fermions and non-commuting observables from classical probabilities
Post on 21-Dec-2015
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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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