Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons. (English) Zbl 0558.22018

Howe, Roger

Applications of group theory in physics and mathematical physics, Proc. Summer Semin., Chicago 1982, Lect. Appl. Math. 21, 179-207 (1985).

[For the entire collection see Zbl 0547.00050.]
This is a very nicely written review on canonical commutation relations, boson algebra, Heisenberg algebra, symplectic group and their special representations, all these here unified under the concept of dual pairs of subgroups (G,G’) of a group S, where G’ is the centralizer of G in S, and vice versa. The unification of so many concepts of interest to both the mathematicians and physicists is remarkable. The physical applications include massless particles, wave equations, Maxwell and Dirac equations, singletons, and dynamical groups of quantum system (the latter not treated in this review).

Reviewer: A.O.Barut

MSC:

22E70

Applications of Lie groups to the sciences; explicit representations

Keywords:

canonical commutation relations; boson algebra; Heisenberg algebra; symplectic group; dual pairs of subgroups; massless particles; wave equations; singletons

Citations:

Zbl 0547.00050