Abstract
In this work we study the Sorkin-Johnston (SJ) vacuum in de Sitter spacetime for free scalar field theory. For the massless theory we find that the SJ vacuum can neither be obtained from the O(4) Fock vacuum of Allen and Folacci nor from the non-Fock de Sitter invariant vacuum of Kirsten and Garriga. Using a causal set discretisation of a slab of 2d and 4d de Sitter spacetime, we find the causal set SJ vacuum for a range of masses m ≥ 0 of the free scalar field. While our simulations are limited to a finite volume slab of global de Sitter spacetime, they show good convergence as the volume is increased. We find that the 4d causal set SJ vacuum shows a significant departure from the continuum Mottola-Allen α-vacua. Moreover, the causal set SJ vacuum is well-defined for both the minimally coupled massless m = 0 and the conformally coupled massless m = mc cases. This is at odds with earlier work on the continuum de Sitter SJ vacuum where it was argued that the continuum SJ vacuum is ill-defined for these masses. Our results hint at an important tension between the discrete and continuum behaviour of the SJ vacuum in de Sitter and suggest that the former cannot in general be identified with the Mottola-Allen α-vacua even for m > 0.
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References
S. Johnston, Feynman propagator for a free scalar field on a causal set, Phys. Rev. Lett.103 (2009) 180401 [arXiv:0909.0944] [INSPIRE].
R.D. Sorkin, Scalar field theory on a causal set in histories form, J. Phys. Conf. Ser.306 (2011) 012017 [arXiv:1107.0698] [INSPIRE].
S. Aslanbeigi and M. Buck, A preferred ground state for the scalar field in de Sitter space, JHEP08 (2013) 039 [arXiv:1306.3231] [INSPIRE].
E. Mottola, Particle creation in de Sitter space, Phys. Rev.D 31 (1985) 754 [INSPIRE].
B. Allen, Vacuum states in de Sitter space, Phys. Rev.D 32 (1985) 3136 [INSPIRE].
T. Garidi, What is mass in de Sitterian physics?, hep-th/0309104 [INSPIRE].
N. Afshordi, M. Buck, F. Dowker, D. Rideout, R.D. Sorkin and Y.K. Yazdi, A ground state for the causal diamond in 2 dimensions, JHEP10 (2012) 088 [arXiv:1207.7101] [INSPIRE].
C.J. Fewster and R. Verch, On a recent construction of ‘vacuum-like’ quantum field states in curved spacetime, Class. Quant. Grav.29 (2012) 205017 [arXiv:1206.1562] [INSPIRE].
M. Brum and K. Fredenhagen, ‘Vacuum-like’ Hadamard states for quantum fields on curved spacetimes, Class. Quant. Grav.31 (2014) 025024 [arXiv:1307.0482] [INSPIRE].
M. Buck, F. Dowker, I. Jubb and R. Sorkin, The Sorkin-Johnston state in a patch of the trousers spacetime, Class. Quant. Grav.34 (2017) 055002 [arXiv:1609.03573] [INSPIRE].
A. Mathur and S. Surya, The Sorkin-Johnston vacuum for a massive scalar field in the 2d causal diamond, arXiv:1906.07952 [INSPIRE].
N. Afshordi, S. Aslanbeigi and R.D. Sorkin, A distinguished vacuum state for a quantum field in a curved spacetime: formalism, features and cosmology, JHEP08 (2012) 137 [arXiv:1205.1296] [INSPIRE].
D.N. Page and X. Wu, Massless scalar field vacuum in de Sitter spacetime, JCAP11 (2012) 051 [arXiv:1204.4462] [INSPIRE].
B. Allen and A. Folacci, The massless minimally coupled scalar field in de Sitter space, Phys. Rev.D 35 (1987) 3771 [INSPIRE].
K. Kirsten and J. Garriga, Massless minimally coupled fields in de Sitter space: O(4) symmetric states versus de Sitter invariant vacuum, Phys. Rev.D 48 (1993) 567 [gr-qc/9305013] [INSPIRE].
N. X, F. Dowker and S. Surya, Scalar field Green functions on causal sets, Class. Quant. Grav.34 (2017) 124002 [arXiv:1701.07212] [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev.D 34 (1986) 373 [INSPIRE].
R.D. Sorkin, Expressing entropy globally in terms of (4D) field-correlations, J. Phys. Conf. Ser.484 (2014) 012004 [arXiv:1205.2953] [INSPIRE].
M. Saravani, R.D. Sorkin and Y.K. Yazdi, Spacetime entanglement entropy in 1 + 1 dimensions, Class. Quant. Grav.31 (2014) 214006 [arXiv:1311.7146] [INSPIRE].
R.D. Sorkin and Y.K. Yazdi, Entanglement entropy in causal set theory, Class. Quant. Grav.35 (2018) 074004 [arXiv:1611.10281] [INSPIRE].
F. Dowker, S. Surya, Nomaan X and Y.K. Yazdi, Entanglement entropy in a de Sitter causal set, in preparation (2019).
R.D. Sorkin, From Green function to quantum field, Int. J. Geom. Meth. Mod. Phys.14 (2017) 1740007 [arXiv:1703.00610] [INSPIRE].
S.P. Johnston, Quantum fields on causal sets, Ph.D. thesis, Imperial Coll., London, U.K. (2010) [arXiv:1010.5514] [INSPIRE].
N.A. Chernikov and E.A. Tagirov, Quantum theory of scalar fields in de Sitter space-time, Ann. Inst. H. Poincaré Phys. Theor.A 9 (1968) 109 [INSPIRE].
I.S. Gradshteyn and I.M. Ryshik, Table of integrals, series and products, 5th ed., Academic Press, London, U.K. (1994).
Wolfram Research Inc., Mathematica, version 11.1, U.S.A. (2017).
L. Bombelli, J. Lee, D. Meyer and R. Sorkin, Space-time as a causal set, Phys. Rev. Lett.59 (1987) 521 [INSPIRE].
S. Surya, Directions in causal set quantum gravity, arXiv:1103.6272 [INSPIRE].
F. Dowker, Causal sets and the deep structure of spacetime, in 100 years of relativity. Space-time structure: Einstein and beyond, A. Ashtekar ed., World Scientific, Singapore (2005), pg. 445 [gr-qc/0508109] [INSPIRE].
S. Johnston, Particle propagators on discrete spacetime, Class. Quant. Grav.25 (2008) 202001 [arXiv:0806.3083] [INSPIRE].
Y.K. Yazdi, Entanglement entropy of scalar fields in causal set theory, Ph.D. thesis, Waterloo U., Waterloo, ON, Canada (2017) [INSPIRE].
R. Bousso, A. Maloney and A. Strominger, Conformal vacua and entropy in de Sitter space, Phys. Rev.D 65 (2002) 104039 [hep-th/0112218] [INSPIRE].
M. Spradlin, A. Strominger and A. Volovich, Les Houches lectures on de Sitter space, in Unity from duality: gravity, gauge theory and strings. Proceedings, NATO Advanced Study Institute, Euro Summer School, 76thsession, Les Houches, France, 30 July-31 August 2001, pg. 423 [hep-th/0110007] [INSPIRE].
J. Hartong, On problems in de Sitter spacetime physics: scalar fields, black holes and stability, Master’s thesis, Groningen U., Groningen, The Netherlands (2004) [INSPIRE].
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Surya, S., X, N. & Yazdi, Y.K. Studies on the SJ vacuum in de Sitter spacetime. J. High Energ. Phys. 2019, 9 (2019). https://doi.org/10.1007/JHEP07(2019)009
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DOI: https://doi.org/10.1007/JHEP07(2019)009