On the Possibility That We Think in a Quantum Probabilistic Manner

8/8/2019 On the Possibility That We Think in a Quantum Probabilistic Manner

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NeuroQuantology|December2010|Vol8|Issue4|SupplementIssue1|PageS347

ConteE.,Onthepossibilitythatwethinkinaquantumprobabilisticmanner

ISSN13035150 www.neuroquantology.com

S3

OriginalArticle

OnthePossibilityThatWeThinkina

QuantumProbabilistic

Manner

ElioConteAbstract

Mydiscussion isarticulatedunder theneurologicalaswellas thepsychological

profile. Iinsistinparticularontheviewthatmentaleventsarise inanalogywith

quantumprobabilityfields. Ireviewsomeresultsobtainedonquantumcognition

discussing indetail those thatweobtainedonquantum interference inmental

states during perceptioncognition in ambiguous figures. Frequently, I usethe

approachto

quantum

mechanics

by

Clifford

algebra.

Iinsist

in

particular

on

two

recent results. The first is thejustification that I obtain of the von Neumann

postulate on quantum measurement and the second relates my Clifford

demonstration on the logical origins of quantummechanics and thus on the

arising feature thatquantummechanics relatesconceptualentities. Thewhole

discussion aims me to support the conclusion that we think in a quantum

probabilisticmanner.

Key Words: quantum cognition, Clifford algebra, quantum probability fields,

logicaloriginsofquantummechanics

NeuroQuantology2010;4:S347

1. Introductory Remar ks 1With the advent of functional brain imagingtechnologies, neuroscience and neuro-psychology have reached satisfactory levelsof understanding and knowledge. It is ofgreat relevance that has been identified brainareas that are involved in a wide variety of brain functions including learning andmemory. On the other hand, the genetic and biochemical approaches offer a constantcontribution in this direction producing step

by step new important advances under theprofile of the investigation, research, andclinic application.

Correspondingauthor:ElioConte

Address: DepartmentofPharmacologyandHumanPhysiology

Tires,CenterforInnovativeTechnologiesforSignalDetectionand

Processing,DepartmentofNeurologicalandPsychiatricSciences,

UniversityofBari,Bari,Italy.SchoolofAdvancedInternational

StudiesonTheoreticalandNonLinearMethodologiesinPhysics,

BariItaly.

Phone:+390805242040

email:[email protected]:June4,2010;finalrevisionreceivedSept

20,2010;acceptedOct15,2010.

These are very valuable studiesproviding knowledge on the functional roleof different brain areas. Howeverneuroscience finds it hard to identify thecrucial link existing between empiricalstudies that are currently described inpsychological terms and the data that ariseinstead as described in neuro-physiological,genetic, and biochemical applications. Thesestudies continue every day with amethodological approach that is well clear. We have a kind of prevailing tendency in which we are inclined to assume that themeasurable properties of the brain throughfunctional imaging technology and biochemical discoveries, should be finallysufficient to achieve an appropriateexplanation of the psychologicallyphenomenology that occurs duringneuropsychological experiments.

I am a physicist that has some mentalreservation on this approach. I am convinced

that intrinsically mental and experientialfunctions such as "feeling" and "knowing"


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and the other basic psychological functionsand attitudes cannot be described exclusivelyin terms of knowledge achieved by thepreviously mentioned approach. Theyrequire in addition an adequate physics inorder to be actually explained. I am

profoundly persuaded about such basicstatement, and I will explain here thereasons of my firm belief. Let me start withsome considerations.

We have some well established ideasabout some foundations on the manner in which biological systems work. Biologicalmatter, including the brain, is ubiquitouslyarranged by protein molecules. According toMonod, allosteric enzymes are logicelements. They interact in information-

processing circuits whose very fundamentalstructure is essentially probabilistic. Intextbooks neurons are described quiteclassically with their tree-like dendrites andspines functioning essentially asdeterministic adding devices. They weightsums of synaptic inputs. However,neurophysiological research has oftenevidenced that such approach could be alsoprofoundly approximated. A single neuroncould be very distant from being a simpleadditive device. According to Crick and to

Koch (Crick & Koch, 1998) it could be a kindof highly complex information-processingstructure. Studies by artificial neuralnetworks are revealing that in a generalframework we could be in presence of a verylarge and highly complex stochastic-computational arrangement about which weknow very little in detail.

Eccles and Beck in 1992 (Beck andEccles, 1992; 2003; Eccles, 1990; Beck, 1996;Margenau, 1950; 1953; Wolf, 1989) obtained by direct calculations that the synapses in

the cortex may respond in a probabilisticmanner to neural excitation; a probabilitythat, given the small dimensions of synapses,should be governed by quantum uncertainty.These authors produced direct estimationsthat still today result very convincing. Thefirst detailed quantum model of quantumconjunction synapse was given by thephysicist, Evan Walker (Walker, 1977). In1970 he proposed a synaptic tunneling modelin which electrons can "quantum tunnel" between adjacent neurons, thereby creatinga virtual neural network overlapping the real

one. It is this virtual nervous system that for Walker produces consciousness and that itcan direct the behavior of the real nervoussystem. In short the real nervous systemoperates by means of synaptic messages while the virtual one operates by means of

quantum tunneling. I think that we arrive toa similar conclusion adopting the view ofEccles and of Margenau. It is the abstractfield of probabilities that in quantummechanics determines events. In his 1992article, Eccles offered plausible argumentsfor mental events causing neural events viathe mechanism of wave function collapse.Conventional operations of the synapsesdepend on the operation of ultimate synapticunits called buttons. Eccles states that, thesesynaptic buttons, when excited by an all-or-nothing nerve impulse, deliver the totalcontent of a single synaptic vesicle, notregularly, but probabilistically. In Eccleswords;

Excitation of synaptic buttons deliveringthe total content of a single synapticvesicle represents the first intrinsically probabilistic event in the brain. Ecclesstudied in detail the problem, evidencingthat a refined physiological analysis of thesynapse shows that the effective structureof each button is a paracrystalline

presynaptic vesicular grid with about 50vesicles. The existence of such a crystallinestructure is suggestive of quantumphysical laws in operation.

Eccles focused attention on these para-crystalline grids as the targets for non-material events. He discussed in detail howthe probability field of quantum mechanics, which carries neither mass nor energy, cannevertheless be envisioned as exertingeffective action at the microlevels ofquantum events. In the event of a sudden

change in the probability field brought on bythe observation of a complementaryobservable, there would be a change in theprobability of emission of one or more of the vesicles. The action of altering theprobability field without changing the energyof the physical system involved can be found by the equation governing the Heisenbergprinciple of uncertainty.

To be clear: for cortical nerveterminals, the observed fraction of action

potential pulses that result in exocytosis isconsiderably less than 100%. This can also


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be modeled classically, but the largeHeisenberg uncertainty in the locations ofthe triggering calcium ions, entails that theclassical uncertainties will carry over tosimilar quantum uncertainties. At this stageof elaboration some different authors

suggested that the sudden change in theprobability field resulting from anobservation could be the mechanism bywhich mental events trigger neural events.

Eccles concluded that calculations based on the Heisenberg uncertaintyprinciple show that the probabilisticemission of a vesicle from the paracrystallinepresynaptic grid could conceivably bemodified by mental intention in the samemanner that mental intention modifies a

quantum wave function.For experimental evidence showing

how mental events influence neural events,Eccles pointed to the papers by Roland etal., (Roland et al., 1980) who recorded, usingradioactive xenon, the regional blood flow(rBF) over a cerebral hemisphere while thesubject was making a complex pattern offinger-thumb movements. They discoveredthat any regional increase in rBF is a reliableindicator of an increased neuronal activity inthat area. Another evidence, using the same

technique of monitoring rBF, showed thatsilent thinking has an action on the cerebralcortex. For example, merely placing onesattention on a finger that was about to betouched, showed that there was an increasein rBF over the postcentral gyrus of thecerebral cortex as well as the mid-prefontalarea. A lot of studies conducted in recentyears by fMRI substantially indicate that thisis the way.

Eccles concluded that non-material

mental events in the brain are at individualmicrosites, the presynaptic vesicular grids ofthe buttons. Each button operates in aprobabilistic manner in the release of asingle vesicle in response to a presynapticimpulse. It is this probability field that Eccles believed to be influenced by mental actionthat is governed in the same way that aquantum probability field undergoes suddenchange when as a result of observation thequantum wave function collapses. By this way we arrive to the following basic

conclusion: mental events cause neuralevents analogously to the manner in which

probability fields of quantum mechanics arecausatively responsible for physical events. As well as Wolf outlines (Beck and Eccles,1992; 2003; Eccles, 1990; Beck, 1996;Margenau, 1950; 1953; Wolf, 1989) Forcompleteness we report here in Fig1 also the

basic scheme as it was used, published,represented and discussed in detail byEccles in all his papers on this matter

Figure 1. (A) Threedimensional construct by J.

SzentAgothaishowingcorticalneuronsofvarioustypes. (B)

Detailed structure of a spine (sp) synapse on a dendrite

(den); st: Axon terminating in synaptic bouton or

presynaptic terminal (pre); sv: synaptic vesicles; c:

presynapticvesiculargrid;d:synapticcleft;e:postsynaptic

membrane; a: spine apparatus; b: spine stalk; m:

mitochondrion.Figure1hasbeentakenProcNatlAcadSciUSA.1992;89(23):1135711361.

All these are important andfundamental elaborations. I have used termsas probability, link of quantum filed ofprobability and mental events, irreducibleindeterminism. When a physicist starts tospeak about structures that admit a so largenumber of abstract entities, he always feelsto be panic-stricken. I am speaking aboutscience or idealizations!


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To be clear: all the scientific theoriesintroduce mathematical models, and all they"approximate" or "idealize" in some mannerour reality. When in such text I use terms asprobability or quantum field of probabilitiesor "equivalence of probability with space of

mental events", or still irreducibleindeterminism, it is here that I see the risk ofidealizations also if, generally speaking, theadmissibility of idealizations in theorizing isand remains of main interest in science. Tocomment the previous conclusions by Walker, Eccles, Beck and Margenau, Icertainly appreciate their highly fascinatingcontent but on the other hand I must be carethat some idealizations certainly avoid therisk to result so extreme as to be consideredphysically inadmissible. As a rule, we needunquestionable verifications to accept anythesis in science. Therefore let us go on stepby step.

First. In order to take seriously inconsideration the possibility that synaptictransmission is realized by quantumtunneling and thus by the foundations andthe rules of quantum mechanics, we have toperform direct calculations. In detail weneed to calculate the MEEP, (miniature endplate potentials) Frequency of vesicle

release as it is obtained by quantummechanics and to compare such obtainedtheoretical results with the existingexperimental data. Only such comparisonmay indicate if we have an agreement or not between experimental and theoretical dataand, in case, such positive result mayorientate in some manner our thesis thatquantum mechanics has a fundamental role.

In quantum tunneling modelformulated by Walker (Walker, 1977) wehave that an electron transfer is made

between two macromolecules, proteins lyingin the presynaptic dark projections of Grayand the postsynaptic density at the cleft. It ispostulated that the charge transferred acrossthe synapse results in raising the protein inthe presynaptic dark projections of Gray athigher energy levels. As consequence,conformational changes in these moleculesforming the vesicle gates in the cleftmembrane alter their size determining themacrogates both to open and to eject a vesicle that is to say its contents realizingsynaptic conjunction. If we have an electron

bound in the molecule that is separated bythe synaptic cleft from a second similarmolecule, its energy Eand the frequency by which the electron attempts to escapefor realizing quantum tunneling are wellknown experimentally. The frequency

by

which the electron makes quantumtunneling from the first to the secondmolecule realizing vesicle release results tobe

= P (1)

being Pthe probability of tunneling that will be function 1( , )oP V V with 0V height of the

potential barrier. For purpose of calculations we assume here it having the value of0.118 eV as it arises from the experimental

data and 1V being instead the presynapticdepolarization. The value, L , of the synaptic

cleft is considered to be of 180 0A as it isobtained from experimental data.

Figure2.InordinatewehavethevaluesofMeppfrequency

(frequencyofvesiclereleaseinsec1)andinabscissathose

ofdepolarizationpotential.Wehave the theoreticalcurve

asgivenbythe(1)withthe(3)andtheexperimentalvalues,

representedbydatapoints,astheywereobtainedbyLiley

in1956.LileystudiedtheappearanceofMEPPs in isolated

rat phrenic nervediaphram preparations. More recently

experimental results were obtained also by H. von

Gersdorffand

coauthors

(private

communication)

and

they

resultverysimilartothosehadbyLileytimeago.


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The performed calculations give thefollowing expression for the potentialbarrier:

= +1 0 0( ) ( )

V VV x x V

L 0 x L (2)

and the value of probability quantumtunneling results after calculations to be

=

3/23/20 1

1

1 0

( )( , ) exp( (1 (1 ) ))o

V E VP V V

V V E,

=

4 2

3

m L (3)

In (1) all the data are known and thus we may compare it with the experimentaldata of vesicle release for different values ofthe depolarization potential 1V .The results

are given in Figure 2.It may be observed that the

agreement between theoretical andexperimental data is excellent,( =2 0.992204R ). I think that a moresatisfying result cannot be asked in acomparison between experimental andtheoretical data. So we may reach the firstconclusion of the present text. I call itEvidence n.1.

Evidence n.1It seems that we may conclude on the role ofquantum mechanics in synapticconjunction. Obviously it is only evidence.The agreement is excellent but is it sufficientsuch result to conclude the argument ofsynaptic conjunction? Certainly, this is not. Ipose to myself the following questions and Ithink that the same neuroscientist willexpect a precise indication about at least twoother basic problems.

a) We have given here preciseindications about the possiblefundamental role of quantummechanics in conjunction synapse between two adjacent neurons, but what is the counterpart of suchconclusion for distant neurons? It isnot the case that I insist here tooutline the importance of such sofundamental question.

b) As basic rough scheme of functioningof a neuron we have the following

scheme.

A neuron takes n-inputs ji . A weight

function is defined, and still a threshold and an output function f , and the complex

mechanism may be summarized as it follows

=

>

=1

1

1

n

j jj

if w i

otherwisef

In (2) we have given the expression ofthe acting potential barrier between the twoadjacent neurons when the action potentialarrives, but may we reconcile suchexpression with the time dependentmechanism that we have just evidenced bythe output function ?f

I attempt to answer to the firstquestion. Science is based on the evidence of

experimental results. I follow Walker in thisargument. There is an experiment that wasperformed by Babich, AL. Jacobson, Bubash,and A. Jacobson in 1965. It was published onScience (Babich et al., 1965). The title of thepaper evidences immediately its content. Itstates: Transfer of a response to nave rats byinjection of ribonucleic acid extracted fromtrained rats. Rats were trained in a Skinnerbox to approach the food cupwhen a distinctclick was sounded. Ribonucleic acid wasextracted from the brains of these rats and

injected into untrained rats. The untrainedrats then manifested a significant tendency(ascompared with controls) to approach thefood cup when the click, unaccompanied byfood, was presented. The conclusion seemsto be: RNA serves to transfer information. Itis well known that it does not affect thesynaptic structures, but it is distributedthroughout the brain. So, the explanationseems evident. RNA molecules develop therole of propagator molecules. Note that theylive within few hundred angstroms of one

another. Electron could use such propagatormolecules as stepping ones, by subsequenttunneling into such propagator moleculesthey could develop the role to enable theelectron to connect two distant synapticmolecules. Their lying within a few hundredangstroms, assures us that the quantumprobability tunneling is not so drasticallydecreased by their presence. We have thepotential barrier with the adjacent ones asdue to such intermediate, propagatormolecules and in this manner information

may be transferred over large distances inthe brain (Fig. 2a).We have here long-range


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quantum mechanical effects in the brainbecause we have to consider such successionof short range quantum effects in the timedependent formalism of quantum mechanicsand the arising result is that we have a finalwave function spreading out until it invades

the entire space that it is allowed to it.

Figure 2a.Quantum tunneling at large distances by RNA

propagatormolecules

These considerations, here arrangedin accord to Walker, seem to give a possibleanswer to the first posed question. HoweverI have some basic comments. The firstquestion is that, as well as I know, theexperiment by the RNA never was repeated with success. But this may be a datum

responding only to my insufficientdocumentation on this matter. The secondcomment is that under a probabilistic profileit does not seem so unreasonable to me thepossibility of a step by step continuedquantum tunneling. Let me do a digressionon this matter.

We know that neural networks areretained the most promising models incognitive neuroscience. In the classicalapproach the neural nets are retained as fullydeterministic systems. The informationprocessing of the neural nets bears theclassic physics with basic determinism andlocality. On the other hand there is analternative approach in which the dynamicsof the neural nets is modeled quantummechanically and in this case it is assumedthat a full quantum mechanical treatment isrequired. I am attempted to consider suchtwo approaches as extreme idealizations.There is instead an alternative approachdeveloped by Dugic and Rakovic (2000) in

which it is proposed that the neural networksare classical physical systems but they are

proposed at the same time to investigatequantum mechanical corrections to be addedto the deterministic dynamics usingquantum-mechanical tunneling. On thegeneral plane this procedure recalls in somemanner a new program of investigating what

is usually called the border region betweenclassical and quantum. These authors havestudied in particular quantum corrections ofan associative (attractor) neural nets. Theyadopt a very interesting strategy since theystart from classical physics models of theassociative neural nets and simultaneouslythey investigate quantum tunneling posingthe question of minimizing the quantumfluctuations that are due to the tunneling.

They consider the physical states of

the associative neural nets represented in theconfiguration (q-V) space. Here, each pointof the horizontal axis represents a uniqueconfiguration, denoted by the vector q. Inthe vertical axis they represent the values ofthe potential energy of each configuration.The vector q having componentsq1,q2,.,qn determines the state of thenetwork as a whole while each qi describesthe state of each constituent neuron or,equivalently, of each synapse in the netGenerally speaking, the (q-V) region has

several potential wells that represent thepatterns and, exhibiting the neural networkquantum mechanical features, we may havetunneling between the patterns during acertain time interval . In brief we have as inphysics a particle oscillating with a givenfrequency around the bottom of the well themoving is driven by external stimuli andresults in the complex process of patternformation and of pattern recognition.

The tunneling between the patterns isa tunneling between bound states; we have asimilar effect to the case of macroscopicquantum tunneling in superconductingdevices. For the formalization the authorsuse Pauli master equations. It may beassumed that only transitions betweenneighbor wells are effective and thusconsidering equations of the following form

+ = + 1 1( 2 )m

m m m

dpW p p p

dt

where the transitions =mnW W are assumed

to be constant and the ratio /tunneling

is

considered, being the time in which the


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energy surface (q-V) remains unchangedand tunneling that one of tunneling effect to

occur. The mp are the tunneling probabilities.

This completes our reasoning onpossible repeated quantum tunneling. Let us

go now back to our previous questions. Thereis still another comment. In (1) with (3) wecalculated the probability of electrontunneling. Actually, the frequency of vesiclerelease meppf will depend on the number m

giving the product of the actual number ofelectrons per molecule in the gate that musttunnel to determine conformational changeand the number of molecule in themacrogate. Indicating by b the number oftransmitting molecules in the postsynaptic

elements for gate, and calling N the numberof gates in the synapse, the meppf frequency of

vesicle release is

=

mepp

Nbf

m.

According to the calculationsperformed by Walker, the presynapticmembrane must have an electric charge

= 1Q CV sufficient to close the gates between

firings of the synapse. Thus it must beQ mNebeing e the electron charge. From

existing experimental data we deduce that 646Q e as Walker calculates and 9m . In

conclusion we have a number of electronsinvolved. Also discharging the hypothesis ofpropagator molecules, it remains as moreefficient the possibility of swapping quantumentanglement and of quantum entanglementas induced by noise. Let me give a very roughmanner to explain quantum entanglement.Two particles are entangled when their statesare correlated (generally exactly opposite) but both uncertain. To take a macroscopicexample but only as general indication ofquantum mechanics, imagine flipping a coinin a way where the flip was truly random,and covering it up before there was anyobservation of which side it landed on. In aquantum sense, the coin is literally bothheads and tails until somebody looks - its wavefunction is a superposition of bothstates. Once somebody observes its state, the wavefunction collapses to one state or theother. Examples of quantum entanglementswapping may be given

Entanglement SwappingIn the last few years many results have beenobtained on entanglement swapping. Wemay have entanglement swapping betweentwo entangled pairs of particles (DesBrandes et al., 2006; Bouda et al., 2001;

2005). The theory shows that if ameasurement is made simultaneously onelements (B) and (D) of the entangled pairs(A) (B) and (C) (D), the entanglement onpairs (A) (B) and (C) (D) collapses, but theelements (A) and (C) become entangledalthough they have never been in contact before. Other example. Entanglementswapping between two entangled gammaand two electrons. Let us admit twoentangled gamma (0) and (1), interactingsimultaneously with two electrons (2) and(3), as example in a crystal. We have theentanglement of the electrons (2) and (3)and the entanglement collapse between (0)and (1). These entangled electrons are thencaptured in the crystal traps and may stay assuch in the traps for months or years atambient temperature.

Still we have (Wu et al., 2004) adifferent paradigm for entanglementgeneration: it is possible to entangle twoparticles that never interact directly by

means of repeated measurements of a thirdsubsystem that interacts with both. Inaddition to its conceptual interest, it isevident that this scheme offers practicaladvantages for long-range entanglementgeneration.

Finally, it has been shown thatentanglement between two qubits can begenerated if the two qubits interact with acommon heat bath in thermal equilibrium,but do not interact directly with each other.In most situations the entanglement is

created for a very short time after theinteraction with the heat bath is switched on,but depending on system, coupling, and heat bath, the entanglement may persist forarbitrarily long times. This is an excellentmechanism that gives new light on thecreation of entanglement (Braun, 2002).

Let us consider that brain contain basic noise. As example thermal excitationsgive origin to conformational change inmolecules and thus cause spontaneously

quantal release. The probability to find a sitein an higher energy state is


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= /E kT p e

and thus the probability that the previously( m ) sites will open at a given instant of timeis

= /m mE kT p e

Scientists regularly discard up to 90percent of the signals from monitoring ofbrain waves, one of the oldest techniques forobserving changes in brain activity. Theydiscard this data as noise because it producesa seemingly irregular patterns. The noisyactivity of the brain at rest and in thebackground when we perform tasks, actuallyrepresents the majority of what the brain isreally doing. The case is when we measurethe cost of running the brain and find that

this background activity accounts for most ofit. There is pioneering research as major stepforward in helping us in understanding howthis background activity is organized.

Still We have Glial Cells in the B rainWhen we consider the cells in the brain theyare often too quick to overlook the mostabundant type of brain cell. Scaffolding cells,otherwise known as glial cells, outnumberother types of neurons in the brain.Oligodendrocytes alone outnumber neurons by an estimated ratio of 3:1. There are fourdifferent types of glial cells oligodendrocytes, astrocytes, microglia, andSchwann Cells each with its own specificattributes, functions, and characteristics.

1. OligodendrocytesOligodendrocytes are what many peopleconsider the classic glial cell. This is the cellthat forms a myelin sheath around axons.The myelin formed by these sheaths (orlaminae) allows an action potential to

propagate faster than if the axon wereunmyelinated because of the time it takes forsodium channels to open. It takes longer forthe channels to open than it does for theaction potential to propagate, so amyelinated axon will transmit a signal atabout 1.7 times the speed of an unmyelinatedaxon.

2.Astrocytes Atrocities have many functions, some of which are unknown. Astrocytes have long

processes, at the ends of which are footpods(or feet). These form tight junctions with one

another and tend to surround blood vessels,forming the blood-brain barrier. Astrocytesalso surround synaptic clefts, scavengingglutamate or other neurotransmitters thatspill out of the synapse. Some peoplehypothesize that astrocytes even form their

own network, feeding off the synapses ofneurons like pyramidal cells.

3.MicrogliaMicroglia clear away dead or dying cells.These small glial cells can become activatedand perform the function of a rudimentaryimmune system in the brain. Microglia areactivated by trauma and help destroyinfectious cells.

4.Schwann CellsSchwann cells form myelin sheaths in theperipheral nervous system (more commonlyknown as the spinal cord). Unlikeoligodendrocytes, each of which canmyelinate more than one axons (in somecases up to 30 different axons), eachSchwann cell only myelinates one axons.This completes our discussion on Evidencen.1 .

T h e s e co n d . We have still the morecomplex problem. It is that we relate thespace of mental states with the field of

probability as previously discussed. Is it anextreme idealization totally excluded fromthe possibility to be admitted in science? Ianswer as it is in my knowledge. We havegiven in the (2) the expression of thepotential barrier when the action potential isarriving to the presynapse. It is obvious thatit is a rough model that of course furnishesan excellent agreement with theexperimental data. According to the classicalscheme of neuron that we have previouslyillustrated, we are actually in presence of

several inputs that are elaborated andcombined by the neuron mechanismaccording to some different weight factors.Thus, in conclusion, the potential given in(2) is a rough, a mean approximation. Wemay refine it considering that the arriving ofthe n-inputs induces in the potential given in(2) some fluctuations that result timedependent. In conclusion, instead ofconsidering the (2) as final expression of thepotential barrier between the two synapsemolecules, we must introduce instead a time-


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dependent potential tunneling containingrandom fluctuations.

In order to account for such feature we may as example write the (2) in thefollowing time-dependent form

=( , ) ( ) ( )V x t k t V x where we assume now that the barrierheight, including ( )k t , is now time-

dependent. We may also consider other moredetailed models in order to account for suchrandom fluctuations in time. On this basis we may refine our calculations on quantumtunneling realizing synapse conjunction.There is a fundamental aspect that we havestill to outline here. Quantum tunneling isan essential feature of quantum processes

signed by irreducible indeterminism.Let us consider a tunneling process.

We have to connect it to a quantum wavefunction state that is the superposition of thetwo possible alternatives

= +1 2tunneling yes tunneling not c c

where2

1c represents the probability that the

tunneling happens and2

2c the probability

that it does not happen.

Now, in this quantum mechanicalapproach it is evident that we have a profound link between the quantum

probability2

and the weight factors

jw that are represented in the more classical

scheme of the neuron given by

=

>

= 1

1

1

n

j jj

if w i

otherwisef

A Quantum Mechanical Model of

NeuronOf course the idea to connect the weightfactors to the quantum wave function arisesalso in studies on artificial neuron networks with quantum mechanical properties(Ventura et al., 1997). The basic idea here isthat we have a direct link with themechanism of synaptic conjunctionexplained in quantum mechanical terms onthe basis of the previous arguments, and thisgives a complete quantum mechanical modelof the human neuron supported also from

the previous experimental results.

In conclusion, the concept to relatethe space of mental states with the filed ofprobability does not represent an extremeidealization.

However, by this way we reach an

impasse. Quantum mechanics has a basicfoundation. It is signed by an irreducibleindeterminism. According to McIntyre(McIntyre, 2007), a paper written time agoon the general problem of Thinkingprobabilistically, the arising problem is toexplain how in our thinking sphere, theexquisite sense of mathematical precision, ofunassailable mathematical truth, of thePlatonic beauty and precision of simplemathematical curves and other deterministicconstructs arise in our brain from the actions

of our tens of billions of interconnectedneurons subsisting in a immeasurablecoexistence of fluctuations and of irreduciblestochasticity and indetermination.

How is it that from a so high complexsystem, governed by stochasticity and byrandom fluctuations, by quantum intrinsicand irreducible indetermination, by a mentalspace conceived in some sense in anisomorphic field of probabilities and, moreprecisely, of quantum probability fields, itarises our thinking and our reasoning that

results to be precise, austere, anddeterministic?, just using McIntyreadjectiveness.

It is true. We are in accord withMcIntyre paper. Our reasoning at the firststage uses an Aristotelian logic and itsfurther developments. An unequivocal logicof thinking on which all we derive theuprightness of our thinking approach,arriving up to the abstract mathematicallevel of our formal advances.

Note that I have used twofundamental terms: determinism and Aristotelian logic. Let me comment aboutsuch two basic statements. By this way we will indicate if previously we used extremeidealizations or not.

The first is the determinism. Are wesure about determinism in our reality and inour thinking?

It is a celebrated statement that theclassical dynamics of physics describes

systems to be fully deterministic. It is still aparadigma that Nature exhibits determinism


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at the macroscopic level of descriptionpertaining to classical physics. The governingequations of classical dynamics derive fromLagrange equations, from variationalprinciples or from Newtons laws of motion.However, we must be careful in admitting

determinism so at all. Determinism does notarise from this contextual physicalframework. Our abstract mathematical levelof reasoning and our precise thinkingtendency to couple our statements withordinary experience, lead directly todeterminism. However, starting from thisframework, often we do not sufficientlyevidence or imprudently it is given silentthat, in order to satisfy the requirement ofdeterminism, we do not use only thepreviously mentioned physics. We add aposteriori a further relevant restriction. We force to coexist the governing equations ofclassical dynamics and given initialconditions with an ad hoc addedmathematical restriction. In order to obtainthat our systems actually exhibit theclaimed determinism, we impose to all suchtheoretical edifice of physics from theoutside, that the differential equationsdescribing a physical system, must satisfythe so called Lipschitz condition with thebasic consequence that all the derivativesthat we introduce at the mathematical and

physical level, must be bounded.2

We are used to admit reality going ina certain conceptual direction as it derivesfrom our macroscopic experiencedreasoning. Really, there is not another wayfor arising determinism. We admit it by anad hoc assumption. It seems to respondmore to and our kind of wishful thinkingthan other. And in fact we pay dear for suchour tendency.

Let me give only one example. It issimple but very convincing. Take a particle ina one-dimensional motion decelerated by afriction force

= = ( )dv

F v m kvdt

(4)

with m and v mass and velocity of theparticle. Invoking the ad hoc assumption

2Itissuchadhocmathematicalrestriction,thatwechoosetoadd

fromthe

outside

that

guarantees

determinism

in

classical

dynamics

since it guarantees the uniqueness of the solutions of the used

differentialequationsthataresubjectedtofixedinitialconditions.

that F must satisfy Lipschitz condition andrestriction we have the solution that iscurrently exposed in textbooks. We have

= ( / )0k m tv v e with =

= >

0( ) 0v

Fk

v ( 0)v

We unrealistically accept that theparticle has 0v for t . We admit thatthe velocity of the particle goes to zero onlyafter an infinite time. This is an abstractionin contrast with all that we actually observe with the experience. However, this is what we accepts and it results to be consolidatedas arising from classical physics about suchsystem. Really the matter does not go in thismanner. We pay dear for such assumedLipschitz condition, and we acceptconsequently that the particle approachesthe equilibrium ( = 0)v after an infinite time while instead really it approaches suchphysical condition in a finite time. Suchphysics describes an unreal situation.

Let us assume instead that the law ofmotion is

= 1F kv k v

with 1n , (5)

being an odd number

For 1 you see that the twoequations, the (4) and the (5), are verysimilar with the only exclusion of a smallneighborhood of the equilibrium point( = 0)v with the fundamental difference thatthis time at this point the Lipschitz conditionis violated. The little but substantialdifference between our usual manner tosolve this problem (the (4)) and the (5) givesenormous differences on the conceptualplane and on the plane of the Newtoniandynamical results. First of all, using the (5),correctly we have now that the time ofapproaching the equilibrium, = 0v , is finiteas it must be. This time results to be

=

1

00

1 (1 )

mvt

k .

In addition, in a finite time themotion of the particle can reach theequilibrium and switch to the singular

solution, and this switch is irreversible. Theequilibrium point = 0v of equation (5)


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S13

represents a terminal attractor which isinfinitely stable and is intersected by all theattracted transients. The uniqueness of thesolution at = 0v is violated and also themotion for < 0t t is totally forgotten. This is

to say that we have irreversibility of thedynamics as required. The conclusion of thisexposition is that if we let go out theLipschitz restriction in some cases, we obtaina correct description of our reality.

So, is determinism an actual foundation or does it represent rather anexpression of our obstinate wishfulthinking?The results to let go out Lipschitzcondition is not new here. It was introducedin literature by Michail Zak and Joseph P.Zbilut (Zak, 1997; Zak et al., 1997; Zbilut,

2004). Systems violating Lipschitzconditions do not represent an abstractmathematical formulation. We have concretecases of Lipschitz violation particularly in thesphere of the biological dynamics. With A.Federici and J.P. Zbilut (Conte et al., 2004;2004; Zbilut et al., 1996; Zimatore et al.,2000; Vena et al., 2004), I was author of anumber of papers showing violation ofLispchitz restriction in the sphere ofbiological dynamics. In a case we showed asthe biological control mechanisms,

formulated by a bicompartment model, violate Lipschitz conditions. The same violation was observed by us during humanrespiration by inspection of tracheal andpulmonary sounds. We gave a model of biological neuron implemented on the basisof a formulation violating Lipschitzcondition. Heartbeat dynamics was analyzed by violation of Lipschitz condition.Previously, other authors gave veryimportant contributions in this direction,and in particular A. Giuliani. For brevity in

(Conte et al., 2004; Zbilut et al., 1996;Zimatore et al., 2000; Vena et al., 2004) weadd only some of such fundamentalcontributing papers but really there is a lot ofsuch important papers that our referencesdoes not take into account as it should benecessary. In short, here is a lot ofascertained violations while we areaccustomed to continue to assumedeterminism as basic universal paradigma atthe foundation of our reality and of ourreasoning without exceptions.

In conclusion, we must be care inaccepting determinism as universalparadigma in our reasoning and thinking.Note that the implications of such possiblefailure are of enormous importance for theargument that we have here in

consideration. Take a differential equation asit was formulated by Zak starting with 1998.Write it as it follows

+ = 0kdx

xdt

(6)

with < 1k

Here the Lipschitz condition fails at theequilibrium point = 0x .

With = 3 / 2

The solutions are=( ) 0x t (7)

and

3/2( ) ( )x t t A (8)

where A is an arbitrary parameter. Theconsequences of losing uniqueness of thesolution are enormous for the problem that we have in consideration here. Consider aparticle located at some summit at rest withthe trivial solution holding for all times

=( ) 0r t .This mass simply remains at rest for

all times (solution (7)). However, it existsalso the other class of solutions given in the(8) so that we may write for any possibleradial direction of the mass that

3/2( ) ( )r t t A for all times >t A

and

=( ) 0r t for all times t A (9)

Note the very important thing. Weare describing the condition of a particle thatis sitting at rest at the summit whereupon atan arbitrary time A it spontaneously movesin a certain arbitrary chosen radial direction.Note the language we are using: arbitrarytime and spontaneously moves. Arbitrarytime and spontaneous movement are twoexpressions that stimulate all ourconsideration in relation to the matter thatwe have here in consideration.


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S14

Arbitrary time spontaneo usmovementWe are adopting here terms as spontaneousmovement after an arbitrary time. Are weusing extreme mathematical modelidealizations or are we remaining instead in

the scheme of classical Newtonian physics?The solutions (9) are in perfect accord withNewton's first and second laws. In theabsence of a net external force, a body is un-accelerated. In fact we obtain that for alltimes t A ,there is a non-zero net force applied, sincethe mass is at positions r> 0 and the massaccelerates in accord with the secondNewton lawF = ma . Finally, when =t A ,the direct computation of the massacceleration from the equation (9) gives us

1/23

( ) ( )4

a t t A (10 )

so that at =t A , the mass is still at the force-free summit, r=0, and the mass acceleration

(0)a is equal to zero. Again we have no force

and no acceleration, as exactly the firstNewton law requires. Acceleration existsonly for times >t A . At time =t A still

acceleration does not exist. So in short weare in perfect accord with Newtons laws. Weare not in the case of extreme idealizationbut on the other hand we are in front of two basic concepts of our reasoning: we havehere a model predicting an arbitrary time ofspontaneous movement and the possibilityof an arising spontaneous movement. Iinterpret such last expression as essentiallyindicating that something happensspontaneously or, that is to say, without acause. Something may happen in our reality

spontaneously, that is to say, without cause.It is the physical model that is able to predictthat the thing may go also in this manner.

Let me clear here. My aim is not todiscuss here the mechanical features of theexample that we have in consideration. Thisargument could be clearly debated on thepure physical level but this is not ourpurpose here. I intend to discuss about theprincipia regarding our reality. We haveexamples in which determinism is violated,causation seems to be violated, and still anexample in which an arbitrary time and

spontaneous arising movement are involved.They do not represent extreme idealizationsas we have seen in detail, and so the arisingconceptual view is perfectly accepted. Wehave also the sphere of spontaneousemergence in our reality, and we must take

care of such conclusion when discussing theinitial Eccles proposal about abstract field ofprobability causing neural events.

We may still comment about themclearing in detail our view point. The variable

= >( ) 0X t A results to be essentially arandom variable in our scheme. Thus wehave a probability ( )P X . We may assert

here that we have a variable expressing a fundamentally random length of time aftergiving origin to a spontaneous movement.

Let us introduce a scheme of Cliffordalgebra as we use it usually in our Cliffordscheme of quantum mechanics. Let usindicate the random X variable by the

3e Clifford basic element so that it is a

dichotomic variable being +3 1e for

0t A and 3 1e for > 0t A . It has a

mean value < >3e that is regulated from its

probability field:

< >= + + + 3 ( 1) ( 1) ( 1) ( 1)e p p (11 )

being +( 1)p the probability for 3e to assume

the value +1 and ( 1)p the probability for

3e to assume the value 1.

So in conclusion. Have we acausationin generation of such spontaneousmovement? I think otherwise. I retain that we have a probability field, as evidenced in(11), which is responsible of suchspontaneous movement. In this manner we

re-obtain the language of quantummechanics that we introduced previouslywhen we outlined that the probability fieldsof quantum mechanics are causativelyresponsible for physical events. It is theabstract field of probabilities that inquantum mechanics determines events. Inthis manner it becomes evident that we mayre-hook Margenau and Eccles positionwhen these authors affirm that mentalevents trigger neural events. Mental eventscausing neural events analogously to the

manner in which probability fields ofquantum mechanics are causatively


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S15

responsible for physical events, as Wolfoutlined. Margenau and Eccles conclusiondo not pertain to extreme idealizationaccording to the examples that we have givenpreviously. I think that we should considerseriously their position when considering the

spontaneous arising and link of mentalevents with neural events.

There is still another importantfeature that we have to outline here. Themodel that we have introduced from the (6)to the (11) evidences the possible existingevents marked by a fundamentally randomlength of time giving origin to spontaneousmovement. It has thus profoundimplications under the psychological profile.We have here a clear indication on the

manner in which it may arises thesubjective experiences of space and of timein humans, and we know that theinvestigation of subjective experiences ofspace and time is at the core ofconsciousness research. For the first time wehave here a mathematical formalism and aconceptual framework explaining it.

Evidence n.2Let us examine now the second theme that we introduced. The argument of the

Aristotelian logic and its furtherdevelopments used in our reasoning.

We have still some comment here. We have eminent mathematicians asMumford and Jaynes who have dedicated alot of their fundamental work to explain the very foundations of the mathematics andthey have also given important contributionsin the field of understanding the origins ofour human functions of knowing andcognition. Munford has been very clear when

explaining that the very foundations ofmathematics should be reformulated on astochastic basis (Mauford et al., 2000).Mumford quotes the importantcontributions of Jaynes in this direction.

I totally agree with these authorswhen reaching the conclusion that Aristotelian logic must be seen as part of probability theory. I may understand thatthis conclusion is shocking but it is so.

Still, the important results do notstop here. There is a further step on. All therules of probability theory can be deduced

from a single primordial idea that involves ina natural way the fact that the information isinvolved. So we have two unavoidablestarting points. The first is that Aristotelianlogic is part of probability theory. The secondis that probability theory can be deduced

from the single primordial idea that involvesthe fact that the information is involved. Thisis obtained not on the basis of assumptionsor elaborations, but instead on the basis of aprecise theorem that is shown inmathematics and it is called the CoxTheorem (Cox, 1961). This theoremdelineates one of the most importantstatements of this paper. Summarizing, itmay be expressed in the following manner:Coxs theorem states that, under certainassumptions, any measure of belief isisomorphic to a probability measure. Notethe presence of terms as "belief" and"probability measure" in the proposition ofthis theorem. The conclusion that arises isclear and evident: We think in a probabilisticmanner and all the previous reservationsthat we introduced about determinism, Aristotelian logic and its furtherdevelopments receive now their completeand significant arrangement.

By the term plausibility we must intend a

thing that seems or is apparently valid,likely, or acceptable or credible. In ourreasoning we have to start from Cox basicpostulates. Cox wanted his system to satisfythe following conditions:

1. Divisibility and comparability - Theplausibility of a statement is a realnumber and is dependent oninformation we have related to thestatement.

2. Common sense - Plausibilities

should vary sensibly with theassessment of plausibilities in themodel.

3. Consistency - Ifthe plausibilityof astatement can be derived in manyways, all the results must be equal.

We may give more convincingexpressions of such statements:

We may say that Coxs probabilitytheory is not defined by precise axioms, butby three "desiderata":

(I) representations of plausibilityare to be given by real numbers;


14/45


15/45




S17

of all evaluating information or ideasrationally, logically. Certainly informationdevelops a central role in our reasoning butalso Feeling, just like Thinking, are matterfor evaluating information. Sensingevaluates information also if takes

information by the senses. Intuiting is a kindof perception, and based on information, works outside of the usual consciousprocesses. It works like sensing but comesfrom the complex integration of largeamounts of information. So we have all suchpsychological functions and, as the readermay verify, each of such functions is marked by the term information. Our reasoningdepends from all such psychologicalfunctions just to account only for thecontributions indicated by Jung.

Arguments, in which it appears soevident the high complex nature of ourreasoning, induce some psychologists oftento consider that the physics is out from thepossibility to describe such mechanisms.Other psychologists remain instead morepossibility and in some manner claim thatphysics, and in particular quantummechanics, could contribute in explainingthe nature of our reasoning. My position isnet. I am convinced that quantum mechanics

is the first "physical theory" of reasoning andof our cognitive processes. I retain thatquantum mechanics is the link betweencognition and reasoning from one hand andthe physical reality from the other hand. I donot forget here the split that occurred between psychology and physical sciencesafter the establishment of psychology as anindependent discipline, and I am convincedthat it contributed to the delay inacknowledging a possible link between suchtwo disciplines. Still, I do not forget that N.

Bohr, in formulating his basic principle ofcomplementarity as foundation of quantummechanics, realized such principle readingJames as well as I do not ignore theconsistent work that arose from thecollaboration of Pauli with Jung. Certainly, Iagree that these are only general argumentsthat may give little contribution to the morearticulated problem to establish thatquantum mechanics is the first theory ofcognition and reasoning, but certainly, onthe other hand, the profound meaning of

contributions of founding fathers as Pauli,Bohr, James, and Jung and their high weight

at the level of their intellectual profile,certainly cannot be ignored at all. Myposition is that quantum mechanics hasorigin in the logics; it is the first "physicaltheory" of cognition and reasoning. Tosupport such thesis we should evidence that

the concepts in our mind may combinefollowing also quantum mechanics. Let usstart introducing an experimental procedure.

Let us elaborate by considering thefollowing experimental situation. We have anabstract or material entity that we call Sthatis constituted by a pair of separated subentities 1S and 2S on which we may perform

four experiments that we call respectively

1a , 2a , 3a , and 4a .Let us still consider that

each of the experiments =( 1,2,3,4)ia i has two

possible outcomes, or ++1 ( )r or 1 ( )r . Still,

continue to admit that some of theseexperiments may be performed together,respectively on 1S and 2S , and we will call

them coincidence experiments ija

=( , 1,2,3,4)i j . The experiment ija has four

possible results, which are

+ +( ) ( )i ja r a r , + + ( ) ( ), ( ) ( ), ( ) ( )i j i j i j a r a r a r a r a r a r (12 )

We may also introduce theexpectation values for such coincidenceexperiments. We call them ijE , and according

to the definition, we have that

= +( 1) (ijE p + +( ) ( )i ja r a r )+(-1)p(

+ + + + +( ) ( )) ( 1) ( ( ) ( ) ( 1) ( ( ) ( ))i j i j i j a r a r p a r a r p a r a r (13 )

Obviously, ijp means the probability

that the coincidence experiment i ja gives the

outcomes i jr r while, generally speaking,

ip will represent the probability that the

single experimenti

a will give outcome

ir = + ( , , )i j

This is a basic scheme that in severalour previous papers we have discussed in theframework of the so called Clifford algebra by which we have realized a rough or "bare bone skeleton" of quantum mechanics(Conte, 2000; 2010; Conte et al., 2006). Wewill not discuss further such elaboration hereaddressing the reader to the above quotedpapers for a close examination.

In the forthcoming steps of this paper we will describe the physical conditions in


16/45




S18

which by using the (12) and the (13), we mayderive the celebrated Bell inequality whichstates explicitly

+ + 13 14 23 24 2E E E E (14 )

Summarizing, we have an entity Sconstituted by two separated componentsentities 1S and 2S . We may perform an

experiment 1a on 1S obtaining as result +r or

r . We may still perform an experiment

2a on 1S still obtaining as result or +r or r .

We may perform an experiment 3a on 2S and

it may be also similar to 1a on 1S with

possible results +r or r , and finally an

experiment 4a on 2S that may be similar to

2

a on1

S with possible results+

r or

r . Now,

the experiment 1a may be performed in

coincidence with the experiments 3a and 4a ,

and thus we denote such coincidenceexperiments by 13a and 14a respectively, and

thus obtaining 13E and 14E . We may also

perform the coincidence experiments 23a

and 24a obtaining 23E and 24E . All such

expectation values are considered in theprevious (14).

In quantum mechanics, we choosethe set of observable propertiesof a quantumentity to which we are interested. Theseconstitute the state of the entity. We alsodefine a state space, which delineates thepossible states of the entity. A quantumentity is described using not just a statespace but also a set of measurementcontexts. The algebraic structure of the statespace is given by the vector space structureof the complex Hilbert space: states arerepresented by unit vectors, and

measurement contexts by self-adjointoperators.

The crucial notion on which we mayfix our consideration is the notion ofquantum entanglement. With reference toentity Sand to the two composingsubentities 1S and 2S , one says that a

quantum entity is entangled if it is acomposite of subentities that no more can befactorized in their components that ofcourse may can be identified only by aseparating measurement. When ameasurement is performed on the entangled

entity, its state changes probabilistically, andthis change of state is called quantumcollapse.

In pure quantum mechanics, if 1H is

the Hilbert space representing the state

space of the first subentity, and 2H theHilbert space representing the state space ofthe second subentity, the state i.e., 1 2H H .

These are standard notions in quantummechanics. Aerts has repeatedly outlined animportant feature at cognitive level (Aerts etal., 2011; 2005a; 2005b; 2000; Gabora et al.,2009; 2007; 2002; Bruza et al., 2009). Hediscusses in detail that the tensor productalways generates new states with newproperties, specifically the entangled states.Thus the space of the composite system isnot the Cartesian product, as in classicalphysics, but the tensor product, and it isused to describe the spontaneous generationof new states with new properties.

Entanglement was recognized earlyas one of the key features of quantummechanics. Entanglement still can bedescribed as the correlation between distinctsubsystems and such correlation cannot becreated by local actions on each subsystemseparately. The advantage given by quantum

entanglement relies on the crucial premisethat it cannot be reproduced by any classicaltheory (Aerts et al., 2011; 2005a; 2005b;2000; Gabora et al., 2009; 2007; 2002;Bruza et al., 2009). Despite the fact that thepossibility of quantum entanglement wasacknowledged almost as soon as quantumtheory was discovered, it is only in recent years that consideration has been given tofinding methods to quantify it, and toanalyze it in the framework of the cognitivelevel (Aerts et al., 2011; 2005a; 2005b;

2000; Gabora et al., 2009; 2007; 2002;Bruza et al., 2009). Historically, the Bellinequalities are seen as a means ofdetermining whether a two quantum statesystem is entangled.

It was known that the larger the violation of the Bell inequality is, the morethe entanglement is present in the system.This led to the perception that to somedegree the Bell inequalities were a measureof entanglement in such systems.

In this manner we arrive to the pointthat we can use the violation of Bell


17/45




S19

inequality as an experimental indication forthe presence of a quantum structure. If Bellinequalities are satisfied for a set ofprobabilities connected to outcomes of thepreviously considered experiments, thereexists a classical Kolmogorovian probability

model. In such model the probability can beexplained as due to a lack of knowledgeabout the precise state of the system underconsideration. If, on the other hand, Bellinequalities are violated, as shown in (Aertset al., 2011; 2005a; 2005b; 2000; Gabora etal., 2009; 2007; 2002; Bruza et al., 2009;Pitowsky, 1989), no such classicalKolmogorovian probability model exists.Quantum states arise as having ontologicalpotentiality and thus intrinsic irreducibleindeterminism. Probabilities in thiscase = 1,2,3)i are involved as non classicaland thus become the non classicalprobabilities, that is to say, the quantumprobabilities that characterize the sphere ofquantum ontological processes. This reasonbecause to examine the (14) is so important.

D. Aerts was the first to consider theopportunity to analyze concepts and theircombination through quantum mechanicsshowing their possibility to violate Bellinequality (Aerts et al., 2011; 2005a; 2005b;

2000; Gabora et al., 2009; 2007; 2002;Bruza et al., 2009 ). We will follow thisscheme but based once again on our basicscheme with Clifford algebra. Let us considertwo Clifford sets based on the following basicelements:

01 02 03( , , )E E E ,

=20 1iE ( = 1,2,3)i , = 0 0 0 0i j j i E E E E ;

=0 0 0i j kE E iE ( i j k )

and cyclic permutation of (1, 2, 3);

10 20 30( , , )E E E ,

=20 1iE ( = 1,2,3)i , = 0 0 0 0i j j i E E E E ;

=0 0 0i j kE E iE ( i j k )

and cyclic permutation of (1, 2, 3); (15 )

0iE and 0iE are basic abstract entities

of our mind representing concepts. Each basic element, according to the (15), mayassume the numerical values or +1 or 1 . Letus admit we ask to a subject to concentratehimself on the class of medicalspecialization. Call A the specializations :

dentist and cardiologist and both suchspecializations are identified by 03E . This is

to say that he considers +03 1E when he

selects dentist while instead he considers 03 1E when he selects cardiologist. Now

introduce the second group ofspecializations, 'A : anaesthetist andurologist. Both such specializations areidentified by 02E , and he considers +02 1E

when he selects anaesthetist while insteadhe considers 02 1E when he selects

urologist. Now we consider conceptsconnected to anatomic structures. Call B:heart and teeth. Both they are identifiedby 30E . The subject considers +30 1E when

he selects heart while instead he selects

30 1E when he selects teeth.

Finally we call 'B : bones andprostate. Both they are identified by 20E . The

subject considers +20 1E when he selects

bones while instead he considers 20 1E

when he selects prostate. In this manner we

have four conceptual groups, ' ', , ,A A B B , and

we may analyze how it is the conceptual behavior of the subject when he combinessuch conceptual groups under the common

conceptual requirement of "to cure adisease". The subject may perform such

combinations: ' ' ' ', , ,AB AB A B A B . In algebraic

terms we have = 03 30AB E E , ='

03 20AB E E ,

=' 02 30A B E E , = 02 20' 'A B E E . According to our

rules of our quantum Clifford algebraicscheme, we may also calculate theexpectation values, writing < >03 30E E ,

< >03 20E E ,

< >02 30E E ,

< >02 20E E ,

and assuming conceptual independence, itresults that each of such expectation valuesmay assume or the +1 or the value 1. Onthe other hand the Bell inequality says that

< >+< > + < >< > 02 20 02 30 03 20 03 30 2E E E E E E E E (16 )

Of course the (16) may be re-written in the

following manner:


18/45




S20

< + > < + > +

< + > < + > +

2202 3002 20

2 2

03 20 03 30

( ) 2( ) 2

2 2

( ) 2 ( ) 22

2 2

E EE E

E E E E (17)

Note that we may have

+03 1E , 30 1E , +20 1E

03 1E , +30 1E ;

02 1E , 20 1E ; (18 )

+02 1E , +30 1E .

Let us observe that with such valuesinserted in the (17), we obtain the final valueof 4 and this is to say that Bell inequality is violated. We have quantum entanglement.

Let us examine under the conceptual profile what actually happens. First let us translatethe case < >= +1AB . Return to our previouslymentioned notion of plausibility. We ask tothe subject what conceptual combinations hefinds plausible:

The dentist cures the heart

The cardiologist cures the teeth

Otherwise let us translate the case < >= 1AB . We ask to the subject what conceptualcombinations he finds plausible.

The dentist cures the teeth

The cardiologist cures the heart.

Without doubts, inspecting both such pair ofsentences the subject should find moreplausible the case that we write

< >03 30E E = < >= 1AB . (19 )

Let us now examine the case < >= +' 1AB . It is

The dentist cures the bonesThe cardiologist cures the prostate

In the case < >= ' 1AB it is

The cardiologist cures the bones

The Dentist cures the prostate.

In this case he should find more plausible

the case < >03 20E E = < >= +' 1AB .

Let us examine now the case < >= +' 1A B . It is

The anaesthetist cures the heart

The urologist cures the teeth.

In the case < >= ' 1A B , it is

The anaesthetistcures the teethThe urologist cures the heart

Also in this case the subject should find moreplausible the case

< >02 30E E = < >=+' 1AB . (20 )

Finally, let us consider the case < >= +' ' 1A B .It is

The anaesthetist cures the bones

The urologist cures the prostate

Instead, in the case < >= ' ' 1A B , it is

The anaesthetist cures the prostate

The urologist cures the bones.

Again in this final case the subjectshould find more plausible the case

< >02 20E E = < >= +' ' 1A B .

Inserting such values in the (16), onefinds that in perfect accord with the (18) theBell inequality is violated, gives value equalto +4, and thus we may conclude that in suchconceptual case the subject combines

concepts and does plausibility, usingquantum entanglement. In conclusion wehave reached the evidence n.3

Evidence n.3Combinations of concepts may followquantum entanglement We re-outline here. Such studies are due toD. Aerts. We have here reformulated thequestion using a proper example using theClifford algebra that usually we engage as

rough bare bone skeleton of quantummechanics.

We have to consider now the problemof the Self. May we introduce amathematical-physical model of the Self?

Also if we mentioned previously thatthe first psychological studies and physicswent both in psychology at the first startingof this discipline, today they are seentogether so infrequently. May be that whenphysics is considered so linked tomathematics as it is the case of the presentelaboration, both fields seem so abstract that


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S21

describing one in terms of the other is seensoon with some prejudice and considerednotable of giving some advantage.

Previously we have outlined insteadthat these two ways, particularly when

physics is supported from strongmathematics, represent two ways of thinkingdeveloped integrally in the same individual.However, Freud, as example, developed hisresults using symbols, analogies, figures inthe world of the arts and of the literature butnever he used mathematics or physics.Instead we have quoted previously eminentfigures of mathematicians that have givenfundamental contributions having had somuch to say about the workings of mind andDescartes gave in my modest opinion the

first psychological legacy to physicalknowledge by his Cogito ergo Sum that insome manner will represent the anticipatedconclusion to which I will arrive examiningsome recent results obtained by me inquantum mechanics.

In this paper I would be able toindicate some result in the direction ofmapping the structure of the self by usingquantum mechanics: To present somemodeling example aiming to match thehuman experience of selfhood. I am

encouraged by this way since previously Igave examples of spontaneous arisingabstract probability fields in accord with thegenuine nature of mental events as they werepostulated by Margenau and Eccles.

In modeling the Self I outline here hisfirst nature that is reflectivity. Self is by itsnature self-referential. It is at once subjectand object, observer and observed of itself aswell as of the others. This attitude has oftenlead psychologists to consider dualistic

theories. Self-observation is the key concepthere. Lefebvre's mathematical approach tosocial psychology is often referred to asreflexive theory - presumably due to thereflexive nature of taking into accountsubjects' self-image(s). I would obtain herethat the centuries-old philosophical andpsychological idea that man has an image ofthe self containing an image of the selfobtains a new advance in the mathematical-physical model of the subject possessingreflection that I outline here. One

assumption underlying the model is that thesubject tends to generate patterns of

behavior such that some kind of similarity isestablished between the subject himself andhis second order image of the self.

Still, quantum mechanics is based onits basic formulation of intrinsic and

irreducible indeterminism. Would psychologists speak about

indetermination or inter-determination?Many disorders of the Self consist in thespreading between the subjective andobjective features of the self. Inhallucinations, as example, dreams,imaginations the subjective and objectivefeatures separate. In the intrinsicundependability of self-observation, a doseof intrinsic and irreducible indeterminationarises for us all and we have unconscious as

relevant counterpart. At the extreme limits we have the whole spectrum ofpsychopathology. So, the importance of amodel arises.

As previously said, we use Cliffordalgebra to represent a bare bone skeleton ofquantum mechanics.

Let us give an example of ourapproach. Let us introduce three basicalgebraic abstract elements ie , = 1,2,3i ,

having the following basic features:

1) =2 1ie and 2) = =i j j i ke e e e ie with

=, , 1,2,3i j k , =ijk permutation of 1, 2, 3 and

= 2 1i . (21 )

We see that the axioms 1) and 2)introduce the two basic requirements that weinvoke for quantum mechanics: potentialityand non commutative. The first axiom in factintroduces an abstract entity, ie , but at the

same time fixes that its square is 1. This is tosay that to each ie with = 1,2,3i , under

particular conditions in such an algebra, maycorrespond or the value +1 or the value -1.For each ie we have the ontological

potentiality to link one of such possiblenumerical values. The second axiomintroduces non commutative for ( 1,2,3)ie i= .

The abstract elements ie are marked

by irreducible, intrinsic indetermination. We may calculate their mean values, < >ie

considering the probabilities for +1 or for -1values, and writing

< >= + + + 1 ( 1) ( 1) ( 1) ( 1)e p p ,


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S23

itself that is ie . We may conclude that, at

least for our present possibilities ofunderstanding what the self is and its self-perception represents, we have for the firsttime identified a basic algebraic scheme andthe corresponding mathematical operationsto represent it.

This completes our brief expositionon self introduced by a bare bone skeleton ofquantum mechanics using the Cliffordalgebra. Note the important interface that weare delineating. As repeatedly outlined weuse a bare bone skeleton of quantummechanics using the Clifford algebra. Thebasic elements are the ie . Note. They do not

represent traditional quantum observables but abstract entities. Of course it istraditionally accepted in standard quantummechanics to connect to the operators ie , the

spin components. We know that previouslyother authors (Hu et al., 2002; 2004)outlined the role of spin as self-referentialvariable and its possible role on the advent ofconsciousness. They introduced the spin-mediated consciousness theory. We willdiscuss in detail such feature in the lastsection entitled "Further Advances", but wemay anticipate here that matter and itsphysical properties must be considered to beinterfaced with cognitive feature. This could be one of the profound reasons because intheir papers in (Hu et al., 2002; 2004) it wasevidenced the so important role for the spin.In particular these authors arrived to giveexplanation of its role and function, givingalso some important neurophysiologicalcorrelates.

Now we may pass to consider a

possible theory of personality. In Jungiantheory, the Self is one of the archetypes. The

coherent whole unifies consciousness andunconscious of a person. The Self, accordingto Jung, is realized as the product ofindividuation, which in Jungian view is theprocess of integrating one's personality.

What distinguishes Jungianpsychology is the idea that there are twocenters of the personality. The ego is thecenter of consciousness, whereas the Self isthe center of the total personality, whichincludes consciousness, the unconscious,and the ego. The Self is both the whole andthe center. While the ego is a self-containedlittle circle off the center contained withinthe whole, the Self accounts for the observedindividual differences in people personality.A large number of personality theorists have

contributed to the field but there is nointegrated theory and we are left with avariety of individual approaches.

Let us return briefly to the question,before mentioned, of basic four psychologicalfunctions of Thinking, Feeling, Sensing,Intuiting and Attitudes (Introversion andExtraversion) as they were considered byJung. Certainly, if I claim here that suchpsychological function are linked and inter-related with attitudes in humans, I do a sogeneral and unspecific statement that all the

psychologists will agree with me. However,an interesting thing could be to advance suchso phenomenological approach only, andattempt to give to the basic fourpsychological functions and to the attitudes atheoretical formulation so that we mayexperiment about, and obtain precise andquantitative results.

As example, a question that I pose tomyself is the following: once again couldpsychological functions be quantum

entangled with attitudes? If such kind ofpossible correlations should be evidenced, Icertainly will obtain first of all a furtherevidence of the effective role explained fromquantum mechanics in brain and mind, and,in addition, a new quantum model of Jungtheory of personality should arise, this time based on the principles of a well definedphysical theory.

Let us indicate me the Feeling by F,the thinking by T, the sensing by S and theIntuition by I. Still I call E the extroversion

and I1 the introversion.

ei

E0i

or

Ei0


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My approach should be well knownto the reader by this time. I introduce nowsome Clifford basic elements. I call theThinking function (T) by 03E . It is a

dichotomous that as previously explained,

may admit values or +1or 1. +03 1E means that the subject is

Thinking. 03 1E means that he is Feeling

(F).

So I have that

= = 03T F E (26 )

This is the quantum scheme for rationalfunctions. Now I introduce the irrationalfunctions. I call the Sensing (S) by theClifford basic element 01E to which again are

linked the values 1 . +01 1E means thatthe subject is Sensing while instead 01 1E

means that he is Intuitive (I). So I have

= = 01S I E . (27)

These are the four psychological functions.Let us now introduce the attitudes of theSelf, calling Eextroversion and

1I introversion. Let us consider another

algebraic Clifford Element

= =1 30E I E (28 )

+30 1E means extroversion, otherwise

30 1E means introversion.

Finally, let us consider another Clifford basicelement representing that accounts for statesof explicit intermediation betweenextraversion E and Introversion 1I . I call it

M, and I pose

= 10M E (29)

with the realization that it assumes

+10 1E when the subject is in a state ofequal superposition of pure extroverted andpure introverted condition while instead wehave 10 1E otherwise.

In this manner we have realized two basic features. The first is that byintroducing the (26), we have fixed that therational functions are opposites from eachother and, considering the (27), we have

admitted that also the irrational functionsare opposites from each other.

Obviously, consider that, using the(26) and the (27), we enter by Cliffordalgebra in a quantum bare bone skeleton of

quantum mechanics. This is to say thatrational as well as irrational functions havean irreducible intrinsic indetermination intheir state. This is to say that the person hasan ontological potentiality, a quantumsuperposition of alternatives, to be TorFbecoming actuallyTor Fwhen his Self issubmitted to direct self or outside directobservation. The same thing happens forpsychological functions Sand I being theperson in a superposition of such states and becoming actuallyS or I. Obviously, the

selection of the state Tor F , and,respectively, S or I is only a matter ofprobability that is enhanced in favoring onepsychological function respect to the other independence of the inner structure of his Selfand of the context in which the self is underdirect observation.

This is the quantum scheme of theapproach. In other terms both superior andinferior functions coexist, and it is only amatter of our inner developed structure andof the instantaneous context that,

probabilistically speaking, one functionresults prevailing on the other in oursubjective dynamics.

Fixed such important conceptualpoints, let us attempt to give soon someresult confirming possibly that we areformulating a theory in a correct direction.Let us calculate the expectation value (meanvalue, of T , F , S, and I). Looking at ourbasic relation of Clifford algebraic scheme ofquantum mechanics given in the (23), we

obtain immediately that< >= cosT , < >= cosF ,

< >=S sen , < >= I sen (30 )

where is an arbitrary angle ranging from to

Let us schematize the results of the(30) in Fig.3. We obtain the behaviors of theexpectation values for such psychologicalfunctions.


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-1.0

-0.6

-0.2

0.2

0.6

1.0

-4 -3 -2 -1 0 1 2 3 4

angle

Figure3.Expectationvaluesofthefourpsychologicalfunctions.

It is easily observed that we obtain eightcorresponding sections:

1) F>I>S>T

2) I>F>T>S

3) I>T>F>S

4) T>I>S>F

5) T>S>I>F

6) S>T>F>I

7) S>F>T>I

8) F>S>I>T

There is no doubt that our approachreproduces perfectly the eight differentproportions that were identified also by Jungtheory when he characterized the superiorand secondary psychological functions of asubject. Remember that he outlined that we

just have them in different proportions. Wehave a superior function which we prefer andit is best developed in us, and a secondaryfunction of which we are aware and we use insupport of our superior function. Thepersonality of a person conflicts if the Selfhas to realize two opponent functions in thesame attitude. Here it is one of theinteresting features of such our resultsobtained by Clifford algebraic scheme ofquantum mechanics.

By using the (30) we may now

experimentally estimate the values of thepossible ratios of , , ,T F S I , and evaluate their

balancing in normal as well in pathologicalconditions. This is an interesting step on.

This last conclusion completes ourexposition on the Jung four psychologicalfunctions as elaborated by a bare boneskeleton of quantum mechanics.

Now the attitudes of the Self. The

different attitudes of the Self may beextraversed or introversed, and they havebeen quantum mechanically expressed by usin the (29) and the (30). According to ourquantum language, as previously for the fourpsychological functions, also here thesituation is now conceptually changed. Wemay have pure extroversed or pureintroversed states but we may also have theontological true potentiality, signed fromirreducible indeterminism, of potentialsuperpositions of extroverted and

introverted states. Here, this feature is alsoenhanced from the presence of the Cliffordalgebraic element

= 10M E

to which we attribute the numerical value of1 if the subject always collapses to apossible state of extraversion or intraversion while it still remains to be +1 if the subjectremains in an uncollapsed state of equalsuperposition of pure introvetrted and pureextroverted states. Also in this case we may

calculated the mean values obtaining


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< >= cosE , < >= cosI ,

< >=M sen (31)

Under the profile of the experimentalinvestigation we may repeat here all that wehave previously outlined for thepsychological functions. We may explore theattitudes of the Self and his balancing. It isrelevant to outline here further theimportance of such acquired possibilitiesunder the basic theoretical profile of theelaboration as well as in the case of analyzingpossible implications under the clinicalprofile.

Now a step one. It may be useful torepeat here the notion of quantumentanglement that we have also prospected

previously. Using very simple terms we maysay that quantum entanglement is a purequantum phenomenon in which the states oftwo or more objects or entities anywayseparated, remain linked together so thatone object can no longer be described without considering its counterpart. Aquantum interconnection maintains betweenthe two components also for any spacedistance separation between the twoseparated objects, leading to a netcorrelation between measurable observable

properties of such two or more components. We need to re outline here that such veryextraordinary property of correlation atdistance relates only quantum entanglementthat is exhibited only from systems subjectedto the principles and to the rules of quantummechanics. We need the previouslymentioned Bell inequality. If it is violated, wehave quantum entanglement.

Our attempt is to verify if or not Jungtheory has a possible quantum formulation.By this way we may admit that humansubjects in some conditions realize quantumentanglement in the sense that psychologicalfunctions are entangled with Self-attitudesand we may write Bell inequality linkingpsychological fu

Documents

On the Possibility That We Think in a Quantum Probabilistic Manner