Christian Fronsdal

Deformation theory and quantization. I. Deformations of symplectic structures

Abstract We present a mathematical study of the differentiable deformations of the algebras associated with phase space. Deformations of the Lie algebra of C∞ functions, defined by the Poisson bracket, generalize the well-known Moyal bracket. Deformations of the algebra of C∞ functions, defined by ordinary multiplication, give rise to noncommutative, associative algebras, isomorphic to the operator algebras of quantum theory. In particular, we study deformations invariant under any Lie algebra of “distinguished observables”, thus generalizing the usual quantization scheme based on the Heisenberg algebra.

Deformation theory and quantization. II. Physical applications

Abstract In the preceding paper general deformations of the structures based on the classical symplectic manifolds were examined. Quantization can be understood as a deformation of the algebra of observables without any need for introducing a Hilbert space. By a slight but crucial restatement of the usual interpretation of classical mechanics we find a framework for the description of both classical and quantum mechanics, within which the continuity of the quantization process is brought out. The spectra of some important physical observables are determined by direct phase space methods; this helps support the belief that a complete and autonomous theory, equivalent to ordinary quantum mechanics in special cases, but capable of wide generalization, can be constructed.

Quantum mechanics as a deformation of classical mechanics

Mathematical properties of deformations of the Poisson Lie algebra and of the associative algebra of functions on a symplectic manifold are given. The suggestion to develop quantum mechanics in terms of these deformations is confronted with the mathematical structure of the latter. As examples, spectral properties of the harmonic oscillator and of the hydrogen atom are derived within the new formulation. Further mathematical generalizations and physical applications are proposed.

On N = 8 supergravity in AdS5 and N = 4 superconformal Yang-Mills theory

Abstract We discuss the spectrum of states of IIB supergravity on AdS 5 × 5 in a manifest SU(2, 2 4 ) invariant setting. The boundary fields are described in terms of N = 4 superconformal Yang-Mills theory and the proposed correspondence between supergravity in AdS5 and superconformal invariant singleton theory at the boundary is formulated in a N = 4 superfield covariant language.

Conformal Maxwell theory as a singleton field theory on AdS, IIB 3-branes and duality

We examine the boundary conditions associated with extended supersymmetric Maxwell theory in five-dimensional anti-de Sitter space. Excitations on the boundary are identical to those of ordinary four-dimensional conformal invariant superelectrodynamics. Extrapolations of these excitations give rise to a five-dimensional topological gauge theory of the singleton type. The possibility of a connection between this phenomenon and the world-volume theory of 3-branes in IIB string theory is discussed.

Deformations of gauge groups. Gravitation

This is a review, and an attempt at completion, of the ’’Gupta program,’’ the ultimate goal of which is either to show that Einstein’s theory of gravitation is the only self‐consistent field theory of interacting, massless, spin‐2 particles in flat space or to discover interesting alternatives. It is useful to notice that the gauge group of general relativity is a deformation (in a mathematically precise sense) of the gauge group associated with the massless, spin‐2 free field. The uniqueness of Einstein’s theory depends on the stability of its gauge group with respect to a class of differentiable deformations. A generalized Gupta program for massless fields of arbitrary spins is proposed.

On Dis and Racs

Abstract We develop a scheme to construct interactions between massless particles of all spins. This scheme is based on two fundamental objects, constituents of massless particles. The theory is a renormalizable local field theory. We also outline possible future developments.

Gauge fields as composite boundary excitations

Abstract We investigate representations of the conformal group that describe “massless” particles in the interior and at the boundary of anti-de Sitter space. It turns out that massless gauge excitations in anti-de Sitter are gauge “current” operators at the boundary. Conversely, massless excitations at the boundary are topological singletons in the interior. These representations lie at the threshold of two “unitary bounds” that apply to any conformally invariant field theory. Gravity and Yang-Mills gauge symmetry in anti-De Sitter is translated to global translational symmetry and continuous R -symmetry of the boundary superconformal field theory.

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felders quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models.

Quarks or singletons

Abstract Singletons and quarks share the most important characteristic of being (locally) unobservable. This opens the door to uncommon statistics. A first stage of deviation from ordinary quantization of singletons leads directly to a formulation of QED as an “effective” theory. A second stage incorporates massive particles. The massless fields are bilinear, the massive ones trilinear, in the “constituent” singleton fields. The scheme strongly suggests a possibility of unifying all interactions.