Infinite Dimensional Analysis, Quantum Probability and Related Topics

We consider the Yang–Mills action functional S_YM on the infinite-dimensional space of connections on a bundle over a compact surface. We find a lower bound for S_YM(ω) in terms of holonomies of the connection ω, the topology of the surface, and the topology of the bundle. An intermediate 'energy inequality' becomes an equality if and only if the connection is a critical point of the Yang–Mills action. Yang–Mills minima can also be understood using these inequalities.

Following the methods of Refs. 1 and 2 we construct here a family of Lie super-algebra representations in terms of quantum stochastic integrals.

The S-transform is studied as a mapping from a space of tensors to a space of functions over a complex space. The range of this transform is characterized in terms of analyticity and growth. These results are applied to a broad class of generalized functions in white noise analysis. These correspond to completions of the Gaussian L²-space which preserve orthogonality of Hermite polynomials. The S-transform is defined for the new generalized functions, and the range of this S-transform is identified in terms of analyticity and growth. Examples of the new spaces of generalized functions are given; these include distributions considered by Kondratiev and Streit, as well as new classes of distributions whose S-transforms have growth bounded by iterated exponentials.

Nuclear Wick algebra of generalized operators generated by quantum white noise is introduced in order to formulate the formal manipulations in free fields at rigorous mathematical level. Solutions of second quantized field equations and various physical observables such as energy and momenta of the fields are realized as generalized operators adn the renormalizations through Wick ordering procedure are investigated from functional analytical point of view.

The contraction coefficient of stochastic mappings between full matrix algebras is introduced with respect to a generalized relative entropy labeled by an operator convex function g. It is proved that the coefficient is actually independent of g, in particular, it can be most conveniently computed by means of the square function.

We study the spaces of Poisson, compound Poisson and Gamma noises as special cases of a general approach to non-Gaussian white noise calculus, see Ref. 18. We use a known unitary isomorphism between Poisson and compound Poisson spaces in order to transport analytic structures from Poisson space to compound Poisson space. Finally we study a Fock type structure of chaos decomposition on Gamma space.

The work of the Brussels–Austin groups over the last six years has demonstrated that for unstable systems, classical or quantum, there exist spectral decompositions of the evolution In terms of resonances and resonance states which appear as eigenvalues and eigenprojections of the evolution operator. These new spectral decompositions are non-trivial only for unstable systems and define an extension of the evolution to suitable dual pairs. Duality here is the states/observables duality and the spectral decompositions extend the algebraic approach to dynamical systems to an intrinsically probabilistic and irreversible formulation. The extended formulation allows for probabilistic prediction and control beyond the traditional local techniques.

A generating function for knot diagrams is constructed. We consider a matrix model with d(N×N) matrices and show that in the limits N→∞ and d→0 the model describes the knot diagrams. The same limit in matrix string theory is also discussed. We speculate that a prototypical M(atrix) without matrix theory exists in void.