# Category: Mathematical quantization

Heisenberg group
In mathematics, the Heisenberg group , named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form under the operation of matrix multiplication. Elements a, b and c can be
Second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quan
Moyal product
In mathematics, the Moyal product (after José Enrique Moyal; also called the star product or Weyl–Groenewold product, after Hermann Weyl and Hilbrand J. Groenewold) is an example of a phase-space star
Geometric quantization
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which ther
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 H
Fredholm module
In noncommutative geometry, a Fredholm module is a mathematical structure used to quantize the differential calculus. Such a module is, up to trivial changes, the same as the abstract elliptic operato
Quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum grou
Noncommutative quantum field theory
In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field the
Fuzzy sphere
In mathematics, the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere
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 succeed
Quantization commutes with reduction
In mathematics, more specifically in the context of geometric quantization, quantization commutes with reduction states that the space of global sections of a line bundle L satisfying the quantization
Canonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest ext
Quantized enveloping algebra
In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra , the quantum enveloping algebra is typically denoted as . Among the appl
Stone–von Neumann theorem
In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between positi
Kontsevich quantization formula
In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebr
Lagrangian foliation
In mathematics, a Lagrangian foliation or polarization is a foliation of a symplectic manifold, whose leaves are Lagrangian submanifolds. It is one of the steps involved in the geometric quantization
Noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutativ
Theta representation
In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under
Quantization of the electromagnetic field
The quantization of the electromagnetic field, means that an electromagnetic field consists of discrete energy parcels, photons. Photons are massless particles of definite energy, definite momentum, a
Lagrangian Grassmannian
In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is 1/2n(n + 1) (where the dimension of V is 2n). It may be
Phase-space formulation
The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position or momentum represen
Quantum affine algebra
In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independ
Dirac–von Neumann axioms
In mathematical physics, the Dirac–von Neumann axioms give a mathematical formulation of quantum mechanics in terms of operators on a Hilbert space. They were introduced by Paul Dirac in 1930 and John
Dirac bracket
The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac to treat classical systems with second class constraints in Hamiltonian mechanics, and to thus allow them to underg
Quantization (physics)
In physics, quantization (in British English quantisation) is the systematic transition procedure from a classical understanding of physical phenomena to a newer understanding known as quantum mechani