# Semiclassical states on Lie algebras

## Abstract

The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.

- Authors:

- King’s College, 133 North River Street, Kingston, Pennsylvania 18702 (United States)

- Publication Date:

- OSTI Identifier:
- 22403124

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 3; Other Information: (c) 2015 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; ALGEBRA; IRREDUCIBLE REPRESENTATIONS; LIE GROUPS; QUANTIZATION; SEMICLASSICAL APPROXIMATION

### Citation Formats

```
Tsobanjan, Artur, E-mail: artur.tsobanjan@gmail.com.
```*Semiclassical states on Lie algebras*. United States: N. p., 2015.
Web. doi:10.1063/1.4914010.

```
Tsobanjan, Artur, E-mail: artur.tsobanjan@gmail.com.
```*Semiclassical states on Lie algebras*. United States. doi:10.1063/1.4914010.

```
Tsobanjan, Artur, E-mail: artur.tsobanjan@gmail.com. Sun .
"Semiclassical states on Lie algebras". United States.
doi:10.1063/1.4914010.
```

```
@article{osti_22403124,
```

title = {Semiclassical states on Lie algebras},

author = {Tsobanjan, Artur, E-mail: artur.tsobanjan@gmail.com},

abstractNote = {The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.},

doi = {10.1063/1.4914010},

journal = {Journal of Mathematical Physics},

number = 3,

volume = 56,

place = {United States},

year = {Sun Mar 15 00:00:00 EDT 2015},

month = {Sun Mar 15 00:00:00 EDT 2015}

}