1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario, Canada, N2L 2Y5
2X, The Moonshot Factory, Mountain View, CA 94043
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We numerically compute renormalized expectation values of quadratic operators in a quantum field theory (QFT) of free Dirac fermions in curved two-dimensional (Lorentzian) spacetime. First, we use a staggered-fermion discretization to generate a sequence of lattice theories yielding the desired QFT in the continuum limit. Numerically-computed lattice correlators are then used to approximate, through extrapolation, those in the continuum. Finally, we use so-called point-splitting regularization and Hadamard renormalization to remove divergences, and thus obtain finite, renormalized expectation values of quadratic operators in the continuum. As illustrative applications, we show how to recover the Unruh effect in flat spacetime and how to compute renormalized expectation values in the Hawking-Hartle vacuum of a Schwarzschild black hole and in the Bunch-Davies vacuum of an expanding universe described by de Sitter spacetime. Although here we address a non-interacting QFT using free fermion techniques, the framework described in this paper lays the groundwork for a series of subsequent studies involving simulation of interacting QFTs in curved spacetime by tensor network techniques.
► BibTeX data
► References
[1] Radiation damping in a gravitational field. Annals of Physics, 9 (2): 220 – 259, 1960. ISSN 0003-4916. https://doi.org/10.1016/0003-4916(60)90030-0.
https://doi.org/https://doi.org/10.1016/0003-4916(60)90030-0
[2] Andreas Albrecht and Paul J. Steinhardt. Cosmology for grand unified theories with radiatively induced symmetry breaking. Phys. Rev. Lett., 48: 1220–1223, Apr 1982. 10.1103/PhysRevLett.48.1220.
https://doi.org/10.1103/PhysRevLett.48.1220
[3] Miguel Alcubierre. Introduction to 3+1 Numerical Relativity. 10.1093/acprof:oso/9780199205677.001.0001.
https://doi.org/10.1093/acprof:oso/9780199205677.001.0001
[4] Bruce Allen. Vacuum states in de Sitter space. Phys. Rev. D, 32: 3136–3149, Dec 1985. 10.1103/PhysRevD.32.3136.
https://doi.org/10.1103/PhysRevD.32.3136
[5] Victor E. Ambruș and Elizabeth Winstanley. Renormalised fermion vacuum expectation values on anti-de Sitter space–time. Physics Letters B, 749: 597 – 602, 2015. ISSN 0370-2693. https://doi.org/10.1016/j.physletb.2015.08.045.
https://doi.org/https://doi.org/10.1016/j.physletb.2015.08.045
[6] Richard Arnowitt, Stanley Deser, and Charles W. Misner. Republication of: The dynamics of general relativity. Gen. Relativ. Gravit., 40, 2008. 10.1007/s10714-008-0661-1.
https://doi.org/10.1007/s10714-008-0661-1
[7] T. Banks, Leonard Susskind, and John Kogut. Strong-coupling calculations of lattice gauge theories: (1 + 1)-dimensional exercises. Phys. Rev. D, 13: 1043–1053, Feb 1976. 10.1103/PhysRevD.13.1043.
https://doi.org/10.1103/PhysRevD.13.1043
[8] T.W. Baumgarte and S.L. Shapiro. Numerical Relativity: Solving Einstein’s Equations on the Computer. Cambridge University Press, 2010. ISBN 9780521514071. 10.1017/CBO9781139193344.
https://doi.org/10.1017/CBO9781139193344
[9] N. D. Birrell and P. C. W. Davies. Quantum Fields in Curved Space. Cambridge University Press, Cambridge, 1982. 10.1017/CBO9780511622632.
https://doi.org/10.1017/CBO9780511622632
[10] Elliot Blommaert. Hamiltonian simulation of free lattice fermions in curved spacetime. Master’s thesis, Ghent University, 2019.
[11] T. S. Bunch and P. C. W. Davies. Covariant point-splitting regularization for a scalar quantum field in a Robertson-Walker Universe with spatial curvature. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 357 (1690): 381–394, 1977. 10.1098/rspa.1977.0174.
https://doi.org/10.1098/rspa.1977.0174
[12] Curtis G. Callan, Steven B. Giddings, Jeffrey A. Harvey, and Andrew Strominger. Evanescent black holes. Phys. Rev. D, 45: R1005–R1009, Feb 1992. 10.1103/PhysRevD.45.R1005.
https://doi.org/10.1103/PhysRevD.45.R1005
[13] Sean M. Carroll. Spacetime and geometry: An introduction to general relativity. 2004. ISBN 0805387323, 9780805387322. 10.1017/9781108770385.
https://doi.org/10.1017/9781108770385
[14] N. A. Chernikov and E. A. Tagirov. Quantum theory of scalar fields in de Sitter space-time. Annales de l’I.H.P. Physique théorique, 9 (2): 109–141, 1968. URL http://www.numdam.org/item/AIHPA_1968__9_2_109_0.
http://www.numdam.org/item/AIHPA_1968__9_2_109_0
[15] S. M. Christensen. Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method. Phys. Rev. D, 14: 2490–2501, November 1976. 10.1103/PhysRevD.14.2490.
https://doi.org/10.1103/PhysRevD.14.2490
[16] S. M. Christensen. Regularization, renormalization, and covariant geodesic point separation. Phys. Rev. D, 17: 946–963, February 1978. 10.1103/PhysRevD.17.946.
https://doi.org/10.1103/PhysRevD.17.946
[17] A J Daley, C Kollath, U Schollwöck, and G Vidal. Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces. Journal of Statistical Mechanics: Theory and Experiment, 2004 (04): P04005, apr 2004. 10.1088/1742-5468/2004/04/p04005.
https://doi.org/10.1088/1742-5468/2004/04/p04005
[18] Ashmita Das, Surojit Dalui, Chandramouli Chowdhury, and Bibhas Ranjan Majhi. Conformal vacuum and the fluctuation-dissipation theorem in a de Sitter universe and black hole spacetimes. Phys. Rev. D, 100: 085002, Oct 2019. 10.1103/PhysRevD.100.085002.
https://doi.org/10.1103/PhysRevD.100.085002
[19] Yves Décanini and Antoine Folacci. Hadamard renormalization of the stress-energy tensor for a quantized scalar field in a general spacetime of arbitrary dimension. Phys. Rev. D., 78 (4): 044025, 2008. ISSN 1550-7998. 10.1103/physrevd.78.044025.
https://doi.org/10.1103/physrevd.78.044025
[20] Daniel Z. Freedman and Antoine Van Proeyen. Supergravity. Cambridge University Press, New York, 2012. 10.1017/CBO9781139026833.
https://doi.org/10.1017/CBO9781139026833
[21] G. W. Gibbons and S. W. Hawking. Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D, 15: 2738–2751, May 1977. 10.1103/PhysRevD.15.2738.
https://doi.org/10.1103/PhysRevD.15.2738
[22] G. W. Gibbons and M. J. Perry. Black holes in thermal equilibrium. Phys. Rev. Lett., 36: 985–987, Apr 1976. 10.1103/PhysRevLett.36.985.
https://doi.org/10.1103/PhysRevLett.36.985
[23] Brian R Greene, Maulik K Parikh, and Jan Pieter van der Schaar. Universal correction to the inflationary vacuum. Journal of High Energy Physics, 2006 (04): 057–057, apr 2006. 10.1088/1126-6708/2006/04/057.
https://doi.org/10.1088/1126-6708/2006/04/057
[24] Alan H. Guth. Inflationary universe: A possible solution to the horizon and flatness problems. Phys. Rev. D, 23: 347–356, Jan 1981. 10.1103/PhysRevD.23.347.
https://doi.org/10.1103/PhysRevD.23.347
[25] Jacques Hadamard. Lectures on Cauchy’s problem in linear partial differential equations. New Haven Yale University Press, 1923.
[26] J. B. Hartle and S. W. Hawking. Path-integral derivation of black-hole radiance. Phys. Rev. D, 13: 2188–2203, Apr 1976. 10.1103/PhysRevD.13.2188.
https://doi.org/10.1103/PhysRevD.13.2188
[27] S. W. Hawking. Black hole explosions? Nature, 248: 379–423. 10.1038/248030a0.
https://doi.org/10.1038/248030a0
[28] W. Israel. Thermo-field dynamics of black holes. Physics Letters A, 57 (2): 107 – 110, 1976. ISSN 0375-9601. https://doi.org/10.1016/0375-9601(76)90178-X.
https://doi.org/https://doi.org/10.1016/0375-9601(76)90178-X
[29] Ted Jacobson. Note on Hartle-Hawking vacua. Phys. Rev. D, 50: R6031–R6032, Nov 1994. 10.1103/PhysRevD.50.R6031.
https://doi.org/10.1103/PhysRevD.50.R6031
[30] P. Jordan and E. Wigner. Über das Paulische äquivalenzverbot. Zeitschrift fur Physik, 47: 631–651, 1928. 10.1007/BF01331938.
https://doi.org/10.1007/BF01331938
[31] Bernard S. Kay and Robert M. Wald. Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Physics Reports, 207 (2): 49 – 136, 1991. ISSN 0370-1573. https://doi.org/10.1016/0370-1573(91)90015-E.
https://doi.org/https://doi.org/10.1016/0370-1573(91)90015-E
[32] John Kogut and Leonard Susskind. Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D, 11: 395–408, Jan 1975. 10.1103/PhysRevD.11.395.
https://doi.org/10.1103/PhysRevD.11.395
[33] R. Laflamme. Geometry and thermofields. Nuclear Physics B, 324 (1): 233 – 252, 1989. ISSN 0550-3213. https://doi.org/10.1016/0550-3213(89)90191-0.
https://doi.org/https://doi.org/10.1016/0550-3213(89)90191-0
[34] Adam G. M. Lewis. Hadamard renormalization of a two-dimensional dirac field. Phys. Rev. D, 101: 125019, Jun 2020. 10.1103/PhysRevD.101.125019.
https://doi.org/10.1103/PhysRevD.101.125019
[35] Adam G.M. Lewis and Guifré Vidal. Matrix product state simulations of quantum fields in curved spacetime.
[36] A.D. Linde. A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Physics Letters B, 108 (6): 389 – 393, 1982. ISSN 0370-2693. https://doi.org/10.1016/0370-2693(82)91219-9.
https://doi.org/https://doi.org/10.1016/0370-2693(82)91219-9
[37] Fannes M., Nachtergaele B., and R.F Werner. Finitely correlated states on quantum spin chains. Comm. Math. Phys., 144: 443490, 1992. 10.1007/BF02099178.
https://doi.org/10.1007/BF02099178
[38] V. Moretti. Comments on the stress-energy operator in curved spacetime. Commun. Math. Phys., 232 (2): 189–221. 10.1007/s00220-002-0702-7.
https://doi.org/10.1007/s00220-002-0702-7
[39] A.-H. Najmi and A. C. Ottewill. Quantum states and the Hadamard form. II. Energy minimization for spin- 1/2 fields. Phys. Rev. D., 30: 2573–2578, December 1984. 10.1103/PhysRevD.30.2573.
https://doi.org/10.1103/PhysRevD.30.2573
[40] Stellan Östlund and Stefan Rommer. Thermodynamic limit of density matrix renormalization. Phys. Rev. Lett., 75: 3537–3540, Nov 1995. 10.1103/PhysRevLett.75.3537.
https://doi.org/10.1103/PhysRevLett.75.3537
[41] Leonard Parker and David Toms. Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge University Press, Cambridge, 2009. 10.1017/CBO9780511813924.
https://doi.org/10.1017/CBO9780511813924
[42] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix product state representations. Quantum Info. Comput., 7 (5): 401–430, July 2007. ISSN 1533-7146. URL http://dl.acm.org/citation.cfm?id=2011832.2011833.
http://dl.acm.org/citation.cfm?id=2011832.2011833
[43] Michael E. Peskin and Daniel V. Schroeder. An Introduction to Quantum Field Theory. Westview Press, Boulder, Colarado, 1995. 10.1201/9780429503559.
https://doi.org/10.1201/9780429503559
[44] Eric Poisson, Adam Pound, and Ian Vega. The motion of point particles in curved spacetime. Living Reviews in Relativity, 14 (1): 7, Sep 2011. ISSN 1433-8351. 10.12942/lrr-2011-7.
https://doi.org/10.12942/lrr-2011-7
[45] Frans Pretorius. Evolution of binary black-hole spacetimes. Phys. Rev. Lett., 95 (12): 121101, 2005a. 10.1103/PhysRevLett.95.121101.
https://doi.org/10.1103/PhysRevLett.95.121101
[46] Frans Pretorius. Numerical relativity using a generalized harmonic decomposition. Class. Quant. Grav., 22 (2): 425, 2005b. 10.1088/0264-9381/22/2/014.
https://doi.org/10.1088/0264-9381/22/2/014
[47] Stefan Rommer and Stellan Östlund. Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group. Phys. Rev. B, 55: 2164–2181, Jan 1997. 10.1103/PhysRevB.55.2164.
https://doi.org/10.1103/PhysRevB.55.2164
[48] Katsuhiko Sato. First-order phase transition of a vacuum and the expansion of the Universe. Monthly Notices of the Royal Astronomical Society, 195 (3): 467–479, 07 1981. ISSN 0035-8711. 10.1093/mnras/195.3.467.
https://doi.org/10.1093/mnras/195.3.467
[49] U. Schollwöck. The density-matrix renormalization group. Rev. Mod. Phys., 77: 259–315, Apr 2005. 10.1103/RevModPhys.77.259.
https://doi.org/10.1103/RevModPhys.77.259
[50] Ulrich Schollwöck. The density-matrix renormalization group in the age of matrix product states. Annals of Physics, 326 (1): 96 – 192, 2011. ISSN 0003-4916. https://doi.org/10.1016/j.aop.2010.09.012. URL http://www.sciencedirect.com/science/article/pii/S0003491610001752. January 2011 Special Issue.
https://doi.org/https://doi.org/10.1016/j.aop.2010.09.012
http://www.sciencedirect.com/science/article/pii/S0003491610001752
[51] Leonard Susskind. Lattice fermions. Phys. Rev. D, 16 (10): 3031–3039, 1976. ISSN 1550-7998. 10.1103/physrevd.16.3031.
https://doi.org/10.1103/physrevd.16.3031
[52] W. G. Unruh. Notes on black-hole evaporation. Phys. Rev. D, 14: 870–892, Aug 1976. 10.1103/PhysRevD.14.870.
https://doi.org/10.1103/PhysRevD.14.870
[53] William G. Unruh and Nathan Weiss. Acceleration radiation in interacting field theories. Phys. Rev. D, 29: 1656–1662, Apr 1984. 10.1103/PhysRevD.29.1656.
https://doi.org/10.1103/PhysRevD.29.1656
[54] Guifré Vidal. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett., 91 (14): 147902, 2003. ISSN 0031-9007. 10.1103/physrevlett.91.147902.
https://doi.org/10.1103/physrevlett.91.147902
[55] Guifré Vidal. Efficient simulation of One-Dimensional quantum Many-Body systems. Phys. Rev. Lett., 93 (4): 040502, 2004. ISSN 0031-9007. 10.1103/physrevlett.93.040502.
https://doi.org/10.1103/physrevlett.93.040502
[56] Robert M Wald. General Relativity. University of Chicago Press, 1984. ISBN 9780226870328. 10.7208/chicago/9780226870373.001.0001.
https://doi.org/10.7208/chicago/9780226870373.001.0001
[57] Robert M Wald. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. The University of Chicago Press, Chicago, 1994.
[58] Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69: 2863–2866, Nov 1992. 10.1103/PhysRevLett.69.2863.
https://doi.org/10.1103/PhysRevLett.69.2863
[59] Steven R. White. Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B, 48: 10345–10356, Oct 1993. 10.1103/PhysRevB.48.10345. URL https://link.aps.org/doi/10.1103/PhysRevB.48.10345.
https://doi.org/10.1103/PhysRevB.48.10345
[60] Steven R. White and Adrian E. Feiguin. Real-time evolution using the density matrix renormalization group. Phys. Rev. Lett., 93: 076401, Aug 2004. 10.1103/PhysRevLett.93.076401.
https://doi.org/10.1103/PhysRevLett.93.076401
[61] Run-Qiu Yang, Hui Liu, Shining Zhu, Le Luo, and Rong-Gen Cai. Simulating quantum field theory in curved spacetime with quantum many-body systems. Phys. Rev. Research, 2: 023107, Apr 2020. 10.1103/PhysRevResearch.2.023107.
https://doi.org/10.1103/PhysRevResearch.2.023107
Cited by
[1] Adam G. M. Lewis, “Hadamard renormalization of a two-dimensional Dirac field”, Physical Review D 101 12, 125019 (2020).
[2] Yue-Zhou Li and Junyu Liu, “On Quantum Simulation Of Cosmic Inflation”, arXiv:2009.10921.
The above citations are from SAO/NASA ADS (last updated successfully 2020-10-28 10:51:27). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch Crossref cited-by data during last attempt 2020-10-28 10:51:25: Could not fetch cited-by data for 10.22331/q-2020-10-28-351 from Crossref. This is normal if the DOI was registered recently.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
Source: https://quantum-journal.org/papers/q-2020-10-28-351/
