Quantum fermions from classical statistics
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Transcript of Quantum fermions from classical statistics
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Quantum Quantum fermionsfermions
from classical from classical statisticsstatistics
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quantum mechanics quantum mechanics can be described by can be described by classical statistics !classical statistics !
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quantumquantum particle particle from from
classical classical probabilities probabilities
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Double slit experimentDouble slit experiment
Is there a classical Is there a classical probabilityprobabilitydensity w(x,t) describing density w(x,t) describing interference ?interference ?
Suitable time evolution Suitable time evolution law :law :local , causal ? local , causal ? Yes !Yes !
Bell’s inequalities ?Bell’s inequalities ?Kochen-Specker Kochen-Specker Theorem ?Theorem ?
Or hidden parameters w(x,Or hidden parameters w(x,αα,t) ? ,t) ? or w(x,p,t) ? or w(x,p,t) ?
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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
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Probabilistic realismProbabilistic realism
Physical theories and lawsPhysical theories and lawsonly describe probabilitiesonly describe probabilities
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Physics only describes Physics only describes probabilitiesprobabilities
Gott würfeltGott würfelt Gott würfelt nichtGott würfelt nicht
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Physics only describes Physics only describes probabilitiesprobabilities
Gott würfeltGott würfelt
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but nothing beyond classical statisticsbut nothing beyond classical statisticsis needed for quantum mechanics !is needed for quantum mechanics !
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fermions from classical fermions from classical statisticsstatistics
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Classical probabilities Classical probabilities for two interfering for two interfering Majorana spinorsMajorana spinors
InterfereInterferencencetermsterms
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Ising-type lattice modelIsing-type lattice model
x : x : points points on on latticelattice
n(x) =1 : particle present , n(x)=0 : n(x) =1 : particle present , n(x)=0 : particle absentparticle absent
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microphysical ensemblemicrophysical ensemble
states states ττ labeled by sequences of occupation labeled by sequences of occupation
numbers or bits nnumbers or bits ns s = 0 or 1= 0 or 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.
s=(x,s=(x,γγ)) probabilities pprobabilities pττ > 0 > 0
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(infinitely) many degrees of (infinitely) many degrees of freedomfreedom
s = ( x , s = ( x , γγ ) )x : lattice points , x : lattice points , γγ : different : different speciesspecies
number of values of s : Bnumber of values of s : Bnumber of states number of states ττ : 2^B : 2^B
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Classical wave functionClassical wave function
Classical wave function q is real , not Classical wave function q is real , not necessarily positivenecessarily positivePositivity of probabilities automatic.Positivity of probabilities automatic.
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Time evolutionTime evolution
Rotation preservesRotation preservesnormalization of normalization of probabilitiesprobabilities
Evolution equation specifies Evolution equation specifies dynamicsdynamicssimple evolution : R simple evolution : R independent of qindependent of q
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Grassmann formalismGrassmann formalism
Formulation of evolution equation in terms of Formulation of evolution equation in terms of action of Grassmann functional integralaction of Grassmann functional integral
Symmetries simple , e.g. Lorentz symmetry Symmetries simple , e.g. Lorentz symmetry for relativistic particlesfor relativistic particles
Result : evolution of classical wave function Result : evolution of classical wave function describes dynamics of Dirac particlesdescribes dynamics of Dirac particles
Dirac equation for wave function of single Dirac equation for wave function of single particle stateparticle state
Non-relativistic approximation : Schrödinger Non-relativistic approximation : Schrödinger equation for particle in potentialequation for particle in potential
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Grassmann wave Grassmann wave functionfunction
Map between classical states and Map between classical states and basis elements of Grassmann algebra basis elements of Grassmann algebra
For every nFor every nss= 0 : g = 0 : g ττ contains contains factor factor ψψss
Grassmann wave Grassmann wave function :function :
s = ( x , s = ( x , γγ ) )
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Functional integralFunctional integral
Grassmann wave function depends on t ,Grassmann wave function depends on t ,
since classical wave function q depends on tsince classical wave function q depends on t
( fixed basis elements of Grassmann ( fixed basis elements of Grassmann algebra )algebra )
Evolution equation for g(t)Evolution equation for g(t)
Functional integralFunctional integral
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Wave function from Wave function from functional integral functional integral
L(t) depends L(t) depends only only on on ψψ(t) and (t) and ψψ(t+(t+εε) )
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Evolution equationEvolution equation
Evolution equation for classical wave Evolution equation for classical wave function , and therefore also for function , and therefore also for classical probability distribution , is classical probability distribution , is specified by action Sspecified by action S
Real Grassmann algebra needed , Real Grassmann algebra needed , since classical wave function is realsince classical wave function is real
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Massless Majorana Massless Majorana spinors in four spinors in four
dimensionsdimensions
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Time evolutionTime evolution
linear in q , non-linear in q , non-linear in plinear in p
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One particle statesOne particle states
One –particle wave One –particle wave function obeysfunction obeysDirac equation Dirac equation
: arbitrary static “vacuum” : arbitrary static “vacuum” statestate
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Dirac spinor in Dirac spinor in electromagnetic fieldelectromagnetic field
one particle state obeys Dirac equation one particle state obeys Dirac equation complex Dirac equation in electromagnetic fieldcomplex Dirac equation in electromagnetic field
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Schrödinger equationSchrödinger equation
Non – relativistic approximation :Non – relativistic approximation : Time-evolution of particle in a potential Time-evolution of particle in a potential
described by standard Schrödinger described by standard Schrödinger equation.equation.
Time evolution of probabilities in classical Time evolution of probabilities in classical statistical Ising-type model generates all statistical Ising-type model generates all quantum features of particle in a potential quantum features of particle in a potential , as interference ( double slit ) or , as interference ( double slit ) or tunneling. This holds if initial distribution tunneling. This holds if initial distribution corresponds to one-particle state.corresponds to one-particle state.
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quantumquantum particle particle from from
classical classical probabilities probabilities
νν
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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
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ii
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Phases and complex Phases and complex structurestructure
introduce complex introduce complex spinors :spinors :
complex wave complex wave function :function :
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hh
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Simple conversion factor Simple conversion factor for unitsfor units
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unitary time evolutionunitary time evolution
νν
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fermions and bosonsfermions and bosons
νν
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[A,B]=C[A,B]=C
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non-commuting non-commuting observablesobservables
classical statistical systems admit many classical statistical systems admit many product structures of observablesproduct structures of observables
many different definitions of correlation many different definitions of correlation functions possible , not only classical functions possible , not only classical correlation !correlation !
type of measurement determines correct type of measurement determines correct selection of correlation function !selection of correlation function !
example 1 : euclidean lattice gauge theoriesexample 1 : euclidean lattice gauge theories example 2 : function observablesexample 2 : function observables
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function observablesfunction observables
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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
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function observablefunction observable
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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
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generalized function generalized function observableobservable
normalizationnormalization
classicalclassicalexpectationexpectationvaluevalue
several species several species αα
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positionposition
classical observable : classical observable : fixed value for every state fixed value for every state ττ
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momentummomentum
derivative observablederivative observable
classical classical observable : observable : fixed value for every fixed value for every state state ττ
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complex structurecomplex structure
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classical product of classical product of position and momentum position and momentum
observablesobservables
commutes !commutes !
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different products of different products of observablesobservables
differs from classical productdiffers from classical product
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Which product describes Which product describes correlations of correlations of
measurements ?measurements ?
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coarse graining of coarse graining of informationinformation
for subsystemsfor subsystems
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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
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quantum density matrixquantum density matrix
density matrix has the properties density matrix has the properties ofof
a quantum density matrixa quantum density matrix
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quantum operatorsquantum operators
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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
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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 !
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classical and quantum classical and quantum dispersiondispersion
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subsystem probabilitiessubsystem probabilities
in contrast :in contrast :
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squared momentumsquared momentum
quantum product between classical observables :quantum product between classical observables :maps to product of quantum operatorsmaps to product of quantum operators
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non – commutativity non – commutativity in classical statisticsin classical statistics
commutator depends on choice of product !commutator depends on choice of product !
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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
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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´)
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quantum mechanics from quantum mechanics from classical statisticsclassical statistics
probability amplitudeprobability amplitude entanglemententanglement interferenceinterference superposition of statessuperposition of states fermions and bosonsfermions and bosons unitary time evolutionunitary time evolution transition amplitudetransition amplitude non-commuting operatorsnon-commuting operators violation of Bell’s inequalitiesviolation of Bell’s inequalities
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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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zwitterszwitters
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zwitterszwitters
no different concepts for classical no different concepts for classical and quantum particles and quantum particles
continuous interpolation between continuous interpolation between quantum particles and classical quantum particles and classical particles is possible particles is possible
( not only in classical limit )( not only in classical limit )
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quantum particle quantum particle classical particleclassical particle
particle-wave particle-wave dualityduality
uncertaintyuncertainty
no trajectoriesno trajectories
tunnelingtunneling
interference for interference for double slitdouble slit
particlesparticles sharp position and sharp position and
momentummomentum classical classical
trajectoriestrajectories
maximal energy maximal energy limits motionlimits motion
only through one only through one slitslit
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quantum particle from quantum particle from classical probabilities in classical probabilities in
phase spacephase space probability distribution in phase space forprobability distribution in phase space for oonene particle particle
w(x,p)w(x,p) as for classical particle !as for classical particle ! observables different from classical observables different from classical
observablesobservables
time evolution of probability distribution time evolution of probability distribution different from the one for classical particledifferent from the one for classical particle
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wave function for classical wave function for classical particleparticle
classical probability classical probability distribution in phase spacedistribution in phase space
wave function for wave function for classical classical particleparticle
depends on depends on positionposition and momentum ! and momentum !
CC
CC
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modification of evolution modification of evolution forfor
classical probability classical probability distributiondistribution
HHWW
HHWW
CC CC
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zwitterzwitter
difference between quantum and difference between quantum and classical particles only through classical particles only through different time evolutiondifferent time evolution
continuous continuous interpolatiointerpolationn
CLCL
QMQMHHWW
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zwitter - Hamiltonianzwitter - Hamiltonian
γγ=0 : quantum – particle=0 : quantum – particle γγ==ππ/2 : classical particle/2 : classical particle
other interpolating Hamiltonians possible !other interpolating Hamiltonians possible !
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HowHow good is quantum good is quantum mechanics ?mechanics ?
small parameter small parameter γγ can be tested can be tested experimentallyexperimentally
zwitter : no conserved microscopic zwitter : no conserved microscopic energyenergy
static state : orstatic state : or
CC
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experiments for experiments for determination ordetermination orlimits on zwitter – limits on zwitter –
parameter parameter γγ ??
lifetime of nuclear spin states > 60 h ( Heil et al.) : lifetime of nuclear spin states > 60 h ( Heil et al.) : γγ < 10 < 10-14-14
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Can quantum physics be Can quantum physics be described by classical described by classical
probabilities ?probabilities ?“ “ No go “ theoremsNo go “ theorems
Bell , Clauser , Horne , Shimony , HoltBell , Clauser , Horne , Shimony , Holt
implicit assumption : use of classical correlation implicit assumption : use of classical correlation function for correlation between measurementsfunction for correlation between measurements
Kochen , SpeckerKochen , Specker
assumption : unique map from quantum assumption : unique map from quantum operators to classical observablesoperators to classical observables