Hilbert Space Theory Research Articles

Relating Berkovits and A ∞ superstring field theories; large Hilbert space perspective

We lift the dynamical field of the A ∞ superstring field theory to the large Hilbert space by introducing a gauge invariance associated with the eta zero mode. We then provide a field redefinition which relates the lifted field to the dynamical field of Berkovits’ superstring field theory in the large Hilbert space. This generalizes the field redefinition in the small Hilbert space described in earlier works, and gives some understanding of the relation between the gauge symmetries of the theories. It also provides a new perspective on the algebraic structure underlying gauge invariance of the Wess-Zumino-Witten-like action.

Finite linear spaces, plane geometries, Hilbert spaces and finite phase space

Finite plane geometry is associated with finite dimensional Hilbert space. The association allows mapping of q-number Hilbert space observables to the c-number formalism of quantum mechanics in phase space. The mapped entities reflect geometrically based line–point interrelation. Particularly simple formulas are involved when use is made of mutually unbiased bases representations for the Hilbert space entries. The geometry specifies a point–line interrelation. Thus underpinning d-dimensional Hilbert space operators (resp. states) with geometrical points leads to operators termed “line operators” underpinned by the geometrical lines. These “line operators”, \(\hat{L}_j;\) (j designates the line) form a complete orthogonal basis for Hilbert space operators. The representation of Hilbert space operators in terms of these operators form the phase space representation of the d-dimensional Hilbert space. Examples for the use of the “line operators” in mapping (finite dimensional) Hilbert space operators onto finite dimensional phase space functions are considered. These include finite dimensional Wigner function and Radon transform and a geometrical interpretation for the involvement of parity in the mappings of Hilbert space onto phase space. Two d-dimensional particles product states are underpinned with geometrical points. The states, \(|L_j\rangle \) underpinned with the corresponding geometrical lines are maximally entangled states (MES). These “line states” provide a complete \(d^2\) dimensional orthogonal MES basis for for the two d-dimensional particles. The complete \(d^2\) dimensional MES i.e. the “line states” are shown to provide a transparent geometrical interpretation to the so-called Mean King Problem and its variant. The “line operators” (resp. “line states”) are studied in detail. The paper aims at self sufficiency and to this end all relevant notions are explained herewith.

White Noise Solution for Nonlinear Stochastic Systems: In memory of A.V. Balakrishnan

Abstract: This paper proposes an alternative theory to the Ito calculus due to Balakrishnan: the white noise theory in Hilbert spaces. The proposed approach extends Blakrishnan’s theory to a new class of nonlinear systems. The method uses the theory of differential geometry to devise a suitable map which transforms the starting system in an equivalent one; then the techniques of white noise theory is applied to this equivalent system. Finally, by means of the inverse map, the existence of a white noise solution for the starting system is proved.

The Resolvent Average of Monotone Operators: Dominant and Recessive Properties

Within convex analysis, a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago. When one considers the subdifferential of the proximal average, a natural averaging operation of the subdifferentials of the averaged functions emerges. In the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in Hilbert spaces, transforming it into the resolvent average. The theory of resolvent averages contains many desirable properties. In particular, we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average. As a consequence, we not only recover a significant part of the theory of proximal averages, but also obtain results and properties which have not been previously available.

Free probability on Hecke algebras and certain group C^{*}-algebras induced by Hecke algebras

In this paper, by establishing free-probabilistic models on the Hecke algebras \(\mathcal{H}\left(GL_{2}(\mathbb{Q}_{p})\right)\) induced by \(p\)-adic number fields \(\mathbb{Q}_{p}\), we construct free probability spaces for all primes \(p\). Hilbert-space representations are induced by such free-probabilistic structures. We study \(C^{*}\)-algebras induced by certain partial isometries realized under the representations.

Representations of quantum SU(2) operators on a local chart

Hilbert space representations of quantum SU(2) by multiplication operators on a local chart are constructed, where the local chart is given by tensor products of square integrable functions on a quantum disc and on the classical unit circle. The actions of generators of quantum SU(2), generators of the opposite algebra, and noncommutative partial derivatives are computed on a Hilbert space basis.

Gravity as a four dimensional algebraic quantum field theory

Based on a family of indefinite unitary representations of the diffeomorphism group of an oriented smooth $4$-manifold, a manifestly covariant $4$ dimensional and non-perturbative algebraic quantum field theory formulation of gravity is exhibited. More precisely among the bounded linear operators acting on these representation spaces we identify algebraic curvature tensors hence a net of local quantum observables can be constructed from $C^*$-algebras generated by local curvature tensors and vector fields. This algebraic quantum field theory is extracted from structures provided by an oriented smooth $4$-manifold only hence possesses a diffeomorphism symmetry. In this way classical general relativity exactly in $4$ dimensions naturally embeds into a quantum framework. Several Hilbert space representations of the theory are found. First a "tautological representation" of the limiting global $C^*$-algebra is constructed allowing to associate to any oriented smooth $4$-manifold a von Neumann algebra in a canonical fashion. Secondly, influenced by the Dougan--Mason approach to gravitational quasilocal energy-momentum, we construct certain representations what we call "positive mass representations" with unbroken diffeomorphism symmetry. Thirdly, we also obtain "classical representaions" with spontaneously broken diffeomorphism symmetry corresponding to the classical limit of the theory which turns out to be general relativity. Finally we observe that the whole family of "positive mass representations" comprise a $2$ dimensional conformal field theory in the sense of G. Segal.

Composition operators induced by injective homomorphisms on infinite weighted graphs

We deal with composition operators induced by injective homomorphisms on infinite weighted graphs from a viewpoint of reproducing kernel Hilbert space theory. The representation formula for their adjoint operators is given in the terms of the frames constructed from reproducing kernels, and de Branges–Rovnyak spaces induced by composition operators are studied.

On the stable Conley index in Hilbert spaces

In this paper, we study Conley theory in Hilbert spaces and make some refinement of the construction of the stable Conley index developed by G\c{e}ba, Izydorek, and Pruszko. For instance, we allow subspaces other than invariant subspaces in the the construction. As a main result, we show that the resulting stable Conley index for a given isolating neighborhood of a flow does not depend on choices in the construction. We also hope that some results can be applied to the construction of Floer homotopy type.

Nonlinear gauge invariance and WZW-like action for NS-NS superstring field theory

We complete the construction of a gauge-invariant action for NS-NS superstring field theory in the large Hilbert space begun in arXiv:1305.3893 by giving a closed-form expression for the action and nonlinear gauge transformations. The action has the WZW-like form and vertices are given by a pure-gauge solution of heterotic string field theory in the small Hilbert space.

Hilbert space theory for relativistic dynamics with reflection. Special cases

We present and study a novel class of one-dimensional Hilbert space eigenfunction transforms that diagonalize analytic difference operators encoding the (reduced) two-particle relativistic hyperbolic Calogero–Moser dynamics. The scattering is described by reflection and transmission amplitudes $t$ and $r$ with function-theoretic features that are quite different from nonrelativistic amplitudes. The axiomatic Hilbert space analysis in the appendices is inspired by and applied to the attractive two-particle relativistic Calogero–Moser dynamics for a sequence of special couplings. Together with the scattering function $u$ of the repulsive case, this leads to a triple of amplitudes $u, t$ and $r$ satisfying the Yang–Baxter equations.

Functional analysis and quantum mechanics: an introduction for physicists

We give an introduction to certain topics from functional analysis which are relevant for physics in general and in particular for quantum mechanics. Starting from some examples, we discuss the theory of Hilbert spaces, spectral theory of unbounded operators, distributions and their applications and present some facts from operator algebras. We do not give proofs, but present examples and analogies from physics which should be useful to get a feeling for the topics considered.

Entries of indefinite Nevanlinna matrices

In the early 1950’s M.G.Krein characterised those entire functions which are an entry of some Nevanlinna matrix, and those pairs of entire functions which are a row of some such matrix. In connection with Pontryagin space versions of Krein’s theory of entire operators and de Branges’ theory of Hilbert spaces of entire functions, an indefinite analogue of Nevanlinna matrices plays a role. In this paper we extend the mentioned characterisations to the indefinite situation and investigate the geometry of associated reproducing kernel Pontryagin spaces. AMS MSC 2010: 46C20, 34B20, 30D10, 30D15

A geometric Hamiltonian description of composite quantum systems and quantum entanglement

Finite-dimensional Quantum Mechanics can be geometrically formulated as a proper classical-like Hamiltonian theory in a projective Hilbert space. The description of composite quantum systems within the geometric Hamiltonian framework is discussed in this paper. As summarized in the first part of this work, in the Hamiltonian formulation the phase space of a quantum system is the Kähler manifold given by the complex projective space P(H) of the Hilbert space H of the considered quantum theory. However the phase space of a bipartite system must be P(H1 ⊗ H2) and not simply P(H1) × P(H2) as suggested by the analogy with Classical Mechanics. A part of this paper is devoted to manage this problem. In the second part of the work, a definition of quantum entanglement and a proposal of entanglement measure are given in terms of a geometrical point of view (a rather studied topic in recent literature). Finally two known separability criteria are implemented in the Hamiltonian formalism.

Polymer quantization versus the Snyder noncommutative space

We study a noncanonical Hilbert space representation of the polymer quantum mechanics. It is shown that the Heisenberg algebra gets some modifications in the constructed setup, from which a generalized uncertainty principle will naturally come out. Although the extracted physical results are the same as those obtained from the standard canonical representation, the noncanonical representation may be notable in view of its possible connection with the generalized uncertainty theories suggested by string theory. In this regard, by considering a Snyder-deformed Heisenberg algebra, we show that since the translation group is not deformed, it can be identified with a polymer-modified Heisenberg algebra. In the classical level, it is shown that the noncanonical Poisson brackets are related to their canonical counterparts by means of a Darboux transformation on the corresponding phase space.

Continuous Atomic Systems for Subspaces

This paper gives new contributions to atomic systems theory in Hilbert spaces. More precisely, we introduce and study the concept of continuous atomic systems for subspaces, and we give some examples to show differences between this and the discrete version. Moreover, we give a new characterization of continuous (resp. discrete) atomic systems. Finally, among other things, we discuss the relationship between continuous (resp. discrete) frame of subspaces (fusion frames) and continuous (resp. discrete) atomic systems. In particular, we show that the perturbation of a Parseval continuous (resp. discrete) frame of subspaces or identity operator, under certain conditions, gives us a continuous (resp. discrete) atomic system.

Computing functions of random variables via reproducing kernel Hilbert space representations

We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations which can be applied to points drawn from the respective distributions. We refer to our approach as kernel probabilistic programming. We illustrate it on synthetic data and show how it can be used for nonparametric structural equation models, with an application to causal inference.

Krein regularization of λϕ3 theory

In this paper, the one-loop self energy of λϕ3 theory is calculated by using Krein regularization in four and six dimensions and the result, which is finite, is compared with the conventional result of λϕ3 theory in Hilbert space. The self energy is calculated in the one-loop approximation and the result is automatically regularized as a result of “Krein Regularization”.

Topology of density matrices

We investigate topological properties of density matrices motivated by the question to what extent phenomena like topological insulators and superconductors can be generalized to mixed states in the framework of open quantum systems. The notion of geometric phases has been extended from pure to mixed states by Uhlmann in [Rep. Math. Phys. 24, 229 (1986)], where an emergent gauge theory over the density matrices based on their pure-state representation in a larger Hilbert space has been reported. However, since the uniquely defined square root $\sqrt{\rho}$ of a density matrix $\rho$ provides a global gauge, this construction is always topologically trivial. Here, we study a more restrictive gauge structure which can be topologically non-trivial and is capable of resolving homotopically distinct mappings of density matrices subject to various spectral constraints. Remarkably, in this framework, topological invariants can be directly defined and calculated for mixed states. In the limit of pure states, the well known system of topological invariants for gapped band structures at zero temperature is reproduced. We compare our construction with recent approaches to Chern insulators at finite temperature.

Interpreting Quantum Theories

Laura Ruetsche, Interpreting Quantum Theories. Oxford: Oxford University Press. Pp. xvii, 377. Most philosophical work on quantum physics has concerned simple systems. And for good reason. Even one or two particle systems can exhibit the striking features we have come to associate with quantum physics—features such as entanglement, interference, and non-locality. But as far as physics goes, such systems barely scrape the surface. In particular, they have only a finite number of degrees of freedom, which means their states are characterized by a finite number of independent parameters. The systems studied in fields such as high energy particle physics and many-body physics, meanwhile, have uncountably many degrees of freedom. They are far more complicated than the usual philosophical fare. This is untroubling as long as the conceptual heart of quantum physics can be effectively condensed down. But what if philosophers’ toy systems are fundamentally different from realistic ones? That would be a problem. And indeed, as Laura Ruetsche convincingly argues in Interpreting Quantum Theories, systems with uncountable degrees of freedom—what Ruetsche collectively calls QM ∞ —are fundamentally different from the systems that have become the stock and trade of philosophy of physics, in ways that change the interpretational project. Ruetsche’s book does two things exceedingly well. The first is to provides a (comparatively) accessible and philosophically-oriented introduction to the algebraic approach to quantum physics, which is essential for understanding QM ∞ . This is a valuable contribution. The algebraic approach is the setting for much recent philosophical work. And yet learning it is notoriously difficult, as even pedagogical texts are aimed at research-level mathematicians. Ruetsche’s book is technically demanding, but it works to bring non-mathematicians along. It is, without a doubt, the best place to enter this literature. Second, the book sets the agenda for future work on QM ∞ , by effectively and judiciously drawing important foundational questions out of a sprawling and difficult physics literature. In this regard, it is a close cousin to John Earman’s excellent Bangs, Crunches, Whimpers, and Shrieks, which has had a significant influence on subsequent philosophy of spacetime physics. 1 Philosophers of physics will be addressing the questions Ruetsche has raised, in many cases for the first time, for years to come. Interpreting Quantum Theories centers on what might be called the problem of inequivalent representations. The quantum theories that philosophers are accustomed to are set in a mathematical structure known as a (separable) Hilbert space, which is an at-most countably infinite dimensional complex vector space. Rays in a Hilbert space represent possible states of a physical system, and self- adjoint operators on that Hilbert space correspond to properties of the system. 1 John Earman, Bangs, Crunches, Whimpers, and Shrieks (Oxford: Oxford University Press, 1995).