In mathematics and physics, deformation quantization roughly amounts to finding a (quantum) algebra whose classical limit is a given (classical) algebra such as a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
or a
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
.
In physics
Intuitively, a
deformation of a mathematical object is a family of the same kind of objects that depend on some parameter(s).
Here, it provides rules for how to deform the "classical" commutative algebra of observables to a quantum non-commutative algebra of observables.
The basic setup in deformation theory is to start with an algebraic structure (say a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
) and ask: Does there exist a one or more parameter(s) family of ''similar'' structures, such that for an initial value of the parameter(s) one has the same structure (Lie algebra) one started with? (The oldest illustration of this may be the realization of
Eratosthenes
Eratosthenes of Cyrene (; ; – ) was an Ancient Greek polymath: a Greek mathematics, mathematician, geographer, poet, astronomer, and music theory, music theorist. He was a man of learning, becoming the chief librarian at the Library of A ...
in the ancient world that a flat Earth was deformable to a spherical Earth, with deformation parameter 1/''R''
⊕.) E.g., one may define a
noncommutative torus as a deformation quantization through a
★-product to implicitly address all convergence subtleties (usually not addressed in formal deformation quantization). Insofar as the algebra of functions on a space determines the geometry of that space, the study of the star product leads to the study of a
non-commutative geometry deformation of that space.
In the context of the above flat phase-space example, the star product (
Moyal product, actually introduced by Groenewold in 1946),
★''ħ'', of a pair of functions in , is specified by
:
where
is the
Wigner–Weyl transform
In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform (after Hermann Weyl and Eugene Wigner) is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödin ...
.
The star product is not commutative in general, but goes over to the ordinary commutative product of functions in the limit of . As such, it is said to define a
deformation of the commutative algebra of .
For the Weyl-map example above, the
★-product may be written in terms of the
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. Th ...
as
:
Here, Π is the
Poisson bivector, an operator defined such that its powers are
:
and
:
where is the
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. Th ...
. More generally,
:
where
is the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
.
Thus, e.g.,
[
] Gaussians compose
hyperbolically,
:
or
:
etc.
These formulas are predicated on coordinates in which the
Poisson bivector is constant (plain flat Poisson brackets). For the general formula on arbitrary
Poisson manifolds, cf. the
Kontsevich quantization formula.
Antisymmetrization of this
★-product yields the
Moyal bracket, the proper quantum deformation of the
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. Th ...
, and the phase-space isomorph (Wigner transform) of the quantum
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
in the more usual Hilbert-space formulation of quantum mechanics. As such, it provides the cornerstone of the dynamical equations of observables in this phase-space formulation.
There results a complete
phase space formulation of quantum mechanics, ''completely equivalent to the Hilbert-space operator representation'', with star-multiplications paralleling operator multiplications isomorphically.
Expectation values in phase-space quantization are obtained isomorphically to tracing operator observables with the density matrix in Hilbert space: they are obtained by phase-space integrals of observables such as the above with the
Wigner quasi-probability distribution effectively serving as a measure.
Thus, by expressing quantum mechanics in phase space (the same ambit as for classical mechanics), the above Weyl map facilitates recognition of quantum mechanics as a
deformation (generalization, cf.
correspondence principle
In physics, a correspondence principle is any one of several premises or assertions about the relationship between classical and quantum mechanics.
The physicist Niels Bohr coined the term in 1920 during the early development of quantum theory; ...
) of classical mechanics, with deformation parameter . (Other familiar deformations in physics involve the deformation of classical Newtonian into relativistic mechanics, with deformation parameter ''v''/''c''; or the deformation of Newtonian gravity into General Relativity, with deformation parameter Schwarzschild-radius/characteristic-dimension. Conversely,
group contraction leads to the vanishing-parameter undeformed theories—
classical limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
s.)
Classical expressions, observables, and operations (such as Poisson brackets) are modified by -dependent quantum corrections, as the conventional commutative multiplication applying in classical mechanics is generalized to the ''noncommutative star-multiplication'' characterizing quantum mechanics and underlying its uncertainty principle.
See also
*
Deligne's conjecture on Hochschild cohomology
*
Poisson manifold
*
Formality theorem; for now, see https://mathoverflow.net/questions/32889/a-few-questions-about-kontsevich-formality
*
Kontsevich quantization formula
References
Further reading
*
* {{cite book , last=Esposito , first=Chiara , url=https://link.springer.com/book/10.1007/978-3-319-09290-4 , title=Formality Theory: From Poisson Structures to Deformation Quantization , publisher=Springer International Publishing , year=2015 , isbn=978-3-319-09289-8 , publication-place=Cham , doi=10.1007/978-3-319-09290-4 , issn=2197-1757
* https://ncatlab.org/nlab/show/deformation+quantization
* https://ncatlab.org/nlab/show/formal+deformation+quantization
Mathematical quantization
Mathematical physics