Moyal product
Encyclopedia
In mathematics
, the Moyal product, named after José Enrique Moyal
, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket
(with a generalization to symplectic manifold
s below).
This particular star product is also sometimes called Weyl
-Groenewold product, as it was introduced by H. J. Groenewold
in 1946, in a trenchant appreciation of Weyl quantization
—Moyal actually appears to not know about it in his celebrated paper, and in his legendary correspondence with Dirac, as adduced in his biography. (The paradoxical popular naming after Moyal, utilized in this stub, appears to have emerged only in the 1970s, in homage to his
flat phase-space quantization
picture.)
s
and 
on 
takes the form
where each
is a certain bidifferential operator of order 
with the following properties. (See below for an explicit formula).
(Deformation of the pointwise product) — implicit in the definition.
(Deformation of the Poisson bracket, called Moyal bracket
.)
(The 1 of the undeformed algebra is also the identity in the new algebra.)
(The complex conjugate is an antilinear antiautomorphism.)
Note that, if one wishes to take functions valued in the real numbers, then an alternative version eliminates the
in condition 2 and eliminates condition 4.
If one restricts to polynomial functions, the above algebra is isomorphic to the Weyl algebra
,
and the two offer alternative realizations of Weyl quantization
of the space of polynomials in
variables (or, the symmetric algebra
of a vector space of dimension
).
To provide an explicit formula, consider a constant Poisson bivector
on 
:
where
is just a complex number for each 
.
The star product of two functions
and 
can then be defined as
where
is the reduced Planck constant, treated as a formal parameter here.
A closed form can be obtained by using the exponential
,
where
is the multiplication map, 
, and the exponential is treated as a power series, 
.
That is, the formula for
is
As mentioned, often one eliminates all occurrences of
above, and the formulas then restrict naturally to real numbers.
Note that if the functions
and 
are polynomials, the above infinite sums become finite (reducing to the ordinary Weyl algebra case).
; and, using the associated Poisson bivector, one may consider the above formula. For it to work globally,
as a function on the whole manifold (and not just a local formula), one must equip the symplectic manifold with a flat symplectic connection
.
More general results for arbitrary Poisson manifolds (where the Darboux theorem does not apply) are given by the Kontsevich quantization formula
.
) is given in the article on Weyl quantization
. Every correspondence prescription between phase space and Hilbert space, however, induces its own proper ∗-product.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Moyal product, named after José Enrique Moyal
José Enrique Moyal
José Enrique Moyal was a mathematical physicist who contributed to aeronautical engineering, electrical engineering and statistics, among other fields...
, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
(with a generalization to symplectic manifold
Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
s below).
This particular star product is also sometimes called Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
-Groenewold product, as it was introduced by H. J. Groenewold
Hilbrand J. Groenewold
Hilbrand Johannes Groenewold was a Dutch theoretical physicist who pioneered the largely operator-free formulation of quantum mechanics in phase space known as phase-space quantization....
in 1946, in a trenchant appreciation of Weyl quantization
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
—Moyal actually appears to not know about it in his celebrated paper, and in his legendary correspondence with Dirac, as adduced in his biography. (The paradoxical popular naming after Moyal, utilized in this stub, appears to have emerged only in the 1970s, in homage to his
flat phase-space quantization
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
picture.)
Definition
The product (for smooth functionSmooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s
and on takes the formwhere each
is a certain bidifferential operator of order with the following properties. (See below for an explicit formula).(Deformation of the pointwise product) — implicit in the definition.
(Deformation of the Poisson bracket, called Moyal bracket
Moyal bracket
In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.The Moyal Bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with Dirac...
.)
(The 1 of the undeformed algebra is also the identity in the new algebra.)
(The complex conjugate is an antilinear antiautomorphism.)
Note that, if one wishes to take functions valued in the real numbers, then an alternative version eliminates the
in condition 2 and eliminates condition 4.If one restricts to polynomial functions, the above algebra is isomorphic to the Weyl algebra
,and the two offer alternative realizations of Weyl quantization
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
of the space of polynomials in
variables (or, the symmetric algebraSymmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
of a vector space of dimension
).To provide an explicit formula, consider a constant Poisson bivector
on :
where
is just a complex number for each .The star product of two functions
and can then be defined aswhere
is the reduced Planck constant, treated as a formal parameter here.A closed form can be obtained by using the exponential
Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
,
where
is the multiplication map, , and the exponential is treated as a power series, .That is, the formula for
is
As mentioned, often one eliminates all occurrences of
above, and the formulas then restrict naturally to real numbers.Note that if the functions
and are polynomials, the above infinite sums become finite (reducing to the ordinary Weyl algebra case).On manifolds
On any symplectic manifold, one can, at least locally, choose coordinates so as make the symplectic structure constant, by Darboux's theoremDarboux's theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry...
; and, using the associated Poisson bivector, one may consider the above formula. For it to work globally,
as a function on the whole manifold (and not just a local formula), one must equip the symplectic manifold with a flat symplectic connection
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
.
More general results for arbitrary Poisson manifolds (where the Darboux theorem does not apply) are given by the Kontsevich quantization formula
Kontsevich quantization formula
In mathematics, the Kontsevich quantization formula describes how to construct an generalized ∗-product operator algebra from a given Poisson manifold. This operator algebra amounts to the deformation quantization of the Poisson algebra...
.
Examples
A simple explicit example of the construction and utility of the ∗-product (for the simplest case of a two-dimensional euclidean phase spacePhase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
) is given in the article on Weyl quantization
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
. Every correspondence prescription between phase space and Hilbert space, however, induces its own proper ∗-product.