- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Impact of nonlinear effective interactions on group field theory quantum gravity condensates

Andreas G. A. Pithis* and Mairi Sakellariadou†

Petar Tomov‡

  • Department of Physics, King’s College London, University of London, Strand, London WC2R 2LS, United Kingdom

  • Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany

  • *andreas.pithis@kcl.ac.uk
  • †mairi.sakellariadou@kcl.ac.uk
  • ‡tomov@mathematik.hu-berlin.de
PDF

Phys. Rev. D 94, 064056 – Published 20 September, 2016

DOI: https://doi.org/10.1103/PhysRevD.94.064056

Abstract

We present the numerical analysis of effectively interacting group field theory models in the context of the group field theory quantum gravity condensate analog of the Gross-Pitaevskii equation for real Bose-Einstein condensates including combinatorially local interaction terms. Thus, we go beyond the usually considered construction for free models. More precisely, considering such interactions in a weak regime, we find solutions for which the expectation value of the number operator N is finite, as in the free case. When tuning the interaction to the strongly nonlinear regime, however, we obtain solutions for which N grows and eventually blows up, which is reminiscent of what one observes for real Bose-Einstein condensates, where a strong interaction regime can only be realized at high density. This behavior suggests the breakdown of the Bogoliubov ansatz for quantum gravity condensates and the need for non-Fock representations to describe the system when the condensate constituents are strongly correlated. Furthermore, we study the expectation values of certain geometric operators imported from loop quantum gravity in the free and interacting cases. In particular, computing solutions around the nontrivial minima of the interaction potentials, one finds, already in the weakly interacting case, a nonvanishing condensate population for which the spectra are dominated by the lowest nontrivial configuration of the quantum geometry. This result indicates that the condensate may indeed consist of many smallest building blocks giving rise to an effectively continuous geometry, thus suggesting the interpretation of the condensate phase to correspond to a geometric phase.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (38)

  1. T. Konopka, F. Markopoulou, and L. Smolin, Quantum graphity, arXiv:hep-th/0611197.
  2. R. De Pietri, L. Freidel, K. Krasnov, and C. Rovelli, Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space, Nucl. Phys. B574, 785 (2000); L. Freidel, Group field theory: An overview, Int. J. Theor. Phys. 44, 1769 (2005); D. Oriti, Approaches to Quantum Gravity, edited by D. Oriti (Cambridge University Press, Cambridge, 2009); D. OritiFoundations of Space and Time, edited by G. Ellis et al. (Cambridge University Press, Cambridge, 2012); A. Baratin and D. Oriti, Ten questions on group field theory (and their tentative answers), J. Phys. Conf. Ser. 360, 012002 (2012); T. Krajewski, Group field theories, Proc. Sci., QGQGS2011 (2011) 005.
  3. D. Oriti, Group field theory as the microscopic description of the quantum spacetime fluid: A new perspective on the continuum in quantum gravity, Proc. Sci., QG-PH (2007) 030; Disappearance and emergence of space and time in quantum gravity, Stud. Hist. Phil. Mod. Phys., 46, 186 (2014).
  4. S. Gielen, D. Oriti, and L. Sindoni, Cosmology from Group Field Theory Formalism for Quantum Gravity, Phys. Rev. Lett. 111, 031301 (2013); Homogeneous cosmologies as group field theory condensates, J. High Energy Phys. 06 (2014) 013.
  5. S. Gielen, Quantum cosmology of (loop) quantum gravity condensates: An example, Classical Quantum Gravity 31, 155009 (2014); G. Calcagni, Loop quantum cosmology from group field theory, Phys. Rev. D 90, 064047 (2014).
  6. S. Gielen, Perturbing a quantum gravity condensate, Phys. Rev. D 91, 043526 (2015); Identifying cosmological perturbations in group field theory condensates, J. High Energy Phys. 08 (2015) 010; S. Gielen and D. Oriti, Quantum cosmology from quantum gravity condensates: Cosmological variables and lattice-refined dynamics, New J. Phys. 16, 123004 (2014); L. Sindoni, Effective equations for GFT condensates from fidelity, arXiv:1408.3095.
  7. D. Oriti, L. Sindoni, and E. Wilson-Ewing, Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates, arXiv:1602.05881.
  8. D. Oriti, L. Sindoni, and E. Wilson-Ewing, Bouncing cosmologies from quantum gravity condensates, arXiv:1602.08271; M. de Cesare and M. Sakellariadou, Accelerated expansion of the Universe without an inflaton and resolution of the initial singularity from GFT condensates, arXiv:1603.01764.
  9. S. Gielen, Emergence of a Low Spin Phase in Group Field Theory Condensates, arXiv:1604.06023.
  10. M. de Cesare, A. G. A. Pithis, and M. Sakellariadou, Cosmological implications of interacting Group Field Theory models: cyclic Universe and accelerated expansion, arXiv:1606.00352.
  11. S. Gielen and L. Sindoni, Quantum cosmology from group field theory condensates: A review, SIGMA 12, 082 (2016).
  12. T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press, Cambridge, 2007); C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, 2007).
  13. C. Rovelli, Zakopane lectures on loop gravity, Proc. Sci., QGQGS2011 (2011) 003; A. Perez, The spin foam approach to quantum gravity, Living Rev. Relativ. 16, 3 (2013); C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2014).
  14. M. P. Reisenberger and C. Rovelli, Spacetime as a Feynman diagram: The connection formulation, Classical Quantum Gravity 18, 121 (2001).
  15. M. Han and M. Zhang, Asymptotics of spinfoam amplitude on simplicial manifold: Euclidean theory, Classical Quantum Gravity 29, 165004 (2012); A. Baratin and D. Oriti, Group Field Theory with Non-Commutative Metric Variables, Phys. Rev. Lett. 105, 221302 (2010).
  16. B. Delamotte, An introduction to the nonperturbative renormalization group, Lect. Notes Phys. 852, 49 (2012); C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993); J. Berges, N. Tetradis, and C. Wetterich, Non-perturbative renormalization flow in quantum field theory and statistical physics, Phys. Rep. 363, 223 (2002); T. R. Morris, The exact renormalisation group and approximate solutions, Int. J. Mod. Phys. A 09, 2411 (1994); A. Wipf, Statistical Approach to Quantum Field Theory: An Introduction (Springer, Berlin, 2013).
  17. P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin, 2D gravity and random matrices, Phys. Rep. 254, 1 (1995).
  18. E. Brezin and J. Zinn-Justin, Renormalization group approach to matrix models, Phys. Lett. B 288, 54 (1992); A. Eichhorn and T. Koslowski, Continuum limit in matrix models for quantum gravity from the functional renormalization group, Phys. Rev. D 88, 084016 (2013); Towards phase transitions between discrete and continuum quantum spacetime from the renormalization group, 90, 104039 (2014).
  19. D. Benedetti, J. Ben Geloun, and D. Oriti, Functional renormalisation group approach for tensorial group field theory: a rank-3 model, J. High Energy Phys. 03 (2015) 084; J. Ben Geloun, R. Martini, and D. Oriti, Functional renormalisation group analysis of a tensorial group field theory on R3, Europhys. Lett. 112, 31001 (2015); Functional renormalisation group analysis of tensorial group field theories on Rd, Phys. Rev. D 94, 024017 (2016); D. Benedetti and V. Lahoche, Functional renormalization group approach for tensorial group field theory: A rank-6 model with closure constraint, Classical Quantum Gravity 33, 095003 (2016).
  20. S. Carrozza, Flowing in group field theory space: A review, SIGMA 12, 070 (2016).
  21. V. Rivasseau, Why are tensor field theories asymptotically free?, Europhys. Lett. 111, 60011 (2015).
  22. L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, 1st ed. (Clarendon, Oxford, 2003); P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonzlez (eds.), Emergent Nonlinear Phenomena in Bose-Einstein Condensates Theory and Experiment (Springer, Berlin, 2008).
  23. N. Bogoliubov, On the theory of superfluidity, J. Phys. (Moscow) 11, 23 (1947); S. Beliaev, Energy spectrum of a non-ideal Bose gas, Sov. Phys. JETP 34, 289 (1958); N. Hugenholtz and D. Pines, Ground-state energy and excitation spectrum of a system of interacting bosons, Phys. Rev. 116, 489 (1959).
  24. H. Ooguri, Topological lattice models in four-dimensions, Mod. Phys. Lett. A 07, 2799 (1992).
  25. L. Freidel and K. Krasnov, A new spin foam model for 4d gravity, Classical Quantum Gravity 25, 125018 (2008); J. Engle, E. Livine, R. Pereira, and C. Rovelli, LQG vertex with finite Immirzi parameter, Nucl. Phys. B799, 136 (2008); J. Ben Geloun, R. Gurau, and V. Rivasseau, EPRL/FK group field theory, Europhys. Lett. 92, 60008 (2010); A. Baratin and D. Oriti, Quantum simplicial geometry in the group field theory formalism: Reconsidering the Barrett-Crane model, New J. Phys. 13, 125011 (2011); W. Kaminksi, M. Kisielowski, and J. Lewandowski, Spin-foams for all loop quantum gravity, Classical Quantum Gravity 27, 095006 (2010); 29, 049502(E) (2012).
  26. D. Oriti, J. P. Ryan, and J. Thürigen, Group field theories for all loop quantum gravity, New J. Phys. 17, 023042 (2015).
  27. D. Oriti, Group field theory as the 2nd quantization of Loop Quantum Gravity, Classical Quantum Gravity 33, 085005 (2016).
  28. D. Oriti, Loop Quantum Gravity, 100 Years of General Relativity, edited by A. Ashtekar and J. Pullin (World Scientific, Singapore, 2014).
  29. F. Strocchi, Symmetry Breaking (Springer, Berlin, 2008); F. StrocchiAn Introduction to Non-Perturbative Foundations of QFT (Oxford University, New York, 2013); N. N. Bogolubov, A. A. Logunov, A. I. Oksak, and I. Todorov, General Principles of Quantum Field Theory (Springer, Berlin, 1990); R. Haag, Local Quantum Physics: Fields, Particles, Algebras (Springer, Berlin, 1996); G. F. Dell’Antonio and S. Doplicher, Total number of particles and Fock representation, J. Math. Phys. (N.Y.) 8, 663 (1967); J. M. Chaiken, Number operators for representations of the canonical comutation relations, Commun. Math. Phys. 8, 164 (1968); Finite-particle representations and states of the canonical commutation relations, Ann. Phys. (N.Y.) 42, 23 (1967).
  30. D. Oriti, D. Pranzetti, J. P. Ryan, and L. Sindoni, Generalized quantum gravity condensates for homogeneous geometries and cosmology, Classical Quantum Gravity 32, 235016 (2015).
  31. A. Ashtekar and P. Singh, Loop quantum cosmology: A status report, Classical Quantum Gravity 28, 213001 (2011); K. Banerjee, G. Calcagni, and M. Martin-Benito, Introduction to loop quantum cosmology, SIGMA 8, 016 (2012).
  32. C. Kiefer, Quantum Gravity, 3rd ed. (Oxford University, New York, 2012).
  33. J. Faraut, Analysis on Lie Groups, An Introduction (Cambridge University Press, Cambridge, 2008).
  34. S. Carrozza, Group field theory in dimension four minus epsilon, Phys. Rev. D 91, 065023 (2015).
  35. T. Gollisch and C. Wetterich, Equation of state for helium-4 from microphysics, Phys. Rev. B 65, 134506 (2002); C. Wetterich, Functional renormalization for quantum phase transitions with non relativistic bosons, 77, 064504 (2008); S. Floerchinger and C. Wetterich, Functional renormalization for Bose-Einstein Condensation, Phys. Rev. A 77, 053603 (2008). N. Dupuis and K. Sengupta, Non-perturbative renormalization group approach to zero-temperature Bose systems, Europhys. Lett. 80, 50007 (2007); C. Castellani, C. Di Castro, F. Pistolesi, and G. C. Strinati, Infrared Behavior of Interacting Bosons at Zero Temperature, Phys. Rev. Lett. 78, 1612 (1997); F. Pistolesi, C. Castellani, C. D. Castro, and G. C. Strinati, Renormalization-group approach to the infrared behavior of a zero-temperature Bose system, Phys. Rev. B 69, 024513 (2004);
  36. R. J. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev. 131, 2766 (1963); J. P. Provost, F. Rocca, and G. Vallee, Coherent states, phase states and condensed states, Ann. Phys. (N.Y.) 94, 307 (1975); J. P. Provost, F. Rocca, G. Vallee, and M. Sirugue, Phase properties of some photon states with nonzero energy density, J. Math. Phys. (N.Y.) 15, 2079 (1974); J. P. Provost, F. Rocca, and G. Vallee, Phase operator and phase transition in a classical Dicke model, Ann. Phys. (N.Y.) 107, 168 (1977); R. Honegger and A. Rapp, General Glauber coherent states on the Weyl algebra and their phase integrals, Physica A (Amsterdam) 167, 945 (1990); R. Honegger and A. Rieckers, The general form of non-Fock coherent boson states, Publ. RIMS, Kyoto Univ. 26, 397 (1990); R. Honegger, The Dynamical Generation of Macroscopic Coherent Light, Appeared in Large-Scale Molecular Systems, edited by W. Gans et al. (Plenum, New York, 1991); The extremal microscopic coherent boson states, Lett. Math. Phys. 28, 155 (1993); R. Honegger and A. Rieckers, Photons in Fock Space and Beyond (World Scientific, Singapore, 2015), Vols. 1–3; J. R. Klauder and B. S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).
  37. B. Dittrich, 100 Years of General Relativity, edited by A. Ashtekar and J. Pullin (World Scientific, Singapore, 2014).
  38. A. Baratin, B. Dittrich, D. Oriti, and J. Tambornino, Non-commutative flux representation for loop quantum gravity, Classical Quantum Gravity 28, 175011 (2011); C. Guedes, D. Oriti, and M. Raasakka, Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups, J. Math. Phys. (N.Y.) 54, 083508 (2013).

Outline

Information

Phys. Rev. D 94, 064056– Published 20 September, 2016
  • Received 16 August 2016

© 2016 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation