple of times in Paris, and I’ll see her again at this conference!)