Abstract
We study the relationship between the quantization of a massless scalar field on the two-dimensional Einstein cylinder and in a spacetime with a time machine. We find that the latter picks out a unique prescription for the state of the zero mode in the Einstein cylinder. We show how this choice arises from the computation of the vacuum Wightman function and the vacuum renormalized stress-energy tensor in the time-machine geometry. Finally, we relate the previously proposed regularization of the zero mode state as a squeezed state with the time-machine warp parameter, thus demonstrating that the quantization in the latter regularizes the quantization in an Einstein cylinder.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Uribe, The first time machine: Enrique Gaspar’s anacronópete, N. Y. Rev. Sci. Fict. 11 (1999) 3565.
C. Sagan, Contact, Touchstone by Simon & Schuster (1997).
K. Thorne, The Science of Interstellar, Norton & Company (2018).
S. Carruth, Primer, StudioCanal (2004).
R. Zemeckis, Back to the future, Amblin Entertainment/Universal Pictures (1985).
K. Godel, An Example of a new type of cosmological solutions of Einstein’s field equations of graviation, Rev. Mod. Phys. 21 (1949) 447 [INSPIRE].
M. S. Morris, K. S. Thorne and U. Yurtsever, Wormholes, Time Machines, and the Weak Energy Condition, Phys. Rev. Lett. 61 (1988) 1446 [INSPIRE].
J. Friedman et al., Cauchy problem in space-times with closed timelike curves, Phys. Rev. D 42 (1990) 1915 [INSPIRE].
V. P. Frolov and I. D. Novikov, Physical Effects in Wormholes and Time Machine, Phys. Rev. D 42 (1990) 1057 [INSPIRE].
V. P. Frolov, Vacuum polarization in a locally static multiply connected space-time and a time machine problem, Phys. Rev. D 43 (1991) 3878 [INSPIRE].
F. Echeverria, G. Klinkhammer and K. S. Thorne, Billiard balls in wormhole space-times with closed timelike curves: Classical theory, Phys. Rev. D 44 (1991) 1077 [INSPIRE].
J. L. Friedman and M. S. Morris, The Cauchy problem for the scalar wave equation is well defined on a class of space-times with closed timelike curves, Phys. Rev. Lett. 66 (1991) 401 [INSPIRE].
P. F. González-Díaz, Kinks, energy conditions and closed timelike curves, Int. J. Mod. Phys. D 9 (2000) 531 [gr-qc/0007065] [INSPIRE].
H. D. Politzer, Path integrals, density matrices, and information flow with closed timelike curves, Phys. Rev. D 49 (1994) 3981 [gr-qc/9310027] [INSPIRE].
J. L. Friedman and M. S. Morris, Existence and uniqueness theorems for massless fields on a class of space-times with closed timelike curves, Commun. Math. Phys. 186 (1997) 495 [gr-qc/9411033] [INSPIRE].
M. Ringbauer, M. A. Broome, C. R. Myers, A. G. White and T. C. Ralph, Experimental simulation of closed timelike curves, Nature Commun. 5 (2014) 4145.
D. D. Solnyshkov and G. Malpuech, Analog time machine in a photonic system, Phys. Rev. B 103 (2021) 054303 [arXiv:2011.11114] [INSPIRE].
D. Deutsch, Quantum mechanics near closed timelike lines, Phys. Rev. D 44 (1991) 3197 [INSPIRE].
S. W. Kim and K. S. Thorne, Do vacuum fluctuations prevent the creation of closed timelike curves?, Phys. Rev. D 43 (1991) 3929 [INSPIRE].
G. Klinkhammer, Vacuum polarization of scalar and spinor fields near closed null geodesics, Phys. Rev. D 46 (1992) 3388 [INSPIRE].
S. V. Krasnikov, On the quantum stability of the time machine, Phys. Rev. D 54 (1996) 7322 [gr-qc/9508038] [INSPIRE].
R. Banach and J. S. Dowker, The Vacuum Stress Tensor for Automorphic Fields on Some Flat Space-times, J. Phys. A 12 (1979) 2545 [INSPIRE].
R. Banach, The Quantum Theory of Free Automorphic Fields, J. Phys. A 13 (1980) 2179 [INSPIRE].
R. Banach and J. S. Dowker, Automorphic field theory: Some mathematical issues, J. Phys. A 12 (1979) 2527 [INSPIRE].
J. S. Dowker, Quantum mechanics and field theory on multiply connected and on homogeneous spaces, J. Phys. A 5 (1972) 936 [INSPIRE].
E. Martín-Martínez and J. Louko, Particle detectors and the zero mode of a quantum field, Phys. Rev. D 90 (2014) 024015 [arXiv:1404.5621] [INSPIRE].
E. Tjoa and E. Martín-Martínez, Zero mode suppression of superluminal signals in light-matter interactions, Phys. Rev. D 99 (2019) 065005 [arXiv:1811.02036] [INSPIRE].
E. Tjoa and E. Martín-Martínez, Vacuum entanglement harvesting with a zero mode, Phys. Rev. D 101 (2020) 125020 [arXiv:2002.11790] [INSPIRE].
D. N. Page and X. Wu, Massless Scalar Field Vacuum in de Sitter Spacetime, JCAP 11 (2012) 051 [arXiv:1204.4462] [INSPIRE].
V. Toussaint and J. Louko, Detecting the massive bosonic zero-mode in expanding cosmological spacetimes, Phys. Rev. D 103 (2021) 105011 [arXiv:2102.04284v1] [INSPIRE].
S.-Y. Lin, C.-H. Chou and B.-L. Hu, Entanglement Dynamics of Detectors in an Einstein Cylinder, JHEP 03 (2016) 047 [arXiv:1508.06221] [INSPIRE].
S. Robles and J. Rodríguez-Laguna, Local quantum thermometry using Unruh-DeWitt detectors, J. Stat. Mech. Theory Exp. 2017 (2017) 033105.
W. G. Brenna, R. B. Mann and E. Martín-Martínez, Anti-Unruh Phenomena, Phys. Lett. B 757 (2016) 307 [arXiv:1504.02468] [INSPIRE].
D. Braun, Entanglement from thermal blackbody radiation, Phys. Rev. A 72 (2005) 062324.
K. Lorek, D. Pecak, E. G. Brown and A. Dragan, Extraction of genuine tripartite entanglement from the vacuum, Phys. Rev. A 90 (2014) 032316 [arXiv:1405.4449] [INSPIRE].
S. Hawking and G. Ellis, The Large Scale Structure of Space-Time, in Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1973).
L.-X. Li and J. R. Gott III, A selfconsistent vacuum for Misner space and the chronology protection conjecture, Phys. Rev. Lett. 80 (1998) 2980 [gr-qc/9711074] [INSPIRE].
R. Emparan and M. Tomašević, Holography of time machines, arXiv:2107.14200 [INSPIRE].
S. V. Krasnikov, Topology change without any pathology, Gen. Rel. Grav. 27 (1995) 529 [INSPIRE].
R. H. Jonsson, D. Q. Aruquipa, M. Casals, A. Kempf and E. Martín-Martínez, Communication through quantum fields near a black hole, Phys. Rev. D 101 (2020) 125005 [arXiv:2002.05482] [INSPIRE].
P. Simidzija, A. Ahmadzadegan, A. Kempf and E. Martín-Martínez, Transmission of quantum information through quantum fields, Phys. Rev. D 101 (2020) 036014 [arXiv:1908.07523] [INSPIRE].
N. Birrell, N. Birrell and P. Davies, Quantum Fields in Curved Space, in Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1984).
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, in Applied Mathematics Series, U.S. Department of Commerce, National Bureau of Standards (1972).
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, San Diego CA U.S.A. (2014).
E. W. Weisstein, Jacobi Theta Functions, (2008) https://mathworld.wolfram.com/JacobiThetaFunctions.html.
H. McKean and V. Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press, Cambridge U.K. (1997) [https://doi.org/10.1017/CBO9781139174879].
NIST, NIST Digital Library of Mathematical Functions, (2020), section 20.1 https://dlmf.nist.gov/.
P. Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, in Graduate Texts in Contemporary Physics, Springer, New York NY U.S.A. (2012).
S. V. Sushkov, Vacuum polarization of a complex automorphic scalar field in two-dimensional space-time with closed null geodesics and the time machine problem, Theor. Math. Phys. 102 (1995) 98 [Teor. Mat. Fiz. 102 (1995) 134] [INSPIRE].
P. Breitenlohner and D. Z. Freedman, Stability in gauged extended supergravity, Ann. Phys. 144 (1982) 249.
C. Dappiaggi and H. R. C. Ferreira, Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions, Phys. Rev. D 94 (2016) 125016 [arXiv:1610.01049] [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: 2108.07274
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
Alonso-Serrano, A., Tjoa, E., Garay, L.J. et al. The time traveler’s guide to the quantization of zero modes. J. High Energ. Phys. 2021, 170 (2021). https://doi.org/10.1007/JHEP12(2021)170
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)170