Abstract
Causally ordered correlation functions of local operators in near-thermal quantum systems computed using the Schwinger-Keldysh formalism obey a set of Ward identities. These can be understood rather simply as the consequence of a topological (BRST) algebra, called the universal Schwinger-Keldysh superalgebra, as explained in our compan-ion paper [1]. In the present paper we provide a mathematical discussion of this topological algebra. In particular, we argue that the structures can be understood in the language of extended equivariant cohomology. To keep the discussion self-contained, we provide a ba-sic review of the algebraic construction of equivariant cohomology and explain how it can be understood in familiar terms as a superspace gauge algebra. We demonstrate how the Schwinger-Keldysh construction can be succinctly encoded in terms a thermal equivariant cohomology algebra which naturally acts on the operator (super)-algebra of the quantum system. The main rationale behind this exploration is to extract symmetry statements which are robust under renormalization group flow and can hence be used to understand low-energy effective field theory of near-thermal physics. To illustrate the general prin-ciples, we focus on Langevin dynamics of a Brownian particle, rephrasing some known results in terms of thermal equivariant cohomology. As described elsewhere, the general framework enables construction of effective actions for dissipative hydrodynamics and could potentially illumine our understanding of black holes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F.M. Haehl, R. Loganayagam and M. Rangamani, Schwinger-Keldysh formalism I: BRST symmetries and superspace, arXiv:1610.01940 [INSPIRE].
J.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2 (1961) 407 [INSPIRE].
L.V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [INSPIRE].
K.-c. Chou, Z.-b. Su, B.-l. Hao and L. Yu, Equilibrium and Nonequilibrium Formalisms Made Unified, Phys. Rept. 118 (1985) 1 [INSPIRE].
N.P. Landsman and C.G. van Weert, Real and Imaginary Time Field Theory at Finite Temperature and Density, Phys. Rept. 145 (1987) 141 [INSPIRE].
J. Maciejko, An introduction to nonequilibrium many-body theory, Lecture Notes, Springer (2007).
A. Kamenev and A. Levchenko, Keldysh technique and nonlinear σ-model: Basic principles and applications, Adv. Phys. 58 (2009) 197 [arXiv:0901.3586] [INSPIRE].
H.A. Weldon, Two sum rules for the thermal n-point functions, Phys. Rev. D 72 (2005) 117901 [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Diagrammar, NATO Sci. Ser. B 4 (1974) 177 [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, The Fluid Manifesto: Emergent symmetries, hydrodynamics and black holes, JHEP 01 (2016) 184 [arXiv:1510.02494] [INSPIRE].
M. Crossley, P. Glorioso and H. Liu, Effective field theory of dissipative fluids, arXiv:1511.03646 [INSPIRE].
R. Kubo, Statistical mechanical theory of irreversible processes. 1. 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. 1., Phys. Rev. 115 (1959) 1342 [INSPIRE].
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, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661 [INSPIRE].
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].
D. Birmingham, M. Blau, M. Rakowski and G. Thompson, Topological field theory, Phys. Rept. 209 (1991) 129 [INSPIRE].
C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
R. Dijkgraaf and G.W. Moore, Balanced topological field theories, Commun. Math. Phys. 185 (1997) 411 [hep-th/9608169] [INSPIRE].
N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, Constraints on Fluid Dynamics from Equilibrium Partition Functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].
K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, The eightfold way to dissipation, Phys. Rev. Lett. 114 (2015) 201601 [arXiv:1412.1090] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Adiabatic hydrodynamics: The eightfold way to dissipation, JHEP 05 (2015) 060 [arXiv:1502.00636] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Chern-Simons terms from thermal circles and anomalies, JHEP 05 (2014) 110 [arXiv:1311.2935] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Topological σ-models & dissipative hydrodynamics, JHEP 04 (2016) 039 [arXiv:1511.07809] [INSPIRE].
M. Blau and G. Thompson, Aspects of N (T ) ≥ 2 topological gauge theories and D-branes, Nucl. Phys. B 492 (1997) 545 [hep-th/9612143] [INSPIRE].
R. Zucchini, Basic and equivariant cohomology in balanced topological field theory, J. Geom. Phys. 35 (2000) 299 [hep-th/9804043] [INSPIRE].
E. Gozzi and M. Reuter, Classical mechanics as a topological field theory, Phys. Lett. B 240 (1990) 137 [INSPIRE].
S. Cordes, G.W. Moore and S. Ramgoolam, Lectures on 2-D Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. Proc. Suppl. 41 (1995) 184 [hep-th/9411210] [INSPIRE].
V.W. Guillemin and S. Sternberg, Supersymmetry and equivariant de Rham theory, Springer Science & Business Media (2013).
V. Mathai and D.G. Quillen, Superconnections, Thom classes and equivariant differential forms, Topology 25 (1986) 85 [INSPIRE].
J. Kalkman, BRST model for equivariant cohomology and representatives for the equivariant Thom class, Commun. Math. Phys. 153 (1993) 447 [INSPIRE].
J. Kalkman, BRST model applied to symplectic geometry, hep-th/9308132 [INSPIRE].
J.H. Horne, Superspace Versions of Topological Theories, Nucl. Phys. B 318 (1989) 22 [INSPIRE].
B.S. DeWitt, Supermanifolds, Cambridge Monographs On Mathematical Physics, Cambridge University Press, Cambridge, U.K. (2012).
H. Basart, M. Flato, A. Lichnerowicz and D. Sternheimer, Deformation theory applied to quantization and statistical mechanics, Lett. Math. Phys. 8 (1984) 483.
H. Basart and A. Lichnerowicz, Conformal symplectic geometry, deformations, rigidity and geometrical (kms) conditions, Lett. Math. Phys. 10 (1985) 167.
M. Bordemann, H. Römer and S. Waldmann, A Remark on formal KMS states in deformation quantization, Lett. Math. Phys. 45 (1998) 49 [math/9801139] [INSPIRE].
M. Bordemann, H. Römer and S. Waldmann, Kms states and star product quantization, Rep. Math. Phys. 44 (1999) 45.
P.C. Martin, E.D. Siggia and H.A. Rose, Statistical Dynamics of Classical Systems, Phys. Rev. A 8 (1973) 423 [INSPIRE].
G. Parisi and N. Sourlas, Random Magnetic Fields, Supersymmetry and Negative Dimensions, Phys. Rev. Lett. 43 (1979) 744 [INSPIRE].
G. Parisi and N. Sourlas, Supersymmetric Field Theories and Stochastic Differential Equations, Nucl. Phys. B 206 (1982) 321 [INSPIRE].
J. Zinn-Justin, Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys. 113 (2002) 1 [INSPIRE].
S.R. Das, G. Mandal and S.R. Wadia, Stochastic Quantization on Two-dimensional Theory Space and Morse Theory, Mod. Phys. Lett. A 4 (1989) 745 [INSPIRE].
H.-K. Janssen, On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties, Z. Physik B 23 (1976) 377.
C. De Dominicis and L. Peliti, Field Theory Renormalization and Critical Dynamics Above t(c): Helium, Antiferromagnets and Liquid Gas Systems, Phys. Rev. B 18 (1978) 353 [INSPIRE].
P. Kovtun, G.D. Moore and P. Romatschke, Towards an effective action for relativistic dissipative hydrodynamics, JHEP 07 (2014) 123 [arXiv:1405.3967] [INSPIRE].
C. Jarzynski, Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach, Phys. Rev. E 56 (1997) 5018 [cond-mat/9707325].
C. Jarzynski, Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett. 78 (1997) 2690 [cond-mat/9610209] [INSPIRE].
G.E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60 (1999) 2721 [cond-mat/9901352].
K. Mallick, M. Moshe and H. Orland, A field-theoretic approach to nonequilibrium work identities, J. Phys. A 44 (2011) 095002 [arXiv:1009.4800] [INSPIRE].
P. Gaspard, Fluctuation relations for equilibrium states with broken discrete symmetries, J. Stat. Mach. 8 (2012) 08021 [arXiv:1207.4409].
P. Gaspard, Time-reversal Symmetry Relations for Fluctuating Currents in Nonequilibrium Systems, Acta Phys. Polon. B 44 (2013) 815 [arXiv:1203.5507].
L.M. Sieberer, A. Chiocchetta, A. Gambassi, U.C. Täuber and S. Diehl, Thermodynamic Equilibrium as a Symmetry of the Schwinger-Keldysh Action, Phys. Rev. B 92 (2015) 134307 [arXiv:1505.00912] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1610.01941
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Haehl, F.M., Loganayagam, R. & Rangamani, M. Schwinger-Keldysh formalism. Part II: thermal equivariant cohomology. J. High Energ. Phys. 2017, 70 (2017). https://doi.org/10.1007/JHEP06(2017)070
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2017)070