I am looking for some application of coadjoint orbits in physics. If you know some of them please let me know.

• I know that the Kirillov method of co-adjoint orbits has lots of uses in finding unitary representations. This has important applications for proving, e.g, the unitary or otherwise nature of a Lagrangian field theory as a quantum theory. Jun 2 '14 at 6:23
• I know enough orbit method, but I am looking for recent results
– user36421
Jun 2 '14 at 6:26

Although groups and their representations were already applied to quantum mechanics almost from the birth of quantum theory, their central role was recognized in its full importance by Eugene Wigner. His work on crystallography, atomic and molecular spectra, relativity (unitary representations of the Poincaré group), led him to appreciate the major importance of groups and their representations in quantum mechanics.

After his pioneering work on the Poincaré group representations, most of his work was deeply connected to groups and their representations. Already in his work on the representations of the Poincaré group, he introduced the method of induced representations. He realized that the Galilean symmetry is realized as a projective (ray) representation on the Schrödinger wave functions. Also together with Inönu he introduced the theory of group contractions (and their representations). There are many people who consider quantization and group representations two faces of the same problem.

Although they appear in other contexts in physics, coadjoint orbits can be thought as the classical phase spaces corresponding to the internal degrees of freedom of quantum particles such as spin, flavor, color etc. This picture allows the treatment of the translational degrees of freedom whose corresponding phase spaces are cotangent bundles and the internal degrees of freedom on the same footing. One important application in which both degrees of freedom coexist and interact is the Wong equations which generalize the Lorentz equation for a particle with a nonAbelian charge such as color:

$$\frac{dx^i}{dt} = p^i$$

$$\frac{dp^i}{dt} = F^a_{ij}(x)T_a(y)p^i$$

$$\frac{dT_a}{dt} = -f^c_{ab}A^b_j(x)T_c(y)$$

Where $A^b_j$ is the Yang-Mills vector potential $F^a_{ij}$ the corresponding field strength, $x^i$ and $p ^i$, the position and momentum coordinates $T_a(y)$ are the Hamiltonian functions on the coadjoint orbits representing the nonAbelian charges and $f^c_{ab}$ the structure constants, and $y$ are the coordinates on the coadjoint orbits. For a deeper discussion please see the following thesis by: Rainer Glaser.

Quantization of coadjoint orbits leads to unitary representations of the corresponding groups. The representations are usually realized as reproducing kernel Hilbert spaces of sections of line bundles. These representations are realized as coherent state representation (please see for example the following article by: Boya, Perelomov and Santander) which makes them especially suited for semiclassical analysis. (please see again the Rainer Glaser thesis).

All coadjoint orbits of compact semsimple Lie groups and some of the coadjoint orbits of the noncompact groups are Kähler. The quantization of these orbits can be achieved by means of the Berezin Toeplitz quantization, please see the following review by Schlichenmaier. Also being Kähler and homogeneous makes these coadjoint orbits accessible to explicit work. Please see also the following article by: Bernatska, and Holod for examples of actual explicit work on semisimple coadjoint orbits. It is important to mention that in order to be able to quantize a compact coadjoint orbit, it needs to be integral, i.e., the flux of its symplectic form through 2-cycles must be quantized. This is the Dirac quantization condition.

The most elementary coadjoint orbit is the two-sphere. Its quantization leads to the theory of spin angular momentum, please see for example the original work by Berezin. Spin systems, which constitute of important models in the theory of magnetism, can be studied using generalizations of these ideas. Please see for example the following article by: Bykov.

The representations associated with integral coadjoint orbits can be obtained as zero modes of a Landau problem of a particle moving on the coadjoint orbit in a magnetic field equal to the symplectic form. The quantization Hilbert space is obtained as the (degenerate) space of the lowest Landau level. Please see, for example, the following lecture notes.

There is an important further application to Yang-Mills and Chern-Simons type of theories called the nonAbelian Stokes theorem in which a Wilson loop can be expressed as a Feynman-path integral over loops on a coadjoint orbit which may be given heuristically as:

$$tr_{\mathcal{H}}T\{exp(i\oint A^{a}(t) T_a)\}= \mathrm{lim}_{m\rightarrow \infty}\int exp\big (i\int _0^T \alpha^{\mathcal{H}}_i\dot{z}^i - \bar{\alpha}^{\mathcal{H}}_i \dot{\bar{z}}^i + \frac{m}{2}g_{i\bar{j}}\dot{z}^i \dot{\bar{z}}^j+A^{a} (t)T^{\mathcal{H}}_a(z, \bar{z})\big) \mathcal{D}z\mathcal{D}\bar{z}$$

Where $\alpha^{\mathcal{H}}$ is the symplectic potential of the coadjoint orbit corresponding to the representation $\mathcal{H}$ (via the Borel-Weil-Bott theorem). The symbol $T_a$ is used for a Lie algebra element in the Wilson loop and also for the corresponding Hamiltonian function inside the path integral. $z$ are the coordinates of the coadjoint orbit. The action describes a particle in a magnetic field which has a distributed charge density as the Hamiltonian function. The limit $m\rightarrow \infty$ is taken to dominate the lowest Landau level over the path integral. This representation has been used for the insertion of Wilson loops in the QCD Path integrals (in the study of confinement) and also the insertion of Wilson loops in the Cher-Simons theory leading to the Jones polynomials.

There are explicitly known classifications of some cases of infinite dimensional coadjoint orbits, mainly, those relevant to string theory. A complete classification of the coadjont orboits of the orientation preserving diffeomorphism group of the circle $Diff^{+}(S^1)$ are given by: Jialing and Pickrell. The classification of coadjoint orbits of loop groups is also known, please see the following lecture notes by Khesin and Wendt.

Coadjoint orbits appear in many more areas and applications in physics. The applications mentioned above are may be the most known in my personal point of view.

• This is an excellent answer, +1. As a side note, usually one does not have to resort to using the fully sophisticated co-adjoint orbits for compact groups since the Peter-Weyl theorem is available. Jun 2 '14 at 23:07

The Wilson loop observables inside 3d Chern-Simons gauge field theory are secretly themselves the quantization of a 1d field theory in terms of coadjoint orbits.

This possibly still surprising-sounding statement was hinted at already on p. 22 of the seminal

• Edward Witten, Quantum Field Theory and the Jones Polynomial Commun. Math. Phys. 121 (3) (1989) 351–399. MR0990772 (project EUCLID)

A detailed discussion of how this works is in section 4 of

• Chris Beasley, Localization for Wilson Loops in Chern-Simons Theory, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) Chern-Simons Gauge Theory: 20 Years After, , AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. (arXiv:0911.2687)

following

• S. Elitzur, Greg Moore, A. Schwimmer, and Nathan Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108–134.

The idea is indicated on the nLab here.

As also discussed there, the statement that there is a coadjoint orbit 1d quantum field theory sort of "inside" 3d Chern-Simons theory has a nice interpretation from a point of view of extended quantum field theory. This we have discussed in section 3.4.5 of

So given the ubiquity of Chern-Simons theory in QFT, and the fact that much of what is interesting about it is encoded in its Wilson loop observables, this means that quantization of coadjoint orbits plays a similarly important role. For instance given that all of rational 2d conformal field theory is dually encoded, via the FRS theorem, by 3d Chern-Simons theory in such a way that CFT field insertions are mapped to the CS Wilson loops, this means that quantized coadjoint orbits are at work behind the scenes in much of 2d CFT.

I may add a few recent works that use coadjoint orbits to better understand the space of solutions of 2+1 gravity. With a cosmological constant you get Virasoro group coadjoint orbits, and without a cosmological constant you get BMS$_3$ coadjoint orbits. These are the symmetries of the spaces of solutions of the corresponding gravitational theories. The works I am talking about are (to list a few):

• arXiv:1403.3835, arXiv:1403.5803, arXiv:1502.00010, arXiv:1502.03108
• arXiv:1403.3367

The first line is about flat space, the second line is about the case with a negative cosmological constant (a work of mine with M. Leston). There should be more works in physics using coadjoint orbits, but these are the ones I know of and that are recent.

I am going to run the risk of having someone else contradict me. My short answer is

No.

There are no longer any applications of co-adjoint orbits to Physics. The topic of co-adjoint orbits belongs to mathematical physics, which is not real Physics.

The work started by Souriau and Kostant on geometric quantisation (I should also mention Michelle Vergne and her students) is about quantisation, which is how to produce a Quantum Mechanical system from a given Classical System. This was interesting back in the 30's, but no longer has any use. What is of interest now are the Quantum Systems which have no Classical analogue, and so cannot be obtained by quantisation at all.

That said, I will eat my words if someone shows me a plausible route from co-adjoint orbits to a completely novel Quantum Field Theory that does not need renormalisation. I have not heard of anything remotely like that, but if there were such an impossible thing, that could be interesting.