# Quantum fermions from classical statistics

Embed Size (px)

description

### Transcript of Quantum fermions from classical statistics

Quantum Quantum fermionsfermions

from classical from classical statisticsstatistics

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

quantumquantum particle particle from from

classical classical probabilities probabilities

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) ?

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 Gott würfelt nichtGott würfelt nicht

Physics only describes Physics only describes probabilitiesprobabilities

Gott würfeltGott würfelt

but nothing beyond classical statisticsbut nothing beyond classical statisticsis needed for quantum mechanics !is needed for quantum mechanics !

fermions from classical fermions from classical statisticsstatistics

Classical probabilities Classical probabilities for two interfering for two interfering Majorana spinorsMajorana spinors

InterfereInterferencencetermsterms

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

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

(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

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.

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

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

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 ) )

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

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) )

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

Massless Majorana Massless Majorana spinors in four spinors in four

dimensionsdimensions

Time evolutionTime evolution

linear in q , non-linear in q , non-linear in plinear in p

One particle statesOne particle states

One –particle wave One –particle wave function obeysfunction obeysDirac equation Dirac equation

: arbitrary static “vacuum” : arbitrary static “vacuum” statestate

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

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.

quantumquantum particle particle from from

classical classical probabilities probabilities

νν

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

ii

Phases and complex Phases and complex structurestructure

introduce complex introduce complex spinors :spinors :

complex wave complex wave function :function :

hh

Simple conversion factor Simple conversion factor for unitsfor units

unitary time evolutionunitary time evolution

νν

fermions and bosonsfermions and bosons

νν

[A,B]=C[A,B]=C

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

function observablesfunction observables

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´)

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

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

zwitterszwitters

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 )

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

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

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

modification of evolution modification of evolution forfor

classical probability classical probability distributiondistribution

HHWW

HHWW

CC CC

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

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 !

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

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

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