Abstract
We consider generating functionals for computing correlators in quantum field theories with random potentials. Examples of such theories include cosmological systems in context of the string theory landscape (e.g. cosmic inflation) or condensed matter systems with quenched disorder (e.g. spin glass). We use the so-called replica trick to define two different generating functionals for calculating correlators of the quantum fields averaged over a given distribution of random potentials. The first generating functional is appropriate for calculating averaged (in-out) amplitudes and involves a single replica of fields, but the replica limit is taken to an (unphysical) negative one number of fields outside of the path integral. When the number of replicas is doubled the generating functional can also be used for calculating averaged probabilities (squared amplitudes) using the in-in construction. The second generating functional involves an infinite number of replicas, but can be used for calculating both in-out and in-in correlators and the replica limits are taken to only a zero number of fields. We discuss the formalism in details for a single real scalar field, but the generalization to more fields or to different types of fields is straightforward. We work out three examples: one where the mass of scalar field is treated as a random variable and two where the functional form of interactions is random, one described by a Gaussian random field and the other by a Euclidean action in the field configuration space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P.W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109 (1958) 1492 [INSPIRE].
A. Altland, A. Kamenev and C. Tian Anderson localization from the replica formalism, Phys. Rev. Lett. 95 (2005) 206601 [cond-mat/0505328].
J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, Cambridge (1996).
J.J. Binney, N.J. Dowrick, A.J. Fisher and M.E.J. Newman, The modern theory of critical phenomena, Clarendon Press, (1992).
M. Tegmark, What does inflation really predict?, JCAP 04 (2005) 001 [astro-ph/0410281] [INSPIRE].
F. Denef and M.R. Douglas, Distributions of nonsupersymmetric flux vacua, JHEP 03 (2005) 061 [hep-th/0411183] [INSPIRE].
D. Marsh, L. McAllister and T. Wrase, The Wasteland of Random Supergravities, JHEP 03 (2012) 102 [arXiv:1112.3034] [INSPIRE].
R. Bousso and J. Polchinski, Quantization of four form fluxes and dynamical neutralization of the cosmological constant, JHEP 06 (2000) 006 [hep-th/0004134] [INSPIRE].
S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].
L. Susskind, The anthropic landscape of string theory, hep-th/0302219 [INSPIRE].
V. Vanchurin, A. Vilenkin and S. Winitzki, Predictability crisis in inflationary cosmology and its resolution, Phys. Rev. D 61 (2000) 083507 [gr-qc/9905097] [INSPIRE].
M. Jain and V. Vanchurin, in progress.
T.C. Bachlechner, On Gaussian Random Supergravity, JHEP 04 (2014) 054 [arXiv:1401.6187] [INSPIRE].
X. Chen, G. Shiu, Y. Sumitomo and S.H.H. Tye, A Global View on The Search for de-Sitter Vacua in (type IIA) String Theory, JHEP 04 (2012) 026 [arXiv:1112.3338] [INSPIRE].
T.C. Bachlechner, D. Marsh, L. McAllister and T. Wrase, Supersymmetric Vacua in Random Supergravity, JHEP 01 (2013) 136 [arXiv:1207.2763] [INSPIRE].
D. Battefeld and T. Battefeld, Multi-Field Inflation on the Landscape, JCAP 03 (2009) 027 [arXiv:0812.0367] [INSPIRE].
T. Battefeld and C. Modi, Local random potentials of high differentiability to model the Landscape, JCAP 03 (2015) 010 [arXiv:1409.5135] [INSPIRE].
J. Frazer and A.R. Liddle, Exploring a string-like landscape, JCAP 02 (2011) 026 [arXiv:1101.1619] [INSPIRE].
J. Frazer and A.R. Liddle, Multi-field inflation with random potentials: field dimension, feature scale and non-Gaussianity, JCAP 02 (2012) 039 [arXiv:1111.6646] [INSPIRE].
A. Zee, Quantum field theory in a nutshell, Princeton University Press, Princeton, U.K. (2010).
L.H. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge, U.K. (1985).
J.S. Schwinger, The Special Canonical Group, Proc. Nat. Acad. Sci. 46 (1961) 1401.
P.M. Bakshi and K.T. Mahanthappa, Expectation value formalism in quantum field theory. 1, J. Math. Phys. 4 (1963) 1 [INSPIRE].
P.M. Bakshi and K.T. Mahanthappa, Expectation value formalism in quantum field theory. 2, J. Math. Phys. 4 (1963) 12 [INSPIRE].
P. Adshead, R. Easther and E.A. Lim, The ‘in-in’ Formalism and Cosmological Perturbations, Phys. Rev. D 80 (2009) 083521 [arXiv:0904.4207] [INSPIRE].
R.D. Jordan, Effective Field Equations for Expectation Values, Phys. Rev. D 33 (1986) 444 [INSPIRE].
S.F. Edwards and P.W. Anderson, Theory of Spin Glasses, J. Phys. F 5 (1975) 965.
C. De Dominicis and I. Giardina, Random fields and spin glasses, Cambridge University Press, Cambridge (2006).
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press, Oxford (1993).
J. Zanella and E. Calzetta, Renormalization group and nonequilibrium action in stochastic field theory, Phys. Rev. E 66 (2002) 036134 [cond-mat/0203566] [INSPIRE].
P.C. Martin, E.D. Siggia and H.A. Rose, Statistical Dynamics of Classical Systems, Phys. Rev. A 8 (1973) 423 [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].
E. Calzetta and B.-L. Hu, Nonequilibrium quantum field theory, Cambridge University Press, Cambridge (2008).
A. Kamenev, Field theory of non-equilibrium systems, Cambridge University Press, Cambridge (2011).
A. Kamenev and M. Mezard, Wigner-Dyson statistics from the replica method, J. Phys. A 32 (1999) 4373 [cond-mat/9901110].
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: 1506.03840v2
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
Jain, M., Vanchurin, V. Generating functionals for quantum field theories with random potentials. J. High Energ. Phys. 2016, 107 (2016). https://doi.org/10.1007/JHEP01(2016)107
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)107