Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’
Abstract
:1. Introduction
2. The Flat FRW Model with Massless Scalar Field
2.1. Classical Relational Dynamics and Internal Time Switches
- (i)
- On , defined by and , we have
- (ii)
- The set is the shared boundary between and , as well as between and . Notice that orbits with are just points in so that the latter is stratified by gauge orbits of different dimension. Since for , no gauge-fixing surface can pierce every such gauge orbit once and only once.
2.2. Reduced Quantization Relative to a Choice of Internal Time
2.3. The Internal-Time-Neutral Dirac Quantization
2.4. Quantum Reduction: From Dirac to Reduced Quantization
2.5. Quantum Internal Time Switches
2.6. Illustration in Concrete States
3. Perspective on the ‘Wave Function of the Universe’
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Hermiticity of the Relational Observable on
Appendix B. Changes of Internal Times in the Quantum Theory
Appendix C. Continuity of the Quantum Relational Dynamics during a Switch
References
- Rovelli, C. Quantum Gravity; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Rovelli, C. What is observable in classical and quantum gravity? Class. Quant. Grav. 1991, 8, 297–316. [Google Scholar] [CrossRef]
- Rovelli, C. Quantum reference systems. Class. Quant. Grav. 1991, 8, 317–332. [Google Scholar] [CrossRef]
- Brown, J.D.; Kuchař, K.V. Dust as a standard of space and time in canonical quantum gravity. Phys. Rev. D 1995, 51, 5600–5629. [Google Scholar]
- Dittrich, B. Partial and complete observables for Hamiltonian constrained systems. Gen. Rel. Grav. 2007, 39, 1891–1927. [Google Scholar] [CrossRef]
- Dittrich, B. Partial and complete observables for canonical General Relativity. Class. Quant. Grav. 2006, 23, 6155–6184. [Google Scholar]
- Dittrich, B.; Tambornino, J. A perturbative approach to Dirac observables and their space-time algebra. Class. Quant. Grav. 2007, 24, 757–784. [Google Scholar] [CrossRef]
- Dittrichm, B.; Tambornino, J. Gauge invariant perturbations around symmetry reduced sectors of general relativity: Applications to cosmology. Class. Quant. Grav. 2007, 24, 4543–4586. [Google Scholar] [CrossRef]
- Tambornino, J. Relational observables in gravity: A review. Symmetry Integr. Geom. 2012, 8, 17–30. [Google Scholar]
- DeWitt, B.S. Quantum theory of gravity. I. The canonical theory. Phys. Rev. 1967, 160, 1113–1148. [Google Scholar] [CrossRef]
- Kuchař, K. Time and interpretations of quantum gravity. Int. J. Mod. Phys. Proc. Suppl. D 2011, 20, 3–86. [Google Scholar] [CrossRef]
- Isham, C. Canonical quantum gravity and the problem of time. In Integrable Systems, Quantum Groups, and Quantum Field Theories; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1993; pp. 157–287. [Google Scholar]
- Anderson, E. The Problem of Time; Springer International Publishing: Cham, Switzerland, 2017; Volume 190. [Google Scholar]
- Rovelli, C. Quantum mechanics without time: A model. Phys. Rev. D 1990, 42, 2638–2646. [Google Scholar] [CrossRef]
- Rovelli, C. Time in quantum gravity: Physics beyond the Schrödinger regime. Phys. Rev. D 1991, 43, 442–456. [Google Scholar] [CrossRef]
- Marolf, D. Almost ideal clocks in quantum cosmology: A brief derivation of time. Class. Quant. Grav. 1995, 12, 2469–2486. [Google Scholar] [CrossRef]
- Marolf, D. Solving the problem of time in minisuperspace: Measurement of Dirac observables. Phys. Rev. 1995, 12, 2469–2486. [Google Scholar]
- Dittrich, B.; Höhn, P.A.; Koslowski, T.A.; Nelson, M.I. Can chaos be observed in quantum gravity? Phys. Lett. B 2017, 769, 554–560. [Google Scholar] [CrossRef]
- Dittrich, B.; Höhn, P.A.; Koslowski, T.A.; Nelson, M.I. Chaos, Dirac observables and constraint quantization. arXiv 2015, arXiv:1508.01947. [Google Scholar]
- Bojowald, M.; Höhn, P.A.; Tsobanjan, A. An Effective approach to the problem of time. Class. Quant. Grav. 2011, 28, 035006. [Google Scholar] [CrossRef]
- Bojowald, M.; Höhn, P.A.; Tsobanjan, A. Effective approach to the problem of time: General features and examples. Phys. Rev. D 2011, 83, 125023. [Google Scholar] [CrossRef]
- Höhn, P.A.; Kubalova, E.; Tsobanjan, A. Effective relational dynamics of a nonintegrable cosmological model. Phys. Rev. D 2012, 86, 065014. [Google Scholar] [CrossRef]
- Thiemann, T. Modern Canonical Quantum General Relativity; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Bojowald, M. Canonical Gravity and Applications: Cosmology, Black Holes and Quantum Gravity; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Ashtekar, A.; Singh, P. Loop Quantum Cosmology: A status report. Class. Quant. Grav. 2011, 28, 213001. [Google Scholar] [CrossRef]
- Banerjee, K.; Calcagni, G.; Martín-Benito, M. Introduction to Loop Quantum Cosmology. Symmetry Integr. Geom. 2012, 8, 016. [Google Scholar] [CrossRef]
- Oriti, D.; Sindoni, L.; Wilson-Ewing, E. Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates. Class. Quant. Grav. 2016, 33, 224001. [Google Scholar] [CrossRef]
- Gielen, S. Emergence of a low spin phase in group field theory condensates. Class. Quant. Grav. 2016, 33, 224002. [Google Scholar] [CrossRef]
- Gielen, S. Group field theory and its cosmology in a matter reference frame. Universe 2018, 4, 103. [Google Scholar] [CrossRef]
- Blyth, W.; Isham, C. Quantization of a Friedmann universe filled with a scalar field. Phys. Rev. D 1975, 11, 768–778. [Google Scholar] [CrossRef]
- Hawking, S. Quantum cosmology. In Relativity, Groups and Topology II, Les Houches Summer School, 1983; DeWitt, B., Stora, R., Eds.; North Holland: Amsterdam, The Netherlands, 1984; p. 333. [Google Scholar]
- Hájíček, P. Origin of nonunitarity in quantum gravity. Phys. Rev. D 1986, 34, 1040. [Google Scholar] [CrossRef]
- Kiefer, C. Wave packets in minisuperspace. Phys. Rev. D 1988, 38, 1761. [Google Scholar] [CrossRef]
- Ashtekar, A.; Corichi, A.; Singh, P. Robustness of key features of loop quantum cosmology. Phys. Rev. D 2008, 77, 024046. [Google Scholar] [CrossRef]
- Bojowald, M. Quantum cosmology. Lect. Notes Phys. 2011, 835, 1–308. [Google Scholar]
- Ashtekar, A.; Pawlowski, T.; Singh, P. Quantum Nature of the Big Bang: An Analytical and Numerical Investigation. I. Phys. Rev. D 2006, 73, 124038. [Google Scholar] [CrossRef]
- Kamenshchik, A.Y.; Tronconi, A.; Vardanyan, T.; Venturi, G. Time in quantum theory, the Wheeler-DeWitt equation and the Born-Oppenheimer approximation. arXiv 2018, arXiv:1809.08083. [Google Scholar] [CrossRef]
- Vanrietvelde, A.; Höhn, P.A.; Giacomini, F.; Castro-Ruiz, E. A change of perspective: Switching quantum reference frames via a perspective-neutral framework. arXiv 2018, arXiv:1809.00556. [Google Scholar]
- Vanrietvelde, A.; Höhn, P.A.; Giacomini, F. Switching quantum reference frames in the N-body problem and the absence of global relational perspectives. arXiv 2018, arXiv:1809.05093. [Google Scholar]
- Höhn, P.A.; Vanrietvelde, A. How to switch between relational quantum clocks. arXiv 2018, arXiv:1810.04153. [Google Scholar]
- Giacomini, F.; Castro-Ruiz, E.; Brukner, Č. Quantum mechanics and the covariance of physical laws in quantum reference frames. Nat. Commun. 2019, 10, 494. [Google Scholar] [CrossRef] [PubMed]
- Hartle, J.B. Time and time functions in parametrized nonrelativistic quantum mechanics. Class. Quant. Grav. 1996, 13, 361–376. [Google Scholar] [CrossRef]
- Hajicek, P. Choice of gauge in quantum gravity. Nucl. Phys. Proc. Suppl. 2000, 80, 1213. [Google Scholar]
- Gambini, R.; Porto, R.A. Relational time in generally covariant quantum systems: Four models. Phys. Rev. D 2001, 63, 105014. [Google Scholar] [CrossRef]
- Bojowald, M.; Halnon, T. Time in quantum cosmology. Phys. Rev. D 2018, 98, 066001. [Google Scholar] [CrossRef]
- Malkiewicz, P. Multiple choices of time in quantum cosmology. Class. Quant. Grav. 2015, 32, 135004. [Google Scholar] [CrossRef]
- Malkiewicz, P. What is Dynamics in Quantum Gravity? Class. Quant. Grav. 2017, 34, 205001. [Google Scholar] [CrossRef]
- Malkiewicz, P. Clocks and dynamics in quantum models of gravity. Class. Quant. Grav. 2017, 34, 145012. [Google Scholar] [CrossRef]
- Malkiewicz, P.; Miroszewski, A. Internal clock formulation of quantum mechanics. Phys. Rev. D 2017, 96, 046003. [Google Scholar] [CrossRef]
- Höhn, P.A. Reflections on the information paradigm in quantum and gravitational physics. J. Phys. Conf. Ser. 2017, 880, 012014. [Google Scholar] [CrossRef]
- Höhn, P.A. Toolbox for reconstructing quantum theory from rules on information acquisition. Quantum 2017, 1, 38. [Google Scholar] [CrossRef]
- Dirac, P.A. Lectures on Quantum Mechanics; Yeshiva University Press: New York, NY, USA, 1964. [Google Scholar]
- Henneaux, M.; Teitelboim, C. Quantization of Gauge Systems; Princeton University Press: Princeton, NJ, USA, 1992. [Google Scholar]
- Hartle, J.B.; Marolf, D. Comparing formulations of generalized quantum mechanics for reparametrization—Invariant systems. Phys. Rev. D 1997, 56, 6247–6257. [Google Scholar] [CrossRef]
- Marolf, D. Refined algebraic quantization: Systems with a single constraint. arXiv 1995, arXiv:gr-qc/9508015. [Google Scholar] [CrossRef]
- Marolf, D. Group averaging and refined algebraic quantization: Where are we now? arXiv 2000, arXiv:gr-qc/0011112. [Google Scholar]
- Ashtekar, A.; Lewandowski, J.; Marolf, D.; Mourao, J.; Thiemann, T. Quantization of diffeomorphism invariant theories of connections with local degrees of freedom. J. Math. Phys. 1995, 36, 6456–6493. [Google Scholar] [CrossRef]
- Haag, R. Local Quantum Physics: Fields, Pparticles, Algebras; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Page, D.N.; Wootters, W.K. Evolution without evolution: Dynamics described by stationary observables. Phys. Rev. D 1983, 27, 2885. [Google Scholar] [CrossRef]
- Pollet, J.; Méplan, O.; Gignoux, C. Elliptic eigenstates for the quantum harmonic oscillator. J. Phys. A Math. Gen. 1995, 28, 7287. [Google Scholar] [CrossRef]
- Hájíček, P. Group quantization of parametrized systems. I. Time levels. J. Math. Phys. 1995, 36, 4612–4638. [Google Scholar] [CrossRef]
- Hájíček, P. Time evolution and observables in constrained systems. Class. Quant. Grav. 1996, 13, 1353–1376. [Google Scholar] [CrossRef]
- Giddings, S.B.; Marolf, D.; Hartle, J.B. Observables in effective gravity. Phys. Rev. D 2006, 74, 064018. [Google Scholar] [CrossRef]
- Bojowald, M.; Paily, G.M. Deformed General Relativity. Phys. Rev. D 2013, 87, 044044. [Google Scholar] [CrossRef]
- Bojowald, M.; Brahma, S.; Reyes, J.D. Covariance in models of loop quantum gravity: Spherical symmetry. Phys. Rev. D 2015, 92, 045043. [Google Scholar] [CrossRef]
- Bojowald, M.; Brahma, S. Covariance in models of loop quantum gravity: Gowdy systems. Phys. Rev. D 2015, 92, 065002. [Google Scholar] [CrossRef]
- Bojowald, M.; Brahma, S. Loops rescue the no-boundary proposal. Phys. Rev. Lett. 2018, 121, 201301. [Google Scholar] [CrossRef] [PubMed]
- Everett, H., III. The Theory of the Universal Wave Function. Ph.D. Thesis, Princeton University, Princeton, NJ, USA, 1956. Available online: http://www-tc.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf (accessed on 10 May 2019).
- Everett, H. Relative state formulation of quantum mechanics. Rev. Mod. Phys. 1957, 29, 454–462. [Google Scholar] [CrossRef]
- Hartle, J.; Hawking, S. Wave function of the universe. Phys. Rev. D 1983, 28, 2960–2975. [Google Scholar] [CrossRef]
- Hawking, S. The quantum state of the universe. Nucl. Phys. B 1984, 239, 257. [Google Scholar] [CrossRef]
- Page, D.N. Hawking’s wave function for the universe. In Quantum Concepts in Space and Time; Penrose, R., Isham, C., Eds.; Clarendon Press: Oxford, UK, 1986; p. 274. [Google Scholar]
- Rovelli, C. Relational quantum mechanics. Int. J. Theor. Phys. 1996, 35, 1637–1678. [Google Scholar] [CrossRef]
- Rovelli, C. Space is blue and birds fly through it. Philos. Trans. R. Soc. A 2018, 376, 20170312. [Google Scholar] [CrossRef] [PubMed]
- Höhn, P.A.; Wever, C.S.P. Quantum theory from questions. Phys. Rev. A 2017, 95, 012102. [Google Scholar] [CrossRef]
- Höhn, P.A. Quantum theory from rules on information acquisition. Entropy 2017, 19, 98. [Google Scholar] [CrossRef]
- Martin-Dussaud, P.; Rovelli, C.; Zalamea, F. The notion of locality in relational quantum mechanics. Found. Phys. 2019, 49, 96–106. [Google Scholar] [CrossRef]
- Koberinski, A.; Müller, M.P. Quantum Theory as a Principle Theory: Insights from an Information-Theoretic Reconstruction; Cambridge University Press: Cambridge, UK, 2018; p. 257. [Google Scholar]
- Brukner, Č. On the quantum measurement problem. In Quantum [Un]Speakables II; Springer: Berlin/Heidelberg, Germany, 2017; pp. 95–117. [Google Scholar]
- Fuchs, C.A.; Stacey, B.C. QBist Quantum Mechanics: Quantum Theory as a Hero’s Handbook. arXiv 2016, arXiv:1612.07308. [Google Scholar]
- Breuer, T. The impossibility of accurate state self-measurements. Philos. Sci. 1995, 62, 197–214. [Google Scholar] [CrossRef]
- Dalla Chiara, M.L. Logical self reference, set theoretical paradoxes and the measurement problem in quantum mechanics. J. Philos. Logic 1977, 6, 331–347. [Google Scholar] [CrossRef]
- Crane, L. Clock and category: Is quantum gravity algebraic? J. Math. Phys. 1995, 36, 6180–6193. [Google Scholar] [CrossRef]
- Markopoulou, F. Planck scale models of the universe. arXiv 2002, arXiv:gr-qc/0210086. [Google Scholar]
- Markopoulou, F. New directions in background independent quantum gravity. arXiv 2007, arXiv:gr-qc/0703097. [Google Scholar]
- Hackl, L.F.; Neiman, Y. Horizon complementarity in elliptic de Sitter space. Phys. Rev. D 2015, 91, 044016. [Google Scholar] [CrossRef]
- Wigner, E.P. Remarks on the mind-body question. In Philosophical Reflections and Syntheses; Springer: Berlin/Heidelberg, Germany, 1995; pp. 247–260. [Google Scholar]
- Deutsch, D. Quantum theory as a universal physical theory. Int. J. Theor. Phys. 1985, 24, 1–41. [Google Scholar] [CrossRef]
- Frauchiger, D.; Renner, R. Quantum theory cannot consistently describe the use of itself. Nat. Commun. 2018, 9, 3711. [Google Scholar] [CrossRef] [PubMed]
1 | In fact, we have included a choice of lapse function . |
2 | For more details of this procedure in a different model, see [40]. |
3 | Please note that generally , despite the form (12). |
4 | We set . |
5 | In the affine momentum representation, states are represented as , where and . The inner product then reads and the configuration observables are represented as and . It is easy to check that this affine representation is equivalent to the canonical one above. |
© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Höhn, P.A. Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’. Universe 2019, 5, 116. https://doi.org/10.3390/universe5050116
Höhn PA. Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’. Universe. 2019; 5(5):116. https://doi.org/10.3390/universe5050116
Chicago/Turabian StyleHöhn, Philipp A. 2019. "Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’" Universe 5, no. 5: 116. https://doi.org/10.3390/universe5050116
APA StyleHöhn, P. A. (2019). Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’. Universe, 5(5), 116. https://doi.org/10.3390/universe5050116