# Least uncertainty principle in deformation quantization

## Abstract

Deformation quantization generally produces families of cohomologically equivalent quantizations of a single physical system. We conjecture that the physically meaningful ones (i) allow enough observable energy distributions, i.e., ones for which no pure quantum state has negative probability, and (ii) reduce the uncertainty in the probability distribution of the resulting quantum states. For the simple harmonic oscillator this principle selects the classic Groenewold-Moyal (or Weyl) product on phase space while for the free particle, in which there is no real quantization, all cohomologically equivalent quantizations are equally good.

- Authors:

- Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395 (United States)

- Publication Date:

- OSTI Identifier:
- 20929633

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 2; Other Information: DOI: 10.1063/1.2456311; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; DEFORMATION; ENERGY SPECTRA; HARMONIC OSCILLATORS; PHASE SPACE; PROBABILITY; QUANTIZATION; UNCERTAINTY PRINCIPLE

### Citation Formats

```
Gerstenhaber, Murray.
```*Least uncertainty principle in deformation quantization*. United States: N. p., 2007.
Web. doi:10.1063/1.2456311.

```
Gerstenhaber, Murray.
```*Least uncertainty principle in deformation quantization*. United States. doi:10.1063/1.2456311.

```
Gerstenhaber, Murray. Thu .
"Least uncertainty principle in deformation quantization". United States.
doi:10.1063/1.2456311.
```

```
@article{osti_20929633,
```

title = {Least uncertainty principle in deformation quantization},

author = {Gerstenhaber, Murray},

abstractNote = {Deformation quantization generally produces families of cohomologically equivalent quantizations of a single physical system. We conjecture that the physically meaningful ones (i) allow enough observable energy distributions, i.e., ones for which no pure quantum state has negative probability, and (ii) reduce the uncertainty in the probability distribution of the resulting quantum states. For the simple harmonic oscillator this principle selects the classic Groenewold-Moyal (or Weyl) product on phase space while for the free particle, in which there is no real quantization, all cohomologically equivalent quantizations are equally good.},

doi = {10.1063/1.2456311},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 48,

place = {United States},

year = {Thu Feb 15 00:00:00 EST 2007},

month = {Thu Feb 15 00:00:00 EST 2007}

}

DOI: 10.1063/1.2456311

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