Heisenberg algebra

Heisenberg algebra

[′hīz·ən·bərg ‚al·jə·brə]
(quantum mechanics)
The Lie algebra formed by the operators of position and momentum.
References in periodicals archive ?
The opening paper reviews in parallel two bosonizations of the CKP hierarchy, one arising from a twisted Heisenberg algebra and the second from an untwisted Heisenberg algebra.
In this case the commutator relations (2)-(3) define a polynomial Heisenberg algebra of degree l-1.
We recall that A(n) denotes the abelian Lie algebra of dimension n and the main results of [2, 5, 6, 7, 11, 18, 19] illustrate that many inequalities on dim M(L) become equalities if and only if L splits in the sums of A(n) and of a Heisenberg algebra H(m) (here m [greater than or equal to] 1 is a given integer).
We consider the generalized Heisenberg algebra, which is a (2n + 1)-dimensional Lie algebra G given, with respect to a basis {[x.sub.1], [x.sub.2],...,[x.sub.2n+1]}, by the following nontrivial brackets:
The invariant (super) algebras with an infinite tower of generators are given by the universal enveloping algebra of a deformed Heisenberg algebra [3-6].
As the main application of quantum groups, q-deformed quantum mechanics [8, 9] was developed by generalizing the standard quantum mechanics which was based on the Heisenberg commutation relation (the Heisenberg algebra).
But we know that this space serves as a representation for the Heisenberg algebra, and hence the algebra of observables, on which the action of the [[??].sub.i] operators for the momentum observables have eigenstates being linear combinations of the basis |[x.sup.i]> states.
The first is the canonical commutation relations of the infinite-dimensional Heisenberg algebra, or oscillator algebra.
The Quesne-Tkachuk algebra is the simplest possible covariant generalization of the Heisenberg algebra that allows for a minimal length [25].
However, putting Lorentz and Heisenberg algebra together should have given us a stable, relativistic quantum theory but unfortunately it did not.