Transitivity and Equicontinuity in Quantum Measure Spaces
Introduction
A mathematical model for quantum statistical mechanics was suggested in the seminal paper by Birkhoff and von Neumann in [6]. Since then significant studies are being made in this direction. The prototype for the mathematical structures known as orthomodular lattices, also called quantum logic, is the family L(H) of all closed subspaces of a Hilbert space H; L(H) is isomorphic to the set P(H) of all projection operators T on H with O ≤ T ≤ I. Owing to this isomorphism, P(H) is called a Hilbert lattice. One of its generalizations is the family B(H) of Hilbert space effects, i.e. the set of all self-adjoint operators A on H satisfying O ≤ A ≤ I. Effects are of significance in representing unsharp measurements or observations on the quantum mechanical system. The concept of unsharpness has hitherto influenced the study of effect algebras, also known as difference posets, that have arisen in stochastic (or phase-space) quantum mechanics (see, e.g. [7, 10, 11, 20, 21, 29, 30, ]). Effect algebras are a generalization of orthomodular lattices and of many structures used in quantum physics [2], in mathematical economics [13], and in functional analysis [30].
By using the notion of a state on a quantum logic one can introduce the entropy of partitions in quantum logic, which is a useful tool in the study of dynamical systems and their isomorphism (see e.g. [6, 11, 15, 16, 22, 23, 25, 33, 34, ]) and has been applied in many other structures. The theory of dynamical systems has been acknowledged as one of the significant breakthroughs in science in the twentieth century. This area is becoming indispensable in types of scientific knowledge, and the techniques developed are being applied to a wide variety of nonlinear problems ranging from physics and economics to ecology and psychology (see, e.g. [9, 13, 24]). The concept of a (metrically) transitive system was initiated by Birkhoff in [5], which may be described as a system (X, f), where f is a self map on a metric space X, having points that eventually move under some iteration of f from one arbitrarily small neighbourhood to any other. Likewise an equicontinuous point may be viewed as a point x in X for which all points in its neighbourhood have their orbits arbitrarily close to the orbit of x. In recent years such notions that involve the internal structure of X served as a basis for a broad classification of dynamical systems by their recurrence properties. Further theory may be seen in [8, 12, 31, 32]. In the present paper we study transitivity and equicontinuity in the realm of dynamics of quantum measure spaces.
The paper is organized as follows: In Section 2, a brief survey of some basic definitions that are required in the present study of quantum dynamical systems (P, μ, ϕ) are collected. Section 3 is devoted to the study of similar and conjugate quantum dynamical systems (P, μ, ϕ) and (Q, v, ψ); the conjugation from ϕ to ψ is a bijection, with certain properties (primarily an orthosupplement and lattice structure preserving conditions) between the corresponding quotient sets and which arose from the equivalence relation given in [14]; see also [27]. It is pertinent to mention here that is eventually made a bounded lattice with an order reversing involution, and so may be used for the mathematical formalism of unsharp structures. It is obtained that similar quantum dynamical systems having join preserving morphisms, are conjugate. In Section 4, notions of transitive and minimal quantum dynamical systems are introduced and studied. It is observed that a minimal quantum dynamical system is transitive. The concepts of chain-transitivity and shadowing, using the notion of a δ-chain, are introduced and it is proved that a transitive system, under a suitably formulated condition (C), is chain-transitive. It is also proved that every chain-transitive system possessing the shadowing property is transitive. In Section 5, we discuss equicontinuity of (P, μ, ϕ) and prove that every equicontinuous system having a condensed set of minimal points is minimal and is therefore transitive.
Section snippets
Preliminaries
2.1 [4, 10, 11]. A difference poset or a D-poset is a poset (P, ≤) that contains the largest element 1P and is endowed with a partially defined binary operation (a, b) → b ⊝ a on P, defined if and only if a ≤ b, a,b ∈ P, such that the following conditions are satisfied for any a,b,c ∈ P with a ≤ b ≤ c:
(D1) b ⊝ a ≤ b;
(D2) b ⊝ (b ⊝ a) = a;
(D3) c ⊝ b ≤ c ⊝ a;
(D4) (c ⊝ a) ⊝ (c ⊝ b) = b ⊝ a.
In any D-poset (P, ≤, ⊝, 1P), we consider 0P:= 1P ⊝ 1P and an operation ⊕, given by a ⊕ b ∈ P ⇔ a ≤ 1P ⊝ b,
Similarity and conjugation
Let (P, μ, ϕ) be a quantum dynamical system. For b ∈ P, ϕ0(b):= b; ϕn(b):= ϕ(ϕn-1(b)) (n ∈ N); and the orbit of b ∈ P, written as orbit(b), is the set {ϕn(b): n ∈ N U {0}}. An element b ∈ P is a periodic point with prime period p, p ∈ N, if ϕp(b) = b and ϕk(b) ≠ b for any k ∈ Jp-1. By a periodic orbit, we mean orbit of a periodic point in P. A point b ∈ P is said to be an almost
periodic point if for any ε ∈ R+, there exists i ∈ N such that μ(bδϕn+j-1(b)) ∞ ε, for some j ∈ Ji+1 and for all n ∈
Definition 4.
A quantum dynamical system (P, μ, ϕ) (or simply, ϕ) is called:
- (1)
transitive if for any b, c ∈ P, and for any ε, δ ∈ R+, there exist n ∈ N and d ∈ P such that μ(b δ d) ∞ δ and μ(ϕn(d) δ c) ∞ ε;
- (2)
minimal if for any b, c ∈ P, and for any ε ∈ R+, there exists n ∈ N such that μ(cδϕn(b)) ∞ ε. In this case, the point ‘b’ is called a minimal point of P.
Proposition 4.1
(i) Let τ: P → Q be a semi-similarity from a quantum dynamical system (P, μ, ϕ) to a
Definition 5.
Let (P, μ, ϕ) be a quantum dynamical system.
- (1)
A point b ∈ P is said to be an equicontinuous point if for any ε ∈ R+, there exists δ ∈ R+ such that for all c ∈ P satisfying μ(bδc) ∞ δ, we have μ(ϕn(b) δϕn(c)) ∞ ε for all . A quantum dynamical system is said to be equicontinuous if all of its points are equicontinuous points.
- (2)
The system (P, μ, ϕ) is said to be uniformly equicontinuous if for any ε ∈ R+, there exists δ ∈ R+ such that for all b, c ∈ P satisfying μ(bδc) ∞ δ, we
References (34)
- et al.
J. Math. Anal. Appl.
(2003) - et al.
J. Econom. Theory
(2002) - et al.
J. Math. Anal. Appl.
(2008) - et al.
J. Math. Anal. Appl.
(2008) Fuzzy Sets and Systems
(2001)- et al.
The Logic of Quantum Mechanics
(1984) - et al.
Internat. J. Theoret. Phys.
(1995) - et al.
Found. Phys.
(1995) Dynamical Systems
- et al.
Ann. of Math.
(1936)
The Quantum Theory of Measurements
Isr. J. Math.
An Introduction to Chaotic Dynamical Systems
New Trends in Quantum Structures
Found. Phys.
Isr. J. Math.
Demonstratio Math.
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The second author acknowledges with gratitude the financial support by Department of Science and Technology (DST), New Delhi, India, under INSPIRE fellowship No. IF160721.