Hilbert bundles with ends

Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.

Hilbert bundles as quantum-classical continua

A hybrid quantum–classical model is proposed whereby a micro-structured (Cosserat-type) continuum is construed as a principal Hilbert bundle. A numerical example demonstrates the possible applicability of the theory.

3D Current Algebra and Twisted K Theory

Equivariant twisted K theory classes on compact Lie groups [Formula: see text] can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra [Formula: see text] using a supersymmetric Wess–Zumino–Witten model. The aim of the present paper is to extend the construction to higher loop algebras using an abelian extension of a 3D current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to 3D gauge transformations but do not define true Hilbert space operators.

FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS IV: MIXED STATES AND EVOLUTION TRANSPORT'S CURVATURE

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. The present fourth part of this series is devoted mainly to the fibre bundle description of mixed quantum states. We show that to the conventional density operator there corresponds a unique density lifting of paths for which the corresponding equations of motion are derived. It is also investigated the bundle description of mixed quantum states in the different pictures of motion. We calculate the curvature of the evolution transport and prove that it is curvature free iff the values of the Hamiltonian operator at different moments commute.

FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS V: INTERPRETATION, SUMMARY AND DISCUSSION

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operator are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. This is the fifth, closing, part of our investigation. We briefly discuss the observer's role in the theory and different realizations of the space–time model used as a base space in the bundle approach to quantum mechanics. The exact conditions for the equivalence of Hilbert bundle and Hilbert space formulations of the theory are pointed out. A table of comparison between both descriptions of nonrelativistic quantum mechanics is presented. We discuss some principal moments of the Hilbert bundle description and show that as a scheme it is more general than the Hilbert space one. Different directions for further research are pointed out.