algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
The Gelfand-Naimark-Segal construction (“GNS construction”) represents a state on a star-algebra over the complex numbers – which a priori is defined purely algebraically as a non-degenerate positive linear function
– by a vector in a complex Hilbert space as the “expectation value”
with respect to some star-representation
of on (a dense subspace of) ; where denotes the inner product on the Hilbert space.
Originally this was considered for C*-algebras and C*-representations (Gelfand-Naimark 43, Segal 47), see for instance (Schmüdgen 90), but the construction applies to general unital star algebras (Khavkine-Moretti 15) as well as to other coefficient rings, such as to formal power series algebras over (Bordemann-Waldmann 96).
The GNS-construction plays a central role in algebraic quantum field theory (Haag 96, Moretti 18, Khavkine-Moretti 15), where plays the role of an algebra of observables and the role of an actual state of a physical system (whence the terminology) jointly constituting the “Heisenberg picture”-perspective of quantum physics; so that the GNS-construction serves to re-construct a corresponding Hilbert space of states as in the Schrödinger picture of quantum physics. In this context the version for C*-algebras corresponds to non-perturbative quantum field theory, while the generalization to formal power series algebras corresponds to perturbative quantum field theory.
under construction
Given
a C*-algebra ;
a state
there exists
of on some Hilbert space
such that is the state corresponding to , in that
for all .
Namely, consider on the underlying complex vector space of the sesquilinear form (inner product)
Since is a “positive functional”, hence taking non-negative values, this is, in general, positive semi-definite. It becomes positive definite on the the quotient vector space
by the subspace of 0-norm elements
In fact, is a left ideal in so that the left multiplication action of on itself descends to an action on the quotient Hilbert space (1)
Therefore, the cyclic vector in question can be taken to be that represented by the unit element :
Hence on this Hilbert space , the original operator-algebraic state is now represented by the tautological density matrix
The GNS construction for -algebras is a special case of a more general construction of Ghez, Lima and Roberts applied to C*-categories (horizontal categorification of -algebras).
Let be a -category. Fix an object and let be a state on the -algebra . Then there exists a -representation
together with a cyclic vector such that for all ,
A C*-algebra is a -category with a single object , where we make the identification . In this case the theorem reduces to the classical GNS construction.
See Functorial Aspects of the GNS Representation.
The original construction for C*-algebras and C*-representations is due to:
Israel Gelfand, Mark Naimark, On the imbedding of normed rings into the ring of operators on a Hilbert space, Matematicheskii Sbornik. 12 (2): 197–217 (1943)
reprinted in:
Robert Doran (ed.), -Algebras: 1943–1993, Contemporary Mathematics 167, AMS 1994 (doi:10.1090/conm/167)
Irving Segal, Irreducible representations of operator algebras, Bull. Am. Math. Soc. 53: 73–88, (1947) (pdf, euclid)
Textbook accounts:
Gerard Murphy, Section 3.4 of: -algebras and Operator Theory, Academic Press 1990 (doi:10.1016/C2009-0-22289-6)
Konrad Schmüdgen, Section 8.3 of: Unbounded operator algebras and representation theory, Operator theory, advances and applications, vol. 37. Birkhäuser, Basel (1990) (doi:10.1007/978-3-0348-7469-4)
Kehe Zhu, Section 14 of: An Introduction to Operator Algebras, CRC Press 1993 (ISBN:9780849378751)
Richard V. Kadison, John R. Ringrose, Theorem 4.5.2 in: Fundamentals of the theory of operator algebras – Volume I: Elementary Theory, Graduate Studies in Mathematics 15, AMS 1997 (ISBN:978-0-8218-0819-1, ZMATH)
in the context of algebraic quantum field theory:
Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras, Texts and Monographs in Physics. Springer (1996).
Valter Moretti, Spectral Theory and Quantum Mechanics :Mathematical Structure of Quantum Theories, Symmetries and introduction to the Algebraic Formulation, 2nd ed. Springer Verlag, Berlin (2018)
Review with an eye towards quantum probability and entropy:
See also
For general unital star-algebras:
in relation with the classical moment problem and the notion of POVM
For formal power series algebras over :
Discussion in terms of universal properties in (higher) category theory is in
Bart Jacobs, Involutive Categories and Monoids, with a GNS-correspondence, Foundations of Physics, July 2012, Volume 42, Issue 7, pp 874–895 (arXiv:1003.4552)
Arthur Parzygnat, From observables and states to Hilbert space and back: a 2-categorical adjunction (arXiv:1609.08975)
Last revised on July 9, 2023 at 08:11:20. See the history of this page for a list of all contributions to it.