Structural Stability Theory and Phase Transitions Models

Global stability theory is introduced as a tool allowing the classification of mathematical models of phase transitions. The point of view is that a topological structure whose stability controls the transition, can be identified in the process of computation of the partition function. In particular we discuss mean field theories and the two dimensional Ising model. Interesting features are disclosed concerning the classification of the instabilities, such as the number of parameters and possible approximations.     

Quantum knitting

We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of “knot invariants,” among which the Jones polynomial plays a prominent role, since it can be associated with observables in topological quantum field theory. Although the problem of computing the Jones polynomial is intractable in the framework of classical complexity theory, it has been recently recognized that a quantum computer is capable of approximating it in an efficient way. The quantum algorithms discussed here represent a breakthrough for quantum computation, since approximating the Jones polynomial is actually a “universal problem,” namely, the hardest problem that a quantum computer can efficiently handle.     

Playing with Numbers, with Fermions and Bosons

We construct nonlinear maps which realize the fermionization of bosons and the bosonization of fermions with the view of obtaining states coding naturally integers in standard or in binary basis. Specifically, with reference to spin systems, we derive raising and lowering bosonic operators in terms of standard fermionic operators and vice versa. The crucial role of multiboson operators in the whole construction is emphasized. Dedicated to Giuseppe Castagnoli for his 65th birthday.     

A novel realization of the Virasoro algebra in number state space

The group of diffeomorphisms is crucial in quantum computing. Representing it by vector fields over a d-manifold, d?2, accounting for both projective action and conformal symmetry at the quantum mechanical level, requires the direct-sum decomposition of tensor product for non-compact algebras, viable only for su(1,1). As a step towards the solution, a realization of the (d=1) Virasoro algebra Vir∼Diff₊(S(1)) in the universal envelope of su(1,1) (and h(1)) is presented, which is simple in the discrete positive series irreducible unitary representation of su(1,1).     

Topology, Formal Languages and Quantum Information

The paper provides a critical overview of the basic conceptual tools and algorithms of quantum information theory underlying the construction of quantum invariants of links and 3-manifolds as well as of their connections with algorithms and algorithmic complexity questions that arise in geometry and quantum gravity models.     

Coherent states and infinite-dimensional lie algebras: An outlook

Summary The possible extension of the notion of generalized coherent state to the case of infinite-dimesional affine Lie algebras is discussed with special attention to the resulting topological structure of the coherent states manifold, and to its connection with the structure of the algebra. The relevance for the solution of nonlinear dynamical systems equations of motion is briefly reviewed. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.