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 or a Poisson algebra.

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) 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 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 f₁, f₂ ∈ C^∞(ℜ²), is specified by

where ${\displaystyle \Phi }$ is the Wigner–Weyl transform.

The star product is not commutative in general, but goes over to the ordinary commutative product of functions in the limit of ħ → 0. As such, it is said to define a deformation of the commutative algebra of C^∞(ℜ²).

For the Weyl-map example above, the ★-product may be written in terms of the Poisson bracket as

Here, Π is the Poisson bivector, an operator defined such that its powers are