Moyal product
E503520
The Moyal product is a noncommutative star product used in deformation quantization to encode quantum mechanical operator multiplication directly on phase-space functions.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
construction in deformation quantization
ⓘ
mathematical concept ⓘ noncommutative product ⓘ star product ⓘ |
| alsoKnownAs |
Groenewold–Moyal product
NERFINISHED
ⓘ
Moyal star product NERFINISHED ⓘ Weyl–Groenewold–Moyal product NERFINISHED ⓘ |
| appearsIn |
noncommutative field theory
ⓘ
phase-space formulation of quantum mechanics ⓘ quantum optics ⓘ |
| codomain | functions on phase space ⓘ |
| constructionMethod | exponential of bidifferential operator involving symplectic form ⓘ |
| definedOn |
flat phase space ℝ^{2n}
ⓘ
symplectic vector spaces ⓘ |
| domain |
Schwartz functions on ℝ^{2n}
ⓘ
functions on phase space ⓘ tempered distributions ⓘ |
| field |
deformation quantization
ⓘ
mathematical physics ⓘ operator algebras ⓘ quantum mechanics ⓘ symplectic geometry ⓘ |
| generalizationOf | pointwise product of functions ⓘ |
| historicalOrigin | introduced in work of José Enrique Moyal in the 1940s ⓘ |
| invariantUnder | linear symplectic transformations of phase space ⓘ |
| mathematicalStructure | formal power series in ℏ of bidifferential operators ⓘ |
| namedAfter | José Enrique Moyal NERFINISHED ⓘ |
| parameter | Planck constant ℏ NERFINISHED ⓘ |
| property |
associative
ⓘ
bilinear ⓘ deformation of pointwise product ⓘ first-order commutator reproduces Poisson bracket ⓘ noncommutative ⓘ reduces to pointwise product as ℏ → 0 ⓘ |
| relatedTo |
Moyal bracket
NERFINISHED
ⓘ
Poisson bracket ⓘ Weyl quantization ⓘ Wigner function NERFINISHED ⓘ star-commutator ⓘ |
| satisfies |
associativity identity (f ⋆ g) ⋆ h = f ⋆ (g ⋆ h)
ⓘ
correspondence principle with classical mechanics ⓘ |
| usedFor |
Wigner–Weyl phase-space formulation of quantum mechanics
NERFINISHED
ⓘ
deformation quantization of classical phase space ⓘ encoding operator multiplication on phase-space functions ⓘ formulating quantum mechanics on phase space ⓘ noncommutative geometry models ⓘ quantization of Poisson manifolds in flat case ⓘ |
Referenced by (1)
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