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Heisenberg group

Index Heisenberg group

In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form \end under the operation of matrix multiplication. [1]

96 relations: Abelian variety, Abstract nonsense, American Mathematical Society, Arc (geometry), Associative algebra, Bilinear form, Bulletin of the American Mathematical Society, Canonical commutation relation, Canonical coordinates, CCR and CAR algebras, Center (group theory), Character theory, Characteristic (algebra), Commutative ring, Commutator subgroup, Connected space, Cotangent bundle, David Mumford, Diffeomorphism, Dihedral group, Equations defining abelian varieties, Exact sequence, Exponential map (Lie theory), Extra special group, Field (mathematics), Fourier analysis, Fourier transform, Frattini subgroup, Free module, Function space, Geodesic, George Mackey, Gromov's theorem on groups of polynomial growth, Group (mathematics), Group cohomology, Group extension, Group representation, Growth rate (group theory), Haar measure, Heisenberg picture, Hermann Weyl, Hilbert space, Holomorphic function, Identity matrix, Integer, Jacobi elliptic functions, Lie algebra, Lie group, Manifold, Mathematics, ..., Matrix multiplication, Matrix representation, Modular arithmetic, Momentum operator, Multiplier (Fourier analysis), Nilpotent group, Non-abelian group, Nondegenerate form, One-form, Order (group theory), Poincaré–Birkhoff–Witt theorem, Poisson bracket, Pontryagin duality, Position operator, Prime number, Projective representation, Quantum mechanics, Real number, Representation of a Lie group, Row and column vectors, Schrödinger picture, Simply connected space, Skew-symmetric matrix, Square-integrable function, Stokes' theorem, Stone–von Neumann theorem, Sub-Riemannian manifold, Subbundle, Symplectic vector space, System of imprimitivity, Tangent bundle, Theta function, Theta representation, Torsion group, Triangular matrix, Unitary representation, Universal enveloping algebra, University of Chicago Press, Upper half-plane, Vector field, Vector space, Viz., Werner Heisenberg, Weyl algebra, Wigner–Weyl transform, Zero matrix. Expand index (46 more) »

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.

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Abstract nonsense

In mathematics, abstract nonsense, general abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra.

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American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

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Arc (geometry)

In Euclidean geometry, an arc (symbol: ⌒) is a closed segment of a differentiable curve.

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Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.

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Bilinear form

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.

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Bulletin of the American Mathematical Society

The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.

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Canonical commutation relation

In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another).

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Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time.

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CCR and CAR algebras

In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions respectively.

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Center (group theory)

In abstract algebra, the center of a group,, is the set of elements that commute with every element of.

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Character theory

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix.

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Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

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Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

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Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

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Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

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Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.

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David Mumford

David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory.

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In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

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Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

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Equations defining abelian varieties

In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve.

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Exact sequence

An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry.

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Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.

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Extra special group

In group theory, a branch of abstract algebra, extra special groups are analogues of the Heisenberg group over finite fields whose size is a prime.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

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Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

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Frattini subgroup

In mathematics, the Frattini subgroup Φ() of a group is the intersection of all maximal subgroups of.

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Free module

In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements.

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Function space

In mathematics, a function space is a set of functions between two fixed sets.

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In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

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George Mackey

George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician.

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Gromov's theorem on groups of polynomial growth

In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Group cohomology

In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology.

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Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group.

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Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

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Growth rate (group theory)

In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group.

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Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

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Heisenberg picture

In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.

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Hermann Weyl

Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

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Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

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An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Jacobi elliptic functions

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that are of historical importance.

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Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

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Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

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Matrix representation

Matrix representation is a method used by a computer language to store matrices of more than one dimension in memory.

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Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

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Momentum operator

In quantum mechanics, the momentum operator is an operator which maps the wave function in a Hilbert space representing a quantum state to another function.

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Multiplier (Fourier analysis)

In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions.

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Nilpotent group

A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with.

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Non-abelian group

In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a.

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Nondegenerate form

In linear algebra, a nondegenerate form or nonsingular form is a bilinear form that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if.

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In linear algebra, a one-form on a vector space is the same as a linear functional on the space.

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Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

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Poincaré–Birkhoff–Witt theorem

In mathematics, more specifically in abstract algebra, in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra.

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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.

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Pontryagin duality

In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups.

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Position operator

In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.

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Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

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Projective representation

In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity; scalar transformations). In more concrete terms, a projective representation is a collection of operators \rho(g),\, g\in G, where it is understood that each \rho(g) is only defined up to multiplication by a constant. These should satisfy the homomorphism property up to a constant: for some constants c(g,h). Since each \rho(g) is only defined up to a constant anyway, it does not strictly speaking make sense to ask whether the constants c(g,h) are equal to 1. Nevertheless, one can ask whether it is possible to choose a particular representative of each family \rho(g) of operators in such a way that the \rho(g)'s satisfy the homomorphism property on the nose, not just up to a constant. If such a choice is possible, we say that \rho can be "de-projectivized," or that \rho can be "lifted to an ordinary representation." This possibility is discussed further below.

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Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Representation of a Lie group

In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry.

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Row and column vectors

In linear algebra, a column vector or column matrix is an m × 1 matrix, that is, a matrix consisting of a single column of m elements, Similarly, a row vector or row matrix is a 1 × m matrix, that is, a matrix consisting of a single row of m elements Throughout, boldface is used for the row and column vectors.

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Schrödinger picture

In physics, the Schrödinger picture (also called the Schrödinger representation) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time.

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Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

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Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition In terms of the entries of the matrix, if aij denotes the entry in the and; i.e.,, then the skew-symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.

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Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

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Stokes' theorem

In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes theorem or the Stokes–Cartan theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

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Stone–von Neumann theorem

In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators.

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Sub-Riemannian manifold

In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold.

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In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right.

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Symplectic vector space

In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form.

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System of imprimitivity

The concept of system of imprimitivity is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations.

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Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.

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Theta function

In mathematics, theta functions are special functions of several complex variables.

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Theta representation

In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics.

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Torsion group

In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which each element has finite order.

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Triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix.

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Unitary representation

In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.

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Universal enveloping algebra

In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.

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University of Chicago Press

The University of Chicago Press is the largest and one of the oldest university presses in the United States.

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Upper half-plane

In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part: The term arises from a common visualization of the complex number x + iy as the point (x,y) in the plane endowed with Cartesian coordinates.

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Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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The abbreviation viz. (or viz without a full stop), short for the Latin italic, is used as a synonym for "namely", "that is to say", "to wit", or "as follows".

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Werner Heisenberg

Werner Karl Heisenberg (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics.

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Weyl algebra

In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form More precisely, let F be the underlying field, and let F be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X.

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Wigner–Weyl transform

In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture.

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Zero matrix

In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero.

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[1] https://en.wikipedia.org/wiki/Heisenberg_group

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