On the deformability of Heisenberg algebras

Abstract

Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of a q-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e. if one works in position-momentum realization, can be mapped on a q-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not posess a proper group of global transformations.

Particles in singular magnetic field

“…It is interesting that in this algebraic approach all commutation relations for particles equipped with arbitrary statistics can be described as a representation of the so-called quantum Weyl algebra W (or Wick algebra) [22,23,27,38]. Similar approach has been also considered by others authors, see [24,25,26,27] and [28,29]. In this attempt the creation and anihilation operators act on certain quadratic algebra A.…”

Embedding Q-deformed Heisenberg algebras into undeformed ones

“…Yes. The affirmative answer is based on the vanishing of the second Hochschild cohomology group of any Weyl algebra [2,3,4]; this allows to prove the existence of f without however providing an explicit construction. The argument is valid not only for deformations of the type (1.6), but for any kind of deformation of A.…”

“…Roughly speaking, the first is: is there a formal realization of the elements of the deformed algebra in terms of elements of the undeformed algebra? The answer is affirmative [2,3,4] but in general the realization is not explicitly known. The second subquestion is: do also the corresponding representation theories coincide?…”

Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics

“…Throughout this paper, we use units wherein = 1 (as well as = m = 1, where and m denote the characteristic length and the particle mass for the quantum mechanical system under consideration). Equation (2.1) admits two important special cases: choosing κ = 0 and α, β > 0 leads to Kempf's commutation relation characterized by nonzero minimal uncertainties ∆X 0 = β/(1 − αβ) and ∆P 0 = α/(1 − αβ) 1 , whereas selecting α = β = 0 gives rise to the qdeformed Heisenberg algebra qXP − q * P X = i with q ≡ 1 − iκ [35].…”

“…and 35) respectively. Here ν is defined by ν = 1 βB g 2 + 4βBr (3.36) and a similar expression applies to ν n in terms of g n and r n , while c j denote some constants.…”