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arXiv:1002.4225 · Journal of Physics A Mathematical and Theoretical

Quantum reality filters

Stan Gudder

Published Nov 11, 2010• 3 citations
arXivHTMLPDF

Abstract

An anhomomorphic logic $\ascript ^*$ is the set of all possible realities for a quantum system. Our main goal is to find the "actual reality" $ϕ_a\in\ascript ^*$ for the system. Reality filters are employed to eliminate unwanted potential realities until only $ϕ_a$ remains. In this paper, we consider three reality filters that are constructed by means of quantum integrals. A quantum measure $μ$ can generate or actualize a $ϕ\in\ascript ^*$ if $μ(A)$ is a quantum integral with respect to $ϕ$ for a density function $f$ over events $A$. In this sense, $μ$ is an "average" of the truth values of $ϕ$ with weights given by $f$. We mainly discuss relations between these filters and their existence and uniqueness properties. For example, we show that a quadratic reality generated by a quantum measure is unique. In this case we obtain the unique actual quadratic reality.

DOI
10.1088/1751-8113/43/48/485301

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An anhomomorphic logic \ascript∗\ascript ^*\ascript∗ is the set of all possible realities for a quantum system. Our main goal is to find the "actual reality" ϕa∈\ascript∗ϕ_a\in\ascript ^*ϕa​∈\ascript∗ for the system. Reality filters are employed to eliminate unwanted potential realities until only ϕaϕ_aϕa​ remains. In this paper, we consider three reality filters that are constructed by means of quantum integrals. A quantum measure μμμ can generate or actualize a ϕ∈\ascript∗ϕ\in\ascript ^*ϕ∈\ascript∗ if μ(A)μ(A)μ(A) is a quantum integral with respect to ϕϕϕ for a density function fff over events AAA. In this sense, μμμ is an "average" of the truth values of ϕϕϕ with weights given by fff. We mainly discuss relations between these filters and their existence and uniqueness properties. For example, we show that a quadratic reality generated by a quantum measure is unique. In this case we obtain the unique actual quadratic reality.
An anhomomorphic logic A * is the set of all possible realities for a quantum system.Our main goal is to find the "actual reality" φ a ∈ A * for the system.Reality filters are employed to eliminate unwanted potential realities until only φ a remains.In this paper, we consider three reality filters that are constructed by means of quantum integrals.A quantum measure µ can generate or actualize a φ ∈ A * if µ(A) is a quantum integral with respect to φ for a density function f over events A. In this sense, µ is an "average" of the truth values of φ with weights given by f .We mainly discuss relations between these filters and their existence and uniqueness properties.For example, we show that a quadratic reality generated by a quantum measure is unique.In this case we obtain the unique actual quadratic reality."Reality is merely an illusion, albeit a very persistent one.
A classical logic exhibits a threefold inner structure comprising an algebra of propositions , a space of "truth values" V , and a distinguished family of mappings ϕ from propositions to truth values. Classically is a Boolean algebra, V = 2 , and the admissible maps ϕ: 2 are homomorphisms . If one admits a larger set of maps, one obtains an anhomomorphic logic that seems better suited to quantal reality (and the needs of quantum gravity). I explain these ideas and illustrate them with three simple examples.
Anhomomorphic logic is a new interpretation of Quantum Theory (due to R. Sorkin). It is a histories formulation (c.f. consistent histories, quantum measure theory). In this approach, reality is a co-event, which is essentially an assignment of a truth value {True, False} to each question. The way this assignment is done mimics classical physics in as much as possible allowing however for sufficient flexibility to accommodate quantum 'paradoxes', as is shown by the analysis of Kochen-Specker theorem. In this contribution, after briefly reviewing the approach, we will examine how probabilistic predictions can arise. The Cournot principle and the use of approximate preclusions will play a crucial role. Facing similar problems in interpreting probability as in classical probability theory, we will resort to the weak form of Cournot principle, where possible realities will be preclusive co-events and the quantum measure is used to obtain predictions. Examples considered, includes the fair coin and the double slit pattern arguably one of the most important paradigms for quantum theory.
Although various schemes for anhomomorphic logics for quantum mechanics have been considered in the past we shall mainly concentrate on the quadratic or grade-2 scheme. In this scheme, the grade-2 truth functions are called coevents. We discuss properties of coevents, projections on the space of coevents and the master observable. We show that the set of projections forms an orthomodular poset. We introduce the concept of precluding coevents and show that this is stronger than the previously studied concept of preclusive coevents. Precluding coevents are defined naturally in terms of the master observable. A result that exhibits a duality between preclusive and precluding coevents is given. Some simple examples are presented.
A discrete quantum process is defined as a sequence of local states ρtρ_tρt​, t=0,1,2,...t=0,1,2,...t=0,1,2,..., satisfying certain conditions on an L2L_2L2​ Hilbert space HHH. If ρ=lim⁡ρtρ=\limρ_tρ=limρt​ exists, then ρρρ is called a global state for the system. In important cases, the global state does not exist and we must then work with the local states. In a natural way, the local states generate a sequence of quantum measures which in turn define a single quantum measure μμμ on the algebra of cylinder sets $\cscript$. We consider the problem of extending μμμ to other physically relevant sets in a systematic way. To this end we show that μμμ can be properly extended to a quantum measure $\mutilde$ on a "quadratic algebra" containing $\cscript$. We also show that a random variable fff can be "quantized" to form a self-adjoint operator $\fhat$ on HHH. We then employ $\fhat$ to define a quantum integral $\int fd\mutilde$. Various examples are given
Several experiments have suggested that the complete indistinguishability of quantum systems is to be considered a real phenomenon. Experiments such as the HOM effect, the existence of BECs, the path identity cases, and many others are physical realities of today. Physicists `explain’ these phenomena using different models, such as Bohmian mechanics, relational quantum mechanics or the QBism approach. Most of these models are constructed within a standard mathematical framework, meaning standard mathematics, where all things can, even if only in principle, be distinguished from one another, so complete indistinguishability can only be mimicked. These approaches may work for physical purposes but are debatable regarding logical and metaphysical questions. If we accept that such questions are at least part of the philosophical aspect of quantum physics, then we are authorised to analyse them and try to offer an alternative. Looking for an alternative, we consider Alain Aspect’s and Anton Zeilinger’s views about quantum realism, opting for Aspect’s acceptance of some kind of reality behind the wave-function, while Zeilinger suggests that everything comes from information. In our opinion, the case is the opposite: it is information that comes from the existence of things. The aim of this paper is to justify why we think that the underlying reality (in Aspect’s sense) can be seen as a quantum ontology of non-individuals.