2000 character limit reached
KMS states on $\mathbb{Z}_2$-crossed products and twisted KMS functionals
Published 23 Feb 2024 in math.OA, math-ph, and math.MP | (2402.15574v2)
Abstract: KMS states on $\mathbb{Z}_2$-crossed products of unital $C*$-algebras $\mathcal{A}$ are characterized in terms of KMS states and twisted KMS functionals of $\mathcal{A}$. These functionals are shown to describe the extensions of KMS states $\omega$ on $\mathcal{A}$ to the crossed product $\mathcal{A} \rtimes \mathbb{Z}_2$ and can also be characterized by the twisted center of the von Neumann algebra generated by the GNS representation corresponding to $\omega$. As a particular class of examples, KMS states on $\mathbb{Z}_2$-crossed products of CAR algebras with dynamics and grading given by Bogoliubov automorphisms are analyzed in detail. In this case, one or two extremal KMS states are found depending on a Gibbs type condition involving the odd part of the absolute value of the Hamiltonian. As an application in mathematical physics, the extended field algebra of the Ising QFT is shown to be a $\mathbb{Z}_2$-crossed product of a CAR algebra which has a unique KMS state.
- Huzihiro Araki and David E Evans “On a C*-algebra approach to phase transition in the two-dimensional Ising model” In Comm. Math. Phys. 91, 1983, pp. 489–503
- “Joint extension of states of subsystems for a CAR system” In Commun. Math. Phys. 237, 2003, pp. 105–122
- “Extension of KMS states and chemical potential” In Comm. Math. Phys. 53.2 Springer, 1977, pp. 97–134
- H. Araki “On the diagonalization of a bilinear Hamiltonian by a Bogoliubov transformation” In Publ. Res. Inst. Math. Sci. Kyoto University, Ser. A 4, 1968, pp. 387–412
- H. Araki “On quasifree states of CAR and Bogoliubov automorphisms” In Publ. Res. Inst. Math. Sci. Kyoto 6, 1971, pp. 385–442 DOI: 10.2977/prims
- “Representations of canonical anticommutation relations” In Helv. Phys. Acta 37, 1964, pp. 136
- “Algebraic Supersymmetry: A case study” In Comm. Math. Phys. 272, 2007, pp. 699–750 DOI: 10.1007/s00220-006-0177-z
- “On the Existence of Equilibrium States in Local Quantum Field Theory” In Comm. Math. Phys. 121, 1989, pp. 255–270
- H. Baumgärtel, M. Jurke and F. Lledó “Twisted duality of the CAR algebra” In J. Math. Phys. 43, 2002, pp. 4158–4179 DOI: 10.1063/1.1483376
- “Modular nuclearity and localization” In Ann. Henri Poincaré 5, 2004, pp. 1065–1080 DOI: 10.1007/s00023-004-0190-8
- “Graded KMS functionals and the breakdown of supersymmetry” In Adv. Theor. Math. Phys. 3, 1999, pp. 615–626 URL: http://arxiv.org/abs/hep-th/9905102
- E Balslev, J Manuceau and A Verbeure “Representations of anticommutation relations and Bogolioubov transformations” In Comm. Math. Phys. 8, 1968, pp. 315–326
- “Operator Algebras and Quantum Statistical Mechanics II” Springer, 1997
- “String- and brane-localized fields in a strongly nonlocal model” In J. Phys. A40, 2007, pp. 2147–2163 DOI: 10.1088/1751-8113
- Vitonofrio Crismale, Rocco Duvenhage and Francesco Fidaleo “C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-Fermi systems and detailed balance” 44 pages In Anal. Math. Phys. 11, 2021 DOI: 10.1007/s13324-020-00412-0
- “C*-crossed-products by an order-two automorphism” In Canadian Mathematical Bulletin 53.1, 2010, pp. 37–50
- “KMS states on crossed products by abelian groups” In Math. Scand. 127.3, 2021 DOI: 10.48550/arXiv.2006.14443
- “Mathematics of Quantization and Quantum Fields” Cambridge University Press, 2022
- S. Doplicher, D. Kastler and D.W. Robinson “Covariance Algebras in Field Theory and Statistical Mechanics ” In Comm. Math. Phys. 3, 1966, pp. 1–28
- Robin Hillier “Super-KMS functionals for graded-local conformal nets” In Ann. Henri Poincaré 16, 2015, pp. 1899–1936
- Arthur Jaffe, Andrzej Lesniewski and Marek Wisniowski “Deformations of super-KMS functionals” In Comm. Math. Phys. 121, 1989, pp. 527–540
- Robert R Kallman “A generalization of free action”, 1969
- Daniel Kastler “Cyclic cocycles from graded KMS functionals” In Comm. Math. Phys., 1989, pp. 345–350
- G. Lechner “On the existence of local observables in theories with a factorizing S-matrix” In J. Phys. A38, 2005, pp. 3045–3056 DOI: 10.1088/0305-4470
- G. Lechner “Construction of Quantum Field Theories with Factorizing S-Matrices” In Comm. Math. Phys. 277, 2008, pp. 821–860 DOI: 10.1007/s00220-007-0381-5
- G. Lechner “Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques” In Advances in Algebraic Quantum Field Theory Springer, 2015, pp. 397–449
- “Tracial states on groupoid C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras and essential freeness”, 2024 DOI: 10.48550/arXiv.2401.15546
- Francis J Murray and J Neumann “On rings of operators” In Annals of Mathematics JSTOR, 1936, pp. 116–229
- B.M. McCoy, C.A. Tracy and T.T. Wu “Two-Dimensional Ising Model as an Exactly Solvable Relativistic Quantum Field Theory: Explicit Formulas for n Point Functions” In Phys. Rev. Lett. 38, 1977, pp. 793–796 DOI: 10.1103/PhysRevLett.38.793
- “The groupoid approach to equilibrium states on right LCM semigroup C∗superscript𝐶∗C^{\ast}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebras” In J. Lond. Math. Soc. 105.1, 2022, pp. 220–250
- Mikio Sato, Tetsuji Miwa and Michio Jimbo “Holonomic quantum fields I” In Publ. Res. Inst. Math. Sci. 14.1, 1978, pp. 223–267
- M. Takesaki “Theory of Operator Algebras II” Springer, 2003
- Klaus Erik Thomsen “An introduction to KMS weights” arXiv:2204.01125, 2023 arXiv:2204.01125 [math.OA]
- Dan Ursu “Characterizing traces on crossed products of noncommutative C*-algebras” In Adv. Math. 391, 2021, pp. 107955 DOI: 10.1016/j.aim.2021.107955
- D.P. Williams “Crossed products of C*-algebras” Providence, Rhode Island: American Mathematical Society, 2007
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.