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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

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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