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Canonical quantization of constrained theories on discrete space-time lattices
General Relativity and Quantum Cosmology · Submitted 29 May 2002
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The whole discussion is framed in such a way as to provide a formalism that would allow a consistent, well defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper.
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The abstract, sentence by sentence
We discuss the canonical quantization of systems formulated on discrete space-times. We start by analyzing the quantization of simple mechanical systems with discrete time. The quantization becomes challenging when the systems have anholonomic constraints. We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations. This allows to construct, for the first time, a canonical formulation for general constrained mechanical systems with discrete time. We extend the analysis to gauge field theories on the lattice. We consider a complete canonical formulation, starting from a discrete action, for lattice Yang--Mills theory discretized in space and Maxwell theory discretized in space and time. After completing the treatment, the results can be shown to coincide with the results of the traditional transfer matrix method. We then apply the method to BF theory, yielding the first lattice treatment for such a theory ever. The framework presented deals directly with the Lorentzian signature without requiring an Euclidean rotation. The whole discussion is framed in such a way as to provide a formalism that would allow a consistent, well defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We discuss the canonical quantization of systems formulated on discrete space-times. | background 0.51 | 0.73 | 0.04 | asserted 0.75 |
| s1 | We start by analyzing the quantization of simple mechanical systems with discrete time. | method 0.80 | 0.81 | 0.02 | asserted 0.89 |
| s2 | The quantization becomes challenging when the systems have anholonomic constraints. | background 0.81 | 0.20 | 0.03 | asserted 1.00 |
| s3 | We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations. | contribution 0.77 | 0.94 | 0.27 | asserted 0.96 |
| s4 | This allows to construct, for the first time, a canonical formulation for general constrained mechanical systems with discrete time. | contribution 0.61 | 0.93 | 0.38 | asserted 0.99 |
| s5 | We extend the analysis to gauge field theories on the lattice. | contribution 0.90 | 0.89 | 0.04 | asserted 0.93 |
| s6 | We consider a complete canonical formulation, starting from a discrete action, for lattice Yang--Mills theory discretized in space and Maxwell theory discretized in space and time. | contribution 0.80 | 0.91 | 0.09 | asserted 0.98 |
| s7 | After completing the treatment, the results can be shown to coincide with the results of the traditional transfer matrix method. | result 0.92 | 0.78 | 0.14 | asserted 0.83 |
| s8 | We then apply the method to BF theory, yielding the first lattice treatment for such a theory ever. | contribution 0.92 | 0.95 | 0.13 | asserted 1.00 |
| s9 | The framework presented deals directly with the Lorentzian signature without requiring an Euclidean rotation. | contribution 0.54 | 0.81 | 0.10 | asserted 1.00 |
| s10 | The whole discussion is framed in such a way as to provide a formalism that would allow a consistent, well defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper. | future work 0.99 | 0.43 | 0.19 | hedged possibility 0.63 |
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Original abstract
We discuss the canonical quantization of systems formulated on discrete space-times. We start by analyzing the quantization of simple mechanical systems with discrete time. The quantization becomes challenging when the systems have anholonomic constraints. We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations. This allows to construct, for the first time, a canonical formulation for general constrained mechanical systems with discrete time. We extend the analysis to gauge field theories on the lattice. We consider a complete canonical formulation, starting from a discrete action, for lattice Yang--Mills theory discretized in space and Maxwell theory discretized in space and time. After completing the treatment, the results can be shown to coincide with the results of the traditional transfer matrix method. We then apply the method to BF theory, yielding the first lattice treatment for such a theory ever. The framework presented deals directly with the Lorentzian signature without requiring an Euclidean rotation. The whole discussion is framed in such a way as to provide a formalism that would allow a consistent, well defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper.
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References
How the literature uses it
5 arXiv papers cite this in 14 sentences
Papers citing it · 7 sentences without a confident label
Builds on it 1 paper
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Generalizing an earlier formalism [ref] (for constant discretization), this work provides a detailed constraint analysis and discussion of Dirac observables for such discrete systems.
Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces · Introduction · 0.98
Cites it as support 1 paper
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Although one could argue that the `problem of implementing the constraints' , e.g. on initial values, has disappeared, choosing the initial values freely, will typically result in solutions with unphysical properties [ref] . * Broken diffeomorphism symmetry leads to inconsistencies for perturbation theory.
Spin foams, Refinement limit and Renormalization · The need for renormalization: restoring diffeomorphism symmetry · 0.96
Cites it as background 4 papers
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For a detailed discussion and introduction to this formalism we refer the reader to [ref] which builds up on and generalizes the earlier works [ref] .[For a related multisymplectic formulation see also the recent [ref] .] An explicit construction of the formalism and a detailed classification of constraints and degrees of freedom for quadratic discrete actions is given in [ref] .
Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces · Review of the classical formalism · 0.96
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In the class of systems which we shall henceforth consider, namely variational discrete systems [ref] , the dynamics does not proceed by a continuous time evolution of the variables, but by a discrete time evolution of the latter on a discrete set of time steps.
Classification of constraints and degrees of freedom for quadratic discrete actions · Preliminaries · 0.96
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Consistent discretisations [ref] were an attempt to discretise theories like general relativity in which the resulting discretised equations were algebraically consistent, that is, they could be solved simultaneously.
Discretisations, Constraints and Diffeomorphisms in Quantum Gravity · Consistent discretisation · 0.86
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