Abstract
We study a non-local scalar quantum field theory in flat spacetime derived from the dynamics of a scalar field on a causal set. We show that this non-local QFT contains a continuum of massive modes in any dimension. In 2 dimensions the Hamiltonian is positive definite and therefore the quantum theory is well-defined. In 4-dimensions, we show that the unstable modes of the non-local d’Alembertian are propagated via the so called Wheeler propagator and hence do not appear in the asymptotic states. In the free case studied here the continuum of massive mode are shown to not propagate in the asymptotic states. However the Hamiltonian is not positive definite, therefore potential issues with the quantum theory remain. Finally, we conclude with hints toward what kind of phenomenology one might expect from such non-local QFTs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Liberati, Tests of Lorentz invariance: a 2013 update, Class. Quant. Grav. 30 (2013) 133001 [arXiv:1304.5795] [INSPIRE].
D. Mattingly, Modern tests of Lorentz invariance, Living Rev. Rel. 8 (2005) 5 [gr-qc/0502097] [INSPIRE].
G. Amelino-Camelia, Quantum-spacetime phenomenology, Living Rev. Rel. 16 (2013) 5 [arXiv:0806.0339] [INSPIRE].
S. Hossenfelder, Minimal length scale scenarios for quantum gravity, Living Rev. Rel. 16 (2013) 2 [arXiv:1203.6191] [INSPIRE].
S. Gielen and D. Oriti, Quantum cosmology from quantum gravity condensates: cosmological variables and lattice-refined dynamics, New J. Phys. 16 (2014) 123004 [arXiv:1407.8167] [INSPIRE].
A. Ashtekar and P. Singh, Loop quantum cosmology: a status report, Class. Quant. Grav. 28 (2011) 213001 [arXiv:1108.0893] [INSPIRE].
A. Barrau, C. Rovelli and F. Vidotto, Fast radio bursts and white hole signals, Phys. Rev. D 90 (2014) 127503 [arXiv:1409.4031] [INSPIRE].
M. Niedermaier and M. Reuter, The asymptotic safety scenario in quantum gravity, Living Rev. Rel. 9 (2006) 5 [INSPIRE].
S. Surya, Directions in causal set quantum gravity, arXiv:1103.6272 [INSPIRE].
J. Henson, The causal set approach to quantum gravity, in Approaches to quantum gravity. Towards a new understanding of space and time, D. Oriti ed., Cambridge University Press (2006) [gr-qc/0601121] [INSPIRE].
D.M.T. Benincasa and F. Dowker, The scalar curvature of a causal set, Phys. Rev. Lett. 104 (2010) 181301 [arXiv:1001.2725] [INSPIRE].
R.D. Sorkin, Does locality fail at intermediate length-scales, in Approaches to quantum gravity. Towards a new understanding of space and time, D. Oriti ed., Cambridge University Press (2006) [gr-qc/0703099] [INSPIRE].
S. Aslanbeigi, M. Saravani and R.D. Sorkin, Generalized causal set d’Alembertians, JHEP 06 (2014) 024 [arXiv:1403.1622] [INSPIRE].
L. Glaser, A closed form expression for the causal set d’Alembertian, Class. Quant. Grav. 31 (2014) 095007 [arXiv:1311.1701] [INSPIRE].
F. Dowker and L. Glaser, Causal set d’Alembertians for various dimensions, Class. Quant. Grav. 30 (2013) 195016 [arXiv:1305.2588] [INSPIRE].
S. Johnston, Correction terms for propagators and d’Alembertians due to spacetime discreteness, arXiv:1411.2614 [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, JHEP 08 (2012) 137 [arXiv:1205.1296] [INSPIRE].
D.G. Barci, L.E. Oxman and M. Rocca, Canonical quantization of nonlocal field equations, Int. J. Mod. Phys. A 11 (1996) 2111 [hep-th/9503101] [INSPIRE].
D. Marolf, Emergent gravity requires (kinematic) non-locality, Phys. Rev. Lett. 114 (2015) 031104 [arXiv:1409.2509] [INSPIRE].
M. Jaccard, M. Maggiore and E. Mitsou, Nonlocal theory of massive gravity, Phys. Rev. D 88 (2013) 044033 [arXiv:1305.3034] [INSPIRE].
D.G. Barci, C.G. Bollini, L.E. Oxman and M.C. Rocca, Nonlocal pseudodifferential operators, hep-th/9606183 [INSPIRE].
N. Barnaby and N. Kamran, Dynamics with infinitely many derivatives: the initial value problem, JHEP 02 (2008) 008 [arXiv:0709.3968] [INSPIRE].
R.L.P.G. do Amaral and E.C. Marino, Canonical quantization of theories containing fractional powers of the d’Alembertian operator, J. Phys. A 25 (1992) 5183 [INSPIRE].
C.G. Bollini and J.J. Giambiagi, Lagrangian procedures for higher order field equations, Rev. Bras. Fis. 17 (1987) 14 [INSPIRE].
I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, series, and products, 7th ed., Academic Press, San Diego U.S.A. (2007).
O.W. Greenberg, Generalized free fields and models of local field theory, Annals Phys. 16 (1961) 158 [INSPIRE].
C.G. Bollini and L.E. Oxman, Propagator for complex mass fields, Int. J. Mod. Phys. A 7 (1992) 6845 [INSPIRE].
J.A. Wheeler and R.P. Feynman, Interaction with the absorber as the mechanism of radiation, Rev. Mod. Phys. 17 (1945) 157 [INSPIRE].
A. Gonzalez Domínguez and S.E. Trione, On the Laplace transforms of retarded, Lorentz invariant functions, Adv. Math. 31 (1979) 51 [INSPIRE].
C.G. Bollini and L.E. Oxman, Unitarity and complex mass fields, Int. J. Mod. Phys. A 8 (1993) 3185 [INSPIRE].
S. Johnston, Feynman propagator for a free scalar field on a causal set, Phys. Rev. Lett. 103 (2009) 180401 [arXiv:0909.0944] [INSPIRE].
A. Belenchia, D. Benincasa and S. Liberati, in preparation.
F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark eds., NIST handbook of mathematical functions, Cambridge University Press, New York U.S.A. (2010).
D.G. Barci and L.E. Oxman, Asymptotic states in nonlocal field theories, Mod. Phys. Lett. A 12 (1997) 493 [hep-th/9611147] [INSPIRE].
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: 1411.6513
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
Belenchia, A., Benincasa, D.M.T. & Liberati, S. Nonlocal scalar quantum field theory from causal sets. J. High Energ. Phys. 2015, 36 (2015). https://doi.org/10.1007/JHEP03(2015)036
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2015)036