# Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

## Abstract

The knowledge of quantum phase flow induced under Weyl's association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phase-space analytic geometry based on the star-product operation can hardly be visualized. The statement 'quantum trajectory belongs to a constraint submanifold' can be changed, e.g., to the opposite by a unitary transformation. Some of the relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the star-composition law and preserves Hamiltonian and constraint star functions.

- Authors:

- Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117259 Moscow (Russian Federation) and Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D-72076 Tuebingen (Germany)
- (Germany)

- Publication Date:

- OSTI Identifier:
- 20929701

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 5; Other Information: DOI: 10.1063/1.2735816; (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; CANONICAL DIMENSION; DIFFERENTIAL EQUATIONS; GEOMETRY; HAMILTONIANS; HILBERT SPACE; MATHEMATICAL EVOLUTION; PHASE SPACE; QUANTUM MECHANICS; SEMICLASSICAL APPROXIMATION; TRANSFORMATIONS

### Citation Formats

```
Krivoruchenko, M. I., Faessler, Amand, and Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D-72076 Tuebingen.
```*Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics*. United States: N. p., 2007.
Web. doi:10.1063/1.2735816.

```
Krivoruchenko, M. I., Faessler, Amand, & Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D-72076 Tuebingen.
```*Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics*. United States. doi:10.1063/1.2735816.

```
Krivoruchenko, M. I., Faessler, Amand, and Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D-72076 Tuebingen. Tue .
"Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics". United States.
doi:10.1063/1.2735816.
```

```
@article{osti_20929701,
```

title = {Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics},

author = {Krivoruchenko, M. I. and Faessler, Amand and Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D-72076 Tuebingen},

abstractNote = {The knowledge of quantum phase flow induced under Weyl's association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phase-space analytic geometry based on the star-product operation can hardly be visualized. The statement 'quantum trajectory belongs to a constraint submanifold' can be changed, e.g., to the opposite by a unitary transformation. Some of the relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the star-composition law and preserves Hamiltonian and constraint star functions.},

doi = {10.1063/1.2735816},

journal = {Journal of Mathematical Physics},

number = 5,

volume = 48,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}