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*
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 , 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,  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.
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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
 and “History of Twentieth Century Physics” , 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 ; 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  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-
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sically a new feature that we are not accustomed to at- member of the ASi algebra that we have discussed tribute to it. Quantum reality starts admitting a role for elsewhere [5,6] logic, thus for cognition, language, semantic not in a k foreseen sense. There is a principle in quantum reality Nnq n (2.2) that we are addressed to evidence in the following man- n0 ner: there are stages of our reality (those engaged from where qn are specific Clifford members having some quantum theory, precisely) in which matter no more may specific properties. be conceived by itself, it no more may be conceived in- Let us consider the case n 0,1 . dependently from the cognition that we have about it. In this case N is given in the following manner This is a new viewpoint that involves mind like entities, Nq01 q (2.3) modulating matter with cognition ab initio in our quan- 01 tum reality. Therefore it opens a new way to acknowl- where q0 and q1 are the following idempotents in Si edge a role of quantum mechanics in cognitive sciences 11 [11,12]. qeq0311; 1e3; (2.4) 22 In previous papers we have investigated such our ap- proach considering indeterminism and quantum interfer- We have ence The aim of the present paper is to add here new 11 11 qeq and e (2.5) results to such thesis. We select to consider here the 03122 223 problem of the quantization. Let us write the mean value of e3 . It is
2. THEORETICAL ELABORATION ep3 111 p1 (2.6) Our basic statement is that quantum reality has its pecu- being p1 and p1 the corresponding probabili- liar features. ties for the abstract entity e3 to assume or the numeri- Instead conventionally formulated quantum mechanics cal value 1 or the numerical value 1. starts always with the use of classical analogies. Our let us admit now that e3 is a cognitive entity. Of approach is different. Our thesis is that by this way we course we know that, according to von Neumann , risk to negate the fundamental nature of quantum reality density operators as well a correspondingly, idempotent that is fixed on three basic and unclassical features. They elements may be considered logic statements. are the integer quanta, the non commutation, and the Let us admit that the cognitive entity, represented by intrinsic-irreducible intedeterminism and quantum inter- e3 is in the condition of absolute certainty that the rep- ference. resented system S to which N is connected, has the Quantization, non commutation and intrinsic and ire- value 1 . This means in the (2.6) that p11 and ducible indetermination are actually evidenced by using p10 . Consequently N will be characterized from the outset of the basic axioms of Clifford algebra. We the discrete integer value n 0 . In the other possible have previously mentioned that, by using such algebraic case, N will be characterized from the discrete integer elaboration, we realized a bare bone skeleton of quantum value n 1. mechanics formulating in particular about the intrinsic- Speaking in general quantum terms, the question of irreducible indetermination shown from quantum reality interest is the immediate connection that we establish and the relevant role of non commutation and quantum between the integer quantized condition of the physical interference. We will not consider here further on such observable that we have identified containing N and statements since they were discussed in detail by us pre- the cognitive task that must be performed. In order to viously [11,12]. ascertain the quantized integer value of N , a cognitive Previously we did not consider the question of the in- task must be performed in the sense that a semantic act is teger quanta and we attempt to derive here a detailed here clearly involved. Of course Orlov  was the first exposition. to identify e as the basic and universal logic operator. Let us sketch the problem remembering that in quan- 3 Still, the aim of the elaboration must be clear here. Cer- tum mechanics some physical quantities may be ex- tainly we do not speak here about human cognition but pressed in the following manner of primitive cognitive entities. A f Nabc,,,, (2.1) The relation of e3 with the basic wave function of quantum mechanics is of course established. where abc,,, may be constants and N assumes only discrete, integer values 0,1,2,3, 4,5, . 1 e p1 3 2 (2.7) N may be conceived to be the following Clifford 2 1
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1 e represented system S to which N is connected, has the p1 3 2 (2.8) 2 2 value 1 . This means in the (2.6) that p11 and p10 . The same reasoning may be developed for being 1 and 2 corresponding selected kets in the E30 , and for E33 . proper Hilbert space. It results evident that by moving in this direction we In conclusion we have given proof of a necessary and are obtaining indication of a new arising scheme of real- sufficient link existing between N and e . 3 ity. It seems that in substance the cognitive entities that We should write we invoke here relate the same concept of existence. Is
A f N e3 ,,,, abc (2.9) this existing condition of reality actually existing? The concept of Existence becomes here itself a variable that with assumes therefore two possible values, indicating yes/not 22 cognitive condition. Existence and cognition result there- qq011001 qq0, q q 1, q 001 q , q q1 (2.10) fore profoundly linked in the scheme of reality that we Let us examine what it happens in the case in which are delineating. The conceptual indication that we sug- we consider N assuming four possible integer values. gest here is that in the basic foundation of our reality ab In this case we need a Clifford algebraic structure initio there are elements of existence defined, not in given at the order n 4 . The four possible combina- terms of some hazy metaphysical concept of existence, tions of the basic primitive idempotent elements are but in the sense that existence, related to cognition, is 1 qEEEE ; represented by abstract entities of the Clifford algebra, 004 0030030and that contains only two possibilities: existence or non- 1 existence. A pure dichotomic variable structured in the qEEEE ; 100030030inner architecture of our reality. Of course consciousness 4 ; (2.11) 1 is awareness and knowledge about something existing. qEEEE ; 204 0030030Certainly we have factors of scale so that a microstruc- 1 ture of our reality employs a limited number of abstract qEEEE . entities and a mechanism of amplification at a macro- 304 0030030 structural level must be expected in order to account for Note that in this case we invoke the basic and univer- awareness as it is usually intended at the level of human sal logic operators ( E03 and E30 ) and the coupling cognition, but it is clear that in any case we are speaking (conjunction) EE33 03E 30EE 30 03 . Obviously, also the about a dynamics that starts as intrinsically conceived in relations like the (2.10) hold in this extended case. the scheme of our reality from its starting ab initio. The 2 idea of course is not new here. We think as example to qqqqqijiiiiij1; ; 0; ; 0,1,2,3; j 0,1, 2,3 i Eddington  and to D. J. Bohm, P. G. Davies, H. J. (2.12) Hiley . Eddington in 1946 argued that within a purely algebraic approach, which he regarded as providing Let us apply now the previous criterium SNii, ,1 that we considered previously. a structural description of physics, there are elements of existence defined, not in terms of some hazy metaphysi- Let us write the mean values of E03 and of E30 and E33 . It is cal concept of existence, but in the sense that existence is represented by a symbol that contains only two possibili- Ep111 p 1; (2.13) 03 ties: existence or non-existence. Just as in our treatment by using Clifford algebra, these authors assumed that the Ep30 111 p 1 ; (2.14) structural concept of existence is represented by an idem- Ep33 111 p 1 ; (2.15) potent of some appropriate algebra and satisfying the conditions given by us in the (2.10) or in the (2.12). being p1 and p1 the corresponding probabili- ties for the abstract entities to assume or the numerical Let us admit now that value 1 or the numerical value 1. EE03 1,30 1 we have in the 2.2qo 1, Let us admit now that E , E , E are cognitive (2.16) 03 30 33 qqq0 entities. As previously said, we know that, according to 123 von Neumann , density operators as well a corre- and the first integer value is obtained. spondingly, idempotent elements may be considered lo- If instead the cognitive performance ascertains that gic statements. EE 1, 1 we have in the 2.2 q 1, Let us admit that the cognitive entity, represented by 03 30 3 (2.17) qqq0 E03 is in the condition of absolute certainty that the 120
Copyright © 2013 SciRes. OPEN ACCESS E. Conte / World Journal of Neuroscience 3 (2013) 157-170 161 and the second integer is obtained. in the (2.12). In the case in which ForEEE 1, 1, 1, we have 003 030 300 (2.30) EE03 1,30 1 we have in the 2.2 q2 1, q1, qqqqqqq0 (2.18) 0 1234567 qqq0 130 and the first integer value is obtained. and the third integer is obtained ForEEE003 1,030 1,300 1, we have Finally, with (2.31) q01, qqqqqqq 12345670 EE1, 1 we have in the 2.2 q 1, 03 30 1 (2.19) and the first integer value is obtained. qqq0 230 ForEEE 1, 1, 1, we have 003 030 300 (2.32) and the fourth integer is obtained. q1, qqqqqqq0 Obviously the case of three integer is trivial and will 1 0234567 not be discussed here. and the second integer value is obtained. The case n 8 proceeds in the same manner. ForEEE003 1,030 1,300 1, we have We need Clifford algebraic elements in SA i )( : (2.33) q21, qqqqqqq 1034567 0 EEEEEEE003,,,,,, 030 300 033 303 330 333 (2.20) and the third integer value is obtained. We may be sure that our Clifford algebraic structure at ForEEE 1, 1, 1, we have the order n = 8 will be 003 030 300 (2.34) qqqqqqqq31, 12045670 EEE001,, 002 003 , EEE 010 ,, 020 030 , EEE 100 ,, 200 300 (2.21) and the fourth integer value is obtained. and related sets providing coupling. In this case they give origin to the following basic ForEEE003 1,030 1,300 1, we have (2.35) Clifford elements q41, qqqqqqq 12305670 1 qEEE111 (2.22) and the fifth integer value is obtained. 0 8 003 030 300 ForEEE003 1,030 1,300 1, we have 1 (2.36) qEEE111 (2.23) q51, qqqqqqq 1234067 0 1 8 003 030 300 and the sixth integer value is obtained. 1 (2.24) qEEE2 111003 030 300 ForEEE 1, 1, 1, we have 8 003 030 300 (2.37) 1 q61, qqqqqqq 12345070 qEEE111 (2.25) 3 8 003 030 300 and the seventh integer value is obtained.
1 ForEEE003 1,030 1,300 1, we have qEEE4 111003 030 300 (2.26) (2.38) 8 q71, qqqqqqq 1234560 0 1 and the eighth integer value is obtained. qEEE5 111003 030 300 (2.27) 8 Corresponding to each value there is a clear condition 1 of semantic awareness that is intrinsically linked. qEEE111 (2.28) 6 8 003 030 300 We may now take a further step on. It is well known that the Clifford ASi , in addition 1 to admits idempotent, also contains nilpotent. qEEE7 111003 030 300 (2.29) 8 Generally speaking, it is known that an element x of a Note the particular alternation in the signs of the ring R is called nilpotent if there exists some positive n idempotent elements arising for each qi with i = 0, 1, integer n such that x = 0.
···, 7. Previously we have considered two idempotent in Si
We have (+,+,+), (–,+,+), (+,–,+), (–,–,+), (+,+,–), written as 1 e3 2 and 1 e3 2. In the same alge-
(–,+,–), (+,–,–), (–,–,–). A combination of all the possible bra two nilpotent can be written as eie12 2 and alternatives: a clear semantic message is contained and it eie12 2 This is at the order n 2 but we may easily is intrinsic to the inner structure of such arising integer generalize them at higher orders. quanta mechanism. The important thing is to observe here that the two Obviously all such qi satisfy the required rules given nilpotent elements may be rewritten linked to idempo-
Copyright © 2013 SciRes. OPEN ACCESS 162 E. Conte / World Journal of Neuroscience 3 (2013) 157-170 tent: 1 b QRSPRS; 22 (2.44) eie1221 e 1 e 3 2 (2.39) withb some given real constant. eie1221 e 1 e 32 bab2 QP PQ RS SR e , where we have used the Clifford representation of the 223 imaginary unity ieee . 123 1 ab22 ba22 These nilpotent elements are the same as the idempo- QRSSRP22, RSSR 444 4 tent elements multiplied by e1 . Still it is instructive to observe that (2.45) Let us now examine RS22 0 . It is ee131212;1 eeee 3113 2 (2.40) 22 R RS S R SR a e3 S RS RS a 0 (2.46) 12ee31 Let us write it explicitly. We obtain that and 22 R RS S R SR a e3 S RS RS a ee13121212;1 e 3 e 1 e 3 ee 132 (2.47) 4 11 11 1212eee31 3 ae 33 e 0 22 22 (2.41). Two important results. What is the reason to have introduced here the notion nn The first is that RS 0 n 2 , starting with nil- of nilpotent that of course is well known in Clifford al- potent elements for R and S, have been reduced again to gebra. The reason is that on the basis of the previously idempotent elements (logic statements). The second is discussed link existing in our view point between idem- that we have obtained a tautology. The (2.47) is always potent elements, logic, semantic, information, and cogni- true in itself, when we consider e 1 as well as tive abstract entities, also on the other hand the existing 3 when we consider e 1. link between idempotent and nilpotent elements, must be 3 The procedure is now well fixed. We may proceed defined also under the profile of the logic, semantic, in- discussing the case at the order n 4 . formation, and cognition delineating what is the meaning We know by now the basic sets of Clifford elements of nilpotent. In our view point, the condition that there that we have to recall (see the (2.20)) and in this case we exists some positive integer n such that xn = 0, under the have logic, semantic, and cognitive profile, means that at this order n we reach an absurdum that our reality cannot 13 13 Ra E iE E E iE admit. 01 02 30 01 02 44 Let us consider now the following two basic nilpotent elements 2 E10 E 01 iE 02 iE 20 E 01 iE 02 4 11 11 Ra e12 eand Sa e12 e , 22ii 22 1313 aEEiE 03 01 02 RS220, RSRSRS 0;nn 0; n11 n 0 (2.42) 44 2 (at the ordern 2 in our case). EiEEiE 4 10 20 01 02 a is some prefixed real constant. (2.48) Note that, in spite of being RSnn 0 (absurdum) ( n 2 in the present case), 13 13 Sa E01 iE 02 EE30 01 iE 02 44 2211 11 2 RS a e33;; SR a e RS SR a e3 22 22 2 E E iE iE E iE (2.43) 10 01 02 20 01 02 4 an idempotent element is instead obtained promptly at 1313 the order n 1. aEE iE 4403 01 02 Let us admit to construct now some variables starting with RS and . In detail, let us introduce the variables 2 EiEEiE QP and (Clifford algebraic elements) in the follow- 4 10 20 01 02 ing manner (2.49)
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and the argument proceeds as in the previous case, this LLij L ji L i L k; i 1,2,3; j 1,2,3; k 1,2,3; i j k . n n n−1 n−1 time at order n 4 and thus having R S = 0; R S (2.54) 0 with n 4 . At just derived previously, at the order n 2 , we In each case nilpotent elements are finally reduced to have the basic Clifford elements previously discussed for idempotent elements indicating logical statements. quantization What is then the interesting feature of the procedure that we have here developed. It is not only in the matter 11 J e12 ie;; J e12 ie to have used pure Clifford members and to have dis- 22 (2.55) cussed about their logic, and thus semantic, cognitive 13 JeJe; 2 feature. The substantial result is that such cognitive fea- z 2430 tures are linked to matter as it is in the thesis of our pa- All the usual commutation relations of standard quan- 12 2 tum mechanics are verified. pers. In fact let us take a in the starting m At the order n 4 , we have im 31 (2.42) and b in the starting (2.44). Consider the J E iE E E iE 2 2201 02 10 01 02 Clifford elements QP and to represent the position 1 and the momentum of a particle signed by the Hamilto- iE20 E 01 iE 02 (2.56) nian 2 31 11222 EiE EiEEiE H PmQ (2.50) 2201 02 01 02 10 20 22m We are examining now the well known case of the 31 JEiEEEiE01 02 10 01 02 harmonic oscillator in standard quantum mechanics. 22 As it is well known, the quantized oscillator energy is 1 iE E iE .(2.57) given by 2 20 01 02 1 31 EN . (2.51) EiE01 02 EiEEiE 01 02 10 20 2 22
In this case it results Again we have obtained the basic result. J and J contain idempotent elements that are expression of logic m NRS (2.52) statement. In fact we have that 2 and the quantized levels are obtained from the (2.46) at 31 J EE01111 03 EEEE 01 03 10 30 ;(2.58) order n 2. The following energy levels are obtained 22 at the order (n = 4), (n = 8), and so on. 31 We have in this case a direct connection between logic J EE01111 03 EEEE 01 03 10 30 .(2.59) 22 statements, semantics, cognition from one hand and a material object as a quantum harmonic oscillator on the Our basic objective is reached also in this case. other hand. Of course, we have to outline here the basic Of course, the procedure of quantization is obtained conceptual foundation that the harmonic oscillator de- following the same procedure outlined in the case of the velops in the whole profile of quantum mechanics start- harmonic oscillator using nilpotent elements that finally ing with the original and initial results of Heisenberg and result expressed by idempotent elements and thus logical arriving also to the most recent applications of the har- statements. monic oscillators in the current days of application of At the order n 2 as well as at the order n 4 we quantum mechanics. obtain the basic relation nn11 n n The same results may be obtained if we study quanti- JJ 0and J 0, J 0 (2.60) zation of orbital angular momentum or the hydrogen that gives origin to the quantization. atom. We have that Relating orbital angular momentum, it is well known that J JiJJJiJx yxy;; (2.61) LQPQPLQPQPLQPQP;;; (2.53) 123322311331221 31 1 J EEEEE ; (2.62) with x 2201 01 10 2 02 20
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31 1 The theory of Fourier and the correspondence princi- J y EEEEE02 01 20 02 10 ; (2.63) ple of Bohr played a vital role in Heisenberg’s develop- 22 2 ment of quantum mechanics. In essence, Heisenberg re- 1 J EE; (2.64) placed the Fourier series by a ‘‘Fourier table’’. In his z 2 03 30 classic paper, each quantum formula was carefully crafted with from the corresponding classical formula . For Heisen- berg, the problem with classical mechanics was not the J x JJJiJJJJJiJJJJJiJzzxyyzzyxxyyxz;;; dynamics, but the kinematics. According to Heisenberg, (2.65) the equations of quantum mechanics are relations be- Note that we have tween observable quantities such as the spectral frequen- cies and intensities, and not the mechanical properties of 2253 1 53 1 the electron motion such as the position and period. In- JEEJEExy10 33;; 10 33 42 2 42 2 (2.66) stead of representing x(t) by a sum of Fourier harmonics, 22225152 JEJJJJ; citcitcit12exp exp 2 3 exp 3 (2.70) zxz4433 y following the basic indications of Born, Pauli and Jordan, When passing In the Clifford algebra N , we have i,1 the dynamical variable x was finally represented by a that for EE1, E 1, it is 03 30 33 matrix of Heisenberg harmonics, J z 32or 32. For E03 1, E30 1; E33 1, citcit11exp 11 12 exp 12 ci11mmexp t J z 12or 12 citcit21 exp21 22 exp 22 ci22mmexp t J z may assume one of the following numbers: jj,1,, j for j 32. J 2 assumes the possible values jj1. citcitcn1122exp n n exp n nm exp i nmt As required in our formulation we have that (2.71) JJnn110; J n 0; J n 0 (2.67) n The Heisenberg harmonic, xnm ci nm exp nmt , is Therefore our basic formulation fixed on nilpotent and associated with the transition nm while the transi- tion amplitude c provides a measure of the intensity idempotent Clifford algebraic elements is again recalled. nm of the light and the transition frequency equals the It remains only a feature that needs to be explained. nm light frequency. In particular, the Heisenberg harmonic When considering J ,,JJ, as said in the (2.65), we x yz x uniquely determines the transition probability A obtain nm nm and the Power Pnm so that a net connection between the J JJJiJJJJJiJJJJJiJ;; x zzxyyzzyxxyyxz quantum mechanical motion of the electron xnm t (the (2.68) state of the electron) and the spectroscopic observable P is strongly established: that do not correspond to the standard basic Clifford al- nm 23 23 gebra AS where in fact we have that ee22 i AxPnm and nm x. (2.72) ee,2 i e being the difference by a factor 2. nm 33nm nm nm i j ijk k 3300cc We gave detailed proof on the existence of the ASi . The new algebra connected to the (2.68) may be demon- Among the key equations of Heisenberg’s famous pa- strated following the same procedure (see the [3,4]) and per that started modern mechanics were a multiplication obtaining in this case the new basic elements rule for quantum-theoretical quantities and a quantum condition that was identical with the Thomas-Kuhn sum 11222 i rule. Within a few weeks after reading Heisenberg’s pa- eeeeeii;;123 eeeeeijjik and 242(2.69) per, Born interpreted the multiplication rule as the rule cyclic permutation ofijk , , , for matrix multiplication and the quantum condition as the statement that each of the diagonal elements of the ijk1, 2, 3; 1, 2, 3; 1, 2, 3. matrix PX XP is equal to i . The reader 1 should well take in mind that Clifford abstract entities Idempotent elements become in this case eˆi . 2 should not be confused with matrices since by this way We may now pass to explore the quantization of the we have only their isomorphic representation, however energy levels for the hydrogen atom. the initial Born interpretation represents the initial step to The history of the first elaboration of quantum me- conceive a bare bone skeleton of quantum mechanics chanics, relating in particular the study of the hydrogen realized by Clifford algebra. The further step, realized by atom, is well known. Pauli , was the analysis of the Hydrogen atom by
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Pauli by using the well known Lorentz-Runge Lentz K12KKKiLKKKKiL 21 3;; 23 32 1 vector [18,19]. (2.79a) KK KK iL In substance he used three matrices 31 13 2 11 Let us introduce still the following basic elements A LP PL LP PL QR1 123323223mZe2 2 1 11 MLKNLK111111;; 11 22 A LP PL LP PL QR1 (2.73) 231131331mZe2 2 2 11 MLKNLK222222;; (2.79b) 11 22 A LP PL LP PL QR1 312212112mZe2 2 3 11 MLKNLK333333; with 22
2222 We have that R QQQ123 (2.74) MMMNNN222222 0 (2.80) They satisfy the following basic properties: 123123 The second important property is that LA11 LA 2 2 LA 3 3 0 2 MMMNNN2 2 2222 and 123123 1 mZ24 e (2.81) 222 2 2222 222222 LLLK123 1 K 2 K 3 1 AAA1231 24 ELLL123 (2.75) 22 mZ e 2 where it results that The basic property that we need to be sure to be in the Clifford algebraic structure Si is that we now have 1 222 21 EH PPP123 ZeR (2.76) M MMMiMNNNNiN;; 2m 12 21 3 12 21 3 (2.82) M MMMiMNNNNiN;; It is trivial to acknowledge the basic meaning of E . 23 32 1 23 32 1
Still we find that the following relations hold. M 31MMMiMNNNNiN 13 2; 31 13 2 (2.83)
AHii,0;,0; ALi LA12 AL 21 iA 3; as we obtained previously in (2.68) and in (2.69). LA AL iA;; LA AL iA We have now given proof that we are in Si . 21 12 3 23 32 1 We have LA32 AL 23 i A 1;; LA 31 AL 13 i A 2 M 22222MM NNN2 (2.84) 2 123123 LA13 AL 31 i A 2;; AA 12 AA 21 i HL 3 (2.77) mZ24 e and 2222 2 M MMM123. (2.85) AA23 AA 32 i24 HL 1; mZ e We may again realize the Clifford algebraic elements 2 AA AA i HL; M MiM12,and M MiM12, (2.86) 31 13mZ24 e 2 and Let us attempt to write Clifford basic elements in M MMiMMiMMMMMM22 22 ASi . 112212 33 Consider the following elements (2.87)
24 12 24 12 and mZ e mZ e K11AK;;2 A2 2222 22 M MMiMMiMMMMMM112212 33 (2.78) 12 mZ24 e with KA33 2 MM iM M iMM M; 31 2 1 23 (2.88) We will obtain that M 31MiMMiMM 2 1 23 M
LK11 LK 2 2 LK 33 0; Since we have found that mZ24 e 1 mZ24 e K 222KK 222LLL2 MMMM2222 (2.89) 1232 123 1234 82 E and finally it results that under the condition E < 0, we write that
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mZ24 e 1 tence, related to the cognitive act, is represented by ab- jj1 (2.90) 82 E 4 stract entities of the Clifford algebra, and it contains only two possibilities: existence or non-existence. A pure di- or chotomic cognitive variable structured ab initio in the 24 inner architecture of our reality. There is ab initio in mZ e 2 14jj 1 n (2.91) quantum reality a variable, we could call it “the factor of 22 E knowledge and existence” that travels with more tradi- with nj21. tional physical variables that identify matter per se and In conclusion, we have that that we are accustomed to use in the traditional approach mZ24 e to reality that we formulate in classical physics. There En22;1,2, 3 (2.92) are stages of our reality in which we no more may sepa- 2n rate matter per se from the cognition and the principle of that is just the usual formula of the energy levels for the existence that we have to attribute to it. hydrogen atom as it is obtained in the standard case of There is still a question that remains to be explained in the usual quantum mechanics. such novel scheme of quantum reality that we delineate. It is instructive to observe that the (2.92) arises from Where is that quantum mechanics prospects so inno- the (2.89) that we have obtained by using the (2.82), the vative peculiarities that of course are totally missing in (2.83), and, in particular the (2.88). Again idempotent traditional classical physics? elements are contained in such basic formulation since, Let us take a step back. J. von Neumann  showed looking at the new basic Clifford scheme given in the that projection operators , satisfying as it is well (2.69) we have expressions as known that 10 , and quantum density matrices can be interpreted as logical statements. 1 MiMM122a 1 M 3nd Let us consider a quantum system S and its quantum 2 (2.93) observable K . k is a state vector for the quantum 1 state in which the observable K is equal to k . The MiMM122 1 M 3 2 density matrix k with k kk represents the logical statement k . It where says “ K k ”. All statements corresponding to mutually 11 commutative observables, constitute a classical logic of M and M (2.94) 2233 propositions where each statement or proposition is rep- resented by its matrix. are still idempotent elements according to the (2.69). This is of course the basic argument that was devel- oped from Y. F. Orlov just in 1993 . The conclusion 3. CONCLUSIONS is what we have previously evidenced by using Clifford The conclusion seems thus unquestionable. algebra. It is that the main quantum phenomena as quan- We have derived quantization as general approach to tization, indeterminism, quantum interference can be con- quantum systems. After we have discussed the general nected at the basic foundations of the theory with a case of the classical quantum harmonic oscillator. Soon purely logic basis, and thus with cognition and by it also after we have also discussed the case of the angular mo- with an intrinsic principle of existence. The only peculiar mentum. Subsequently we have given a rapid look at the nature is that in this elaboration, the statements are rep- initial quantization procedure as it was formulated ini- resented by projectors, that is to say, as algebraic coun- tially by Heisenberg, Born, Pauli, Jordan. Still, using the terpart, as idempotent elements that of course are iso- Lorentz-Runge Lentz vector that of course was used also morphic to Hermitean matrices. by Pauli, we have performed the analysis of hydrogen Generally speaking, let K be an observable with a atom energy levels. According to standard formulation of set of possible numerical values (quantum numbers, ei- quantum mechanics, we have covered a rather large genvalues ), kk12,, , and let the connected physical system be in state k . The logical statement is spectrum of interest in this discipline. Always we have i ki found the same result. Idempotent elements are involved. : “The system is in state k ”, (2.95) Since, as previously said, idempotent elements are rep- ki i resentative of logical statements and thus of cognition that means that and semantics, we conclude that in the basic foundation =K=k“”, (2.96) of our quantized basic reality ab initio there are elements ki i of existence defined, not in terms of some hazy meta- It describes the real situation in this case and therefore physical concept of existence, but in the sense that exis- it is true.
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As it is well known, generalizing we arrive to write the and move toward a wall with two slits. Let us admit that most general relation of quantum mechanics we install a device that runs as detection screen. It is posed behind the wall and in this manner we may record Kk (2.97) iki i whether or not the electron hits at a point x along the wall. Tr 1; 1 (2.98) kkii Let us examine different experimental cases. Close the i first slit, the slit 1. The probability px with which In the (2.97) K is an operator-observable, connected the electron hits different positions x is given by a directly to observable features of matter. are in- ki shaped distribution with the maximum at x 12d that stead logic statements, thus connected to cognition. The is the position on the screen directly from slit 2. (2.97) clearly explains that such two basic features, mat- Now we open the slit 2 and close the slit 1. than ter from one hand and cognition from the other hand, are px has a shaped distribution with maximum at the indissolubly connected from its starting in the theory. point x 12d .We call px2 the probability the Matter cannot be conceived per se but in relation to the particle hits pint x when slit 1 is closed. It went through cognition that it is possible to have about it. Logic state- the slit 2. Similarly we call px1 the probability the ments, i.e. cognitive elements are quantum observ- ki particle hits pint x when slit 2 is closed. ables themselves, nonlocal by nature, variables them- Now we open both the slits. The probability distribu- selves in the dynamics of our reality and commuting with tion px becomes with a maximum centred at x 0 the corresponding quantum observables. The truths of and it has the well known superimposed interference logical statements about numerical values of quantum fringes that we well know. Call this probability distribu- observables are quantum observables themselves and are tion for two open slits with px1,2. This is the prob- represented in quantum mechanics by density matrices of ability the particle reaches x given it can travel through pure states. In this manner a new framework of quantum slit 1 or slit 2. reality arises in which ab initio information, cognition It is also known that we expect some relation among and principle of existence are structured in it. Matter px 1,2 , px 1 , and px2 . does no go on by only in its dynamics but it is constantly In fact, if we use the classical theory of probability we coupled to an actual principle of existence and to cogni- have that tion. px 1,2 px 1, 1,2 p 1 1,2 We have thus two new principles that in our view . (2.99) point delineate new possibilities linking matter to cogni- px2, 1,2 p 2 1,2 tive primitive processes. The first principle is that logic, cognition, semantic As correctly outlined from Bordley where is it the er- acts are intrinsically structured in the basic scheme of our ror that we perform at this stage of the usual discussions? reality as it relates quantum mechanics. The error is that we assume the following relations to The second important principle is that in this scheme hold: cognition, here intended as logic statement, does not re- px 1, 1,2 px 1 and px 2, 1,2 px 2 main an abstract entity as we are accustomed to admit (2.100) about cognitive entities, but becomes a quantum observ- able itself as explained previously. This is the crucial error that we commit. We are thus in presence of a new approach that has The (2.100) are in evident violation of the whole definite implications also for cognitive sciences. Here the model that we have delineated in the present paper. starting point is a new physical model in which cognition, We cannot admit that also if intended as primitive cognitive entity, is contained px 1, 1,2 px 1 (2.101) ab initio as basic founding principle in the dynamics of reality. In fact in our model we have spoken about a and we cannot admit that “factor of knowledge” that in quantum reality goes on px 2, 1,2 px 2 (2.102) travelling with the dynamics of the matter. Have we probing evidences in psychology that could and the basic reason is that the above mentioned equa- support such view point? tions, on the basis of the arguments previously outlined, Let us start with some simple example, considering in contain a basic difference. This difference is the “knowl- particular some important papers that years ago were edge factor” (thus the logic statement and thus the primi- discussed by R. F. Bordley . tive cognition act) that characterizes There is a basic and well known experiment in quan- px 1, 1,2 respect to px 1 and px2, 1,2 tum mechanics. Electrons are produced from a source respect to px 2 . Relating available information, that
Copyright © 2013 SciRes. OPEN ACCESS 168 E. Conte / World Journal of Neuroscience 3 (2013) 157-170 is knowledge and thus cognition features, the two rela- states that a compound experiment in which first is re- tions in the (2.101) and in the (2.102) cannot be admitted solved the uncertainty of the decision maker about an at some stages of our reality. intermediate outcome , p0 , and then, contingent The basic reason is that we cannot ignore the cognitive on the intermediate outcome , is resolved the uncer- feature that, as a quantum variables, is structured ab ini- tainty of the decision maker on , pL , is reduci- tio in our reality so that the two experimental conditions ble to a simple experiment in which directly it is resolved responding respectively to px1, 1,2 and to px 1 , the uncertainty of the decision maker about . and, respectively, to px2, 1,2 and px2, are to- The central question is that a vast number of literature tally different.  evidences that the way we actually choose among
This evidence concludes in some manner our exposi- experiments, does not minimize uL . We know that tion. It remains only a feature to be discussed. many theories have attempted to overcome such basic Let us see the problem. It is as following. Speaking difficulties as Kahneman-Tversky , Hogarth-Einborn now at a general level involving directly human cogni- , Chew approach , Fishburn model . tion and decisions, have we some experimental evidence Segal has evidenced that the basic violation is con- that at such cognitive level we have an human behaviour tained in the (2.104) . that confirms such our model? The answer to such ques- If we denote the information the subject as I prior to tion is affirmative. receive the experiment, we have ppI00 . Since We intend to recall here the words of R. F. Bordley the decision maker becomes aware of the experiment, the that in our opinion wrote an excellent paper  in 1997 starting background information I changes, arriving to taking the focus of the question. the new condition LI, . In this manner First of all we have to observe about a possible anal- pL becomes pLI , and pL be- ogy. He says that just as physicists usually consider comes pLL ,,I . physical systems undergoing trajectories for which the In conclusion, “the factor knowledge” becomes fun- action is an extremum, also scholars in the psychology or damental and unavoidable in cognition of human beings social sciences retain that human beings make those just as it was previously outlined by us in (2.99), in choices that lead to consequences having the highest (2.100), in (2.101 ) and (2.102). possible value or action. The action associated with an To be clear. In human cognition we cannot have experiment that has 50% chance of giving apples and pppL 0 L (2.105) 50% of giving pears, is equal to the average of the action associated with apples and the action associated with but it is necessarily pears. However, experiments in cognitive psychology have evidenced that subjects appear inconsistent with ppL 0 pL (2.106) this approach in the sense that they appear to perform decisions that cannot be modelled with any action func- exactly as we find in the present formulation of our the- tion. Generally speaking, if in a psychological gamble, ory that the (2.99) no more holds if we claim to insert in the action associated with the pay off is u , the- it the (2.100). ory states that the subject choose the pay off for Of course in the past years we submitted the (2.106) to which u is the smallest. Theory makes predictions a number of experimental verification and confirmation also in the case in which one cannot be guaranteed of at human perceptive-cognitive level [28-50]. Always we getting a given pay off. Here we introduce the probability found confirmation. We do not discuss in detail such
pL of getting pay off given the occurrence of experiments here for brevity but we suggest the reader to the event . L states for the offered experiment. If the examine the results that are reported by us in the quoted probability of state is called p0 , the assigned references. Reassuming, we may say that we investigated action is at perceptive-cognitive level by using ambiguous figures. Still we examined the case of semantic conflict by using uupL L (2.103) n the well known Stroop effect. Still we considered the case of so called cognitive anomalies by using conjunc- where p is usually defined as L tion fallacy. We also examined experimental situations at pppL 0 L (2.104) cognitive level to demonstrate Bell’s inequality violation in mental states. All such results give experimental and Here is the mandatory point that relates the thesis of clinical evidence supporting the theory and also indicate our paper. a possible way for future applications in neuropsychol- We have here the following situation. L represents ogy. They have the advantage to be now based on a di- the decision maker’s state of knowledge. The (2.104) rect and robust theoretical formulation. Finally, we have
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