Papers
Topics
Authors
Recent
Search
2000 character limit reached

Matter relative to quantum hypersurfaces

Published 24 Aug 2023 in quant-ph, gr-qc, and hep-th | (2308.12912v2)

Abstract: We explore the canonical description of a scalar field as a parameterized field theory on an extended phase space that includes additional embedding fields that characterize spacetime hypersurfaces $\mathsf{X}$ relative to which the scalar field is described. This theory is quantized via the Dirac prescription and physical states of the theory are used to define conditional wave functionals $|\psi_\phi[\mathsf{X}]\rangle$ interpreted as the state of the field relative to the hypersurface $\mathsf{X}$, thereby extending the Page-Wootters formalism to quantum field theory. It is shown that this conditional wave functional satisfies the Tomonaga-Schwinger equation, thus demonstrating the formal equivalence between this extended Page-Wootters formalism and standard quantum field theory. We also construct relational Dirac observables and define a quantum deparameterization of the physical Hilbert space leading to a relational Heisenberg picture, which are both shown to be unitarily equivalent to the Page-Wootters formalism. Moreover, by treating hypersurfaces as quantum reference frames, we extend recently developed quantum frame transformations to changes between classical and nonclassical hypersurfaces. This allows us to exhibit the transformation properties of a quantum field under a larger class of transformations, which leads to a frame-dependent particle creation effect.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (104)
  1. K. V. Kuchař, “Time and interpretations of quantum gravity,” Int. J. of Mod. Phys. D 20, 3 (2011).
  2. C. J. Isham, “Canonical Quantum Gravity and the Problem of Time,” in Integrable Systems, Quantum Groups, and Quantum Field Theories, edited by L. A. Ibort and M. A. Rodríguez (Springer Netherlands, Dordrecht, 1993) pp. 157–287.
  3. L. Smolin, “The Case for Background Independence,” in The Structural Foundations of Quantum Gravity, edited by D. Rickles, S. French,  and J. T. Saatsi (Oxford University Press, 2006) pp. 196–239.
  4. C. Rovelli, “What Is Observable in Classical and Quantum Gravity?” Class. Quantum Gravity 8, 297 (1991a).
  5. C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, 2004).
  6. C. Rovelli, “Quantum reference systems,” Class. Quantum Gravity 8, 317 (1991b).
  7. B. S. DeWitt, “Quantum Theory of Gravity. I. The Canonical Theory,” Phys. Rev. 160, 1113 (1967).
  8. A. Ashtekar and J. Lewandowski, “Background Independent Quantum Gravity: A Status Report,” Class. Quantum Gravity 21, R53 (2004).
  9. K. Banerjee, G. Calcagni,  and M. Martin-Benito, “Introduction to loop quantum cosmology,” SIGMA 8, 016 (2012).
  10. S. Gielen and L. Menéndez-Pidal, ‘‘Unitarity, clock dependence and quantum recollapse in quantum cosmology,” Class. Quantum Gravity 39, 075011 (2022).
  11. D. Poulin, “Toy model for a relational formulation of quantum theory,” Int. J. of Theor. Phys. 45, 1189 (2006).
  12. S. D. Bartlett, T. Rudolph,  and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. of Mod. Phys. 79, 555 (2007).
  13. R. M. Angelo, N. Brunner, S. Popescu, A. J. Short,  and P. Skrzypczyk, “Physics within a quantum reference frame,” J. of Phys. A 44, 145304 (2011).
  14. P. A. M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Sciencem Yeshiva University, New York, 1964).
  15. Karel Kuchař, “Canonical quantization of gravity,” in Relativity, Astrophysics and Cosmology, edited by Werner Israel (Springer Netherlands, Dordrecht, 1973) pp. 237–288.
  16. C. J. Isham and K. V. Kuchař, “Representations of Space-time Diffeomorphisms. 2. Canonical Geometrodynamics,” Annals Phys. 164, 316 (1985a).
  17. C. J. Isham and K. V. Kuchař, “Representations of Space-time Diffeomorphisms. 1. Canonical Parametrized Field Theories,” Annals Phys. 164, 288 (1985b).
  18. K. V. Kuchař, “Parametrized scalar field on ℝ×S1ℝsuperscript𝑆1\mathbb{R}\times{S}^{1}blackboard_R × italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT: Dynamical pictures, spacetime diffeomorphisms, and conformal isometries,” Phys. Rev. D 39, 1579 (1989a).
  19. K. V. Kuchař, “Dirac constraint quantization of a parametrized field theory by anomaly-free operator representations of spacetime diffeomorphisms,” Phys. Rev. D 39, 2263 (1989b).
  20. A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao,  and T. Thiemann, “A Manifestly Gauge-Invariant Approach to Quantum Theories of Gauge Fields,” arXiv:hep-th/9408108  (1994).
  21. C. G. Torre and M. Varadarajan, “Quantum fields at any time,” Phys. Rev. D 58, 064007 (1998).
  22. C. G. Torre and M. Varadarajan, “Functional evolution of free quantum fields,” Class. Quantum Gravity 16, 2651 (1999).
  23. A. Ashtekar, J. Lewandowski,  and H. Sahlmann, “Polymer and Fock representations for a scalar field,” Class. Quantum Gravity 20, L11 (2002).
  24. M. Varadarajan, “Dirac quantization of parametrized field theory,” Phys. Rev. D 75, 044018 (2007).
  25. C. Kiefer, Quantum Gravity, 3rd ed. (Oxford University Press, 2012).
  26. C. Anastopoulos, M. Lagouvardos,  and K. Savvidou, “Gravitational effects in macroscopic quantum systems: a first-principles analysis,”  38, 155012 (2021).
  27. S. Weinberg, The Quantum Theory of Fields: Foundations, Vol. I (Cambridge University Press, 1995).
  28. S. Tomonaga, “On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields,” Prog. Theor. Phys. 1, 27 (1946).
  29. Z. Koba, T. Tati,  and S. Tomonaga, “On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. II: Case of Interacting Electromagnetic and Electron Fields,” Prog. Theor. Phys. 2, 101 (1947).
  30. J. Schwinger, “Quantum Electrodynamics. I. A Covariant Formulation,” Phys. Rev. 74, 1439 (1948).
  31. J. Schwinger, “Quantum Electrodynamics. II. Vacuum Polarization and Self-Energy,” Phys. Rev. 75, 651 (1949).
  32. Don N. Page and W. K. Wootters, “Evolution without evolution: Dynamics described by stationary observables,” Phys. Rev. D 27, 2885 (1983).
  33. W. K. Wootters, ““time” replaced by quantum correlations,” Int. J. of Theor. Phys. 23, 701 (1984).
  34. A. R. H. Smith and M. Ahmadi, “Quantizing time: Interacting clocks and systems,” Quantum 3, 160 (2019).
  35. A. R. H. Smith and M. Ahmadi, “Quantum clocks observe classical and quantum time dilation,” Nat. Commun. 11, 5360 (2020).
  36. P. A. Höhn, A. R. H. Smith,  and M. P. E. Lock, “Trinity of relational quantum dynamics,” Phys. Rev. D 104, 066001 (2021a).
  37. P. A. Höhn, A. R. H. Smith,  and M. P. E. Lock, “Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings,” Front. Phys. 9, 181 (2021b).
  38. A.-C. de la Hamette and T. D. Galley, “Quantum reference frames for general symmetry groups,” Quantum 4, 367 (2020).
  39. A.-C. de la Hamette, T. D. Galley, P. A. Höhn, L. Loveridge,  and M. P. Müller, “Perspective-neutral approach to quantum frame covariance for general symmetry groups,” arXiv:2110.13824 [gr-qc]  (2021).
  40. P. A. Höhn and A. Vanrietvelde, “How to switch between relational quantum clocks,” New J. Phys.  (2020).
  41. A. Vanrietvelde, P. A. Höhn,  and F. Giacomini, “Switching quantum reference frames in the N-body problem and the absence of global relational perspectives,” Quantum 7, 1088 (2023).
  42. P. A. Höhn, ‘‘Switching internal times and a new perspective on the ‘wave function of the universe’,” Universe 5, 116 (2019).
  43. A. Vanrietvelde, P. A. Höhn, F. Giacomini,  and E. Castro-Ruiz, “A change of perspective: switching quantum reference frames via a perspective-neutral framework,” Quantum 4, 225 (2020).
  44. L. Loveridge, T. Miyadera,  and P. Busch, “Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics,” Found. Phys. 48, 135 (2018).
  45. E. Castro-Ruiz, F. Giacomini, A. Belenchia,  and Č. Brukner, “Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems,” Nat. Commun. 11, 2672 (2020).
  46. F. Giacomini, “Spacetime Quantum Reference Frames and superpositions of proper times,” Quantum 5, 508 (2021).
  47. Anne-Catherine de la Hamette, Stefan L. Ludescher,  and Markus P. Müller, “Entanglement-Asymmetry Correspondence for Internal Quantum Reference Frames,” Phys. Rev. Lett. 129, 260404 (2022), arXiv:2112.00046 [quant-ph] .
  48. A. Ballesteros, F. Giacomini,  and G. Gubitosi, “The group structure of dynamical transformations between quantum reference frames,” Quantum 5, 470 (2021).
  49. S. A. Ahmad, T. D. Galley, P. A. Höhn, M. P. E. Lock,  and A. R. H. Smith, “Quantum relativity of subsystems,” Phys. Rev. Lett. 128, 170401 (2022).
  50. T. Carette, J. Głowacki,  and L. Loveridge, “Operational Quantum Reference Frame Transformations,”  (2023), arXiv:2303.14002 [math-ph] .
  51. J. Waldron J. Głowacki, L. Loveridge, “Quantum Reference Frames on Finite Homogeneous Spaces,”  (2023), arXiv:2302.05354 [quant-ph] .
  52. L. C. Barbado, E. Castro-Ruiz, L. Apadula,  and Č. Brukner, “Unruh effect for detectors in superposition of accelerations,” Phys. Rev. D 102, 045002 (2020).
  53. Michael Suleymanov, Ismael L. Paiva,  and Eliahu Cohen, “Non-relativistic spatiotemporal quantum reference frames,”   (2023), arXiv:2307.01874 [quant-ph] .
  54. F. Giacomini and A. Kempf, “Second-quantized Unruh-DeWitt detectors and their quantum reference frame transformations,” Phys. Rev. D 105, 125001 (2022).
  55. V. Kabel, A.-C. de la Hamette, E. Castro-Ruiz,  and Č. Brukner, “Quantum conformal symmetries for spacetimes in superposition,” arXiv:2207.00021 [quant-ph]  (2022).
  56. Christophe Goeller, Philipp A. Höhn,  and Josh Kirklin, “Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance,”  (2022), arXiv:2206.01193 [hep-th] .
  57. S. Carrozza, S. Eccles,  and P. A. Höhn, “Edge modes as dynamical frames: charges from post-selection in generally covariant theories,”   (2022), arXiv:2205.00913 [hep-th] .
  58. V. Kabel, Č. Brukner,  and W. Wieland, “Quantum Reference Frames at the Boundary of Spacetime,”   (2023), arXiv:2302.11629 [gr-qc] .
  59. A. Kaya, “Schrödinger from Wheeler-DeWitt: The Issues of Time and Inner Product in Canonical Quantum Gravity,”   (2023), arXiv:2211.11826 [gr-qc] .
  60. T. Thiemann, Modern canonical quantum general relativity (Cambridge University Press, 2008).
  61. R. M. Wald, General relativity (University of Chicago press, 2010).
  62. K. Kuchar, “Conditional symmetries in parametrised field theories,” J. Math. Phys. 23, 1647–1661 (1982).
  63. J. Glimm and A. Jaffe, Quantum physics: a functional integral point of view (Springer-Verlag New York, 1981).
  64. R. Ticciati, Quantum Field Theory for Mathematicians, Encyclopedia of Mathematics and its Applications (Cambridge University Press, 1999).
  65. A. Perelomov, Generalized Coherent States and Their Applications (Springer Berlin Heidelberg, 1986).
  66. S. Twareque Ali and G. A. Goldin, “Quantization, coherent states and diffeomorphism groups,” in Differential Geometry, Group Representations, and Quantization, Lecture Notes in Physics, edited by J.-D. Hennig, W. Lücke,  and J. Tolar (Springer, Berlin, Heidelberg, 1991) pp. 145–178.
  67. B. Dittrich, “Partial and complete observables for canonical general relativity,” Class. Quantum Gravity 23, 6155 (2006).
  68. B. Dittrich, “Partial and complete observables for Hamiltonian constrained systems,” Gen. Relativ. Gravit. 39, 1891 (2007).
  69. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Statistics and Probability, Vol. 1 (North-Holland, Amsterdam, 1982).
  70. P. Busch and P. Mittelstaedt P. J. Lahti, The Quantum Theory of Measurement (Springer-Verlag, 1991).
  71. D. Page, “Time as an inaccessible observable,” NSF-ITP-89-18  (1989).
  72. L. Doplicher, “Generalized Tomonaga-Schwinger equation from the Hadamard formula,” Phys. Rev. D 70, 064037 (2004).
  73. H. Wakita, “Integration of the Tomonaga-Schwinger equation,” Commun. Math. Phys. 50, 61 (1976).
  74. H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, 2002).
  75. Z. Koba, “On the integrability condition of Tomonaga-Schwinger equation,” Prog. Theor. Phys. 5, 139 (1950).
  76. L. Van Hove, On Certain Unitary Representations of an Infinite Group, Vol. 61 (Académie royale de Belgique, 1951).
  77. M. J. Gotay, H. B. Grundling,  and G. M. Tuynman, “Obstruction results in quantization theory,” J. of Nonlin. Science 6, 469 (1996).
  78. M. J. Gotay, “On the Groenewold-Van Hove problem for ℝ2⁢nsuperscriptℝ2𝑛\mathbb{R}^{2n}blackboard_R start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT,” J. Math. Phys. 40, 2107 (1999).
  79. S. A. Hojman, K. Kuchar,  and C. Teitelboim, “Geometrodynamics Regained,” Annals Phys. 96, 88 (1976).
  80. K. Kuchar, “Kinematics of Tensor Fields in Hyperspace. 2.” J. Math. Phys. 17, 792 (1976a).
  81. K. Kuchar, “Dynamics of Tensor Fields in Hyperspace. 3.” J. Math. Phys. 17, 801 (1976b).
  82. T. Fulton, F. Rohrlich,  and L. Witten, “Physical consequences of a co-ordinate transformation to a uniformly accelerating frame,” Il Nuov. Cim. 26, 652 (1962).
  83. W. R. Wood, G. Papini,  and Y. Q. Cai, “Conformal transformations and maximal acceleration,” Il Nuov. Cim. B 104, 653 (1989).
  84. S. D. Bartlett, T. Rudolph, R. W. Spekkens,  and P. S. Turner, “Quantum communication using a bounded-size quantum reference frame,” New J. Phys. 11, 063013 (2009).
  85. A. R. H. Smith, M. Piani,  and R. B. Mann, “Quantum reference frames associated with noncompact groups: The case of translations and boosts and the role of mass,” Phys. Rev. A 94, 012333 (2016).
  86. A. R. H. Smith, ‘‘Communicating without shared reference frames,” Phys. Rev. A 99, 052315 (2019).
  87. L. Chataignier, “Construction of quantum Dirac observables and the emergence of WKB time,” Phys. Rev. D 101, 086001 (2020).
  88. L. Chataignier, “Relational observables, reference frames, and conditional probabilities,” Phys. Rev. D 103, 026013 (2021).
  89. J. Foo, C. S. Arabaci, M. Zych,  and R. B. Mann, “Quantum superpositions of Minkowski spacetime,” Phys. Rev. D 107, 045014 (2023).
  90. J. Foo, C. S. Arabaci, M. Zych,  and R. B. Mann, “Quantum Signatures of Black Hole Mass Superpositions,” Phys. Rev. Lett. 129, 181301 (2022).
  91. J. D. Brown and K. V. Kuchař, “Dust as a standard of space and time in canonical quantum gravity,” Phys. Rev. D 51, 5600 (1995).
  92. Johannes Tambornino, “Relational Observables in Gravity: a Review,” SIGMA 8, 017 (2012).
  93. K. Giesel, S. Hofmann, T. Thiemann,  and O. Winkler, “Manifestly Gauge-Invariant General Relativistic Perturbation Theory. I. Foundations,” Class. Quantum Gravity 27, 055005 (2010).
  94. K. Giesel and T. Thiemann, “Algebraic quantum gravity (AQG). IV. Reduced phase space quantisation of loop quantum gravity,” Class. Quantum Gravity 27, 175009 (2010).
  95. V. Husain and T. Pawlowski, “Time and a physical Hamiltonian for quantum gravity,” Phys. Rev. Lett. 108, 141301 (2012).
  96. V. Giovannetti, S. Lloyd,  and L. Maccone, “Geometric Event-Based Quantum Mechanics,” New J. Phys. 25, 023027 (2023).
  97. N. L. Diaz, J. M. Matera,  and R. Rossignoli, “Spacetime Quantum Actions,” Phys. Rev. D 103, 065011 (2021a), arXiv:2010.09136 [quant-ph] .
  98. N. L. Diaz, J. M. Matera,  and R. Rossignoli, “Path Integrals from Quantum Action Operators,”   (2021b), arXiv:2111.05383 [quant-ph] .
  99. W. Donnelly and L. Freidel, “Local subsystems in gauge theory and gravity,” JHEP 09, 102 (2016).
  100. A. J. Speranza, “Local phase space and edge modes for diffeomorphism-invariant theories,” JHEP 02, 021 (2018).
  101. A. J. Speranza, “Geometrical tools for embedding fields, submanifolds, and foliations,”   (2019), arXiv:1904.08012 [gr-qc] .
  102. L. Freidel, M. Geiller,  and D. Pranzetti, “Edge modes of gravity. Part I. Corner potentials and charges,” JHEP 11, 026 (2020).
  103. L. Freidel, M. Geiller,  and W. Wieland, “Corner symmetry and quantum geometry,”   (2023), arXiv:2302.12799 [hep-th] .
  104. L. Ciambelli, R. G. Leigh,  and P.-C. Pai, “Embeddings and Integrable Charges for Extended Corner Symmetry,” Phys. Rev. Lett. 128 (2022).
Citations (7)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up