Hilbert Bundle Research Articles

Lagrangian and Hamiltonian dynamics for probabilities on the statistical bundle

We provide an Information-Geometric formulation of accelerated natural gradient on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. In a non-parametric formalism, we consider the full set of positive probability functions on a finite sample space, and we provide a specific expression for the tangent and cotangent spaces over the statistical manifold, in terms of a Hilbert bundle structure that we call the Statistical Bundle. In this setting, we compute velocities and accelerations of a one-dimensional statistical model using the canonical dual pair of parallel transports and define a coherent formalism for Lagrangian and Hamiltonian mechanics on the bundle. We show how our formalism provides a consistent framework for accelerated natural gradient dynamics on the probability simplex, paving the way for direct applications in optimization.

Einstein's quantum elevator: Hermitization of non-Hermitian Hamiltonians via a generalized vielbein formalism

The formalism for non-Hermitian quantum systems sometimes blurs the underlying physics. We present a systematic study of the vielbein-like formalism which transforms the Hilbert space bundles of non-Hermitian systems into the conventional ones, rendering the induced Hamiltonian to be Hermitian. In other words, any non-Hermitian Hamiltonian can be "transformed" into a Hermitian one without altering the physics. Thus we show how to find a reference frame (corresponding to Einstein's quantum elevator) in which a non-Hermitian system, equipped with a non-trivial Hilbert space metric, reduces to a Hermitian system within the standard formalism of quantum mechanics.

Bergman bundles and applications to the geometry of compact complex manifolds

We introduce the concept of Bergman bundle attached to a hermitian manifold X, assuming the manifold X to be compact - although the results are local for a large part. The Bergman bundle is some sort of infinite dimensional very ample Hilbert bundle whose fibers are isomorphic to the standard L² Hardy space on the complex unit ball; however the bundle is locally trivial only in the real analytic category, and its complex structure is strongly twisted. We compute the Chern curvature of the Bergman bundle, and show that it is strictly positive. As a potential application, we investigate a long standing and still unsolved conjecture of Siu on the invariance of plurigenera in the general situation of polarized families of compact Kahler manifolds.

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.

Schatten classes for Hilbert modules over commutative C⁎-algebras

We define Schatten classes of adjointable operators on Hilbert modules over abelian C⁎-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a C⁎-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm–continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the C⁎-valued Fredholm determinant and operator zeta functions, and propose a notion of p-summable unbounded Kasparov cycles in the commutative setting.

Non-associative magnetic translations from parallel transport in projective Hilbert bundles

The non-associativity of translations in a quantum system with magnetic field background has received renewed interest in association with topologically trivial gerbes over Rn. The non-associativity is described by a 3-cocycle of the group Rn with values in the unit circle S1. The gerbes over a space M are topologically classified by the Dixmier–Douady class which is an element of H3(M,Z). However, there is a finer description in terms of local differential forms of degrees d=0,1,2,3 and the case of the magnetic translations for n=3 the 2-form part is the magnetic field B with non zero divergence. In this paper we study a quantum field theoretic construction in terms of n-component fermions on a circle. The non associativity arises when trying to lift the translation group action on the 1-particle system to the second quantized system.

Linear instability for periodic orbits of non-autonomous Lagrangian systems

Inspired by the classical Poincaré criterion about the instability of orientation preserving minimizing closed geodesics on surfaces, we investigate the relation intertwining the instability and the variational properties of periodic solutions of a non-autonomous Lagrangian on a finite dimensional Riemannian manifold. We establish a general criterion for a priori detecting the linear instability of a periodic orbit on a Riemannian manifold for a (maybe not Legendre convex) non-autonomous Lagrangian simply by looking at the parity of the spectral index, which is the right substitute of the Morse index in the framework of strongly indefinite variational problems and defined in terms of the spectral flow of a path of Fredholm quadratic forms on a Hilbert bundle.

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.

A Dirichlet Problem in Noncommutative Potential Theory

We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.

Generally covariant quantum theory

This paper represents a concise introduction to the quantum theory of point particles in a time orientable curved spacetime part of which was presented in the DICE conference in Castiglioncello, Italy. It is manifest covariant with a unique section of vacuum states as part of a Hilbert bundle.

Induced C*-Complexes in Metaplectic Geometry

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of the Kostant's symplectic spinor bundle. Defining a Hilbert $C^*$-structure on this bundle for a suitable $C^*$-algebra, we obtain an elliptic $C^*$-complex in the sense of Mishchenko--Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert $C^*$-modules. The paper can serve as a guide for handling of differential complexes and PDEs on Hilbert bundles

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.

Multiplicity-freeness of Unitary Representations in Sections of Holomorphic Hilbert Bundles

Abstract We prove several results asserting that the action of a Banach–Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity-free. These results require the existence of compatible anti-holomorphic bundle maps and certain multiplicity-freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,\sigma )$-weakly visible action (generalizing T. Koboyashi’s visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for group actions on homogeneous bundles for larger groups. We illustrate these general results by several examples related to operator groups and von Neumann algebras.

A Hilbert bundle description of differential K-theory

We give an infinite dimensional description of the differential K-theory of a manifold M. The generators are triples [H,A,ω] where H is a Z2-graded Hilbert bundle on M, A is a superconnection on H and ω is a differential form on M. The relations involve eta forms. We show that the ensuing group is the differential K-group Kˇ0(M). In addition, we construct the pushforward of a finite dimensional cocycle under a proper submersion with a Riemannian structure. We give the analogous description of the odd differential K-group Kˇ1(M). Finally, we give a model for twisted differential K-theory.

Contact Quantization: Quantum Mechanics = Parallel transport

Quantization together with quantum dynamics can be simultaneously formulated as the problem of finding an appropriate flat connection on a Hilbert bundle over a contact manifold. Contact geometry treats time, generalized positions and momenta as points on an underlying phase-spacetime and reduces classical mechanics to contact topology. Contact quantization describes quantum dynamics in terms of parallel transport for a flat connection; the ultimate goal being to also handle quantum systems in terms of contact topology. Our main result is a proof of local, formal gauge equivalence for a broad class of quantum dynamical systems—just as classical dynamics depends on choices of clocks, local quantum dynamics can be reduced to a problem of studying gauge transformations. We further show how to write quantum correlators in terms of parallel transport and in turn matrix elements for Hilbert bundle gauge transformations, and give the path integral formulation of these results. Finally, we show how to relate topology of the underlying contact manifold to boundary conditions for quantum wave functions.

The Unitary Group in Its Strong Topology

The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for applications in topology. In addition, it is shown that U(H) in its strong topology is metrizable and contractible if H is separable. As an application Hilbert bundles are classified by homotopy.

C*-algebras of 2-groupoids

We define topological 2-groupoids and study locally compact 2-groupoids with 2-Haar systems. We consider quasi-invariant measures on the sets of 1-arrows and unit space and build the corresponding vertical and horizontal modular functions. For a given 2-Haar system, we construct the vertical and horizontal full \Cst -algebras of a 2-groupoid and show that they are independent of the choice of the 2-Haar system, up to strong Morita equivalence. We make a correspondence between their bounded representations on Hilbert spaces and those of the 2-groupoid on Hilbert bundles. We show that representations of certain closed 2-subgroupoids are induced to representations of the 2-groupoid and use regular representation to build the vertical and horizontal reduced \Cst -algebras of the 2-groupoid. We establish strong Morita equivalence between \Cst -algebras of the 2-groupoid of composable pairs and those of the 1-arrows and unit space. We describe the reduced \Cst -algebras of r-discrete principal 2-groupoids and find their ideals and masa's.

Gauge theory for spectral triples and the unbounded Kasparov product

We explore factorizations of noncommutative Riemannian spin geometries over commutative base manifolds in unbounded KK-theory. After setting up the general formalism of unbounded KK-theory and improving upon the construction of internal products, we arrive at a natural bundle-theoretic formulation of gauge theories arising from spectral triples. We find that the unitary group of a given noncommutative spectral triple arises as the group of endomorphisms of a certain Hilbert bundle; the inner fluctuations split in terms of connections on, and endomorphisms of, this Hilbert bundle. Moreover, we introduce an extended gauge group of unitary endomorphisms and a corresponding notion of gauge fields. We work out several examples in full detail, to wit Yang–Mills theory, the noncommutative torus and the \theta -deformed Hopf fibration over the two-sphere.

Elliptic complexes over C∗-algebras of compact operators

For a $C^*$-algebra $A$ of compact operators and a compact manifold $M,$ we prove that the Hodge theory holds for $A$-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective $A$-Hilbert bundles over $M.$ For these $C^*$-algebras, we get also a topological isomorphism between the cohomology groups of an $A$-elliptic complex and the space of harmonic elements. Consequently, the cohomology groups appear to be Banach spaces. We prove as well, that if the Hodge theory holds for a complex in the category of Hilbert $A$-modules and continuous adjointable Hilbert $A$-module homomorphisms, the complex is self-adjoint parametrix possessing.

Hilbert quasimodular forms and vector bundles

Hilbert quasimodular forms are quasimodular forms of several variables. We construct vector bundles over a Hilbert modular variety whose sections can be identified with Hilbert quasimodular forms.