Extended MacMahon–Schwingerʼs Master Theorem and conformal wavelets in complex Minkowski space

Abstract

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D 4 = SO(4, 2)/(SO(4) × SO(2)) of the conformal group SO(4, 2) (locally isomorphic to SU (2, 2)) in 1+3 dimensions. The manifold D 4 can be mapped one-to-one onto the future tube domain C 4 + of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L 2 h (D 4 , dν λ ) and L 2 h (C 4 + , dν λ ) of square integrable holomorphic functions with scale dimension λ and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a λextension of Schwinger's Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitaryrepresentation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of L 2 h (D 4 , dν λ ). SMT is related to MacMahon's Master Theorem (MMT) and an extension of both in terms of Louck's SU (N ) solid harmonics is also provided for completeness. Convergence conditions are also studied.MSC: 42C, 43A, 22E70, 05A10, 81R

On the oscillator realization of conformal U(2, 2) quantum particles and their particle-hole coherent states

“…This theorem has been proved in [28] in the context of conformal wavelets. Here we only point out that, replacing X = Z ′ † Z in (22) and using determinant and Wigner's D-matrix properties [8], one easily realizes that that (22) reproduces (24).…”

“…, where there is an excess of 2κ − 1 a-type over b-type quanta. However, these are still not the CS (28) we are dealing with in this article. Actually, the CS (28) will be related to massive conformal particles (see later in Sec.…”

“…We have used the Iwasawa decomposition of an element g given in (11,12) and denoted by |dz| and |dZ| the Lebesgue measures on C and C 4 , respectively (see [28] for more explicit expressions of this measure). The group U(2, 2) is represented on L 2 (U(2, 2), dµ) as (left-action) [U(g ′ )ψ](g) = ψ(g ′−1 g).…”

“…The case of positive-mass unireps. of U(2, 2) was already discussed in, for example, [25,26,27], and more recently by us in [28,29,30], but their oscillator realizations have not been studied as thoroughly as the simpler case of ladder (most degenerate) representations. An inspiring article on this subject has been recently published in [31], in the context of deformation quantization.…”

Massive conformal particles with non-Abelian charges from free U(2N,2N)-twistor dynamics: Quantization and coherent states

“…Proof: It is easy to see that U n (U)U n (U ′ ) = U n (UU ′ ) (group homomorphism). Irreducibility is related to the fact that the constant function φ(Z) = 1 is mapped to det(A † +B † Z) −n (the multiplier), which can be expanded in a complete basis of homogeneous polynomials of arbitrary homogeneity degree in the 4N 2 complex entries of Z (see [26] for a relation of this expansion with the MacMahon-Schwinger's Master Theorem). In order to prove unitarity, i.e.…”