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Nonlocal correlations and causation: conflictor peaceful coexistence?

Lars-Goran JohanssonUppsala University

Lars-Goran Johansson Uppsala University Nonlocal correlations and causation: conflict or peaceful coexistence?

Nonlocal correlations

day Arthur Bertiemon red greentues green redwed yellow bluethurs green red

fri green redsat yellow bluesun red greenmon blue yellowtues red greenwed blue yellow

Table : Tie colour each day for two persons living at different places



Observations: The sequence of colours of each person’s tie israndom. No algorithm, shorter than the sequence, canproduce the sequence.But Arthur and Bertie always choose complementary colours.Their tie colours are strictly correlated.How could this be?



Reichenbach’s principle: A true correlation between to typesof events, A and B, can occur in three ways.A-events cause B-events, orB-events cause A-events, orA- and B-events have a common cause.Applied to this case:Arthur sends signals to BertieBertie sends signals to ArthurArthur and Bertie have agreed in advance of following acommon rule



Suppose that the sequence is unlimited. Since it is random it isnot possible to have an algorithm for producing the sequenceshorter than the sequence. But they cannot agree on anunlimited sequence.Suppose further that we can control that no signals gobetween Arthur and Bertie.If these conditions are fulfilled, the tie colour correlation isnon-local.Is this situation possible?



My verdict is that this is an impossible situation.A correlation in an unlimited sequence of pairs of eventswithout any mechanism would be a ’cosmic coincidence’. Idon’t believe such things exist.But quantum theory predicts such correlations!


Nonlocal correlations in quantum theory

Arthur and Bertie are two electrons in a singlet state. It can beany distance between them.A two-electron state is a singlet state if the total spin, in anychosen direction, is zero.Tie colour is electron spinBlue/Yellow is spin up/spin down in, say x-direction.Red/Green is spin up/spin down in y-direction.Arthur and Bertie travel in opposite directions in the z-direction.



Quantum theory says that a pair of electrons in a singlet statehave, when measured, perfectly anti-correlated spins in arandomly chosen direction.Quantum theory also says that electrons can have a definitespin in only one direction at each point of time.Choose randomly a direction for a spin measurement of anelectron: The result is either ’spin up’ or ’spin down’.



Since the choice is random and a definite spin is possible inonly one direction, the electrons cannot have the measuredspin before the measurement.The electrons get definite spin values as a result of theinteraction with the measurement device.The result of the measurement is random and unpredictable;The sequence cannot be produced by any algorithm shorterthan the sequence.This is beyond doubt confirmed by experiment.



One might guess that some kind of signal is sent between thetwo correlated electrons.According to relativity theory no signal can go faster than thespeed of light.But the correlation is established within such a short time thatno signal could have travelled from one to the other electron.This is well confirmed by Aspect’s and other’s experiments.But this is no violation of relativity!No signal is sent between the correlated electrons; no quantityof any conserved quantity is transmitted from the one to theother.



The dilemma:On the one hand: I don’t accept that the colours of two men’sties could be nonlocally correlated as described in the thoughtexperiment; I accept Reichenbach’s principle.On the other, I accept that quantum theory predicts nonlocalcorrelations between pairs of particles in singlet states, and Iaccept that quantum theory is extremely well tested andconfirmed on this point.So there must be a relevant difference between the thoughtexperiment with the ties and the real case of nonlocalcorrelations between particles in singlet states.The crucial difference is that of being in a singlet state.


Particles and events in the quantumdomain

My conclusion: The events ’A gets spin up’ and ’B gets spindown’ are not two different events, if A and B are two particlesin a singlet state.Question 1: What are the principles of individuation of particlesand events?Question 2:What is a singlet state?Question 3: What is spin?Question 4: How does an electron look like?I’ll begin with spin.


Quantum spin-half objects

z

√3/2

1/√2

1/2

Figure : A model of spin-half


Quantum spin-half objects

Electrons can have well defined spins in only one direction ateach moment of time; one can think of this as that the spinvector rotates around an axis.So the definite spin value cannot be had before measurement;the value must come into existence at the moment ofmeasurement.Measuring spin is forcing the particle to perform an internalrotation around a well defined direction in space defined by anexternal magnetic field; it is not a determination of apreexisting value.This cannot be if the particle is a point object; internal rotationrequires spatial extension.


Singlet states

A singlet state of two electrons (or other fermions) is a state inwhich the total spin is zero, no matter how far from each otherthey are.Since an electron (or any other fermion) always show, uponmeasurement, a spin value equal to ±1/2 (in units of h/2π),this must mean that one of them has, when measured, spin+1/2 and the other −1/2.This is true no matter which direction we chose whenperforming the measurement. According to quantummechanics, the alignment of spin values happensinstantaneously, or at least faster than any signal betweenthem. And experiments confirm the theory.Is it possible to explain?


How does an electron look like?

We usually assume that electrons are very small particleshaving a diameter less than 10−14 m or so.This is certainly wrong!Interference experiments show convincingly that electronssometimes can have macroscopic extension in space.This is so because interference pattern requires that eachelectron passes both slits.


Double slit experiment


How does an electron look like?

The interference pattern for light and for electrons arestructurally similar. The only difference is that one need muchnarrower slit distance with electrons.The explanation in both cases is interference of coherentwaves emerging from the two slits.The conclusion is that both light and electrons propagate as(more or less) extended waves.Further tests show convincingly that the interference patternoccurs even if only one electron is present at each point oftime; so the interference is not the result of interaction betweenelectrons.


Electrons propagate as waves

Conclusions:Each electron must pass through both slits.Electrons propagate as waves.There is no limit for the extension of electron waves indirections perpendicular to direction of propagation.How does it look like in the direction of propagation?


Take the wave model seriously!

The indeterminacy relations, ∆vx∆x ≥ ~, tell us that if aquantum object has a well defined velocity in one direction, ithas long extension in this direction, and vice versa.A mathematical model of the object fitting this fact is a wavepacket composed of an infinite number of waves.If the bulk of the wave packet in real space is narrow, itscomponent waves have all possible momenta.Each component wave has a well defined momentum, hence itis infinitely extended in real space.


A wave packet-10

-7,5 -5

-2,5 0

2,5 5

7,5 10

-0,5

0

0,5

1

1,5

Figure : Plot of y = exp(−x2/2)


Fourier expansion

The fourier expansion of a function of a spatial coordinate givesus its resolution in component functions in momentum space.

f(x) =√

2π

∫F(k )cos(kx)dk

exp(− x2

2 ) =√

2π

∫ ∞−∞

exp(− k2

2 )cos(kx)dk

Thus, we can view a wave packet of gaussian form as aninfinite sum of infinitely extended cosine waves with differentwave numbers


A wave packet in momentum space

-4,8 -4

-3,2

-2,4

-1,6

-0,8 0

0,8

1,6

2,4

3,2 4

4,8-3

-2

-1

0

1

2

3

Figure : Plot of y = exp(− k2

2 )cos(50k)


Fourier expansion of two wave packets

In the direction of propagation a wave packet is a wave with a’bulk’, i.e. almost the entire wave is confined within a limitedextension.This means that the wave has a fairly well defined position inthe direction of propagation.Fourier analysis of this wave packet tells us that it is composedof an infinite number of partial waves with all possible wavenumbers, i.e., momenta (p=hk)Each such partial wave has a definite wave number; hence it isinfinitely extended in space.It follows that two wave packets at different places hasoverlapping partial waves.If the overlapping partial waves of two wave packets arecoherent, the two wave packets will behave as one object!


Intensity = probability for detection

State functions in quantum theory are complex waves.The intensity distribution of such a complex wave representsthe probability for detection.It is not possible to interprete the probability for detection at acertain place as the probability that the particle was therebefore the detection! See interference experiments!The intensity distribution must be interpreted as a descriptionof the distribution of an extended object.This extended object however interacts at one point (narrowlydefined region) in space.


Quantisation of interaction

The foundational postulate of quantum mechanics is:Exchange of conserved quantities between matter andradiation field is quantized!Quantisation means that state changes are discrete jumps, nomatter how fine grained analysis we make.Quantisation also means that interaction processes areindependent of each other.If a quantum system is described by a function which cannotbe separated into terms with incoherent phases, then it willinteract as one object.


Two electron wave packets

The wave function for a singlet state consists of two coherentone-electron functions.The coherence of the two wave functions means that thesystem behaves as one indivisible unit during interactions withother objects, such as measurement devices.This means that the two electrons in the singlet state is oneobject independent of its spatial extension.But how can two particles act as one indivisible object?


Individuals in the quantum domain

Quantum particles are not individual objects.The fact that we sometimes can count the number of particlesin a system doesn’t mean that they are individual objects.Counting is possible if we can determine the total quantity ofe.g. charge and know the charge of each particle.This condition is fulfilled for electrons, protons, photons, etc.We should look upon these things rather as definite portions ofquantities.This is not just a matter of metaphysics; from Fermi-Diracstatistics (fermions) and Bose-Einstein statistics (bosons) wecan infer that quantum particles are not individual objects.If two such portions make up a singlet state they jointly interactwith the environment as one unit.


Causation and interaction with singletstates

We cannot perform a measurement on one electron in a singletstate and by this interaction send a signal to the other electron!Why?Sending a signal from one place to another requires that wecan identiy the state of one object at the first place andindependently of this identification identify the state of anotherobject at the other place.But this is not possible with a pair of particles in a singlet state,because they are not two distinct objects.


A thought experiment: the rigid body

A rigid body is a body where the propagation of a quantity ofmomentum hitting one part of the body spreads instantly to theentire body, no matter its spatial extension.Suppose two observers could observe its momentum bylooking at differnt portions of the body without in any wayaffecting it.They would always observe exactly the same state at the sametime of the body.From their observations one would conclude that they haveobserved the same body in the same state at all times.They could never see parts of the body being in differentstates.


A thought experiment: the rigid body

Conclusion 1: Independently of the dimensions of the rigidbody, we would be forced to conclude that it has no parts.Conclusion 2: A singlet state consisting of two electronsbehaves as a rigid body when it exchanges spin!A singlet system does not behave as one object in some otherinteractions,viz., those represented by operators that commutewith the spin operator.Conclusion 3: individuation of objects depends on type ofinteractions done.


Ontology

The conclusion that individuation of objects, and hence theontology, depends on which type of interactions a systemundergoes may seem ad hoc. But it is not!The same conclusion, albeit completely general, was arrived atby Quine in his ”Things and their place in theories”.His general conclusion was:1. All individuation of objects is theory dependent.2. Criteria for identity and individuation are connected to thegeneral terms.The core argument is that any theory can be translated intoanother theory that assumes entirely different objects, mappingtruths onto truths.


Causation

Causes are relations between events.An event is a state change of an object.individuation of events depends, among other things, onindividuation of objects.If A causes B, then A and B must be different events.


Causation

If A and B are event descriptions that describe events thatoccur simultaneously at the same object, they are differentdescriptions of the same event.The descriptions ’Measuring the spin of the left electron ofsinglet state S at time t’ and ’Measuring the spin of the rightelectron of singlet state S at time t’ are descriptions of eventsthat occur at the same time with the same object.Therefore there cannot be any causal relation between thereferents of these two event descriptions.Compare coin tossing: the events ’landing head up’ andlanding tail down’ are not two events, but one, if thedescriptions refer to the same toss.


SummaryThere is no reason to give up Reichenbachs principle.If A has a causal effect on B, then a portion of some conservedquantity must have been transmitted from A to B.This cannot go faster than light.Nonlocal correlations between the states of two electrons in asinglet state is no violation of causation.A singlet state pair of electrons is one object in respect of spin;therefore a spin measurement on one electron in a singlet pairis in fact at the same time a measurement of the otherelectron’s spin.A causal relation betweeen two events requires that therereally are two different events!General holism does not follow; It is false that everything isconnected to everything else!


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