Abstract We show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.
Abstract We show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.
A general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals. In this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional Hilbert space H. We study operator representations for decoherence functionals on this space of histories. We first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on H⊗H, an infinite dimensional analogue of the Isham–Linden–Schreckenberg representation for finite dimensions. Since this excludes many decoherence functionals of physical interest, we then identify the large and physically important class of decoherence functionals which can be represented, canonically, by bounded operators on H⊗H.
We show that quantum measures and integrals appear naturally in any L2-Hilbert space H. We begin by defining a decoherence operator D(A, B) and it’s associated q-measure operator µ(A) = D(A, A) on H. We show that these operators have certain positivity, additivity and continuity properties. If ρ is a state on H, then Dρ(A, B) = tr [ρD(A, B)] and µρ(A) = Dρ(A, A) have the usual properties of a decoherence functional and q-measure, respectively. The quantization of a random variable f is defined to be a certain self-adjoint operator fb on H. Continuity and additivity properties of the map f 7→ fb are discussed. It is shown that if f is nonnegative, then fb is a positive operator. A quantum integral is de- fined by R f dµρ = tr(ρfb). A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.
Quantum mechanics describes the physical state of a microscopic system using vectors in a complex Hilbert space rather than deterministic numerical variables. This mathematical formulation provides a rigorous framework for understanding quantum superposition, probability, and measurement. The Hilbert space structure allows quantum states to be represented as normalized vectors, while inner products define transition probabilities through the Born rule. Linear operators acting on these vectors correspond to measurable physical observables such as position, momentum, and energy. This paper presents a concise mathematical treatment of quantum states in Hilbert space, emphasizing vector representation, superposition, normalization, and projection-based measurement. In addition, the mathematical formulation underlying the concept of quantum immortality is briefly discussed from the perspective of the Many-Worlds Interpretation. The study demonstrates how linear algebra serves as the fundamental language of modern quantum mechanics and provides the theoretical basis for advanced areas including quantum computation, quantum information theory, and quantum communication.
Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.
In this thesis we will explore the development of a realist quantum theory based on the decoherence functional using the co-event interpretation of Quantum Measure Theory. The Sum-Over-Histories theory of quantum mechanics will provide the bedding for a Hilbert-space-free stochastic-like theory that can accommodate spacetime-like objects, and can therefore be applied to quantum gravity and cosmology, as well as give an alternative perspective on quantum phenomena. The primitive objects of the theory are histories, which give different accounts of a system’s evolution, and the decoherence functional, which sums the quantum interference between these histories. Quantum Measure Theory and Generalised Quantum Mechanics (a theory close to Decoherent Histories) then give alternative interpretations of the decoherence functional’s relation to reality. In these theories, the decoherence functional is mathematically constrained in analogue to probability measures. However, one of the conditions, called weak positivity, can be lost under composition of isolated systems. We will extend this composition argument to take the case for a stronger condition of strong positivity for decoherence functionals. The bulk of the report will then focus on the co-event interpretation of Quantum Measure Theory, where co-events give full accounts of which events do or do not occur for a given system. The quantum nature of reality is expressed through the breaking of classical logic within these co-event descriptions. We will focus on evolving co-event schemes, which dynamically construct co-events to describe the reality of an evolving system in tandem with its progression. The evolving co-event schemes will be shown to reproduce classical logic when they are applied to classical systems. Moreover, similar to classical stochastic theory, these schemes will be shown to be invariant under the inclusion or exclusion of non-interfering histories. We will also explore a number of outstanding problems for these schemes, and will propose some potential modifications.
1. Introduction.In quantum field theory we must consider the Hilbert space having non-countable bases which corresponds to a sequence of non-negative integers (n, n.,...).Since we can construct one-to-one mapping from the set of the sequences (n, n,...) onto the points in 0, 1 interval [8, we can identify these bases to [0, 1 interval.Let 7 denote a point in 0, 1 interval and let r be the element of the Hilbert space which corresponds to y.The element of this Hilbert space is usually represented by the formulae flCrrdt( 7) and Cr, in [3], E4] and [6], where C, Cr are constants, and d/(') is a measure on E0, 1].By single dz(r), however, we cannot represent every element of this Hilbert space.That is to say, by a continuous measure dz('), we cannot represent the element of the second form.On the other hand by the second form, we cannot represent the element of the first form.