Functional evolution of free quantum fields
Open accessAbstract
We consider the problem of evolving the state of a quantum field between any two (in general, curved) Cauchy surfaces. Classically, this dynamical evolution is represented by a canonical transformation on the phase space for the field theory. We show that this canonical transformation cannot, in general, be unitarily implemented on the Fock space for free quantum fields on flat spacetimes of dimension greater than 2. We do this by considering time evolution of a free Klein-Gordon field on a flat spacetime (with toroidal Cauchy surfaces) starting from a flat initial surface and ending on a generic final surface. The associated Bogolubov transformation is computed; it does not correspond to a unitary transformation on the Fock space. This means that functional evolution of the quantum state as originally envisioned by Tomonaga, Schwinger and Dirac is not a viable concept. Nevertheless, we demonstrate that functional evolution of the quantum state can be satisfactorily described using the formalism of algebraic quantum field theory. We discuss possible implications of our results for canonical quantum gravity.
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A different choice of time-slicing might lead to a different quantum theory, as is known in quantum field theory.
“a different choice might lead to a different quantum theory, as it is known to do in quantum field theory [258].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [258]
In “B. Quantum Field Theory” (p. 8), the paper says that evolution between Cauchy surfaces can yield states whose Hilbert-space representations are not unitarily equivalent. That supports the possibility of a difference arising from time-slicing, but the paper does not say that the resulting representations are different quantum theories; it describes evolution between arbitrary surfaces within one algebraic formulation (p. 6). (LLM, conf=0.84)
Experimental — accuracy not yet measuredFor field theories, even in flat spacetime, different time-slicings do not always evolve an initial wave function to the same final wave function.
“For field theories, even in flat spacetime, this will not always happen [258].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [258]
The Discussion (pp. 14–15) says that evolution between certain Cauchy surfaces in flat spacetime cannot be implemented unitarily in the usual Fock-space Schrödinger picture. It does not show that two time-slicings produce different final wave functions from the same initial wave function; rather, it says that the failure to define the evolution makes that question of foliation independence academic. (LLM, conf=0.86)
Experimental — accuracy not yet measured
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Cites
Lectures on Non-Perturbative Canonical Gravity
Ashtekar, A. (1991) · Advanced Series in Astrophysics and Cosmology · doi:10.1142/1321 · Cited by 872 · see citing works
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Research works printed but not linked: 5
- [3][3] K. V. Kuchaˇ r, in General Relativity and Relativistic Astrophysics , Proceedings of the Fourth Canadian Conference on General Relativity and Relativistic Astrophysics, edited by G. Kunstatter, D. Vincent, and J. Williams (World Scientific, Singapore, 1992). -
- [10][10] The literature on algebraic quantum field theory is quite extensive. See, for example, O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics , (Springer-Verlag, New York 1981); R. Haag, Local Quantum Physics , (Springer, Berlin, 1996), and references therein. -
- [11][11] D. Shale, Trans. Am. Math. Soc. 103 , 149 (1962). -
- [12][12] R. M. Wald, Ann. Phys. 118 , 490 (1979); M. Reed and B. Simon, Methods of Modern Mathematical Physics III: Scattering Theory , (Academic Press, New York 1979); R. Honegger and A. Rieckers, J. Math. Phys. 37 , 4292 (1996). -
- [14][14] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics , (University of Chicago Press, Chicago 1994). -
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Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
- Authors
- Torre, C. G., & Varadarajan, M.
- Affiliations
- Utah State University (US)
- Raman Research Institute (IN)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1999-07-20
- Volume
- 16
- Issue
- 8
- Pages
- 2651-2668
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2123782066
- Cited by
- 138
- References
- 1
- Citations by year
- 2026: 1 · 2025: 18 · 2024: 5 · 2023: 6 · 2022: 1 · 2021: 6 · 2020: 3 · 2019: 17 · 2018: 2 · 2017: 1 · 2015: 6 · 2014: 3 · 2013: 4 · 2012: 2
- FWCI
- 3.98
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Digital Commons - USU (Utah State University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (9732636, 9600616)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- free quantum fields, Cauchy surfaces, canonical transformations, Fock space, Klein-Gordon field, algebraic quantum field theory, canonical quantum gravity, curved spacetime
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