Renzo Cirelli

A FUNCTIONAL REPRESENTATION FOR NON-COMMUTATIVE C*-ALGEBRAS

We obtain a representation of non commutative C*-algebras as function algebras on the pure state space, with a convenient product. This construction is an extension of the classical functional representation of commutative C*-algebras.

The orbit space of the action of gauge transformation group on connections

Abstract The behaviour of orbits of the action of the group of smooth gauge transformations on connections for a principal bundle P(M, G) is discussed with and without compactness assumption on M and G. In the case of compact M and with suitable conditions on G a stratification structure for the space of orbits is established. A natural tame weak Riemannian metric is given on each stratum.

The group of gauge transformations as a Schwartz–Lie group

The group of gauge transformations of a smooth principal bundle P(M,G) over a not necessarily compact manifold M and with a not necessarily compact structure group G is proved to be a Schwartz–Lie group. Its Lie algebra and exponential map are discussed.

QUANTUM PHASE SPACE FORMULATION OF SCHRÖDINGER MECHANICS

We develop a geometrical approach to Schrodinger quantum mechanics, alternative to the usual one, which is based on linear and algebraic structures such as Hilbert spaces, operator algebras, etc. The starting point of this approach is the Kahler structure possessed by the set of the pure states of a quantum system. The Kahler manifold of the pure states is regarded as a “quantum phase space”, conceptually analogous to the phase space of a classical hamiltonian system, and all the constituents of the conventional formulation, in particular the algebraic structure of the observables, are reproduced using a suitable “Kahler formalism”. We also show that the probabilistic character of the measurement process in quantum mechanics and the uncertainty principle are contained in the geometrical structure of the quantum phase space. Finally, we obtain a characterization for quantum phase spaces which can be interpreted as a statement of uniqueness for Schrodinger quantum mechanics.

Smoothness of the action of the gauge transformation group on connections

The NLF–Lie group structure of the group G of the gauge transformations, defined as the group of sections of the bundle P[G] associated to the principal bundle P(M,G), is discussed. Other current definitions of the group of gauge transformations are shown to admit a nontrivial smooth structure only in the case of compact G. The space C of principal connections, as well, is given the structure of local affine NLF‐manifold, after identifications of connections with sections of a convenient vector bundle on M. Finally, the smoothness of the action of G on C is proved in general. In the case of compact M, the group G becomes a tame Frechet–Lie group and the action a tame smooth action.

An algebraic representation of continuous superselection rules

From the logic approach to quantum and classical mechanics, the W*−algebraic approach is deduced in dependence of a suitable ’’prestate.’’ An algebraic representation of the logic description is in fact constructed in a framework in which continuous superselection rules can be present. Logic propositions, observables, and states are represented by decomposable projections, decomposable self−adjoint operators, and normal states in a direct integral of Hilbert spaces. In this representation each algebraic term becomes the representative of a homologous logic one and the expectation values as well as the superselection rules are conserved. When a principle of ’’undistinguishability’’ is taken into account, the representation is faithful. In the classical case, the representation results in Koopman’s formalism.

From a ’’laboratory’’ Galilei−Hilbert bundle to an algebra of observables

After a schematic examination of the ’’physical’’ representations of the Galilei group, a quantum mechanical description is drawn from a ’’laboratory’’ one in the case of a free elementary system with spin, assuming that the kinematical group is the Galilei group. In fact, the ’’laboratory states’’ of the system naturally fit in a Galilei−Hilbert bundle. Then, according to the general theory of the unitary representations of groups in the framework of Hilbert bundles which is outlined in this paper, a unitary representation of the Galilei group is constructed which is shown to contain all the physical Galilei representations. The quantum mechanical description which has been obtained in this way gives rise to an algebra of observables by means of a general procedure which connects a unitary representation of a Lie group with a Hilbert representation of the corresponding Lie algebra. The inner energy is shown to be a superobservable for the system under consideration. Moreover, the kinematical properties o...

G‐Hilbert bundles

A notion of Hilbert bundle is proposed which leads to the construction of a ’’big’’ Hilbert space H starting from a family of Hilbert spaces. For this, such a family is equipped with a suitable structure, called Borel field structure. A meaningful relationship is established between the Borel structures which can be defined on the union of the Hilbert spaces of the family and the Borel field structures with which the family can be equipped. For a topological group G, the structure of G‐Hilbert bundle is defined linking in a suitable way a Hilbert bundle with actions of G. In the framework of a G‐Hilbert bundle, a continuous unitary representation of G in H can be constructed. The transitive G‐Hilbert bundles which are often used in the theory of induced representations of groups are shown to be a subclass of the class of the G‐Hilbert bundles which are proposed in this paper.