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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 81: Quantum Theory

Introduction

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.

History

See the article on Quantum theory at St Andrews.

Applications and related fields

Subfields

• 81P: Axiomatics, foundations, philosophy
• 81Q: General mathematical topics and methods in quantum theory
• 81R: Groups and algebras in quantum theory
• 81S: General quantum mechanics and problems of quantization
• 81T: Quantum field theory; related classical field theories
• 81U: Scattering theory, see also 47A40
• 81V: Applications to specific physical systems

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.

Textbooks, reference works, and tutorials

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)

Software and tables

Quantum Chemistry software.

General Atomic and Molecular Electronic Structure System (GAMESS)

Other web sites with this focus

• Preprint server at Los Alamos has hierarchies for Quantum Algebra and Topology and Quantum Physics.
• Here are the AMS and Goettingen resource pages for area 81.
• The Net Advance of Physics
• Quantum lie algebras"

Selected Topics at this site

• What are Grand Unified Theories (GUTs)?
• Formal solutions of the Schrödinger equation
• What is renormalization in mathematical physics?
• Summary: hierarchy of elementary particles (standard model)
• Sources of software for computational chemistry