Extended Hořava Gravity with Physical Ground-State Wavefunction
Abstract
:1. Introduction
- (i)
- The theory is power-counting renormalizable, in the sense that the candidate theory should be renormalizable in the UV regime;
- (ii)
- The theory has a healthy IR behavior, namely, the theory should be free of ghosts and does not have strong coupling;
- (iii)
- The theory can be well quantized in the sense that the theory has a stable vacuum state (physical ground state). Generally speaking, in accordance with the symmetry of the theory, the number of allowed terms in the action is large. As we will see in Section 5, to ensure the theory has a stable vacuum, the allowed terms in the actions should be carefully selected such that the probability density is finite and normalized. In other words, to have a physical ground state, the theory should have a finite (and normalized) probability density. To ensure there is a finite probability density, one has to select suitable terms in the action such that the Euclidean action of the theory is positive definite. We should emphasize that this selection is not unique. What we will adopt in the present paper is one of the choices.
2. Anisotropic Theory of Gravity
3. Power-Counting Renormalizable
4. IR Behavior
5. Quantization of the Theory
5.1. Brief Review of Stochastic Quantization
5.2. Quantization of BPS Model
5.3. Stochastic Quantization of Our Model
6. Conclusions and Discussions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Horava, P. Quantum Gravity at a Lifshitz Point. Phys. Rev. D 2009, 79, 084008. [Google Scholar] [CrossRef] [Green Version]
- Zhang, T.; Shu, F.W.; Tang, Q.W.; Du, D.H. Constraints on Hořava—Lifshitz gravity from GRB 170817A. Eur. Phys. J. C 2020, 80, 1062. [Google Scholar] [CrossRef]
- Albert, A.; Alfaro, R.; Alvarez, C.; Angeles Camacho, J.R.; Arteaga-Velazquez, J.C.; Arunbabu, K.P.; Avila Rojas, D.; Ayala Solares, H.A.; Baghmanyan, V.; Belmont-Moreno, E.; et al. Constraints on Lorentz Invariance Violation from HAWC Observations of Gamma Rays above 100 TeV. Phys. Rev. Lett. 2020, 124, 131101. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Acciari, V.A.; Ansoldi, S.; Antonelli, L.A.; Engels, A.A.; Baack, D.; Babić, A.; Banerjee, B.; de Almeida, U.B.; Barrio, J.A.; González, J.B.; et al. Bounds on Lorentz invariance violation from MAGIC observation of GRB 190114C. Phys. Rev. Lett. 2020, 125, 021301. [Google Scholar] [CrossRef]
- Barvinsky, A.O.; Blas, D.; Herrero-Valea, M.; Sibiryakov, S.M.; Steinwachs, C.F. Renormalization of Hořava gravity. Phys. Rev. D 2016, 93, 064022. [Google Scholar] [CrossRef] [Green Version]
- Blas, D.; Pujolas, O.; Sibiryakov, S. Consistent Extension of Horava Gravity. Phys. Rev. Lett. 2010, 104, 181302. [Google Scholar] [CrossRef] [Green Version]
- Dutta, S.; Saridakis, E.N. Observational constraints on Horava-Lifshitz cosmology. J. Cosmol. Astropart. Phys. 2010, 1, 13. [Google Scholar] [CrossRef] [Green Version]
- Dutta, S.; Saridakis, E.N. Overall observational constraints on the running parameter λ of Horava-Lifshitz gravity. J. Cosmol. Astropart. Phys. 2010, 5, 13. [Google Scholar] [CrossRef] [Green Version]
- Blas, D.; Pujolas, O.; Sibiryakov, S. Models of non-relativistic quantum gravity: The Good, the bad and the healthy. J. High Energy Phys. 2011, 4, 18. [Google Scholar] [CrossRef] [Green Version]
- Barausse, E.; Jacobson, T.; Sotiriou, T.P. Black holes in Einstein-aether and Horava-Lifshitz gravity. Phys. Rev. D 2011, 83, 124043. [Google Scholar] [CrossRef] [Green Version]
- Yagi, K.; Blas, D.; Barausse, E.; Yunes, N. Constraints on Einstein-Æther theory and Hořava gravity from binary pulsar observations. Phys. Rev. D 2014, 89, 084067, Erratum in Phys. Rev. D 2014, 90, 069902; Erratum in Phys. Rev. D 2014, 90, 069901. [Google Scholar] [CrossRef] [Green Version]
- Bonetti, M.; Barausse, E. Post-Newtonian constraints on Lorentz-violating gravity theories with a MOND phenomenology. Phys. Rev. D 2015, 91, 084053, Erratum in Phys. Rev. D 2016, 93, 029901. [Google Scholar] [CrossRef] [Green Version]
- Gümrükçüoğlu, A.E.; Saravani, M.; Sotiriou, T.P. Hořava gravity after GW170817. Phys. Rev. D 2018, 97, 024032. [Google Scholar] [CrossRef] [Green Version]
- Gong, Y.; Hou, S.; Papantonopoulos, E.; Tzortzis, D. Gravitational waves and the polarizations in Hořava gravity after GW170817. Phys. Rev. D 2018, 98, 104017. [Google Scholar] [CrossRef] [Green Version]
- Ramos, O.; Barausse, E. Constraints on Hořava gravity from binary black hole observations. Phys. Rev. D 2019, 99, 024034. [Google Scholar] [CrossRef] [Green Version]
- Frusciante, N.; Benetti, M. Cosmological constraints on Hořava gravity revised in light of GW170817 and GRB170817A and the degeneracy with massive neutrinos. arXiv 2020, arXiv:2005.14705. [Google Scholar]
- Zhu, T.; Wu, Q.; Wang, A.; Shu, F.W. U(1) symmetry and elimination of spin-0 gravitons in Horava-Lifshitz gravity without the projectability condition. Phys. Rev. D 2011, 84, 101502. [Google Scholar] [CrossRef] [Green Version]
- Zhu, T.; Shu, F.W.; Wu, Q.; Wang, A. General covariant Horava-Lifshitz gravity without projectability condition and its applications to cosmology. Phys. Rev. D 2012, 85, 044053. [Google Scholar] [CrossRef] [Green Version]
- Shu, F.W.; Lin, K.; Wang, A.; Wu, Q. Lifshitz spacetimes, solitons, and generalized BTZ black holes in quantum gravity at a Lifshitz point. J. High Energy Phys. 2014, 4, 056. [Google Scholar] [CrossRef] [Green Version]
- Lin, K.; Shu, F.W.; Wang, A.; Wu, Q. High-dimensional Lifshitz-type spacetimes, universal horizons, and black holes in Hořava-Lifshitz gravity. Phys. Rev. D 2015, 91, 044003. [Google Scholar] [CrossRef] [Green Version]
- Frusciante, N.; Raveri, M.; Vernieri, D.; Hu, B.; Silvestri, A. Hořava Gravity in the Effective Field Theory formalism: From cosmology to observational constraints. Phys. Dark Univ. 2016, 13, 7–24. [Google Scholar] [CrossRef] [Green Version]
- Shin, S.; Park, M.I. On gauge invariant cosmological perturbations in UV-modified Hořava gravity. J. Cosmol. Astropart. Phys. 2017, 12, 33. [Google Scholar] [CrossRef] [Green Version]
- Paliathanasis, A.; Leon, G. Cosmological solutions in Hořava-Lifshitz gravity. arXiv 2019, arXiv:1903.10821. [Google Scholar]
- Bandyopadhyay, T.; Debnath, U. Bouncing cosmology for entropy corrected models in Hořava Lifshitz gravity and fractal universe. Eur. Phys. J. Plus 2020, 135, 613. [Google Scholar] [CrossRef]
- Hořava, P. Spectral dimension of the universe in quantum gravity at a Lifshitz point. Phys. Rev. Lett. 2009, 102, 161301. [Google Scholar] [CrossRef] [Green Version]
- Cai, R.-G.; Cao, L.-M.; Ohta, N. Topological Black Holes in Horava-Lifshitz Gravity. Phys. Rev. D 2009, 80, 024003. [Google Scholar] [CrossRef] [Green Version]
- Volovik, G.E. z = 3 Lifshitz-Horava model and Fermi-point scenario of emergent gravity. JETP Lett. 2009, 89, 525–528. [Google Scholar] [CrossRef] [Green Version]
- Piao, Y.S. Primordial Perturbation in Horava-Lifshitz Cosmology. Phys. Lett. B 2009, 681, 1–4. [Google Scholar] [CrossRef]
- Cai, R.G.; Hu, B.; Zhang, H.B. Dynamical Scalar Degree of Freedom in Horava-Lifshitz Gravity. Phys. Rev. D 2009, 80, 041501. [Google Scholar] [CrossRef] [Green Version]
- Ghodsi, A. Toroidal solutions in Horava Gravity. Int. J. Mod. Phys. A 2011, 26, 925–934. [Google Scholar] [CrossRef] [Green Version]
- Kobakhidze, A. On the infrared limit of Horava’s gravity with the global Hamiltonian constraint. Phys. Rev. D 2010, 82, 064011. [Google Scholar] [CrossRef] [Green Version]
- Wu, P.; Yu, H.W. Emergent universe from the Horava-Lifshitz gravity. Phys. Rev. D 2010, 81, 103522. [Google Scholar] [CrossRef] [Green Version]
- Boehmer, C.G.; Lobo, F.S.N. Stability of the Einstein static universe in IR modified Horava gravity. Eur. Phys. J. C 2010, 70, 1111–1118. [Google Scholar] [CrossRef]
- Park, M.I. Remarks on the Scalar Graviton Decoupling and Consistency of Horava Gravity. Class. Quant. Grav. 2011, 28, 015004. [Google Scholar] [CrossRef]
- Cai, R.G.; Wang, A. Singularities in Horava-Lifshitz theory. Phys. Lett. B 2010, 686, 166–174. [Google Scholar] [CrossRef] [Green Version]
- Chaichian, M.; Nojiri, S.; Odintsov, S.D.; Oksanen, M.; Tureanu, A. Modified F(R) Horava-Lifshitz gravity: A way to accelerating FRW cosmology. Class. Quant. Grav. 2010, 27, 185021, Erratum: Class. Quant. Grav. 2012, 29, 159501. [Google Scholar] [CrossRef]
- Huang, Y.; Wang, A.; Wu, Q. Stability of the de Sitter spacetime in Horava-Lifshitz theory. Mod. Phys. Lett. A 2010, 25, 2267–2279. [Google Scholar] [CrossRef] [Green Version]
- Bellorin, J.; Restuccia, A. On the consistency of the Horava Theory. Int. J. Mod. Phys. D 2012, 21, 1250029. [Google Scholar] [CrossRef] [Green Version]
- Cai, R.G.; Hu, B.; Zhang, H.B. Scalar graviton in the healthy extension of Hořava-Lifshitz theory. Phys. Rev. D 2011, 83, 084009. [Google Scholar] [CrossRef] [Green Version]
- Wang, A.; Wu, Q. Stability of spin-0 graviton and strong coupling in Horava-Lifshitz theory of gravity. Phys. Rev. D 2011, 83, 044025. [Google Scholar] [CrossRef] [Green Version]
- Kimpton, I.; Padilla, A. Matter in Horava-Lifshitz gravity. J. High Energy Phys. 2013, 4, 133. [Google Scholar] [CrossRef] [Green Version]
- Lopes, D.V.; Mamiya, A.; Pinzul, A. Infrared Horava–Lifshitz gravity coupled to Lorentz violating matter: A spectral action approach. Class. Quant. Grav. 2016, 33, 045008. [Google Scholar] [CrossRef]
- Ramazanov, S.; Arroja, F.; Celoria, M.; Matarrese, S.; Pilo, L. Living with ghosts in Hořava-Lifshitz gravity. J. High Energy Phys. 2016, 6, 020. [Google Scholar] [CrossRef] [Green Version]
- Bramberger, S.F.; Coates, A.; Magueijo, J.; Mukohyama, S.; Namba, R.; Watanabe, Y. Solving the flatness problem with an anisotropic instanton in Hořava-Lifshitz gravity. Phys. Rev. D 2018, 97, 043512. [Google Scholar] [CrossRef] [Green Version]
- Coates, A.; Melby-Thompson, C.; Mukohyama, S. Revisiting Lorentz violation in Hořava gravity. Phys. Rev. D 2019, 100, 064046. [Google Scholar] [CrossRef] [Green Version]
- Barvinsky, A.O.; Herrero-Valea, M.; Sibiryakov, S.M. Towards the renormalization group flow of Horava gravity in (3 + 1) dimensions. Phys. Rev. D 2019, 100, 026012. [Google Scholar] [CrossRef] [Green Version]
- Wang, A. Hořava gravity at a Lifshitz point: A progress report. Int. J. Mod. Phys. D 2017, 26, 1730014. [Google Scholar] [CrossRef] [Green Version]
- Charmousis, C.; Niz, G.; Padilla, A.; Saffin, P.M. Strong coupling in Horava gravity. J. High Energy Phys. 2009, 8, 070. [Google Scholar] [CrossRef] [Green Version]
- Blas, D.; Pujolas, O.; Sibiryakov, S. On the Extra Mode and Inconsistency of Horava Gravity. J. High Energy Phys. 2009, 10, 029. [Google Scholar] [CrossRef] [Green Version]
- Nastase, H. On IR solutions in Horava gravity theories. arXiv 2009, arXiv:0904.3604. [Google Scholar]
- Sotiriou, T.P.; Visser, M.; Weinfurtner, S. Phenomenologically viable Lorentz-violating quantum gravity. Phys. Rev. Lett. 2009, 102, 251601. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Kehagias, A.; Sfetsos, K. The Black hole and FRW geometries of non-relativistic gravity. Phys. Lett. B 2009, 678, 123–126. [Google Scholar] [CrossRef] [Green Version]
- Li, M.; Pang, Y. A Trouble with Horava-Lifshitz Gravity. J. High Energy Phys. 2009, 8, 015. [Google Scholar] [CrossRef] [Green Version]
- Farkas, S.; Martinec, E.J. Gravity from the Extension of Spatial Diffeomorphisms. J. Math. Phys. 2011, 52, 062501. [Google Scholar] [CrossRef] [Green Version]
- Blas, D.; Pujolàs, O.; Sibiryakov, S. Comment on ‘Strong coupling in extended Horava-Lifshitz gravity. Phys. Lett. B 2010, 688, 350. [Google Scholar] [CrossRef]
- Papazoglou, A.; Sotiriou, T.P. Strong coupling in extended Hořava-Lifshitz gravity. Phys. Lett. B 2010, 685, 197. [Google Scholar] [CrossRef] [Green Version]
- Parisi, G.; Wu, Y.S. Perturbation Theory without Gauge Fixing. Sci. Sin. 1981, 24, 483. [Google Scholar]
- Huffel, H.; Kelnhofer, G. QED revisited: Proving equivalence between path integral and stochastic quantization. Phys. Lett. B 2004, 588, 145–150. [Google Scholar] [CrossRef] [Green Version]
- Shu, F.W.; Wu, Y.S. Stochastic Quantization of Hořava gravity. arXiv 2009, arXiv:0906.1645. [Google Scholar]
- Joyce, A.; Jain, B.; Khoury, J.; Trodden, M. Beyond the Cosmological Standard Model. Phys. Rept. 2015, 568, 1–98. [Google Scholar] [CrossRef] [Green Version]
- Platania, A.; Wetterich, C. Non-perturbative unitarity and fictitious ghosts in quantum gravity. Phys. Lett. B 2020, 811, 135911. [Google Scholar] [CrossRef]
- Kostelecky, V.A. Gravity, Lorentz violation, and the standard model. Phys. Rev. D 2004, 69, 105009. [Google Scholar] [CrossRef] [Green Version]
- Kostelecky, A.V.; Tasson, J.D. Matter-gravity couplings and Lorentz violation. Phys. Rev. D 2011, 83, 016013. [Google Scholar] [CrossRef] [Green Version]
- Pospelov, M.; Shang, Y. On Lorentz violation in Horava-Lifshitz type theories. Phys. Rev. D 2012, 85, 105001. [Google Scholar] [CrossRef] [Green Version]
- Mattingly, D. Modern tests of Lorentz invariance. Living Rev. Rel. 2005, 8, 5. [Google Scholar] [CrossRef] [Green Version]
- Kostelecky, V.A.; Russell, N. Data Tables for Lorentz and CPT Violation. arXiv 2008, arXiv:0801.0287. [Google Scholar]
- Eichhorn, A.; Platania, A.; Schiffer, M. Lorentz invariance violations in the interplay of quantum gravity with matter. Phys. Rev. D 2020, 102, 026007. [Google Scholar] [CrossRef]
- Parisi, G.; Sourlas, N. Supersymmetric Field Theories and Stochastic Differential Equations. Nucl. Phys. B 1982, 206, 321–332. [Google Scholar] [CrossRef]
- Floratos, E.; Iliopoulos, J. Equivalence of Stochastic and Canonical Quantization in Perturbation Theory. Nucl. Phys. B 1983, 214, 392. [Google Scholar] [CrossRef]
- Rumpf, H. Stochastic Quantization of Einstein Gravity. Phys. Rev. D 1986, 33, 942. [Google Scholar] [CrossRef]
- Damgaard, P.H.; Huffel, H. Stochastic Quantization. Phys. Rept. 1987, 152, 227. [Google Scholar] [CrossRef] [Green Version]
- Fukai, T.; Okano, K. Stochastic Quantization of Linearized Euclidean Gravity and No Ghost Feynman Rules. Prog. Theor. Phys. 1985, 73, 790. [Google Scholar] [CrossRef] [Green Version]
- Orlando, D.; Reffert, S. On the Renormalizability of Horava-Lifshitz-type Gravities. Class. Quant. Grav. 2009, 26, 155021. [Google Scholar] [CrossRef] [Green Version]
- Jacobson, T. Extended Horava gravity and Einstein-aether theory. Phys. Rev. D 2010, 81, 101502, Erratum: Phys. Rev. D 2010, 82, 129901. [Google Scholar] [CrossRef] [Green Version]
- Bogdanos, C.; Saridakis, E.N. Perturbative instabilities in Horava gravity. Class. Quant. Grav. 2010, 27, 075005. [Google Scholar] [CrossRef] [Green Version]
- D’Odorico, G.; Saueressig, F.; Schutten, M. Asymptotic Freedom in Hořava-Lifshitz Gravity. Phys. Rev. Lett. 2014, 113, 171101. [Google Scholar] [CrossRef] [Green Version]
- Garattini, R.; Saridakis, E.N. Gravity’s Rainbow: A bridge towards Hořava–Lifshitz gravity. Eur. Phys. J. C 2015, 75, 343. [Google Scholar] [CrossRef]
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Shu, F.-W.; Zhang, T. Extended Hořava Gravity with Physical Ground-State Wavefunction. Symmetry 2021, 13, 100. https://doi.org/10.3390/sym13010100
Shu F-W, Zhang T. Extended Hořava Gravity with Physical Ground-State Wavefunction. Symmetry. 2021; 13(1):100. https://doi.org/10.3390/sym13010100
Chicago/Turabian StyleShu, Fu-Wen, and Tao Zhang. 2021. "Extended Hořava Gravity with Physical Ground-State Wavefunction" Symmetry 13, no. 1: 100. https://doi.org/10.3390/sym13010100