Wigner-Weyl isomorphism for quantum mechanics on Lie groups

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My profile My library Metrics Alerts. Sign in. INFN -- Italy. Articles Cited by. Journal of High Energy Physics 08 , , Journal of mathematical physics 46 1 , , Modern Physics Letters A 18 33n35 , , Journal of High Energy Physics 09 , , Reviews in Mathematical Physics 18 08 , , Journal of Geometry and Physics 56 4 , , Journal of Physics: Conference Series 53 1 , , Communications in mathematical physics 1 , , Reviews in Mathematical Physics 23 06 , , Reviews in Mathematical Physics 25 05 , , Quantum Groups and Noncommutative Spaces, , Journal of Geometry and Physics 62 7 , , Letters in Mathematical Physics 2 , , Reports on Mathematical Physics 68 3 , , And that is in the classical theory as well.

Lie algebra in physics

But it's not as physically clear from the beginning from that point of view. The current framework for quantum mechanics motivates the study of projective anti unitary representations of the symmetry groups of the physical system.

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One can get around this theorem by considering not Lie groups but Lie supergroups, but that's another story. It was Mackey who extended Wigner's method and placed it firmly in the "right" mathematical context. Let us start with finite groups. They factor through faithful representations of a group known as Wigner's little group. But there's more.

And the nice surprise is that these partial differential equations are precisely the linearised free field equations for the corresponding particles: the Klein-Gordon, Dirac, Weyl, Maxwell, My understanding is that this started with the Dirac wave equation.

Moyal Formulation of Quantum Mechanics

This was a relativistic equation for an electron. However it happened to also introduce the idea that a point particle could have an internal state space. This was a successful theory and was taken up and imitated when it came to probing the structure of the nucleus. In particular, it gives plenty of references and historical notes. As a physicist, I may be able to give a different perspective on this question. In particular, many of the responses so far have been about quantum mechanics and quantum field theory which involve Lie groups , but if the question is, "Why is the particle content of physics theories derived from Lie groups?

It's about their geometry, which can be discussed separately from quantum effects. In , Kaluza and Klein attempted to unify electromagnetism with gravity by proposing a theory of General Relativity with 5 dimensions 4 spatial. Since we don't macroscopically experience this extra degree of freedom, they proposed that it is topologically like a cylinder with a small radius, so small that the extra degree of freedom can't be probed as a direction.

This degree of freedom does, however, allow us to encode classical electromagnetism as part of the geometry of space-time. We'll see in a moment that while this formulation isn't exactly right, it does show how the differential geometry concepts of General Relativity can be used in particle physics theories, leading to a unification of all four forces at a classical level. It's the quantization of gravity that's the hard part. The Dirac Lagrangian.

This difference is perhaps related to the reason Kaluza and Klein's original theory didn't work? That is, the classical photon field is the "curvature" of the fiber bundle in the same sense that gravitation is the curvature of space-time.

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Moreover, this picture unifying the geometry of electromagnetism with the geometry of gravity also works for all the other known forces. The Yang-Mills idea wasn't popular at first because it didn't seem to describe the nuclear strong force but that was based on a wrong assumption that the nuclear force is a Yukawa interaction. I'm skipping over many important contributions for brevity. By the mid's or early 's, depending on who I ask, this became known as the Standard Model of particle physics because of its experimental success.

The structures of the groups are directly responsible for the charges and interactions of the matter fields quarks and leptons , but the matter fields themselves are not derived from the groups supersymmetry might change that part of the picture. There is a direct analogy between these group connections the gluon, W, Z, and photon and the space-time connection in General Relativity which we could call a graviton field, if you wish.

I have said nothing at this point about the quantization of all of these fields, which further complicates the picture, especially in the case of gravity! By the way, I would love to know more about the curvature of fiber bundles, in order to understand the above at a deeper mathematical level.

On Wigner Functions for NC Quantum mechanics:a group representation based construction.

If you have any suggested reading, I'm interested. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 9 years, 2 months ago. Active 9 years, 2 months ago. Viewed 5k times. Makhalan Duff Makhalan Duff 2, 29 29 silver badges 71 71 bronze badges.

They are, in fact, very much like the gauged symmetries! That's how the gauged symmetries were discovered!! If you don't believe me, check out any of the early literature on the topic, or any of the technical histories of physics books out there, or any intro nuclear physics text they rely heavily on this! The gauged case is obviously more complicated, but spin in quantum mechanics is where these ideas came from , and not realizing that is missing the quantum mechanical point of view of gauge theories entirely!

Many physicists aren't even aware of the gauge geometry And conserved charges don't directly have anything to do with representations. A conserved charge comes from a symmetry of a PDE, there are no representations that need to be involved here. The representations are what show up very clearly in the quantum mechanical picture of how states transform.

For an explicit example of this, see any intro nuclear physics text, which all rely heavily on this. Induced representations Let us start with finite groups.


Wigner-Weyl isomorphism for quantum mechanics on Lie groups - Publications of the IAS Fellows

For simple Lie groups, it's fine just to work with the universal cover, but I like to emphasize this because that's where the central extension arises in the affine case. This is reflected in various quantization conditions Bohr-Sommerfeld, Maslov index, Lere, etc. In fact, in an earlier draft I did mention Bergmann's and Wigner's theorems on projective and anti-unitary reps, but decided in the end to cut the post in size.

I'll edit, though, for the sake of precision. Thanks for keeping me honest. This is a different Clifford to Clifford algebras.