## Proceedings of the International Conference on Geometry, Integrability and Quantization

- Geom. Integrability & Quantization
- Proceedings of the Fourth International Conference on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov and Gregory L. Naber, eds. (Sofia: Coral Press Scientific Publishing, 2003), 11 - 41

### Deformation Quantization in Quantum Mechanics and Quantum Field Theory

#### Abstract

We discuss deformation quantization in quantum mechanics and quantum field theory. We begin with a discussion of the mathematical question of deforming the commutative algebra of functions on a manifold into a non-commutative algebra by use of an associative product. We then apply these considerations to the commutative algebra of observables of a classical dynamical system, which may be deformed to the non-commutative algebra of quantum observables. This is the process of deformation quantization, which provides a canonical procedure for finding the measurable quantities of a quantum system. The deformation quantization approach is illustrated, first for the case of a simple harmonic oscillator, then for an oscillator coupled to an external source, and finally for a quantum field theory of scalar bosons, where the well-known formula for the number of quanta emitted by a given external source in terms of the Poisson distribution is reproduced.

The relation of the star product method to the better-known methods involving the representation of observables as linear operators on a Hilbert space, or the representation of expectation values as functional integrals, is analyzed. The final lecture deals with a remarkable formula of Cattaneo and Felder, which relates Kontsevich’s star product to an expectation value of a product of functions on a Poisson space, and indicates how this formula may be interpreted.

#### Article information

**Dates**

First available in Project Euclid:
12 June 2015

**Permanent link to this document**

https://projecteuclid.org/
euclid.pgiq/1434148595

**Digital Object Identifier**

doi:10.7546/giq-4-2003-11-41

**Mathematical Reviews number (MathSciNet)**

MR1977559

**Zentralblatt MATH identifier**

1039.53104

#### Citation

Hirshfeld, Allen. Deformation Quantization in Quantum Mechanics and Quantum Field Theory. Proceedings of the Fourth International Conference on Geometry, Integrability and Quantization, 11--41, Coral Press Scientific Publishing, Sofia, Bulgaria, 2003. doi:10.7546/giq-4-2003-11-41. https://projecteuclid.org/euclid.pgiq/1434148595

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