Abstract
We present a detailed and self-contained analysis of the universal SchwingerKeldysh effective field theory which describes macroscopic thermal fluctuations of a relativistic field theory, elaborating on our earlier construction [1]. We write an effective action for appropriate hydrodynamic Goldstone modes and fluctuation fields, and discuss the symmetries to be imposed. The constraints imposed by fluctuation-dissipation theorem are manifest in our formalism. Consequently, the action reproduces hydrodynamic constitutive relations consistent with the local second law at all orders in the derivative expansion, and captures the essential elements of the eightfold classification of hydrodynamic transport of [2]. We demonstrate how to recover the hydrodynamic entropy and give predictions for the non-Gaussian hydrodynamic fluctuations.
The basic ingredients of our construction involve (i) doubling of degrees of freedom a la Schwinger-Keldysh, (ii) an emergent gauge U(1)T symmetry associated with entropy which is encapsulated in a Noether current a la Wald, and (iii) a BRST/topological supersymmetry imposing the fluctuation-dissipation theorem a la Parisi-Sourlas. The overarching mathematical framework for our construction is provided by the balanced equivariant cohomology of thermal translations, which captures the basic constraints arising from the Schwinger-Keldysh doubling, and the thermal Kubo-Martin-Schwinger relations. All these features are conveniently implemented in a covariant superspace formalism. An added benefit is that the second law can be understood as being due to entropy inflow from the Grassmann-odd directions of superspace.
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F.M. Haehl, R. Loganayagam and M. Rangamani, Topological σ-models & dissipative hydrodynamics, JHEP 04 (2016) 039 [arXiv:1511.07809] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Adiabatic hydrodynamics: The eightfold way to dissipation, JHEP 05 (2015) 060 [arXiv:1502.00636] [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, JHEP 09 (2017) 095 [arXiv:1511.03646] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Schwinger-Keldysh formalism. Part I: BRST symmetries and superspace, JHEP 06 (2017) 069 [arXiv:1610.01940] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Schwinger-Keldysh formalism. Part II: thermal equivariant cohomology, JHEP 06 (2017) 070 [arXiv:1610.01941] [INSPIRE].
P. Glorioso and H. Liu, The second law of thermodynamics from symmetry and unitarity, arXiv:1612.07705 [INSPIRE].
K. Jensen, N. Pinzani-Fokeeva and A. Yarom, Dissipative hydrodynamics in superspace, JHEP 09 (2018) 127 [arXiv:1701.07436] [INSPIRE].
P. Gao and H. Liu, Emergent Supersymmetry in Local Equilibrium Systems, JHEP 01 (2018) 040 [arXiv:1701.07445] [INSPIRE].
P. Glorioso, M. Crossley and H. Liu, Effective field theory of dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current, JHEP 09 (2017) 096 [arXiv:1701.07817] [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].
P.C. Martin, E.D. Siggia and H.A. Rose, Statistical Dynamics of Classical Systems, Phys. Rev. A 8 (1973) 423 [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Inflow Mechanism for Hydrodynamic Entropy, Phys. Rev. Lett. 121 (2018) 051602 [arXiv:1803.08490] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
V.E. Hubeny, S. Minwalla and M. Rangamani, The fluid/gravity correspondence, in Black holes in higher dimensions, pp. 348–383 (2012) [arXiv:1107.5780] [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].
H.A. Weldon, Two sum rules for the thermal n-point functions, Phys. Rev. D 72 (2005) 117901 [INSPIRE].
M. Geracie, F.M. Haehl, R. Loganayagam, P. Narayan, D.M. Ramirez and M. Rangamani, Schwinger-Keldysh superspace in quantum mechanics, Phys. Rev. D 97 (2018) 105023 [arXiv:1712.04459] [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].
F.M. Haehl, R. Loganayagam and M. Rangamani, The eightfold way to dissipation, Phys. Rev. Lett. 114 (2015) 201601 [arXiv:1412.1090] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661 [INSPIRE].
A.H. Taub, General Relativistic Variational Principle for Perfect Fluids, Phys. Rev. 94 (1954) 1468 [INSPIRE].
B. Carter, Elastic Perturbation Theory in General Relativity and a Variation Principle for a Rotating Solid Star, Commun. Math. Phys. 30 (1973) 261.
B. Carter, Covariant Theory of Conductivity in Ideal Fluid or Solid Media, Lect. Notes Math. 1385 (1989) 1.
D. Nickel and D.T. Son, Deconstructing holographic liquids, New J. Phys. 13 (2011) 075010 [arXiv:1009.3094] [INSPIRE].
S. Dubovsky, L. Hui, A. Nicolis and D.T. Son, Effective field theory for hydrodynamics: thermodynamics and the derivative expansion, Phys. Rev. D 85 (2012) 085029 [arXiv:1107.0731] [INSPIRE].
S. Dubovsky, L. Hui and A. Nicolis, Effective field theory for hydrodynamics: Wess-Zumino term and anomalies in two spacetime dimensions, Phys. Rev. D 89 (2014) 045016 [arXiv:1107.0732] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya and M. Rangamani, Non-dissipative hydrodynamics: Effective actions versus entropy current, JHEP 02 (2013) 153 [arXiv:1211.1020] [INSPIRE].
O. Saremi and D.T. Son, Hall viscosity from gauge/gravity duality, JHEP 04 (2012) 091 [arXiv:1103.4851] [INSPIRE].
F.M. Haehl and M. Rangamani, Comments on Hall transport from effective actions, JHEP 10 (2013) 074 [arXiv:1305.6968] [INSPIRE].
M. Geracie and D.T. Son, Effective field theory for fluids: Hall viscosity from a Wess-Zumino-Witten term, JHEP 11 (2014) 004 [arXiv:1402.1146] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Effective actions for anomalous hydrodynamics, JHEP 03 (2014) 034 [arXiv:1312.0610] [INSPIRE].
P. Romatschke, Relativistic Viscous Fluid Dynamics and Non-Equilibrium Entropy, Class. Quant. Grav. 27 (2010) 025006 [arXiv:0906.4787] [INSPIRE].
S. Bhattacharyya, Constraints on the second order transport coefficients of an uncharged fluid, JHEP 07 (2012) 104 [arXiv:1201.4654] [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].
K. Jensen, R. Loganayagam and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088 [arXiv:1207.5824] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, JHEP 05 (2014) 134 [arXiv:1310.7024] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Chern-Simons terms from thermal circles and anomalies, JHEP 05 (2014) 110 [arXiv:1311.2935] [INSPIRE].
S. Bhattacharyya, Entropy current and equilibrium partition function in fluid dynamics, JHEP 08 (2014) 165 [arXiv:1312.0220] [INSPIRE].
S. Bhattacharyya, Entropy Current from Partition Function: One Example, JHEP 07 (2014) 139 [arXiv:1403.7639] [INSPIRE].
S. Grozdanov and J. Polonyi, Viscosity and dissipative hydrodynamics from effective field theory, Phys. Rev. D 91 (2015) 105031 [arXiv:1305.3670] [INSPIRE].
S. Endlich, A. Nicolis, R.A. Porto and J. Wang, Dissipation in the effective field theory for hydrodynamics: First order effects, Phys. Rev. D 88 (2013) 105001 [arXiv:1211.6461] [INSPIRE].
T. Hayata, Y. Hidaka, T. Noumi and M. Hongo, Relativistic hydrodynamics from quantum field theory on the basis of the generalized Gibbs ensemble method, Phys. Rev. D 92 (2015) 065008 [arXiv:1503.04535] [INSPIRE].
S. Floerchinger, Variational principle for theories with dissipation from analytic continuation, JHEP 09 (2016) 099 [arXiv:1603.07148] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Two roads to hydrodynamic effective actions: a comparison, arXiv:1701.07896 [INSPIRE].
M. Hongo, Path-integral formula for local thermal equilibrium, Annals Phys. 383 (2017) 1 [arXiv:1611.07074] [INSPIRE].
M. Hongo, Nonrelativistic hydrodynamics from quantum field theory: (I) Normal fluid composed of spinless Schrödinger fields, arXiv:1801.06520 [INSPIRE].
K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom, An entropy current in superspace, arXiv:1803.07070 [INSPIRE].
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].
R. Haag, N.M. Hugenholtz and M. Winnink, On the Equilibrium states in quantum statistical mechanics, Commun. Math. Phys. 5 (1967) 215 [INSPIRE].
F.M. Haehl, R. Loganayagam, P. Narayan, A.A. Nizami and M. Rangamani, Thermal out-of-time-order correlators, KMS relations and spectral functions, JHEP 12 (2017) 154 [arXiv:1706.08956] [INSPIRE].
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].
R.P. Feynman and F.L. Vernon Jr., The Theory of a general quantum system interacting with a linear dissipative system, Annals Phys. 24 (1963) 118 [INSPIRE].
A.O. Caldeira and A.J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121 (1983) 587.
G. ’t Hooft and M.J.G. Veltman, Diagrammar, NATO Sci. Ser. B 4 (1974) 177 [INSPIRE].
B.S. DeWitt, Supermanifolds, CambridgE Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K. (2012).
G. Parisi and N. Sourlas, Supersymmetric Field Theories and Stochastic Differential Equations, Nucl. Phys. B 206 (1982) 321 [INSPIRE].
P. Gaspard, Fluctuation relations for equilibrium states with broken discrete symmetries, J. Stat. Mech. 8 (2012) 08021 [arXiv:1207.4409].
P. Gaspard, Time-reversal Symmetry Relations for Fluctuating Currents in Nonequilibrium Systems, Acta Phys. Pol. B 44 (2013) 815 [arXiv:1203.5507].
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.
S. Cordes, G.W. Moore and S. Ramgoolam, Large N 2-D Yang-Mills theory and topological string theory, Commun. Math. Phys. 185 (1997) 543 [hep-th/9402107] [INSPIRE].
M. Blau and G. Thompson, N = 2 topological gauge theory, the Euler characteristic of moduli spaces and the Casson invariant, Commun. Math. Phys. 152 (1993) 41 [hep-th/9112012] [INSPIRE].
R. Dijkgraaf and G.W. Moore, Balanced topological field theories, Commun. Math. Phys. 185 (1997) 411 [hep-th/9608169] [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].
C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
J. Zinn-Justin, Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys. 113 (2002) 1 [INSPIRE].
C. Jarzynski, Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett. 78 (1997) 2690 [cond-mat/9610209] [INSPIRE].
S. Bhattacharyya et al., Local Fluid Dynamical Entropy from Gravity, JHEP 06 (2008) 055 [arXiv:0803.2526] [INSPIRE].
C.G. Callan Jr. and J.A. Harvey, Anomalies and Fermion Zero Modes on Strings and Domain Walls, Nucl. Phys. B 250 (1985) 427 [INSPIRE].
H. Basart, M. Flato, A. Lichnerowicz and D. Sternheimer, Deformation theory applied to quantization and statistical mechanics, Lett. Math. Phys. 8 (1984) 483.
M. Bordemann, H. Romer 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, Rept. Math. Phys. 44 (1999) 45.
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Haehl, F.M., Loganayagam, R. & Rangamani, M. Effective action for relativistic hydrodynamics: fluctuations, dissipation, and entropy inflow. J. High Energ. Phys. 2018, 194 (2018). https://doi.org/10.1007/JHEP10(2018)194
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DOI: https://doi.org/10.1007/JHEP10(2018)194