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.
I included this long introduction about Wigner although Wigner himself (as far as I know) never worked on coadjoint orbits. But all his important work (which actually covered all major areas of quantum physics) is intimately connected to coadjoint orbits. In fact coadjoint orbits may serve as a unifying principle of all the seemingly separate models on which Wigner worked: Wigner's classification of the representations of the Poincaré group is actually a classification of the Poincaré coadjoint orbits and their quantization, please see for example the following work: by Cariñena, Gracia-Bondia, Lizzi, Marmo and Vitale. Also, the projective representations that Wigner dealt with appear naturally in the coadjoint orbit picture, since, for example, an integral coadjoint orbit is a smooth projective variety (please see for example the following article by: Schlichenmaier. Also, the representations in the Inönu-Wigner contractions are also at least partially) related to quantizations of coadjoint orbits, please see the following two articles by Benjamin Cahen.
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 post imported from StackExchange Physics at 2014-08-06 18:58 (UCT), posted by SE-user David Bar Moshe
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