Proceedings of the International Conference on Geometry, Integrability and Quantization

Deformation Quantization in Quantum Mechanics and Quantum Field Theory

Allen Hirshfeld

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

Source
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

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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