Abstract
The theory of modular flow has proved extremely useful for applying thermodynamic reasoning to out-of-equilibrium states in quantum field theory. However, the standard proofs of the fundamental theorems of modular flow use machinery from Fourier analysis on Banach spaces, and as such are not especially transparent to an audience of physicists. In this article, I present a construction of modular flow that differs from existing treatments. The main pedagogical contribution is that I start with thermal physics via the KMS condition, and derive the modular operator as the only operator that could generate a thermal time-evolution map, rather than starting with the modular operator as the fundamental object of the theory. The main technical contribution is a new proof of the fundamental theorem stating that modular flow is a symmetry. The new proof circumvents the delicate issues of Fourier analysis that appear in previous treatments, but is still mathematically rigorous.
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References
H. Araki and E.J. Woods, Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas, J. Math. Phys. 4 (1963) 637.
R. Haag, N.M. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics, Commun. Math. Phys. 5 (1967) 215 [INSPIRE].
E. Witten, Why does quantum field theory in curved spacetime make sense? And what happens to the algebra of observables in the thermodynamic limit?, arXiv:2112.11614 [INSPIRE].
H. Araki, Type of von Neumann algebra associated with free field, Prog. Theor. Phys. 32 (1964) 956.
K. Fredenhagen, On the modular structure of local algebras of observables, Commun. Math. Phys. 97 (1985) 79 [INSPIRE].
M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Springer (2006) [https://doi.org/10.1007/BFb0065832].
M. Takesaki, Theory of operator algebras II, Springer (2003) [https://doi.org/10.1007/978-3-662-10451-4].
S. Strǎtilǎ and L. Zsidó, Lectures on von Neumann algebras, Cambridge University Press, Cambridge, U.K. (2019) [https://doi.org/10.1017/9781108654975].
V.S. Sunder, An invitation to von Neumann algebras, Springer (2012) [https://doi.org/10.1007/978-1-4613-8669-8].
E. Witten, APS medal for exceptional achievement in research: invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
A.C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, Phys. Rev. D 85 (2012) 104049 [Erratum ibid. 87 (2013) 069904] [arXiv:1105.3445] [INSPIRE].
R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Proof of a quantum Bousso bound, Phys. Rev. D 90 (2014) 044002 [arXiv:1404.5635] [INSPIRE].
T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for deformed half-spaces and the averaged null energy condition, JHEP 09 (2016) 038 [arXiv:1605.08072] [INSPIRE].
J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
S. Hollands and K. Sanders, Entanglement measures and their properties in quantum field theory, arXiv:1702.04924 [INSPIRE].
H. Casini, E. Teste and G. Torroba, Modular hamiltonians on the null plane and the Markov property of the vacuum state, J. Phys. A 50 (2017) 364001 [arXiv:1703.10656] [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP 07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A general proof of the quantum null energy condition, JHEP 09 (2019) 020 [arXiv:1706.09432] [INSPIRE].
Y. Chen, X. Dong, A. Lewkowycz and X.-L. Qi, Modular flow as a disentangler, JHEP 12 (2018) 083 [arXiv:1806.09622] [INSPIRE].
T. Faulkner, M. Li and H. Wang, A modular toolkit for bulk reconstruction, JHEP 04 (2019) 119 [arXiv:1806.10560] [INSPIRE].
N. Lashkari, Constraining quantum fields using modular theory, JHEP 01 (2019) 059 [arXiv:1810.09306] [INSPIRE].
R. Jefferson, Comments on black hole interiors and modular inclusions, SciPost Phys. 6 (2019) 042 [arXiv:1811.08900] [INSPIRE].
F. Ceyhan and T. Faulkner, Recovering the QNEC from the ANEC, Commun. Math. Phys. 377 (2020) 999 [arXiv:1812.04683] [INSPIRE].
N. Lashkari, Modular zero modes and sewing the states of QFT, JHEP 21 (2020) 189 [arXiv:1911.11153] [INSPIRE].
Y. Chen, Pulling out the island with modular flow, JHEP 03 (2020) 033 [arXiv:1912.02210] [INSPIRE].
T. Faulkner, S. Hollands, B. Swingle and Y. Wang, Approximate recovery and relative entropy I: general von Neumann subalgebras, Commun. Math. Phys. 389 (2022) 349 [arXiv:2006.08002] [INSPIRE].
S. Leutheusser and H. Liu, Emergent times in holographic duality, Phys. Rev. D 108 (2023) 086020 [arXiv:2112.12156] [INSPIRE].
E. Witten, Gravity and the crossed product, JHEP 10 (2022) 008 [arXiv:2112.12828] [INSPIRE].
V. Chandrasekaran, R. Longo, G. Penington and E. Witten, An algebra of observables for de Sitter space, JHEP 02 (2023) 082 [arXiv:2206.10780] [INSPIRE].
V. Chandrasekaran, G. Penington and E. Witten, Large N algebras and generalized entropy, JHEP 04 (2023) 009 [arXiv:2209.10454] [INSPIRE].
S. Leutheusser and H. Liu, Subalgebra-subregion duality: emergence of space and time in holography, arXiv:2212.13266 [INSPIRE].
G. Penington and E. Witten, Algebras and states in JT gravity, arXiv:2301.07257 [INSPIRE].
O. Parrikar and V. Singh, Canonical purification and the quantum extremal shock, JHEP 08 (2023) 155 [arXiv:2302.14318] [INSPIRE].
K. Furuya, N. Lashkari, M. Moosa and S. Ouseph, Information loss, mixing and emergent type III1 factors, JHEP 08 (2023) 111 [arXiv:2305.16028] [INSPIRE].
K. Jensen, J. Sorce and A. Speranza, Generalized entropy for general subregions in quantum gravity, arXiv:2306.01837 [INSPIRE].
S. Ali Ahmad and R. Jefferson, Crossed product algebras and generalized entropy for subregions, arXiv:2306.07323 [INSPIRE].
M.S. Klinger and R.G. Leigh, Crossed products, extended phase spaces and the resolution of entanglement singularities, arXiv:2306.09314 [INSPIRE].
J. Kudler-Flam, S. Leutheusser and G. Satishchandran, Generalized black hole entropy is von Neumann entropy, arXiv:2309.15897 [INSPIRE].
A. Van Daele, A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras, J. Funct. Anal. 15 (1974) 378.
L. Zsidó, A proof of Tomita’s fundamental theorem in the theory of standard von Neumann algebras, Rev. Roumaine Math. Pures Appl. 20 (1975) 609.
M. Rieffel and A. Van Daele, A bounded operator approach to Tomita-Takesaki theory, Pacific J. Math. 69 (1977) 187.
R. Longo, A simple proof of the existence of the modular automorphisms in approximately finite dimensional von Neumann algebras, Pacific J. Math. 75 (1978) 199.
S.L. Woronowicz, Operator systems and their application to the Tomita-Takesaki theory, J. Operator Theor. 2 (1979) 169.
J. Sorce, A short proof of Tomita’s theorem, to appear.
D. Buchholz, K. Fredenhagen and C. D’Antoni, The universal structure of local algebras, Commun. Math. Phys. 111 (1987) 123 [INSPIRE].
I. Cioranescu and L. Zsidó, Analytic generators for one-parameter groups, Tohoku Math. J. 28 (1976) 327.
O. Bratteli and D.W. Robinson, Operator algebras and quantum statistical mechanics, Springer (2012) [https://doi.org/10.1007/978-3-662-03444-6].
W. Rudin, Functional analysis, McGraw-Hill, U.S.A. (1991).
J.B. Conway, A course in operator theory, American Mathematical Soc., U.S.A. (2000).
J. Sorce, Notes on the type classification of von Neumann algebras, arXiv:2302.01958 [INSPIRE].
J. von Neumann, Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Annalen 102 (1930) 370.
H. Reeh and S. Schlieder, Bemerkungen zur unitäräquivalenz von lorentzinvarianten feldern, Nuovo Cim. 22 (1961) 1051 [INSPIRE].
A. Strohmaier, The Reeh-Schlieder property for quantum fields on stationary space-times, Commun. Math. Phys. 215 (2000) 105 [math-ph/0002054] [INSPIRE].
A. Strohmaier, R. Verch and M. Wollenberg, Microlocal analysis of quantum fields on curved space-times: analytic wavefront sets and Reeh-Schlieder theorems, J. Math. Phys. 43 (2002) 5514 [math-ph/0202003] [INSPIRE].
J. Weidmann, Linear operators in Hilbert spaces, Springer (2012) [https://doi.org/10.1007/978-1-4612-6027-1].
N. Dunford and J.T. Schwartz, Linear operators, part 1: general theory, John Wiley & Sons, U.S.A. (1988).
K. Yosida, Functional analysis, Springer (2012) [https://doi.org/10.1007/978-3-642-61859-8].
J. Sorce, Vector integration, https://sorcenotes.blogspot.com/2023/08/vector-integration.html, August 2023.
J. Sorce, Stone’s theorem, https://sorcenotes.blogspot.com/2023/08/stones-theorem.html, August 2023.
R. Kubo, Statistical mechanical theory of irreversible processes. I. General theory and simple applications in magnetic and conduction problems, J. Phys. Soc. Jap. 12 (1957) 570 [INSPIRE].
P.C. Martin and J.S. Schwinger, Theory of many particle systems. I, Phys. Rev. 115 (1959) 1342 [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, JHEP 03 (2021) 178 [arXiv:1905.00577] [INSPIRE].
Acknowledgments
I thank Brent Nelson for comments on a companion paper that improved the presentation of section 4. I also thank Şerban Strǎtilǎ and László Zsidó for writing their wonderful book [8], where I learned most of the techniques of operator analysis that were used in this paper. Part of this work was completed at the long term workshop YITP-T-23-01 held at YITP, Kyoto University. Financial support was provided by the AFOSR under award number FA9550-19-1-0360, by the DOE Early Career Award, and by the Templeton Foundation via the Black Hole Initiative.
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Sorce, J. An intuitive construction of modular flow. J. High Energ. Phys. 2023, 79 (2023). https://doi.org/10.1007/JHEP12(2023)079
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DOI: https://doi.org/10.1007/JHEP12(2023)079