A Clifford Algebraic Analysis Gives Mathematical Explanation Of
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World Journal of Neuroscience, 2013, 3, 157-170 WJNS http://dx.doi.org/10.4236/wjns.2013.33021 Published Online August 2013 (http://www.scirp.org/journal/wjns/) A Clifford algebraic analysis gives mathematical explanation of quantization of quantum theory and delineates a model of quantum reality in which information, primitive cognition entities and a principle of existence are * intrinsically represented ab initio Elio Conte1,2 1School of International Advanced Studies on Applied Theoretical and Non Linear Methodologies of Physics, Bari, Italy 2Department of Neurosciences and Sense Organs, University of Bari “Aldo Moro”, Bari, Italy Email: [email protected] Received 7 April 2013; revised 20 May 2013; accepted 24 June 2013 Copyright © 2013 Elio Conte. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT tistics of subatomic particles. Finally, the Copenhagen interpretation became widely accepted but with profound The thesis of this paper is that Information, Cognition reservations of some distinguished scientists and, in par- and a Principle of Existence are intrinsically struc- ticular, A. Einstein who prospected the general and al- tured in the quantum model of reality. We reach such ternative view point of the hidden variables, originating a evidence by using the Clifford algebra. We analyze quantization in some traditional cases of quantum large debate about the conceptual foundations of the the- mechanics and, in particular in quantum harmonic ory that has received in the past years renewed strength- oscillator, orbital angular momentum and hydrogen ening with Bell theorem [1], and still continues in the atom. The results are confirmed analyzing human present days. By 1930, quantum mechanics was further cognition behavior that evidences a very consistent unified and formalized by the work of David Hilbert, agreement with the basic quantum mechanical foun- Paul Dirac and John von Neumann, [2] with a greater dations. emphasis placed on measurement in quantum mechanics, the nature of reality and of its knowledge, involving the Keywords: Information; Quantum Cognition; Principle debate also a large body of epistemological and philoso- of Existence; Quantum Mechanics; Quantization; phical interest. Another feature that has always charac- Clifford Algebra; Cognitive Sciences terized the debate on quantum mechanics has been that one to identify what is the best mathematics that we should use in order to prospect quantum reality. 1. INTRODUCTION Conventionally formulated quantum mechanics starts The earliest versions of quantum mechanics were for- always with the combined standard mathematical, well mulated in the first decade of the 20th century following known, description from one hand and the use of classi- about the same time the basic discoveries of physics as cal physical analogies on the other hand. the atomic theory and the corpuscular theory of light that Our position is that by this way we risk to negate the was basically updated by Einstein. Early quantum theory fundamental nature of quantum reality that is fixed on was significantly reformulated in the mid-1920s by Werner some basic and unclassical features. They are the integer Heisenberg, Max Born and Pascual Jordan, who created quanta, the non commutation, the intrinsic-irreducible matrix mechanics, Louis de Broglie and Erwin Schrod- intedeterminism and quantum interference. It is possible inger who introduced wave mechanics, and Wolfgang to demonstrate that quantization, non commutation, intrin- Pauli and Satyendra Nath Bose who introduced the sta- sic and irreducible indetermination, and quantum inter- ference may be also obtained in a rough scheme due to *A preliminary and partial version of the present paper, based on the mathematical and physical foundations, is going to be published on the outset of the basic axioms of Clifford algebra. NeuroQuantology and on Advanced Studies in Theoretical Physics. First, let us follow the illuminating thinking of P. OPEN ACCESS 158 E. Conte / World Journal of Neuroscience 3 (2013) 157-170 Dirac. of a quantum system plus measuring apparatus devel- As previously said, P. A. M. Dirac contributed at the oped at the order n = 4 of the considered Clifford alge- highest level to the final formulation of quantum me- bras and of the corresponding density matrix in standard chanics. In his “The Development of Quantum Theory” quantum mechanics. In each of such cases examined, we [3] and “History of Twentieth Century Physics” [4], he found that the passage from the algebra Si to Ni,1 , wrote: considered during the quantum measurement of the sys- “I saw that non commutation was really the dominant tem, actually describes the collapse of the wave function. characteristic of Heisenberg’s new theory. It was really Therefore we concluded that the actual quantum meas- more important than Heisenberg’s idea of building up the urement has as counterpart in the Clifford algebraic de- theory in terms of quantities closely connected with ex- scription, the passage from the Si to the Ni,1 Clifford perimental results. So I was led to concentrate on the algebras, reaching in this manner the objective to refor- idea of non commutation. I was dealing with these new mulate von Neumann postulate on quantum measurement variables, the quantum variables, and they seemed to be and proposing a self-consistent formulation of quantum some very mysterious physical quantities and I invented theory. We reached also another objective. The com- a new word to describe them. I called them q-numbers bined use of the Si Clifford algebra and the Ni,1 di- and the ordinary variables of mathematics I called c- hedral Clifford algebra, also accomplishes to another numbers to distinguish them… Then I proceed to build basic requirement that the advent of quantum mechanics up a theory of these q-numbers. Now, I did not know strongly outlined. Heisenberg initial view point was to anything about the real nature of these q-numbers. Heisen- modify substantially our manner to look at the reality. He berg’s matrices, I thought, were just an example of q- replaced numbers by actions as also outlined by Stapp numbers, may be q-numbers were really something more [7]; a number represents the manner in which the dy- general. All that I knew about q-numbers was that they namics of a given object has happened. Heisenberg re- obeyed an algebra satisfying the ordinary axioms except placed such standard view point requiring instead that we for the commutative axiom of multiplication. I did not have to explicit the mathematical action (let us remember bother at all about finding a precise mathematical nature that the notion of operator will be subsequently adopted), of q-numbers”. and this action becomes the mathematical counterpart of Our approach may be reassumed as it follows. the physical corresponding action whose outcome will Initiating with 2010 [5,6] we started giving proof of give a number as final determination. Such double fea- two existing Clifford algebras, the Si that has isomor- tures of standard quantum mechanics represent of course phism with that one of Pauli matrices and the Ni,1 a basic and conceptually profound innovation in our where Ni stands for the dihedral Clifford algebra. manner to conceive reality and the methodology to in- The salient feature is that we showed that the Ni,1 vestigate it. It is clearly synthesized in our Clifford alge- may be obtained from the Si algebra when we attribute braic formulation by using from one hand the Clifford a numerical value (+1 or −1) to one of the basic elements Si and, as counterpart, the Ni,1 dihedral Clifford al- eee123,, of the Si . We utilized such result to advance gebra. a criterium under which the Si algebra has as counter- Generally speaking, our general position is that quan- part the description of quantum systems that in standard tization, non commutation, intrinsic-irreducible indeter- quantum mechanics are considered in absence of obser- mination and quantum variables as new “mysterious vation and quantum measurement while the Ni,1 attend physical quantities”, also if in a rough scheme, may be when a quantum measurement is performed on such sys- actually described and due to the outset of the basic axi- tem with advent of wave function collapse. oms of Clifford algebra. This is the reason because we The physical content of the criterium is that the quan- started in 1972 to attempt to formulate a bare bone skele- tum measurement and wave function collapse induce the ton of quantum mechanics by using Clifford algebra and passage in the considered quantum system from the Si on this basis we have obtained also some other interest- to Ni,1 or to the Ni,1 algebras, where each algebra ing results. Rather recently, as example, we have ob- has of course its proper rules of commutation. On this tained a very interesting feature that could be related to basis we re-examined the von Neumann postulate on quantum reality. It is well known that J. von Neumann [2] quantum measurement, and we gave a proper justifica- constructed a matrix logic on the basis of quantum me- tion of such postulate by using the Si . algebra. We also chanics. In [8-10] we inverted the demonstration, we studied some direct applications of the above mentioned showed that quantum mechanics may be constructed criterium to some cases of interest in standard quantum from logic. This feature may represent a turning point. In mechanics, analyzing in particular a two state quantum fact, the evidence is that we have indication about the system, the case of time dependent interaction of such logical origin of quantum mechanics and by this way we system with a measuring apparatus and finally the case are induced to conclude that quantum reality has intrin- Copyright © 2013 SciRes.