Stochastic Processes and Their Representations in Hilbert Space

Methods of -representations in Hilbert space are applied to study of systems of n subspaces in a linear space. It is proved that the problem of description of n-transitive subspaces in a finite-dimensional linear space is -wild for n 5.

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous representations in a nested scale of Hilbert spaces. We also construct a couple of examples illustrative of the key features of group representations in rigged Hilbert spaces. Finally, we establish a simple criterion for the integrability of an operator Lie algebra in a rigged Hilbert space.

Index theory and second quantization of boundary value problems

In this book, modeling and estimation problems of random processes are treated in a unified geometric framework. For this, we need some basic facts about the Hilbert space theory of stochastic vector processes that have finite second order moments and are stationary in the wide sense. Such a process \(\{y(t)\}_{t\in \mathbb{Z}}\) is a collection of real or complex-valued random variables \(y_{k}(t),k = 1,2,\ldots,m,t \in \mathbb{Z}\), which generate a Hilbert space H with inner product $$\displaystyle{\langle \xi,\eta \rangle =\mathop{ \mathrm{E}}\nolimits \{\xi \bar{\eta }\},}$$ where bar denotes conjugation. This Hilbert space is endowed with a shift, i.e., a unitary operator \(\mathcal{U}:\, \mathbf{H} \rightarrow \mathbf{H}\) with the property that $$\displaystyle{y_{k}(t + 1) = \mathcal{U}y_{k}(t),\quad k = 1,2,\ldots,t \in \mathbb{Z}.}$$ In this chapter we introduce some basic geometric facts for such Hilbert spaces. Although we shall assume that the reader has some knowledge of elementary Hilbert space theory, for the benefit of the reader, some relevant facts are collected in Appendix B.2.

This chapter is almost independent of the previous ones and it can be read accordingly. For the first four sections only elementary point set theoretical and measure theoretical background is presupposed. Thus Section 1 can be read immediately and before starting Sections 2, 3 or 4 it is sufficient to be familiar with the contents of Section III.7. The remaining two sections require a casual acquaintence with the fundamental concepts of representations in Hilbert spaces. A partial reading of Sections IV.1, 2 and 3 is satisfactory for this purpose. In Section 5 we shall use also some basic facts about v. Neumann algebras A. In particular we need that A is its second commutant and that A is weakly closed. All of this is discussed and proved in Section III.6. The text contains detailed references to this background material at the places where it is used. See also the preliminary remarks on the last page.

We consider C*-algebras associated with simple unimodal one-dimensional ambiguous dynamical systems \(\left( {f,\mathbb{R}} \right)\) with certain special restrictions. For these algebras, we present a complete classification of irreducible representations in Hilbert spaces and describe the dual space. As an example, we consider the one-parameter family fμ(x) = μx(1 − x).

We consider the connection of the separating functions ρr with locally scalar representations of graphs and with spectral theory of graphs.

Quantum Natural Language Processing (QNLP) represents a novel interdisciplinaryparadigm that integrates Quantum Computing with Natural Language Processing toenhance semantic modeling and computational efficiency. Despite the remarkable successof transformer-based architectures such as BERT and GPT, classical NLP systems requirelarge-scale computational resources and extensive parameter tuning. This paper proposesa hybrid quantum–classical architecture employing parameterized quantum circuits (PQCs)integrated with classical embedding layers for text classification tasks. The frameworkutilizes classical preprocessing, quantum state encoding, variational optimization, andclassical post-measurement classification. Experimental results on benchmark sentimentanalysis datasets demonstrate competitive accuracy with significantly reduced parametercomplexity. The findings indicate that hybrid QNLP models provide enhanced semanticcompositionality and efficient representation in Hilbert space while remaining feasiblewithin Noisy Intermediate-Scale Quantum (NISQ) constraints

Atmospheric oscillations—V. The propagator matrix and the transmission of an electrostatic potential along the geomagnetic field lines

An example is given where the linear filtering theory in Hilbert spaces [8] is applied to solving a smoothing problem for hereditary systems with observation delays. As a particular case, we find again the new results from A. Bagchi (1976). This problem is transposed into a filtering problem in Hilbert spaces, using the formalism introduced by M. C. Delfour and S. K. Mitter (1975) in their investigations of hereditary systems. The filter equations then yield a system of equations—stochastic linear equations and Ricatti integro-differential equations—which allows us to find the smoothed state.

In this article I want to discuss a local and manifestly gauge-invariant formulation of quantum electrodynamics which has been developed by R. Menikoff and myself.(1,2) The manifest gauge invariance is achieved by using the electromagnetic field strengths, rather than potentials, to describe the electromagnetic field, and by using local currents, rather than canonical fields, to describe the matter. In order to represent this theory in Hilbert space, we study the continuous unitary representations of the group obtained from the exponentiated currents and electromagnetic field strengths. These operators can be represented on a Hilbert space having positive norm, so that the necessity for an indefinite metric does not arise here, and the equations of motion hold as operator equations in this Hilbert space.

Weakly closed, commutative, symmetric rings of operators on a Hilbert space are examined and their structure is ex ploited to characterize normal operators in terms of the rings they generate. In the first chapter, notation is established, and the less common theorems from Hilbert space theory not to be used in later sections are briefly reviewed. In Chapter II, the structure of weakly closed, commu tative, symmetric rings with a cyclic vector is specified. This theorem is then used in conjunction with a strucure called a canonical decomposition system to establish that every weakly closed, commutative, symmetric ring or opera tors acting on a separable space and containing the iden tity is spatially isomorphic to a "divect sum" of L rings of a certain type. In Chapter III, two sets of necessary and sufficient conditions are given for two normal operators on a separ able Hilbert space to be unitarily equivalent. The concept of a spectral class of a normal operator is introduced -a natural generalization of the concept of an eigen value. Each spectral class is assigned a multiplicity, and it is shown that two normal operators are unitarily equivalent iff they have the same spectral classes with the same respective multiplicities. A multiplicity is iv also defined for weakly continuous positive functionals o n the symmetric ring generated by a normal operator in the weak topology, and it is shown that this multiplicity function determines the operator. This multiplicity theory is compared w i t h a previous theory. v CHAPTER I INTRODUCTION We shall be concerned in this paper with rings of operators on a Hilbert space, in particular, weakly closed, commutative, symmetric rings. All Hilbert spaces to be considered will be complex Hilbert spaces. If H is a Hilbert space and x,y e H, then (x(y) will denote the inner product of x and y, and 1 Jxi| will denote the norm of x. We shall consider only bounded linear operators in this paper, and therefore we shall use the word "operator" solely to mean a bounded linear operator. The set of all operators on the Hilbert space H will be denoted by B(H). If A s B(H), then A* will denote the adjoint of A, We employ the usual concepts of isometric, unitary, hermitian, normal, and positive definite operators. Knowledge of the weak, strong, strongest, and norm topologies for B(H) is assumed. The norm topology is the strongest of the four; hence, any set of operators closed in one of the topologies is closed in the norm topology. By a ring of operators we shall mean a set of opera tors forming a ring in the usual algebraic sense which is also a vector space over the complex numbers. A ring R of operators is symmetric if A e R whenever A e R. from the theory of Banach algebras, we know that any norm closed, 1 we define an operator A a e BCLg) by A ftf = af* Note that two Ly,, functions which differ only on a set of measure zero give rise to the same operator. The ring of operators thus obtained is a maximal, commutative, symmetric ring. If E is a weakly closed, commutative, symmetric ring with maximal ideal space M, then the fact that E is closed under sups gives rise to some strong topological properties of M. In this case, M is totally disconnected, and the closure of every open set is open. Thus, M contains many sets which are both open and closed, and such sets we call clopen. It is easy to see that projections in E correspond v i a the Gelfand transform to the characteristic functions of clopen sets. If U is a clopen subset of M, then we will let pu denote the corresponding projection in E. pu (m) -JtjjCm) = the characteristic function of U. Throughout the remainder of this discussion, E will denote a weakly closed commutative, symmetric subring of B(H) with identity, and M will denote the maximal ideal space of E. Corresponding to each vector 5 e H, there is a measure u on M defined by The topological properties of M carry strong implications for the resulting measure space. If S = M is measurable, then there exists a clopen set U such that m (S\UUU\S) * 0. If ? is cyclic for E ' , then u is supported on all of M, in ( A $ 0 = J A (m)du(m)

We extend the discussion of projective group representations in quaternionic Hilbert space that was given in our recent book. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. The multi-centrality and centrality assumptions are also analyzed in generator terms, and implications of this analysis are discussed.