Abstract
We write down a Schwinger-Keldysh effective field theory for non-relativistic (Galilean) hydrodynamics. We use the null background construction to covariantly couple Galilean field theories to a set of background sources. In this language, Galilean hydrodynamics gets recast as relativistic hydrodynamics formulated on a one dimension higher spacetime admitting a null Killing vector. This allows us to import the existing field theoretic techniques for relativistic hydrodynamics into the Galilean setting, with minor modifications to include the additional background vector field. We use this formulation to work out an interacting field theory describing stochastic fluctuations of energy, momentum, and density modes around thermal equilibrium. We also present a translation of our results to the more conventional Newton-Cartan language, and discuss how the same can be derived via a non-relativistic limit of the effective field theory for relativistic hydrodynamics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P.C. Martin, E.D. Siggia and H.A. Rose, Statistical Dynamics of Classical Systems, Phys. Rev. A 8 (1973) 423 [INSPIRE].
Y. Pomeau and P. Resibois, Time Dependent Correlation Functions and Mode-Mode Coupling Theories, Phys. Rept. 19 (1974) 63.
I.M. De Schepper, H. Van Beyeren and M.H. Ernst, The nonexistence of the linear diffusion equation beyond Fick’s law, Physica 75 (1974) 1.
P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].
L. Landau and E. Lifshitz, Statistical Physics, Part 1, vol. 5 of Course of Theoretical Physics, Butterworth-Heinemann, Oxford (1980).
L. Landau and E. Lifshitz, Hydrodynamic fluctuations, Sov. Phys. JETP 5 (1957) 512.
P.C. Hohenberg and B.I. Halperin, Theory of Dynamic Critical Phenomena, Rev. Mod. Phys. 49 (1977) 435 [INSPIRE].
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].
I. Khalatnikov, V. Lebedev and A. Sukhorukov, Diagram technique for calculating long wave fluctuation effects, Phys. Lett. A 94 (1983) 271 [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. Grozdanov and J. Polonyi, Viscosity and dissipative hydrodynamics from effective field theory, Phys. Rev. D 91 (2015) 105031 [arXiv:1305.3670] [INSPIRE].
M. Crossley, P. Glorioso and H. Liu, Effective field theory of dissipative fluids, JHEP 09 (2017) 095 [arXiv:1511.03646] [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. Glorioso and H. Liu, The second law of thermodynamics from symmetry and unitarity, arXiv:1612.07705 [INSPIRE].
P. Gao and H. Liu, Emergent Supersymmetry in Local Equilibrium Systems, JHEP 01 (2018) 040 [arXiv:1701.07445] [INSPIRE].
P. Glorioso, H. Liu and S. Rajagopal, Global Anomalies, Discrete Symmetries, and Hydrodynamic Effective Actions, JHEP 01 (2019) 043 [arXiv:1710.03768] [INSPIRE].
P. Gao, P. Glorioso and H. Liu, Ghostbusters: Unitarity and Causality of Non-equilibrium Effective Field Theories, JHEP 03 (2020) 040 [arXiv:1803.10778] [INSPIRE].
K. Jensen, N. Pinzani-Fokeeva and A. Yarom, Dissipative hydrodynamics in superspace, JHEP 09 (2018) 127 [arXiv:1701.07436] [INSPIRE].
K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom, A panoply of Schwinger-Keldysh transport, SciPost Phys. 5 (2018) 053 [arXiv:1804.04654] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Effective Action for Relativistic Hydrodynamics: Fluctuations, Dissipation, and Entropy Inflow, JHEP 10 (2018) 194 [arXiv:1803.11155] [INSPIRE].
H. Liu and P. Glorioso, Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics, PoS TASI2017 (2018) 008 [arXiv:1805.09331] [INSPIRE].
E. Wang and U.W. Heinz, A Generalized fluctuation dissipation theorem for nonlinear response functions, Phys. Rev. D 66 (2002) 025008 [hep-th/9809016] [INSPIRE].
X. Chen-Lin, L.V. Delacrétaz and S.A. Hartnoll, Theory of diffusive fluctuations, Phys. Rev. Lett. 122 (2019) 091602 [arXiv:1811.12540] [INSPIRE].
P. Glorioso, L.V. Delacrétaz, X. Chen, R.M. Nandkishore and A. Lucas, Hydrodynamics in lattice models with continuous non-Abelian symmetries, arXiv:2007.13753 [INSPIRE].
M.J. Landry, The coset construction for non-equilibrium systems, JHEP 07 (2020) 200 [arXiv:1912.12301] [INSPIRE].
P. Glorioso and D.T. Son, Effective field theory of magnetohydrodynamics from generalized global symmetries, arXiv:1811.04879 [INSPIRE].
S.E. Gralla and N. Iqbal, Effective Field Theory of Force-Free Electrodynamics, Phys. Rev. D 99 (2019) 105004 [arXiv:1811.07438] [INSPIRE].
E. Cartan, Sur les varietes a connexion affine et la theorie de la relativite generalisee (premiere partie) (Suite), Annales Sci. Ecole Norm. Sup. 41 (1924) 1.
E. Cartan, Sur les varietes a connexion affine et la theorie de la relativite generalisee (premiere partie), Annales Sci. Ecole Norm. Sup. 40 (1923) 325.
K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, SciPost Phys. 5 (2018) 011 [arXiv:1408.6855] [INSPIRE].
K. Jensen, Aspects of hot Galilean field theory, JHEP 04 (2015) 123 [arXiv:1411.7024] [INSPIRE].
K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [INSPIRE].
K. Jensen, Anomalies for Galilean fields, SciPost Phys. 5 (2018) 005 [arXiv:1412.7750] [INSPIRE].
G. Festuccia, D. Hansen, J. Hartong and N.A. Obers, Torsional Newton-Cartan Geometry from the Noether Procedure, Phys. Rev. D 94 (2016) 105023 [arXiv:1607.01926] [INSPIRE].
E. Bergshoeff, A. Chatzistavrakidis, L. Romano and J. Rosseel, Newton-Cartan Gravity and Torsion, JHEP 10 (2017) 194 [arXiv:1708.05414] [INSPIRE].
E.A. Bergshoeff, J. Hartong and J. Rosseel, Torsional Newton-Cartan geometry and the Schrödinger algebra, Class. Quant. Grav. 32 (2015) 135017 [arXiv:1409.5555] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan Geometry and Lifshitz Holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].
J. Hartong, N.A. Obers and M. Sanchioni, Lifshitz Hydrodynamics from Lifshitz Black Branes with Linear Momentum, JHEP 10 (2016) 120 [arXiv:1606.09543] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger Invariance from Lifshitz Isometries in Holography and Field Theory, Phys. Rev. D 92 (2015) 066003 [arXiv:1409.1522] [INSPIRE].
M. Geracie, K. Prabhu and M.M. Roberts, Fields and fluids on curved non-relativistic spacetimes, JHEP 08 (2015) 042 [arXiv:1503.02680] [INSPIRE].
M. Geracie, K. Prabhu and M.M. Roberts, Physical stress, mass, and energy for non-relativistic matter, JHEP 06 (2017) 089 [arXiv:1609.06729] [INSPIRE].
C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys. A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].
R. Banerjee and P. Mukherjee, Milne boost from galilean gauge theory, Phys. Lett. B 778 (2018) 303 [arXiv:1710.10882] [INSPIRE].
R. Banerjee and P. Mukherjee, Torsional Newton-Cartan geometry from Galilean gauge theory, Class. Quant. Grav. 33 (2016) 225013 [arXiv:1604.06893] [INSPIRE].
N. Banerjee, S. Dutta and A. Jain, Null Fluids - A New Viewpoint of Galilean Fluids, Phys. Rev. D 93 (2016) 105020 [arXiv:1509.04718] [INSPIRE].
J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [INSPIRE].
A. Adams, K. Balasubramanian and J. McGreevy, Hot Spacetimes for Cold Atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [INSPIRE].
C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [INSPIRE].
C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann Structures and Newton-cartan Theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].
C. Duval, G.W. Gibbons and P. Horvathy, Celestial mechanics, conformal structures and gravitational waves, Phys. Rev. D 43 (1991) 3907 [hep-th/0512188] [INSPIRE].
B. Julia and H. Nicolai, Null Killing vector dimensional reduction and Galilean geometrodynamics, Nucl. Phys. B 439 (1995) 291 [hep-th/9412002] [INSPIRE].
A. Jain, Galilean Anomalies and Their Effect on Hydrodynamics, Phys. Rev. D 93 (2016) 065007 [arXiv:1509.05777] [INSPIRE].
N. Banerjee, S. Dutta and A. Jain, Equilibrium partition function for nonrelativistic fluids, Phys. Rev. D 92 (2015) 081701 [arXiv:1505.05677] [INSPIRE].
N. Banerjee, S. Dutta and A. Jain, First Order Galilean Superfluid Dynamics, Phys. Rev. D 96 (2017) 065004 [arXiv:1612.01550] [INSPIRE].
N. Banerjee, S. Atul Bhatkar and A. Jain, Second order Galilean fluids and Stokes’ law, Phys. Rev. D 97 (2018) 096018 [arXiv:1711.09076] [INSPIRE].
A. Jain, A universal framework for hydrodynamics, Ph.D. thesis, Durham University, CPT, 6, 2018.
M. Hassaine and P.A. Horvathy, Field dependent symmetries of a nonrelativistic fluid model, Annals Phys. 282 (2000) 218 [math-ph/9904022] [INSPIRE].
P.A. Horvathy and P.-M. Zhang, Non-relativistic conformal symmetries in fluid mechanics, Eur. Phys. J. C 65 (2010) 607 [arXiv:0906.3594] [INSPIRE].
M.L. Bellac, Thermal Field Theory, Cambridge Monographs on Mathematical Physics 3 (2011) [INSPIRE].
L. Landau and E. Lifshitz, Fluid Mechanics, Teoreticheskaia fizika, Pergamon Press (1959).
R. Loganayagam, Anomaly Induced Transport in Arbitrary Dimensions, arXiv:1106.0277 [INSPIRE].
P. Kovtun, First-order relativistic hydrodynamics is stable, JHEP 10 (2019) 034 [arXiv:1907.08191] [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].
C. Arzt, Reduced effective Lagrangians, Phys. Lett. B 342 (1995) 189 [hep-ph/9304230] [INSPIRE].
S. Bhattacharyya, Entropy Current from Partition Function: One Example, JHEP 07 (2014) 139 [arXiv:1403.7639] [INSPIRE].
S. Bhattacharyya, Entropy current and equilibrium partition function in fluid dynamics, JHEP 08 (2014) 165 [arXiv:1312.0220] [INSPIRE].
J. Armas and A. Jain, Viscoelastic hydrodynamics and holography, JHEP 01 (2020) 126 [arXiv:1908.01175] [INSPIRE].
J. Armas and A. Jain, Hydrodynamics for charge density waves and their holographic duals, Phys. Rev. D 101 (2020) 121901 [arXiv:2001.07357] [INSPIRE].
S. Grozdanov and N. Poovuttikul, Generalized global symmetries in states with dynamical defects: The case of the transverse sound in field theory and holography, Phys. Rev. D 97 (2018) 106005 [arXiv:1801.03199] [INSPIRE].
S. Grozdanov, D.M. Hofman and N. Iqbal, Generalized global symmetries and dissipative magnetohydrodynamics, Phys. Rev. D 95 (2017) 096003 [arXiv:1610.07392] [INSPIRE].
J. Armas, J. Gath, A. Jain and A.V. Pedersen, Dissipative hydrodynamics with higher-form symmetry, JHEP 05 (2018) 192 [arXiv:1803.00991] [INSPIRE].
J. Armas and A. Jain, One-form superfluids & magnetohydrodynamics, JHEP 01 (2020) 041 [arXiv:1811.04913] [INSPIRE].
J. Armas and A. Jain, Magnetohydrodynamics as superfluidity, Phys. Rev. Lett. 122 (2019) 141603 [arXiv:1808.01939] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Two roads to hydrodynamic effective actions: a comparison, arXiv:1701.07896 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2008.03994
Rights and permissions
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.
About this article
Cite this article
Jain, A. Effective field theory for non-relativistic hydrodynamics. J. High Energ. Phys. 2020, 208 (2020). https://doi.org/10.1007/JHEP10(2020)208
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2020)208