fermions and non-commuting observables from classical probabilities

Fermions and Fermions and non-commuting non-commuting

observables fromobservables fromclassical probabilitiesclassical probabilities

quantum mechanics quantum mechanics can be described by can be described by classical statistics !classical statistics !

statistical picture of the statistical picture of the worldworld

basic theory is not deterministicbasic theory is not deterministic basic theory makes only statements basic theory makes only statements

about probabilities for sequences of about probabilities for sequences of events and establishes correlationsevents and establishes correlations

probabilism is fundamental , not probabilism is fundamental , not determinism !determinism !

quantum mechanics from classical statistics :quantum mechanics from classical statistics :not a deterministic hidden variable theorynot a deterministic hidden variable theory

Probabilistic realismProbabilistic realism

Physical theories and lawsPhysical theories and lawsonly describe probabilitiesonly describe probabilities

Physics only describes Physics only describes probabilitiesprobabilities

Gott würfeltGott würfelt

fermions from classical fermions from classical statisticsstatistics

microphysical ensemblemicrophysical ensemble

states states ττ labeled by sequences of labeled by sequences of

occupation numbers or bits noccupation numbers or bits ns s = 0 = 0 or 1or 1

ττ = [ n = [ nss ] = ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc.[0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc.

probabilities pprobabilities pττ > 0 > 0

Grassmann functional Grassmann functional integralintegral

action :action :

partition function :partition function :

Grassmann wave Grassmann wave functionfunction

observablesobservables

representation as functional integralrepresentation as functional integral

particle numbersparticle numbers

time evolutiontime evolution

d=2 quantum field theoryd=2 quantum field theory

time evolution of time evolution of Grassmann wave functionGrassmann wave function

Lorentz invarianceLorentz invariance

what is an atom ?what is an atom ?

quantum mechanics : isolated objectquantum mechanics : isolated object quantum field theory : excitation of quantum field theory : excitation of

complicated vacuumcomplicated vacuum classical statistics : sub-system of classical statistics : sub-system of

ensemble with infinitely many ensemble with infinitely many degrees of freedomdegrees of freedom

one - particle wave one - particle wave functionfunction

from coarse graining from coarse graining of microphysical of microphysical

classical statistical classical statistical ensembleensemble

non – commutativity in classical statisticsnon – commutativity in classical statistics

microphysical ensemblemicrophysical ensemble

states states ττ labeled by sequences of labeled by sequences of

occupation numbers or bits noccupation numbers or bits ns s = 0 = 0 or 1or 1

ττ = [ n = [ nss ] = ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc.[0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc.

probabilities pprobabilities pττ > 0 > 0

function observablefunction observable

function observablefunction observable

ss

I(xI(x11)) I(xI(x44))I(xI(x22)) I(xI(x33))

normalized difference between occupied and empty normalized difference between occupied and empty bits in intervalbits in interval

generalized function generalized function observableobservable

normalizationnormalization

classicalclassicalexpectationexpectationvaluevalue

several species several species αα

positionposition

classical observable : classical observable : fixed value for every state fixed value for every state ττ

momentummomentum

derivative observablederivative observable

classical classical observable : observable : fixed value for every fixed value for every state state ττ

complex structurecomplex structure

classical product of classical product of position and momentum position and momentum

observablesobservables

commutes !commutes !

different products of different products of observablesobservables

differs from classical productdiffers from classical product

Which product describes Which product describes correlations of correlations of

measurements ?measurements ?

coarse graining of coarse graining of informationinformation

for subsystemsfor subsystems

density matrix from coarse density matrix from coarse graininggraining

• position and momentum observables position and momentum observables use only use only small part of the information contained small part of the information contained in in ppττ ,,

• relevant part can be described by relevant part can be described by density matrixdensity matrix

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

quantum density matrixquantum density matrix

density matrix has the properties density matrix has the properties ofof

a quantum density matrixa quantum density matrix

quantum operatorsquantum operators

quantum product of quantum product of observablesobservables

the productthe product

is compatible with the coarse grainingis compatible with the coarse graining

and can be represented by operator productand can be represented by operator product

incomplete statisticsincomplete statistics

classical productclassical product

is not computable from information whichis not computable from information which

is available for subsystem !is available for subsystem ! cannot be used for measurements in the cannot be used for measurements in the

subsystem !subsystem !

classical and quantum classical and quantum dispersiondispersion

subsystem probabilitiessubsystem probabilities

in contrast :in contrast :

squared momentumsquared momentum

quantum product between classical observables :quantum product between classical observables :maps to product of quantum operatorsmaps to product of quantum operators

non – commutativity non – commutativity in classical statisticsin classical statistics

commutator depends on choice of product !commutator depends on choice of product !

measurement correlationmeasurement correlation

correlation between measurements correlation between measurements of positon and momentum is given of positon and momentum is given by quantum productby quantum product

this correlation is compatible with this correlation is compatible with information contained in subsysteminformation contained in subsystem

coarse coarse graininggraining

from fundamental from fundamental fermions fermions

at the Planck scaleat the Planck scaleto atoms at the Bohr to atoms at the Bohr

scalescale

p([np([nss])])

ρρ(x , x´)(x , x´)

conclusionconclusion

quantum statistics emerges from classical quantum statistics emerges from classical statisticsstatistics

quantum state, superposition, quantum state, superposition, interference, entanglement, probability interference, entanglement, probability amplitudeamplitude

unitary time evolution of quantum unitary time evolution of quantum mechanics can be described by suitable mechanics can be described by suitable time evolution of classical probabilitiestime evolution of classical probabilities

conditional correlations for measurements conditional correlations for measurements both in quantum and classical statisticsboth in quantum and classical statistics

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