M. Flato

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.

One massless particle equals two Dirac singletons

The ‘remarkable representations of the 3+2 de Sitter group’, discovered by Dirac, later called singleton representations and here denoted Di and Rac, are shown to possess the following truly remarkable property: Each of the direct products Di ⊗ Di, Di ⊗ Rac, and Rac ⊗ Rac decomposes into a direct sum of unitary, irreducible representations, each of which admits an extension to a unitary, irreducible representation of the conformal group SO(4, 2). Therefore, in de Sitter space, every state of a free, ‘massless’ particle may be interpreted as a state of two free singletons — and vice versa. The term ‘massless’ is associated with a set of particle-like representations of SO(3, 2) that, besides the noted conformal extension, exhibit other phenomena typical of masslessness, especially gauge invariance.

Deformation quantization and Nambu mechanics

Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ℝ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

Closed Star Products and Cyclic Cohomology

We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.

Representations of conformal supersymmetry

The superalgebras of (generalized) conformal supersymmetry have some very interesting unitarizable representations that contain only massless representations of the conformal subalgebra, in spite of contrary claims that have recently been made Castell, L., and Heidenreich, W., Phys. Rev.D25, 1745 (1982)., Castell, L. and Kunemund, Th., Phys. Rev.D26, 1485 (1982)..

The hidden group structure of quantum groups: strong duality, rigidity and preferred deformations

A notion of well-behaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeld-type and their duals, algebras of coefficients of compact semi-simple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyal-product-like deformations are naturally found for all FRT-models on coefficients andC∞-functions. Strong rigidity (Hbi2={0}) under deformations in the category of bialgebras is proved and consequences are deduced.

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.

Deformations of Poisson brackets, Dirac brackets and applications

After a short review of results which we recently obtained on deformations of Lie algebras associated with symplectic manifolds, we discuss physical applications and treat some examples with deformed Poisson brackets. We make explicit a connection between classical and quantum mechanics, and the theory of Dirac brackets for second class constraints, from the viewpoint of deformation theory. Finally we discuss the general Dirac constraints formalism.