Higher- U (2,2)-spin fields and higher-dimensional 𝒲-gravities: quantum AdS space and radiation phenomena

Abstract

A physical and geometrical interpretation of previously introduced tensor operator algebras of U (2, 2) in terms of algebras of higher-conformal-spin quantum fields on the anti-de Sitter space AdS 5 is provided. These are higher-dimensional W-like algebras and constitute a potential gauge guide principle towards the formulation of induced conformal gravities (Wess-Zumino-Witten-like models) in realistic dimensions. Some remarks on quantum (Moyal) deformations are given and potentially tractable versions of noncommutative AdS spaces are also sketched. The role of conformal symmetry in the microscopic description of Unruh's and Hawking's radiation effects is discussed. , tensor operator algebras. * This seems to be an important and general feature of quantum gauge theories as opposite to their classical counterparts. For example, see Refs. [2,3,4] for a cohomological (Higgs-less) generation of mass in Yang-Mills theories through non-trivial representations of the gauge group; in this proposal, gauge modes become also physical and the corresponding extra field degrees of freedom are transferred to the vector potentials (longitudinal components) to form massive vector bosons.algebras w have also a space origin as (area preserving) diffeomorphisms and Poisson algebras of functions on symplectic manifolds (e.g. cylinder). There is a group-theoretic structure underlying their quantum (Moyal [7]) deformations (collectively denoted by W), according to which W algebras are just particular members of a one-parameter family L ρ (sl(2, R)) -in the notation of the present paper-of non-isomorphic infinite-dimensional Lie-algebras of SL(2, R) tensor operators -see later on Eq. (6). The connection with the theory of higher-spin gauge fields in (1+1)-and (2+1)-dimensional anti-de Sitter space AdS [8] -homogeneous spaces of SO(1, 2) ∼ SL(2, R) and SO(2, 2) ∼ SL(2, R) × SL(2, R), respectively-is then apparent in this group-theoretical context. The AdS spaces are arousing an increasing interest as asymptotic background spaces in (super)gravity theories, essentially sparked off by Maldacena's conjecture [9], which establishes a correspondence of holographic type between field theories on AdS and conformal field theories on the boundary (locally Minkowski). The AdS space plays also an important role in the above mentioned attempts to understand the microscopic source of black hole entropy.This scenario constitutes a suitable approach for our purposes. Indeed, the (3+1)-dimensional generalization of the previous constructions is just straight-forward when considering the infinitedimensional Lie algebras L ρ (so(4, 2)) of SO(4, 2)-tensor operators (where ρ is now a threedimensional vector). They can be regarded as infinite enlargements of the (finite) conformal symmetry SO(4, 2) in 3+1 dimensions which incorporate the subalgebra diff(4) of diffeomorphisms [the higher-dimensional analogue of the Virasoro algebra diff(S 1 )] of the four-dimensional space-time manifold (locally Minkowski), in addition to interacting fields with all SO(4, 2)...

Thermal Vacuum Radiation in Spontaneously Broken Second-Quantized Theories on Curved Phase Spaces of Constant Curvature

“…The more involved case of U (4) can be found in [7] and that of U (2, 2) in [15]. We shall give here only the explicit expressions of the lengths for U (4) (positive sign) and U (1, 3) (negative sign):…”

“…. , N, the expression of the probability distribution (15), for the Hamiltonian given by (65), can be again factorized as P (E) = Ω(E)e −E/kB T . Moreover, the expectation value of the total energy (17) in the excited vacuum (12) can now be written as: n0 (x) ± | 2 = 2s|θ| 2 ωe (µ2− ω)/kB T + · · · + ωe (µN − ω)/kB T 1 ± e (µ2− ω)/kB T ± · · · ± e (µN − ω)/kB T ,…”

On generalized Yang-Mills theories and extensions of the standard model in Clifford (tensorial) spaces

“…It has the same form as the Baker integral representation of the star product. We should emphasize that one must not confuse VasilievÕs defomation of the SO (3, 2) algebra using the AdS throat-size as deformation parameter, with the Moyal star products in phase spaces whose the deformation parameter is the Planck constant h. Calixto has recently studied higher-dim extensions of W 1 symmetries based on higher-spin U (2, 2) fields in AdS spaces that are very relevant to radiation phenomena [6,26,34].…”

Generalized higher-spin algebras and symbolic calculus on flag manifolds

“…The structure constants for (17) can be obtained through the scalar product f n lm (c) = ψ c n |{ψ c l , ψ c m } P , with integration measure (18), when the set {ψ c n } is chosen to be orthonormal. To each function ψ ∈ C ∞ (O C ), one can assign its Hamiltonian vector field H ψ ≡ {ψ, ·} P , which is divergence-free and preserves de natural volume form…”

“…an infinite extension ("promotion or analytic continuation" in the sense of [13]) of the finite-dimensional conformal symmetry SU (2, 2) ∼ SO(4, 2) in 3+1D. Also, W ∞ (2, 2) was interpreted as a higher-conformal-spin extension of the diffeomorphism algebra diff(4) of vector fields on a 4-dimensional manifold (just as W ∞ is a higher-spin extension of the Virasoro diff(1) algebra), thus constituting a potential gauge guide principle towards the formulation of of induced conformal gravities (Wess-Zumino-Witten-like models) in realistic dimensions [18]. For completeness, let as say that W ∞ -algebras also appear as central extensions of the algebra of (pseudo-)differential operators on the circle [19], and higher-dimensional analogues have been constructed in that context [20]; however, we do not find a clear connection with our construction.…”