Quantum processes, space-time representation and b

arXiv:quant-ph/0304137v27May2003QuantumProcesses,Space-timeRepresentationandBrainDynamicsSisirRoy1,2andMenasKafatos11CenterforEarthObservingandSpaceResearch,SchoolofComputationalSciencesGeorgeMasonUniversity,Fairfax,VA22030USA2PhysicsandAppliedMathematicsUnit,IndianStatisticalInstitute,Calcutta700108INDIAReceived;accepted1e-mail:[e-mail];[e-mail]2e-mail:[e-mail]–2–ABSTRACTTherecentcontroversyofapplicabilityofquantumformalismtobraindynamicsiscriticallyanalyzed.TheprerequisitesforapplicabilityofanytypeofquantumformalismorquantumfieldtheoryistoinvestigatewhethertheanatomicalstructureofbrainpermitsanykindofsmoothgeometricHilbertstructureorfourdimensionalMinkowskianstructure.Thepresentunderstandingofbrainfunctionclearlydeniesanykindofspace-timerepresentationintheMinkowskiansense.However,threedimensionalspaceandonedimensionaltimecanbeassignedtotheneuro-manifoldandtheconceptofprobabilisticgeometryisshowntobeanappropriateframeworktounderstandthebraindynamics.Thepossibilityofquantumstructureforbraindynamicsisalsodiscussedinthisframework.Subjectheadings:Minkowskigeometry,Hilbertspace,quantumformalism,quantumfieldtheory,functionalgeometry,probabilisticgeometryPACSNo.:87.16.Ka,87.15.-v,03.67Lx–3–1.IntroductionTheapplicabilityofquantumformalismtobraindynamicshasraisedlotofinterestamongthescientificcommunity(Tegmark2000;Haganetal.2002).Severalauthors(Stapp1990;1993;Hameroffetal.1996;Riccardietal.1967;Umezawa1993;Alfinitoet.al.2000;Jibuetal.1996)claimedthatquantumprocessesandcollapseofwavefunctioninthebrainareofimportancetohelpustounderstandinformationprocessingandhigherordercognitiveactivitiesofbrain.Evenbefore,Pribram(1991)hadproposedhisholographicmodeltohelpinunderstandingtheinformationprocessinginthebrain.However,themostfundamentalissuewhichshouldbesolvedbeforeapplyinganyquantumapproachhasnotbeenaddressedbyanyoftheaboveauthors.Oneoftheprerequisitesofapplyinganyformofquantummechanicstobraindynamics(inthenon-relativisticdomain)istoinvestigatewhethertheanatomicalstructureofbrainpermitsassigninganykindofsmoothgeometricnotionlikedistancefunction,ororthogonalityrelationbetweenthevectorsintheneuro-manifold.Forapplyingquantumfieldtheoreticmodeltomemoryfunctionorspontaneoussymmetrybreaking,oneneedstoconstructspace-timegeometryinMinkowskiansenseoverthisneuro-manifold.ThenitisnecessarytolookintotheplausibilityofindeterminacyrelationwithPlanck’sconstanthoranyotherconstant,say,abrainconstantatanylevelofbrainfunctions.Wethinkoneshouldaddresstheseissuesbeforeapplyinganykindofquantumformalismtounderstandtheinformationprocessingandhigherordercognitiveactivities(Royetal.2003).Theplanofthispaperisasfollows:AtfirstwewillanalyzetheanatomicalstructureofbrainanditsrelationtoEuclideanornon-Euclideandistanceandthenthepossibilityofassigningspace-time(fourdimensional)representation.PellioniszandLlinas(1982;1985)haveshownthatourpresentunderstandingofbrainfunctiondoesnotpermittoassignanyspace-timerepresentationintheusualfourdimensionalform.TheyconsideredtensornetworktheorywheretheyassignedametrictensorgijtotheCentralNervousSystem–4–(CNS).However,forglobalactivitiesofthebraini.e.,todefinethemetrictensoroverthewholeneuro-manifold,thisraisesalotofdifficulties.Forexample,somecorticalareasarenon-linearorrough,sothetensornetworktheorybecomesverymuchcomplicatedandalmostintractabaletosolvetherelevantmathematicalequations.Inoneofourrecentpapers(Royetal.2002),weproposedthatthestatisticaldistancefunctionmaybeconsideredovertheentireneuro-manifoldconsideringtheselectivitypropertiesofneurons(Hubel1995).Inthispaper,weshallshowthatthestatisticaldistancefunctionandthestatisticalmetrictensoraughttobeconsidered,astheyareexpectedtobeveryimportantconceptstounderstandtheabovementionedissues.ThenthepossibilityofHilbertspacestructureandquantumprocessesarediscussedwithrespecttostructureofneuro-manifold.2.FunctionalGeometryandSpace-timeRepresentationTheinternalizationofexternalgeometriesintotheCentralNervousSystem(CNS)andthereciprocalissuehavecreatedlotofinterestforthelasttwodecades.Thecentraltenetofthe(PellioniszandLlinas(1982)hypothesisisthatbrainisatensorialsystemusingvectoriallanguage.Thishypothesisisbasedontheconsiderationofcovariantsensoryandcontravariantvectorsrepresentingmotorbehaviour.Here,CNSactsasthemetrictensoranddeterminestherelationshipbetweenthecontravariantandcovariantvectors.ThecontravariantobservabletheoremhasbeendiscussedinthecontextofMinkowskiangeometryaswellasinstochasticspace-timeandquantumtheory.Itcanbestatedthatmeasurementsofdynamicalvariablesarecontravariantcomponentsoftensors.Thismeansthatwheneverameasurementcanbereducedtoadisplacementinaparticularcoordinatesystem,itcanberelatedtocontravariantcomponentsofthecoordinatesystem.Tomakeanobservationofadynamicalvariableaspositionormomentum,themeasurementisusually–5–doneintheformareadingofameterorsomethingsimilartothat.Throughaseriesofcalculationsonecanreducethedatumtoadisplacementofacoordinatesystem.Margenau(1959)analyzedthisissueandclaimedthattheabovereductioncangiverisetoameasurementif