Functional Integrals and Probabilistic Amplitudes

Functional integrals and probabilistic amplitudes. Brief historical notes. The reconstruction of quantum mechanics from path integrals. The Feynman formulation. Definition and properties of the coherent states and the Bargmann representation.

Interpolations of Bargmann Type Measures

In this paper, we shall discuss Bargmann type measures on C for several classes of probability measures on R. The unified interpolation expressions include not only the classical Bargmann measure and its q-deformation, but also their t-deformations and dilations. As a special case, we get conditions on existence and an explicit form of the Bargmann representation for the free Meixner family of probability measures.

A polynomial class of u(2) algebras

A r-parameter u{κ1,κ2,…,κr}(2) algebra is introduced. Finite unitary representations are investigated. This polynomial algebra reduces via a contraction procedure to the generalized Weyl–Heisenberg algebra 𝒜{κ1,κ2,…,κr} [M. Daoud and M. Kibler, J. Phys. A: Math. Theor.45 (2012) 244036]. A pair of nonlinear (quadratic) bosons of type 𝒜κ ≡ 𝒜{κ1=κ,κ2=0,…,κr=0} is used to construct, à la Schwinger, a one parameter family of (cubic) uκ(2) algebra. The corresponding Hilbert space is constructed. The analytical Bargmann representation is also presented.