81: Quantum Theory
Quantum Theory: study of solutions of the Schrödinger (differential) equation! Also includes a good deal of Lie group theory and quantum group theory, theory of distributions and topics from Functional analysis, Yang-Mills problems, Feynman diagrams, and so on.
See the article on Quantum theory at St Andrews.
This is among the largest areas in the Math Reviews database; 81T (Quantum field theory) is among the largest of the 3-digit areas, and 81T13, 81T30, and 81T40 are (each!) among the largest of the 5-digit areas.
During 1980-1990 this field was subdivided a little differently, although many of the parts of the earlier and present system correspond (as is corroborated by the diagram): the old 81B is roughly the present 81P, 81C=81Q, 81D=81S, 81E=81T, 81F=81U, and 81G is roughly 81V; the section 81R is distinctly new, and the old sections 81H through 81N (primarily applications of quantum theory to other parts of physics) were dropped.
(Prior to 1980, the classification simply mirrored the entire MSC, categorizing papers according the mathematical tools involved.)
Browse all (old) classifications for this area at the AMS.
Brydges, David C.: "What is a quantum field theory?", Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 31--40. MR84a:81023
Ramond, Pierre: "Field theory: a modern primer", Second edition. Frontiers in Physics, 74. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990. 329 pp. ISBN 0-201-54611-6 MR92b:81001 (First edition MR84h:81077)
Polchinski, Joseph: "What is string theory?", Géométries fluctuantes en mécanique statistique et en théorie des champs (Les Houches, 1994), 287--422; North-Holland, Amsterdam, 1996. MR98b:81212
Glimm, James; Jaffe, Arthur: "What is renormalization?", Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), pp. 401--411; Amer. Math. Soc., Providence, R.I., 1973. MR49#11992
Kastler, D.: "State of the art of Alain Connes' version of the standard model", Quantum and non-commutative analysis (Kyoto, 1992), 53--71, Math. Phys. Stud., 16, Kluwer Acad. Publ., Dordrecht, 1993. MR95d:81155
Rovelli, Carlo: "What is a gauge transformation in quantum mechanics?" Phys. Rev. Lett. 80 (1998), no. 21, 4613--4616. MR99c:81126
Doebner, H.-D.; Hennig, J. D.; Lücke, W.: "Mathematical guide to quantum groups", Quantum groups (Clausthal, 1989), 29--63, Lecture Notes in Phys., 370; Springer, Berlin, 1990. MR94a:81055
Bilal, A.: "What is W-geometry?", Phys. Lett. B 249 (1990), no. 1, 56--62. MR91k:81053
Streater, R. F.; Wightman, A. S.: "PCT, spin and statistics, and all that", Reprint of the second edition. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. 207 pp. ISBN 0-201-09410-X MR91d:81055 (1st edition, MR28#4807)
Quantum Chemistry software.
General Atomic and Molecular Electronic Structure System (GAMESS)