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Minisuperspace model of Machian resolution of Problem of Time. I. Isotropic case

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Abstract

A local resolution to the Problem of Time that is Machian and was previously demonstrated for relational mechanics models is here shown to work for a more widely studied quantum cosmological model. I.e., closed isotropic minisuperspace GR with minimally coupled scalar field matter. This resolution uses work firstly along the lines of Barbour’s at the classical level. Secondly, it uses a Machianized version of the semiclassical approach to quantum cosmology (the resolution given is not more than semiclassical). Finally, it uses a Machianized version of a combined semiclassical histories timeless records scheme along the lines of Halliwell’s work. This program’s goal is the treatment of inhomogeneous perturbations about the present paper’s model. This draws both from this paper’s minisuperspace work and from qualitative parallels with relational particle mechanics, since both have nontrivial notions of inhomogeneity/structure (clumping) as well as nontrivial linear constraints.

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Notes

  1. The third form’s changes in configuration \(\mathrm{d}Q^\mathsf{C }\) are held to be more primary than these velocities through not involving the label time.

  2. In this paper, for further Machian reasons independent from the present SSec’s discussion of Mach’s Principle, the spatial topological manifold \(\Sigma \) is taken to be compact without boundary. This is to avoid undue influence of boundary or asymptotic physics, and is a criterion also argued for by Einstein. It is also a simpler case to handle mathematically. Moreover, the simplest particular such \(\Sigma \) is the \(\mathbb {S}^3\) adopted by Halliwell–Hawking and hence also in the present paper.

  3. I use round brackets for function dependence, square brackets for functional dependence and \(( \; ; \; ]\) for mixed function dependence (before the semicolon) and functional dependence (after the semicolon). \(h_{ab}\) is the spatial 3-metric, with determinant \(h\), Ricci scalar Ric\((x; h]\) and GR configuration space metric \(M^{abcd} := h^{ac}h^{bd} - h^{ab}h^{cd}\). The present paper’s \(\mathbb {S}^3\) closed-universe case requires the cosmological constant term \(\Lambda \) for nontrivial solutions to exist. \(\phi \) is a minimally coupled scalar field. See Sect. 2.5 for the meaning of the \(F\) subscript.

  4. One says ‘of the Lagrangian’ in the reparametrization-invariant formulation, but parametrization-irrelevant and arc element conceptualizations use Jacobi–Synge arc elements instead of Lagrangians: \(\mathrm{d}s\) in place of \(L \, \mathrm{d}\lambda \) as the integrand of the action.

  5. In the ADM formulation, \(\mathcal{H}\) is well-known to arise from variation with respect to the lapse. In relational product-type actions like (1), there is no longer any lapse. However the ensuing mystery of how to arrive at \(\mathcal{H}\) in this case is straightforwardly mopped up by Dirac’s observation that such an action guarantees a primary constraint.

  6. In the case of the relativistic particle, a Klein–Gordon type wave equation arises instead. However, despite GR’s redundant configuration space of spatial 3-metrics Riem(\(\Sigma \)) being of indefinite signature, a Klein–Gordon interpretation cannot be pinned upon quantum GR. This is due to a lack of Killing vector structures that additionally manage to be compatible with GR’s potential [37]. Thus my ‘guarantees’ precludes this alternative.

  7. As argued in [9], this necessitates a curious indirect procedure in making such an approximation. I.e., one can not simply compare the sizes of the various energy terms, but must rather [9] assess this at the level of the resulting force terms upon variation.

  8. See [45] for an outline of how to render the entirety of the Principles of Dynamics compatible with Temporal Relationalism.

  9. Whilst [59, 60] also consider a ‘Multiple Choice Problem’, these references do not make any mention of the Groenewold–Van Hove phenomenon. Thus ‘Multiple Choice Problem’ in [59, 60] is not meant in the specific sense of the Kuchař and Isham notion of ‘Multiple Choice Problem’ that I follow in the present program.

  10. Here \(M_{\mathrm{h}\mathrm{h}}\) is the \(\mathrm{h}\mathrm{h}\) component of the configuration space metric with inverse \(N^{\mathrm{h}\mathrm{h}}\).

  11. The simplest models to manifest these are the relational quadrilateral and the non-diagonal Bianchi IX minisuperspace.

  12. Here \(R\) a region of configuration space \(\mathsf Q \) with corresponding characteristic function \(f_R\). \(\mathbf{n}\) is the normal in configuration space. \(P\) is a prefactor function, the detailed form of which is given in [91]. \(\theta \) is a step function and \(\epsilon \) is a small number. 0 and f subscripts denote ‘initial’ and ‘final’.

References

  1. Kuchař, K.V.: In: Kunstatter, G., Vincent, D., Williams, J. (eds.) Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics. World Scientific, Singapore (1992)

    Google Scholar 

  2. Isham, C.J.: In: Ibort, L.A., Rodríguez, M.A. (eds.) Integrable Systems, Quantum Groups and Quantum Field Theories. Kluwer, Dordrecht (1993). gr-qc/9210011

  3. Anderson, E.: In: Frignanni, V.R. (ed.) Classical and Quantum Gravity: Theory, Analysis and Applications. Nova, New York (2012). arXiv:1009.2157; Invited Review in Annalen der Physik, 524, 757 (2012). arXiv:1206.2403; arXiv:1310.1524

  4. Anderson, E.: arXiv:1111.1472

  5. Barbour, J.B., Bertotti, B.: Proc. R. Soc. Lond. A 382, 295 (1982)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Barbour, J.B.: Class. Quantum Gravity 11, 2853 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  7. Barbour, J.B., Foster, B.Z., Murchadha, N.Ó.: Class. Quantum Gravity 19, 3217 (2002). gr-qc/0012089

  8. Anderson, E.: arXiv:1205.1256

  9. Anderson, E.: arXiv:1209.1266

  10. Anderson, E.: Class. Quantum Gravity 29, 235015 (2012). arXiv:1204.2868

  11. Anderson, E.: Class. Quantum Gravity 31, (2014) 025006, arXiv:1305.4685; Invited Seminar at ’XXIX-th International Workshop on High Energy Physics: New Results and Actual Problems in Particle and Astroparticle Physics and Cosmology’, Moscow 2013, Accepted for Conference Proceedings. arXiv:1306.5812

  12. Anderson, E.: Invited Seminar at ‘Do we need a Physics of Passage’ Conference, Cape Town, December 2012. arXiv:1306.5816

  13. Anderson, E.: Int. J. Mod. Phys. D 23, 1450014 (2014). arXiv:1202.4186

  14. Anderson, E., Kneller, S.A.R.: Relational Quadrilateralland. II. The Quantum Theory. Accepted by Int. J. Mod. Phys. D. arXiv:1303.5645

  15. Misner, C.W.: In: Klauder, J. (ed.) Magic Without Magic: John Archibald Wheeler. Freeman, San Francisco (1972)

    Google Scholar 

  16. Ryan, M.P.: Hamiltonian Cosmology. Lecture Notes in Physics, vol. 13. Springer, Berlin (1972)

    Google Scholar 

  17. Blyth, W.F., Isham, C.J.: Phys. Rev. D11, 768 (1975)

    ADS  MathSciNet  Google Scholar 

  18. Hartle, J.B., Hawking, S.W.: Phys. Rev. D 28, 2960 (1983)

    Google Scholar 

  19. Wiltshire, D.L.: In: Robson, B., Visvanathan, N., Woolcock, W.S. (eds). Cosmology: The Physics of the Universe. World Scientific, Singapore (1996). gr-qc/0101003; Isotropic models with scalar fields are also often used in inflationary and further cosmological modelling (e.g. quintessence)

  20. Halliwell, J.J., Hawking, S.W.: Phys. Rev. D 31, 1777 (1985)

    Google Scholar 

  21. Ade, P.A.R. et al.: The Planck satellite data continues to conform with inflationary theory. arXiv:1303.5082

  22. Anderson, E.: Problem of time in slightly inhomogeneous cosmology (forthcoming)

  23. Anderson, E.: Configuration space geometry in slightly inhomogeneous cosmology . arXiv:1403.7583

  24. Anderson, E.: Reduced formulation of slightly inhomogeneous cosmology and the Problem of Time (forthcoming)

  25. Anderson, E.: Slightly inhomogeneous semiclassical quantum cosmology (forthcoming)

  26. Rindler, W.: Relativity, Special, General and Cosmological. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

  27. Anderson, E.: Minisuperspace model of Machian resolution of Problem of Time. II. Bianchi IX (forthcoming)

  28. Anastopoulos, C., Savvidou, N.: Class. Quantum Gravity 22, 1841 (2005). gr-qc/0410131

  29. Halliwell, J.J.: In: Gibbons, G.W., Shellard, E.P.S., Rankin, S.J. (eds). The Future of Theoretical Physics and Cosmology (Stephen Hawking 60th Birthday Festschrift volume). Cambridge University Press, Cambridge (2003). arXiv:gr-qc/0208018

  30. Halliwell, J.J.: Phys. Rev. D 80, 124032 (2009). arXiv:0909.2597

  31. Halliwell, J.J.: J. Phys. Conf. Ser. 306, 012023 (2011). arXiv:1108.5991

  32. Barbour, J.B.: The End of Time. Oxford University Press, Oxford (1999)

    Google Scholar 

  33. Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1949)

    MATH  Google Scholar 

  34. Baierlein, R.F., Sharp, D., Wheeler, J.A.: Phys. Rev. 126, 1864 (1962)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Anderson, E.: Class. Quantum Gravity 25, 175011 (2008). arXiv:0711.0288

  36. Dirac, P.A.M.: Lectures on Quantum Mechanics. Yeshiva University, New York (1964)

    Google Scholar 

  37. Kuchař, K.V.: In: Isham, C.J., Penrose, R., Sciama, D.W. (eds.) Quantum Gravity 2: A Second Oxford Symposium. Clarendon, Oxford (1981)

    Google Scholar 

  38. Rovelli, C.: fqxi ‘Nature of Time’ Essay Competition: Community First Prize. arXiv:0903.3832

  39. Barbour, J.B.: fqxi ‘Nature of Time’ Essay Competition: Juried First Prize. arXiv:0903.3489

  40. Clemence, G.M.: Rev. Mod. Phys. 29, 2 (1957)

    Article  ADS  Google Scholar 

  41. Wheeler, J.A.: In: DeWitt, B.S., DeWitt, C.M. (eds.) Groups, Relativity and Topology. Gordon and Breach, New York (1963)

    Google Scholar 

  42. Belasco, E.P., Ohanian, H.C.: J. Math. Phys. 10, 1503 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  43. Bartnik, R., Fodor, G.: Phys. Rev. D 48, 3596 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  44. Teitelboim, C.: Ann. Phys. N.Y. 79, 542 (1973)

    Google Scholar 

  45. Anderson, E., Mercati, F.: arXiv:1311.6541

  46. Anderson, E., Barbour, J.B., Foster, B.Z., Ó Murchadha, N.: Class. Quantum Gravity 20, 157 (2003). gr-qc/0211022

  47. Barbour, J.B.: In: Proceedings of the Conference “Quantum Field Theory and Gravity (Regensburg, 2010). arXiv:1105.0183

  48. Gryb, S.B.: Shape Dynamics and Mach’s Principles: Gravity from Conformal Geometrodynamics. Ph.D. Thesis, University of Waterloo, Canada 2011. arXiv:1204.0683

  49. Crane, L.: hep-th/9301061

  50. Crane, L.: J. Math. Phys. 36, 6180 (1995). arXiv:gr-qc/9504038

  51. Rovelli, C.: Int. J. Theor. Phys. 35, 1637 (1996). quant-ph/9609002v2

  52. Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  53. Bell, J.S.: In: Isham, C.J., Penrose, R., Sciama, D.W. (eds.) Quantum Gravity 2. A Second Oxford Symposium. Carendon, Oxford (1981)

    Google Scholar 

  54. Wheeler, J.A.: In: DeWitt, C., Wheeler, J.A. (eds.) Battelle Rencontres: 1967 Lectures in Mathematics and Physics. Benjamin, New York (1968)

    Google Scholar 

  55. Hojman, S.A., Kuchař, K.V., Teitelboim, C.: Ann. Phys. N.Y. 96, 88 (1976)

    Google Scholar 

  56. Anderson, E.: Stud. Hist. Phil. Mod. Phys. 38, 15 (2007). gr-qc/0511070

  57. Anderson, E.: In: Christiansen, M.N., Rasmussen, T.K. (eds.) Classical and Quantum Gravity Research. Nova, New York (2008). arXiv:0711.0285

  58. Gotay, M.J.: In: Marsden, J., Wiggins, S. (eds.) Mechanics: From Theory to Computation (Essays in Honor of Juan-Carlos Simó). J. Nonlinear Sci. vol. 171. Springer, New York (2000). math-ph/9809011

  59. Bojowald, M., Hoehn, P.A., Tsobanjan, A.: Class. Quantum Gravity 28, 035006 (2011). arXiv:1009.5953

  60. Bojowald, M., Hoehn, P.A., Tsobanjan, A.: Phys. Rev. D 83, 125023 (2011). arXiv:1011.3040

  61. Anderson, E.: Class. Quantum Gravity 27, 045002 (2010). arXiv:0905.3357

  62. Kiefer, C.: Quantum Gravity. Clarendon, Oxford (2004)

    MATH  Google Scholar 

  63. Banks, T.: Nucl. Phys. B 249, 322 (1985)

    Article  ADS  Google Scholar 

  64. Padmanabhan, T., Singh, T.P.: Class. Quantum Gravity 7, 411 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  65. Brout, R., Venturi, G.: Phys. Rev. D 39, 2436 (1989)

    Article  ADS  Google Scholar 

  66. Kiefer, C.: Class. Quantum Gravity 4, 1369 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  67. Kiefer, C., Kramer, M.: Phys. Rev. Lett. 108, 021301 (2012)

    Article  ADS  Google Scholar 

  68. Casadio, R.: Int. J. Mod. Phys. D 9, 511 (2000). gr-qc/9810073

  69. Massar, S., Parentani, R.: Phys. Rev. D 59, 123519 (1999). gr-qc/9812045

  70. Alberghi, G.L., Casadio, R., Gruppuso, A.: Phys. Rev. D 61, 084009 (2000). gr-qc/9912095

  71. Alberghi, G.L., Appignani, C., Casadio, R., Sbisá, F., Tronconi, A.: Phys. Rev. D 77, 044002 (2008). arXiv:0708.0483

  72. Zeh, H.D.: The Physical Basis of the Direction of Time. Springer, Berlin (1989)

    Book  Google Scholar 

  73. Kamenshchik, A.Y., Tronconi, A., Venturi, G.: Accepted in Phys. Lett. B. arXiv:1305.6138

  74. Marolf, D.: Class. Quantum Gravity 12, 1441 (1995). gr-qc/9409049

  75. Halliwell, J.J.: Phys. Rev. D 36, 3626 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  76. DeWitt, B.S.: Phys. Rev. 160, 1113 (1967)

    Article  ADS  MATH  Google Scholar 

  77. Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I.-O., Zeh, H.D.: Decoherence and the Appearance of a Classical World in Quantum Theory. Springer, Berlin, Heidelberg (1996)

    Book  MATH  Google Scholar 

  78. Anderson, E.: Class. Quantum Gravity 28, 185008 (2011). arXiv:1101.4916

  79. Hájíček, P.: Phys. Rev. D 34, 1040 (1986)

    Article  MathSciNet  Google Scholar 

  80. Zeh, H.D.: Phys. Lett. A 126, 311 (1988)

    Article  ADS  Google Scholar 

  81. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics. Pergamon, New York (1965)

    MATH  Google Scholar 

  82. Barbour, J.B.: Phys. Rev. D 47, 5422 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  83. Zeh, H.D.: Phys. Lett. A 116, 9 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  84. Barbour, J.B.: Class. Quantum Gravity 11, 2875 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  85. Anderson, E.: Int. J. Mod. Phys. D18, 635 (2009). arXiv:0709.1892

  86. Anderson, E.: In: O’Loughlin, , Stanič, S., Veberič, D. (eds.) Proceedings of the Second Conference on Time and Matter. University of Nova Gorica Press, Nova Gorica, Slovenia (2008). arXiv:0711.3174

  87. Isham, C.J., Linden, N.: J. Math. Phys. 36, 5392 (1995). gr-qc/9503063

    Google Scholar 

  88. Gell-Mann, M., Hartle, J.B.: Phys. Rev. D 47, 3345 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  89. Halliwell, J.J.: Phys. Rev. D 60, 105031 (1999). quant-ph/9902008

  90. Doering, A., Isham, C.: To appear in Coecke, R. (ed.) New Structures in Physics. arXiv:0803.0417

  91. Halliwell, J.J., Thorwart, J.: Phys. Rev. D 65, 104009 (2002). gr-qc/0201070

  92. Halliwell, J.J., Yearsley, J.M.: Phys. Rev. D 86, 024016 (2012). arXiv:1205.3773

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Acknowledgments

I thank those close to me for support. Jeremy Butterfield, George Ellis, Jonathan Halliwell, Sophie Kneller, Matteo Lostaglio,Flavio Mercati, David Sloan and the Anonymous Referees for comments and discussions. John Barrow, Jeremy Butterfield, Marc Lachièze-Rey, Malcolm MacCallum, Claus Kiefer, Don Page, Reza Tavakol and Juan Valiente–Kroon for help with my career. This work was carried out whilst I was funded in 2011 and 2012 by a grant from the Foundational Questions Institute (FQXi) Fund, a donor-advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-RFP3-1101 to the FQXi. I thank also Theiss Research and the CNRS for administering this grant. Some of this work was typed up whilst receiving hospitality from the Institute of High Energy Physics at Protvino, Moscow.

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    Edward Anderson

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Anderson, E. Minisuperspace model of Machian resolution of Problem of Time. I. Isotropic case. Gen Relativ Gravit 46, 1708 (2014). https://doi.org/10.1007/s10714-014-1708-0

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