Reality conditions inducing transforms for quantum gauge field theory and quantum gravity
Open accessAbstract
The algebraic form of the Hamiltonian or Hamiltonian constraint of various (field) theories simplifies considerably if one uses certain complex-valued phase space variables. We show, for a general theory, that if we prescribe first a canonical complexification and second representation of the canonical commutation relations in which the real connection is diagonal, then there is only one choice of a holomorphic representation which incorporates the correct reality conditions and keeps the Hamiltonian (constraint) algebraically simple. We derive a canonical algorithm to obtain this holomorphic representation and in particular explicitly compute it for quantum gravity in terms of a Wick rotation transform.
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A corresponding analysis has been done for the Reissner-Nordstrøm model.
“A corresponding analysis has been performed for the Reissner-Nordstrøm model [26].”
From Symmetry reduction for quantized diffeomorphism-invariant theories of connections (2000) · cites this paper as [26]
The reduced phase space is two-dimensional, with canonical variable p proportional to electric charge and conjugate = -B dx ω.
“The reduced phase space is two dimensional with the canonical variable p, which is proportional to the electric charge and the canonically conjugate = -B dx ω.”
From Symmetry reduction for quantized diffeomorphism-invariant theories of connections (2000) · cites this paper as [26]
The canonical variables of the symmetry reduced theory are the radial components p and iω of the electric field and the U(1) connection form.
“The canonical variables of the symmetry reduced theory are given by the radial components p and iω of the electric field and the U(1) connection form, respectively.”
From Symmetry reduction for quantized diffeomorphism-invariant theories of connections (2000) · cites this paper as [26]
These variables obey the Gauß constraint p≈0, which forces p to be constant.
“They are subject to the Gauß constraint p ≈ 0 which enforces p to be constant.”
From Symmetry reduction for quantized diffeomorphism-invariant theories of connections (2000) · cites this paper as [26]
The complications of the complex connection can be overcome by starting with Euclidean gravity and Wick rotating to the Lorentzian regime.
“the first possibility is to start with Euclidean gravity, in which the Wheeler-DeWitt constraint is also a polynomial [4], and to Wick rotate the Euclidean theory into the Lorentzian regime [5].”
From A length operator for canonical quantum gravity (1998) · cites this paper as [5]
The volume is quantized with a discrete spectrum, and the quantum of volume is of order the Planck volume.
“the volume is quantized with a discrete spectrum, the quantum of volume being indeed of order 3 p (see [20,21])!”
From Quantum spin dynamics (QSD): V. Quantum gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories (1998) · cites this paper as [21]
The Lorentzian hamiltonian constraint C L [ N ] can be written as the sum of two terms: C L [ N ] = C [ N ] + V [ N ], where C L [ N ] is the Euclidean Hamiltonian constraint.
“The Lorentzian hamiltonian constraint C L [ N ] can be written as the sum of two terms: C L [ N ] = C [ N ] + V [ N ] [42], where C L [ N ] is the Euclidean Hamiltonian constraint.”
From “Sum over surfaces” form of loop quantum gravity (1997) · cites this paper as [42]
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Cites
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- Authors
- Thiemann, T.
- Affiliations
- Harvard University (US)
- Corresponding author
- Thiemann, T
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1996-06-01
- Volume
- 13
- Issue
- 6
- Pages
- 1383-1403
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2031166591
- Cited by
- 102
- References
- 7
- Citations by year
- 2026: 2 · 2025: 2 · 2024: 4 · 2023: 2 · 2022: 2 · 2021: 6 · 2020: 1 · 2019: 3 · 2018: 3 · 2017: 2 · 2015: 5 · 2014: 2 · 2013: 2
- FWCI
- 8.95
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Keywords
- reality conditions, holomorphic representation, canonical commutation relations, Hamiltonian constraint, quantum gauge theory, quantum gravity, Wick rotation, constraint algebra
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