On the theory of unitary representations of theSL(2C) group

Abstract

The irreducible unitary representation of the noncompact SL(2, C) is discussed by a method based on the use of the homogeneous functions. The mat¡ elements of the finite transformations are calculated explicitly. The method is very eonvenient for applications in physics and the results of GELFAND and NAIMAI~K are obtained in a very simple way.

Twistorial phase space for complex Ashtekar variables

“…(A.17)Canonical basis. The canonical basis vectors f (ρ,jo) j,m in the (−j o − 1 + iρ, j o + 1 + iρ) irreducible unitary representation space of SL(2, C)[25,24,23] diagonalise a complete set of commuting observables:Since singular configurations H = 0 have already been removed, we get the desired constraint C = H − H = 0. (B.7)…”

Analog of the Peter-Weyl expansion for Lorentz group

“…Let us use the explicit expression for the matrix coefficients in (1). The first explicit expression of the principal series matrix coefficients D (k,ρ) jn,j ′ m , k ∈ Z, ρ ∈ C was obtained by Duc and Hieu in 1967 [1], formula (4.11):…”

“…Let us use the explicit expression for the matrix coefficients in (1). The first explicit expression of the principal series matrix coefficients…”

“…In this paper the sum convergence proof is the main and the most challenging task. To prove convergence we use the following: a) the principal series matrix coefficients expression via the hypergeometric functions [1], b) formula (4.11), the Watson's asymptotic of the hypergeometric functions 2 F 1 (a, b, c, z), when all three parameters tend to infinity [5], and c) the D'Alemebert-Cauchy convergence ratio test. The paper is organized as follows.…”

Complex Ashtekar Variables and Reality Conditions for Holst’s Action

“…In this section we check if all the constraints (23,24,25,25,26) are preserved under the time evolution defined by (30). As a preliminary step to calculate this one first proves the following list of Poisson brackets:…”