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 taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group").
The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to n-dimensional systems, and most generally, to any symplectic vector space.
In mathematics, the Heisenberggroup H {\displaystyle H} , named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ( 1...
Werner Karl Heisenberg (pronounced [ˈvɛʁnɐ kaʁl ˈhaɪzn̩bɛʁk] ; 5 December 1901 – 1 February 1976) was a German theoretical physicist, one of the main pioneers...
is Lagrangian. A Heisenberggroup can be defined for any symplectic vector space, and this is the typical way that Heisenberggroups arise. A vector space...
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a...
U(1) (the circle group of unit modulus complex numbers). As an abstract group, the Heisenberggroup is the three-dimensional Lie group of triples (x, ξ...
of the Heisenberggroup (fixing its center) form the symplectic group, so at first sight this seems to give an action of the symplectic group on H {\displaystyle...
asserting the uniqueness of the Schrödinger representation of the Heisenberggroup. The fields of probability and statistics frequently use the normal...
The quantum Heisenberg model, developed by Werner Heisenberg, is a statistical mechanical model used in the study of critical points and phase transitions...
abelian normal subgroup of the group, and the corresponding quotient is the circle group. The finite Heisenberggroup H3,p of order p3 is metabelian....
translations on the Heisenberggroup of the reals, identified with 3-dimensional Euclidean space. This is a discrete cocompact group of affine transformations...
In particular, taking n = 3 yields the Heisenberggroup H, an example of a non-abelian infinite nilpotent group. It has nilpotency class 2 with central...
In group theory, a branch of abstract algebra, extraspecial groups are analogues of the Heisenberggroup over finite fields whose size is a prime. For...
representations of a certain group. In several joint papers, Deninger and Wilhelm Singhof studied quotients of the n-dimensional Heisenberggroup H by the standard...
discrete subgroup of the Heisenberggroup. This invariance is presented in the article on the theta representation of the Heisenberggroup. If F is a quadratic...
the Heisenberggroup. The point stabilizer is O(2, R). The group G has 2 components, and is a semidirect product of the 3-dimensional Heisenberggroup by...
mathematics, the theta representation is a particular representation of the Heisenberggroup of quantum mechanics. It gains its name from the fact that the Jacobi...
quantum Heisenberg chain. The recipe can be applied for every translationally invariant one-dimensional lattice. DMRG is a renormalization-group technique...
the Jacobi group, introduced by Eichler & Zagier (1985), is the semidirect product of the symplectic group Sp2n(R) and the Heisenberggroup R1+2n. The...
Heisenberg algebra, the Lie algebra of the Heisenberggroup, by setting the central element of the Heisenberg algebra (namely [X,Y]) equal to the unit of...
on the Heisenberggroup have been also studied, like minimal surfaces in the Heisenberggroup, the Bernstein problem in the Heisenberggroup and curvature...
[{\hat {x}},{\hat {p}}]=i\hbar } is called the Heisenberggroup. This group can be realized as the group of 3 × 3 {\displaystyle 3\times 3} upper triangular...
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