deformationquantization

Deformation quantization is a mathematical approach to quantize a classical system by deforming the algebra of classical observables. Instead of replacing functions by operators on a Hilbert space, one constructs a noncommutative product, called a star product, on the space of smooth functions on a manifold, depending on a deformation parameter usually identified with Planck's constant ħ.

More precisely, on a Poisson manifold M, one seeks an associative product * on C∞(M)[[ħ]] of the form f * g = fg + ħ C1(f,g) + ħ2 C2(f,g) + ... with Ci bidifferential operators. The product should reduce to the ordinary pointwise product when ħ = 0. The first-order commutator satisfies f * g − g * f = iħ {f,g} + O(ħ2), tying the deformation to the Poisson bracket.

Historically, deformation quantization was developed in the late 1970s as an algebraic framework for quantization. The Moyal–Weyl product provides a concrete example on R2n with a constant Poisson structure. For general symplectic manifolds, Fedosov constructed a geometric quantization. In 1997, Kontsevich proved that every finite-dimensional Poisson manifold admits a star product, via his formality theorem, and furnished a classification up to equivalence by Poisson cohomology.

Equivalence of star products is realized by formal series of differential operators, yielding isomorphic deformations of the function algebra. Different choices correspond to distinct quantizations related by gauge-like transformations, with the set of equivalence classes often governed by Poisson cohomology.

Deformation quantization provides a rigorous bridge between classical and quantum mechanics and has applications in noncommutative geometry, representation theory, and theoretical physics as an alternative to operator-based quantization.