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 Hermitian inner product on the Hilbert space.
Originally and typically by default this is considered for C*-algebras and C*-representations (Gelfand & Naimark 1943, Segal 1947), see for instance (Schmüdgen 1090), but the construction applies to general unital star algebras (Khavkine& Moretti 2015) as well as to other coefficient rings, such as to formal power series algebras over (Bordemann & Waldmann 1996).
The GNS-construction plays a central role in algebraic quantum field theory (cf. Haag 1996, Moretti 2017, Khavkine & Moretti 2015), 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 in the sense of
a linear map on the underlying vector space
which takes star-involution to complex conjugation
and satisfies “positivity in the sense that
there exists
of on some Hilbert space
such that is the pure state corresponding to , in that
for all .
Consider on the underlying complex vector space of the sesquilinear form (inner product)
By the “positivity”-condition (3) on , the pairing is positive semi-definite, and therefore it satisfies the Cauchy-Schwarz inequality:
However, (4) will in general not be definite, in that there may be a non-trivial linear subspace of 0-norm elements:
and hence we have to quotient out by this null-space in order to produce the desired Hilbert space.
Observe that this null space (6) is not just a linear subspace but a left ideal for the algebra structure, as follows by the Cauchy-Schwarz inequality (5):
Similarly, the Cauchy-Schwarz inequality (5) implies that any inner product with a null vector vanishes:
Therefore first of all descends to the quotient vector space
because
where in the last step we used (8), and from this:
where in the first and last step we used (2).
Now descended to the quotient, the pairing becomes positive definite by construction, so that it defines a Hermitian – by (2) – inner product on , making it the desired Hilbert space.
Finally, again by the left-ideal property (7), the left multiplication action of on itself also descends to an action on the quotient Hilbert space (9):
and so the cyclic vector in question is that represented by the unit element :
Hence on this Hilbert space , the original operator-algebraic state is now represented by the tautological density matrix
Notice in summary that this GNS construction constitutes a kind of operator-state correspondence modulo null operators:
The GNS construction for -algebras is a special case of a more general construction of Ghez, Lina & Roberts 1985, Prop. 1.9 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 Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, Springer (2017) [doi:10.1007/978-3-319-70706-8]
Review with an eye towards quantum probability and entropy:
See also
In the generality of -categories (and with an eye towards -categories):
For general unital star-algebras:
and 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 January 14, 2024 at 09:21:04. See the history of this page for a list of all contributions to it.