Alessandro Manià

The Lie group of automorphisms of a principle bundle

Abstract A convenient structure of Lie group to the entire group Aut P of G-automorphisms of a principal G-bundle without any assumption of compactness on the structure group G or on the base manifold. Its Lie algebra and the exponential map are illustrated. Some relevant principal bundles are discussed having Aut P or its subgroup Gau P of gauge transformations as structure group.

The pure state space of quantum mechanics as Hermitian symmetric space

Abstract The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kahler manifold. The classical principles of quantum mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are discussed in this non-linear geometrical context.

On differential structure for projective limits of manifolds

We investigate the differential calculus defined by Ashtekar and Lewandowski on projective limits of manifolds by means of cylindrical smooth functions and compare it with the C∞ calculus proposed by Frohlicher and Kriegl in a more general context. For products of connected manifolds, a Boman theorem is proved, showing the equivalence of the two calculi in this particular case. Several examples of projective limits of manifolds are discussed, arising in String Theory and in loop quantization of Gauge Theories.

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.

On the nonlinear extension of quantum superposition and uncertainty principles

Abstract We show that nonlinear extensions of quantum mechanics exist in which (extensions of) quantum superposition and uncertainty principles hold.

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.

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.

Quantum logic and operational quantum mechanics

Abstract We characterize the class of the μ -complete F -spaces with unit corresponding to the observables of a quantum logic. We show that, conversely, every μ -complete F -space satisfying Axiom I and Axiom II corresponds to a quantum logic. The latter class of F -spaces generalizes that of “spectral F -spaces” introduced by Alfsen and Shultz and by Edwards.

Inductive Construction of the Loop Transform for Abelian Gauge Theories

We construct the loop transform in the case of Abelian gauge theories as a unitary operator given by the inductive limit of Fourier transforms on tori. We also show that its range, i.e. the space of kinematical states of the quantum loop representation, is the Hilbert space of square integrable complex valued functions on the group of hoops.

The G-central decomposition of states of statistical systems in the algebraic and in the operational description

We consider a group of automorphisms of a compact convex set X. Using simplical measure techniques we find a decomposition of points of X into “G-primary” points. The decomposition of Guichardet and Kastler for quasi-invariant states of a separable C∗- algebra is generalized.