X y p z >0 p z

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  • xypz>0pz
  • what is an atom ?quantum mechanics : isolated objectquantum field theory : excitation of complicated vacuumclassical statistics : sub-system of ensemble with infinitely many degrees of freedom

  • quantum mechanics can be described by classical statistics !

  • quantum mechanics from classical statisticsprobability amplitudeentanglementinterferencesuperposition of statesfermions and bosonsunitary time evolutiontransition amplitudenon-commuting operatorsviolation of Bells inequalities

  • essence of quantum mechanics description of appropriate subsystems of classical statistical ensembles

    1) equivalence classes of probabilistic observables 2) incomplete statistics 3) correlations between measurements based on conditional probabilities 4) unitary time evolution for isolated subsystems

  • statistical picture of the worldbasic theory is not deterministicbasic theory makes only statements about probabilities for sequences of events and establishes correlationsprobabilism is fundamental , not determinism !quantum mechanics from classical statistics :not a deterministic hidden variable theory

  • Probabilistic realismPhysical theories and lawsonly describe probabilities

  • Physics only describes probabilitiesGott wrfelt

  • Physics only describes probabilities Gott wrfeltGott wrfelt nicht

  • Physics only describes probabilities Gott wrfelt Gott wrfelt nichthumans can only deal with probabilities

  • probabilistic PhysicsThere is one realityThis can be described only by probabilities

    one droplet of water 1020 particleselectromagnetic fieldexponential increase of distance between two neighboring trajectories

  • probabilistic realismThe basis of Physics are probabilities for predictions of real events

  • laws are based on probabilitiesdeterminism as special case : probability for event = 1 or 0

    law of big numbersunique ground state

  • conditional probability sequences of events( measurements ) are described by conditional probabilitiesboth in classical statisticsand in quantum statistics

  • w(t1) not very suitable for statement , if here and nowa pointer falls down:

  • Schrdingers catconditional probability :if nucleus decaysthen cat dead with wc = 1 (reduction of wave function)

  • one - particle wave functionfrom coarse graining of microphysical classical statistical ensemblenon commutativity in classical statistics

  • microphysical ensemblestates labeled by sequences of occupation numbers or bits ns = 0 or 1 = [ ns ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,] etc.probabilities p > 0

  • function observable

  • function observablesI(x1)I(x4)I(x2)I(x3)normalized difference between occupied and empty bits in interval

  • generalized function observablenormalizationclassicalexpectationvalueseveral species

  • positionclassical observable : fixed value for every state

  • momentumderivative observableclassical observable : fixed value for every state

  • complex structure

  • classical product of position and momentum observablescommutes !

  • different products of observablesdiffers from classical product

  • Which product describes correlations of measurements ?

  • coarse graining of informationfor subsystems

  • density matrix from coarse graining position and momentum observables use only small part of the information contained in p ,

    relevant part can be described by density matrix

    subsystem described only by information which is contained in density matrix coarse graining of information

  • quantum density matrixdensity matrix has the properties of a quantum density matrix

  • quantum operators

  • quantum product of observablesthe productis compatible with the coarse grainingand can be represented by operator product

  • incomplete statisticsclassical product

    is not computable from information which is available for subsystem !cannot be used for measurements in the subsystem !

  • classical and quantum dispersion

  • subsystem probabilitiesin contrast :

  • squared momentumquantum product between classical observables :maps to product of quantum operators

  • non commutativity in classical statisticscommutator depends on choice of product !

  • measurement correlationcorrelation between measurements of positon and momentum is given by quantum productthis correlation is compatible with information contained in subsystem

  • coarse graining

    from fundamental fermions at the Planck scaleto atoms at the Bohr scalep([ns])(x , x)

  • quantum particle from classical probabilities in phase space

  • quantum particleandclassical particle

  • quantum particle classical particleparticle-wave dualityuncertainty

    no trajectories

    tunneling

    interference for double slitparticlessharp position and momentumclassical trajectories

    maximal energy limits motiononly through one slit

  • double slit experiment

  • double slit experimentone isolated particle ! no interaction between atoms passing through slitsprobability distribution

  • double slit experimentIs there a classical probability distribution in phase space ,and a suitable time evolution , which can describe interference pattern ?

  • quantum particle from classical probabilities in phase spaceprobability distribution in phase space for one particle w(x,p) as for classical particle ! observables different from classical observables

    time evolution of probability distribution different from the one for classical particle

  • quantum mechanics can be described by classical statistics !

  • quantum mechanics from classical statisticsprobability amplitudeentanglementinterferencesuperposition of statesfermions and bosonsunitary time evolutiontransition amplitudenon-commuting operatorsviolation of Bells inequalities

  • quantum physicswave functionprobabilityphase

  • Can quantum physics be described by classical probabilities ? No go theorems

    Bell , Clauser , Horne , Shimony , Holt

    implicit assumption : use of classical correlation function for correlation between measurements

    Kochen , Specker

    assumption : unique map from quantum operators to classical observables

  • zwittersno different concepts for classical and quantum particles continuous interpolation between quantum particles and classical particles is possible ( not only in classical limit )

  • classical particle without classical trajectory

  • quantum particle classical particleparticle-wave dualityuncertainty

    no trajectories

    tunneling

    interference for double slitparticle wave dualitysharp position and momentumclassical trajectories

    maximal energy limits motiononly through one slit

  • no classical trajectoriesalso for classical particles in microphysics :

    trajectories with sharp position and momentum for each moment in time are inadequate idealization !

    still possible formally as limiting case

  • quantum particle classical particle

    quantum - probability -amplitude (x)

    Schrdinger - equationclassical probability in phase space w(x,p)

    Liouville - equation for w(x,p) ( corresponds to Newton eq. for trajectories )

  • quantum formalism forclassical particle

  • probability distribution for one classical particleclassical probability distributionin phase space

  • wave function for classical particleclassical probability distribution in phase spacewave function for classical particle depends on position and momentum ! CC

  • wave function for oneclassical particle

    real depends on position and momentum square yields probabilityCCsimilarity to Hilbert space for classical mechanicsby Koopman and von Neumann in our case : real wave function permits computationof wave function from probability distribution( up to some irrelevant signs )

  • quantum laws for observablesCC

  • xypz>0pz
  • time evolution of classical wave function

  • Liouville - equationdescribes classical time evolution of classical probability distributionfor one particle in potential V(x)

  • time evolution of classical wave functionCCC

  • wave equationmodified Schrdinger - equationCC

  • wave equationCCfundamenal equation for classical particle in potential V(x)replaces Newtons equations

  • particle - wave dualitywave properties of particles :

    continuous probability distribution

  • particle wave dualityexperiment if particle at position x yes or no :discrete alternative

    probability distribution for findingparticle at position x :continuous110

  • particle wave dualityAll statistical properties of classical particles

    can be described in quantum formalism !

    no quantum particles yet !

  • modification of Liouville equation

  • evolution equationtime evolution of probability has to be specified as fundamental lawnot known a priori Newtons equations with trajectories should follow only in limiting case

  • zwitterssame formalism for quantum and classical particlesdifferent time evolution of probability distribution

    zwitters : between quantum and classical par