# Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces

## Abstract

A temporally varying discretization often features in discrete gravitational systems and appears in lattice field theory models subject to a coarse graining or refining dynamics. To better understand such discretization changing dynamics in the quantum theory, an according formalism for constrained variational discrete systems is constructed. While this paper focuses on global evolution moves and, for simplicity, restricts to flat configuration spaces R{sup N}, a Paper II [P. A. Höhn, “Quantization of systems with temporally varying discretization. II. Local evolution moves,” J. Math. Phys., e-print http://arxiv.org/abs/arXiv:1401.7731 [gr-qc].] discusses local evolution moves. In order to link the covariant and canonical picture, the dynamics of the quantum states is generated by propagators which satisfy the canonical constraints and are constructed using the action and group averaging projectors. This projector formalism offers a systematic method for tracing and regularizing divergences in the resulting state sums. Non-trivial coarse graining evolution moves lead to non-unitary, and thus irreversible, projections of physical Hilbert spaces and Dirac observables such that these concepts become evolution move dependent on temporally varying discretizations. The formalism is illustrated in a toy model mimicking a “creation from nothing.” Subtleties arising when applying such a formalism to quantum gravity models are discussed.

- Authors:

- Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5 (Canada)

- Publication Date:

- OSTI Identifier:
- 22306098

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; HILBERT SPACE; LATTICE FIELD THEORY; PROPAGATOR; QUANTIZATION; QUANTUM GRAVITY; QUANTUM STATES; VARIATIONAL METHODS

### Citation Formats

```
Höhn, Philipp A., E-mail: phoehn@perimeterinstitute.ca.
```*Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces*. United States: N. p., 2014.
Web. doi:10.1063/1.4890558.

```
Höhn, Philipp A., E-mail: phoehn@perimeterinstitute.ca.
```*Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces*. United States. doi:10.1063/1.4890558.

```
Höhn, Philipp A., E-mail: phoehn@perimeterinstitute.ca. Fri .
"Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces". United States.
doi:10.1063/1.4890558.
```

```
@article{osti_22306098,
```

title = {Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces},

author = {Höhn, Philipp A., E-mail: phoehn@perimeterinstitute.ca},

abstractNote = {A temporally varying discretization often features in discrete gravitational systems and appears in lattice field theory models subject to a coarse graining or refining dynamics. To better understand such discretization changing dynamics in the quantum theory, an according formalism for constrained variational discrete systems is constructed. While this paper focuses on global evolution moves and, for simplicity, restricts to flat configuration spaces R{sup N}, a Paper II [P. A. Höhn, “Quantization of systems with temporally varying discretization. II. Local evolution moves,” J. Math. Phys., e-print http://arxiv.org/abs/arXiv:1401.7731 [gr-qc].] discusses local evolution moves. In order to link the covariant and canonical picture, the dynamics of the quantum states is generated by propagators which satisfy the canonical constraints and are constructed using the action and group averaging projectors. This projector formalism offers a systematic method for tracing and regularizing divergences in the resulting state sums. Non-trivial coarse graining evolution moves lead to non-unitary, and thus irreversible, projections of physical Hilbert spaces and Dirac observables such that these concepts become evolution move dependent on temporally varying discretizations. The formalism is illustrated in a toy model mimicking a “creation from nothing.” Subtleties arising when applying such a formalism to quantum gravity models are discussed.},

doi = {10.1063/1.4890558},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 15 00:00:00 EDT 2014},

month = {Fri Aug 15 00:00:00 EDT 2014}

}