From Noncommutative Phase Space to Hilbert Space

Journal Article · Wed Oct 03 04:00:00 EDT 2007 · AIP Conference Proceedings

DOI:https://doi.org/10.1063/1.2803828· OSTI ID:21035978

Muthukumar, B ^[1]

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1. Saha Institute of Nuclear Physics, 1/AF, Bidhan nagar, Kolkata-700 064 (India)

In the Hamiltonian formulation, classical mechanics employs the commutative algebra of functions that are defined on phase space a point of which could be represented using Dirac delta distributions. In the absence of such a concrete existence of the notion of point in the quantum domain, we show how the quantum mechanical formalism emerges by replacing the commutative algebra by a noncommutative algebra of functions and introducing the quantum condition. The noncommutativity is achieved by deforming the product of two functions and by introducing the Moyal brackets, the basic Moyal brackets between two spatial coordinates and between position and momentum coordinates being noncommutative. The only other input we make use of is the quantum condition which is motivated from its classical analogue--the square of Dirac delta distributions.

• OSTI ID: 21035978
• Journal Information: AIP Conference Proceedings, Journal Name: AIP Conference Proceedings Journal Issue: 1 Vol. 939; ISSN APCPCS; ISSN 0094-243X
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
CLASSICAL MECHANICS
COMMUTATION RELATIONS
COORDINATES
DISTRIBUTION
GAUGE INVARIANCE
HAMILTONIANS
HILBERT SPACE
PHASE SPACE
QUANTUM FIELD THEORY
QUANTUM MECHANICS
STRING MODELS
STRING THEORY