nLab superselection theory

Idea

In (algebraic) quantum field theory, superselection theory is concerned with identifying superselection sectors: Namely conservation laws imply that in an isolated system the total value of certain charges (e.g. electric charges, magnetic charges) cannot change due to any interactions taking place in the system, and a superselection sector is specified by prescribing a number to each of these charges, it is the space of all quantum states of the system with these values for the charges.

There are two complementary viewpoints about superselection sectors:

In the picture of Wick, Wigner & Wightman 1952, all possible quantum states of a system form one big Hilbert space and superselection sectors are subspaces of this Hilbert space, and no measurement, that is no observable, can map a state in one superselection sector to another. Further, superpositions of states in different sectors do not exist in physical reality.

Mathematically speaking this means that superselection sectors are the irreducible representations of a given algebra of observables inside a possibly larger space of quantum states.

An example would be a superposition of a state containing one electron with a state containing two electrons.

In the AQFT approach to quantum field theory the observables of the theory are self-adjoint elements of a $C^\ast$-algebra algebra of quantum observables. A concrete physical system is a state of this algebra, which is accompanied, via the GNS construction, with a representation of the algebra. A superselection sector from this viewpoint is an equivalence class of unitarily equivalent representations, see at representation of a C-star algebra.

Examples

Usually, in AQFT not all representations of the algebra of observables are considered to be physically relevant, so that superselection theory starts with the statement of conditions that representations have to fulfill in order to be admissible, and only those are considered. Example:

• DHR superselection theory

Related concepts

• quantum parallelism

• quantum superposition

• quantum interference

• entanglement

References

The notion originates (in a context similar to the Wigner classification of fundamental particles) in:

• Gian-Carlo Wick, Eugene Wigner, Arthur Wightman: The intrinsic parity of elementary particles, Phys. Rev. 88 (1952) 101-105 [doi;10.1103/PhysRev.88.101, pdf]

Brief early survey:

• Raymond F. Streater, section 2.3 of: Outline of axiomatic relativistic quantum field theory, Rep. Prog. Phys. 38 (1975) 771 [doi:10.1088/0034-4885/38/7/001]

Lecture notes:

• Klaus Fredenhagen: Superselection Sectors, lecture notes, Hamburg (1994/5) [pdf, pdf]

Seminar notes:

• Tadashi Fujimoto: Outline of superselectionrule on Algebraic Quantum Field Theory (2021) [pdf]

See also at AQFT, QFT and Haag-Kastler axioms and DHR superselection theory.

References that make explicit the identification of superselection sections with irreducible representations of the algebra of quantum observables:

• John E. Roberts, p. 110 (5 of 14) in: Local cohomology and superselection structure, Comm. Math. Phys. 51 2 (1976) 107-119 [euclid:cmp/1103900345]

• Jürg Fröhlich, Fabrizio Gabbiani, Pieralberto Marchetti, below Def. 2.1, p 16 of: Braid statistics in three-dimensional local quantum field theory, in: H.C. Lee (ed.) Physics, Geometry and Topology, NATO ASI Series 238, Springer (1990) [doi:10.1007/978-1-4615-3802-8_2, pdf]

  (in a context of braid group statistics)

• Klaas Landsman, p. 48 (4 of 28): Quantization and Superselection Sectors I. Transformation Group $C^\ast$ Algebras, Reviews in Mathematical Physics 02 01 (1990) 45-72 [doi:10.1142/S0129055X9000003X, pdf]

• Klaas Landsman, p. 73-4 (1-2 of 32) in: Quantization and Superselection Sectors II. Dirac Monopole and Aharonov-Bohm effect, Reviews in Mathematical Physics 02 01 (1990) 73-104 [doi:10.1142/S0129055X90000041, pdf]

• David John Baker, p. 273 (12 of 24): Identity, Superselection Theory, and the Statistical Properties of Quantum Fields, Philosophy of Science 80 2 (2013) 262-285 [doi:10.1086/670296]

Discussion via vertex operator algebras:

• Haisheng Li: The Physics Superselection Principle in Vertex Operator Algebra Theory, Journal of Algebra 196* (1997) 436-457 [doi;10.1006/jabr.1997.7126]

Discussion in the context of holographic entanglement entropy:

• Horacio Casini, Marina Huerta, Javier M. Magan, Diego Pontello, Entanglement entropy and superselection sectors I. Global symmetries [arXiv:1905.10487]

In the context of the Aharonov-Bohm effect:

• Claudio Dappiaggi, Giuseppe Ruzzi, Ezio Vasselli: Aharonov-Bohm superselection sectors, Lett Math Phys 110 (2020) 3243–3278 [doi;10.1007/s11005-020-01335-4, arXiv:, 1912.05297]

Discussion in relation to asymptotic symmetries of gauge theories:

• A. P. Balachandran, V. P. Nair, A. Pinzul, A. F. Reyes-Lega, S. Vaidya: Superselection, boundary algebras, and duality in gauge theories, Phys. Rev. D 106 (2022) 025001 [doi:10.1103/PhysRevD.106.025001, arXiv:2112.08631]