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NeuroQuantology

On the Logical Origins of Quantum Mechanics Demonstrated By Using Clifford Algebra

Elio Conte

Published May 24, 2011• 21 citations
Publisher

Abstract

We review a rough scheme of quantum mechanics using the Clifford algebra. Following the steps previously published in a paper by another author [31], we demonstrate that quantum interference arises in a Clifford algebraic formulation of quantum mechanics. In 1932 J. von Neumann showed that projection operators and, in particular, quantum density matrices can be interpreted as logical statements. In accord with a previously obtained result by V. F Orlov , in this paper we invert von Neumann’s result. Instead of constructing logic from quantum mechanics , we construct quantum mechanics from an extended classical logic. It follows that the origins of the two most fundamental quantum phenomena , the indeterminism and the interference of probabilities, lie not in the traditional physics by itself but in the logical structure as realized here by the Clifford algebra.

DOI
10.14704/nq.2011.9.2.397

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We review a rough scheme of quantum mechanics using the Clifford algebra. Following the steps previously published in a paper by another author [31], we demonstrate that quantum interference arises in a Clifford algebraic formulation of quantum mechanics. In 1932 J. von Neumann showed that projection operators and, in particular, quantum density matrices can be interpreted as logical statements. In accord with a previously obtained result by V. F Orlov , in this paper we invert von Neumann’s result. Instead of constructing logic from quantum mechanics , we construct quantum mechanics from an extended classical logic. It follows that the origins of the two most fundamental quantum phenomena , the indeterminism and the interference of probabilities, lie not in the traditional physics by itself but in the logical structure as realized here by the Clifford algebra.
Abstract: We review a rough scheme of quantum mechanics using the Clifford algebra. Following the steps previously published in a paper by another author [31], we demonstrate that quantum interference arises in a Clifford algebraic formulation of quantum mechanics. In 1932 J. von Neumann showed that projection operators and, in particular, quantum density matrices can be interpreted as logical statements. In accord with a previously obtained result by V. F Orlov, in this paper we invert von Neumann’s result. Instead of constructing logic from quantum mechanics, we construct quantum mechanics from an extended classical logic. It follows that the origins of the two most fundamental quantum phenomena, the indeterminism and the interference of probabilities, lie not in the traditional physics by itself but in the logical structure as realized here by the Clifford algebra.
We review our approach to quantum mechanics adding also some new interesting results. We start by giving proof of two important theorems on the existence of the and Clifford algebras. This last algebra gives proof of the von Neumann basic postulates on the quantum measurement explaining thus in an algebraic manner the wave function collapse postulated in standard quantum theory. In this manner we reach the objective to expose a self-consistent version of quantum mechanics. We give proof of the quantum like Heisenberg uncertainty relations, the phenomenon of quantum Mach Zender interference as well as quantum collapse in some cases of physical interest We also discuss the problem of time evolution of quantum systems as well as the changes in space location. We also give demonstration of the Kocken-Specher theorem, and also we give an algebraic formulation and explanation of the EPR . By using the same approach we also derive Bell inequalities. Our formulation is strongly based on the use of idempotents that are contained in Clifford algebra. Their counterpart in quantum mechanics is represented by the projection operators that are interpreted as logical statements, following the basic von Neumann results. Using the Clifford algebra we are able to invert such result. According to the results previously obtained by Orlov in 1994, we are able to give proof that quantum mechanics derives from logic. We show that indeterminism and quantum interference have their origin in the logic.
The main quantum phenomena-indeterminism and interference-are explained on a purely logical basis by extending classical mathematical logic. In this extension, statements are represented by Hermitian matrices that, it is postulated, can undergo unitary transformations of truth values (the principle of relativity of truth values). The formula for quantum probability is deduced. A new method of quantization is then introduced by the additional postulate that invariants of translations in some space can be described by statements represented by matrices that are invariants of the same translations in the same space. The formula p̂ = −ih∂/∂q is deduced (up to an unknown multiplicative constant). The connection between the existence of the continuum and the existence of quantum mechanics is demonstrated.
My discussion is articulated under the neurological as well as the psychological profile. I insist in particular on the view that mental events arise in analogy with quantum probability fields. I review some results obtained on quantum cognition discussing in detail those that we obtained on quantum interference in mental states during perception-cognition in ambiguous figures. Frequently, I use the approach to quantum mechanics by Clifford algebra. I insist in particular on two recent results. The first is the justification that I obtain of the von Neumann postulate on quantum measurement and the second relates my Clifford demonstration on the logical origins of quantum mechanics and thus on the arising feature that quantum mechanics relates conceptual entities. The whole discussion aims me to support the conclusion that we think in a quantum probabilistic manner.
The most peculiar, specifically quantum, features of quantum mechanics --- quantum nonlocality, indeterminism, interference of probabilities, quantization, wave function collapse during measurement --- are explained on a logical-geometrical basis. It is shown that truths of logical statements about numerical values of quantum observables are quantum observables themselves and are represented in quantum mechanics by density matrices of pure states. Structurally, quantum mechanics is a result of applying non-Abelian symmetries to truth operators and their eigenvectors --- wave functions. Wave functions contain information about conditional truths of all possible logical statements about physical observables and their correlations in a given physical system. These correlations are logical, hence nonlocal, and exist when the system is not observed. We analyze the physical conditions and logical and decision-making operations involved in the phenomena of wave function collapse and unpredictability of the results of measurements. Consistent explanations of the Stern-Gerlach and EPR-Bohm experiments are presented."