Nonlocal correlations and causation: conflict or peaceful coexistence?
Lars-G ¨oran Johansson Uppsala University
Nonlocal correlations
day Arthur Bertie
mon red green
tues green red wed yellow blue thurs green red
fri green red sat yellow blue
sun red green
mon blue yellow
tues red green
wed blue yellow
Table : Tie colour each day for two persons living at different places
Nonlocal correlations
Observations: The sequence of colours of each person’s tie is random. No algorithm, shorter than the sequence, can
produce the sequence.
But Arthur and Bertiealwayschoose complementary colours.
Their tie colours are strictly correlated.
How could this be?
Nonlocal correlations
Reichenbach’s principle: A true correlation between to types of events, A and B, can occur in three ways.
A-events cause B-events, or B-events cause A-events, or
A- and B-events have a common cause.
Applied to this case:
Arthur sends signals to Bertie Bertie sends signals to Arthur
Arthur and Bertie have agreed in advance of following a common rule
Nonlocal correlations
Suppose that the sequence is unlimited. Since it is random it is not possible to have an algorithm for producing the sequence shorter than the sequence. But they cannot agree on an unlimited sequence.
Suppose further that we can control that no signals go between Arthur and Bertie.
If these conditions are fulfilled, the tie colour correlation is non-local.
Is this situation possible?
Nonlocal correlations
My verdict is that this is an impossible situation.
A correlation in an unlimited sequence of pairs of events without any mechanism would be a ’cosmic coincidence’. I don’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 be any distance between them.
A two-electron state is a singlet state if the total spin, in any chosen direction, is zero.
Tie colour is electron spin
Blue/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.
Nonlocal correlations in quantum theory
Quantum theory says that a pair of electrons in a singlet state have,when measured, perfectly anti-correlated spins in a randomlychosen direction.
Quantum theory also says that electrons can have a definite spin in only one direction at each point of time.
Choose randomly a direction for a spin measurement of an electron: The result is either ’spin up’ or ’spin down’.
Nonlocal correlations in quantum theory
Since the choice is random and a definite spin is possible in only one direction, the electrons cannot have the measured spin before the measurement.
The electrons get definite spin values as a result of the interaction with the measurement device.
The result of the measurement is random and unpredictable;
The sequence cannot be produced by any algorithm shorter than the sequence.
This is beyond doubt confirmed by experiment.
Nonlocal correlations in quantum theory
One might guess that some kind of signal is sent between the two correlated electrons.
According to relativity theory no signal can go faster than the speed of light.
But the correlation is established within such a short time that no 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 quantity of any conserved quantity is transmitted from the one to the other.
Nonlocal correlations in quantum theory
The dilemma:
On the one hand: I don’t accept that the colours of two men’s ties could be nonlocally correlated as described in the thought experiment; I accept Reichenbach’s principle.
On the other, I accept that quantum theory predicts nonlocal correlations between pairs of particles in singlet states, and I accept that quantum theory is extremely well tested and confirmed on this point.
So there must be a relevant difference between the thought experiment with the ties and the real case of nonlocal correlations between particles in singlet states.
The crucial difference is that of being in a singlet state.
Particles and events in the quantum domain
My conclusion: The events ’A gets spin up’ and ’B gets spin down’ are not two different events, if A and B are two particles in a singlet state.
Question 1: What are the principles of individuation of particles and 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 at each moment of time; one can think of this as that the spin vector rotates around an axis.
So the definite spin value cannot be had before measurement;
the value must come into existence at the moment of measurement.
Measuring spin is forcing the particle to perform an internal rotation around a well defined direction in space defined by an external magnetic field; it is not a determination of a
preexisting value.
This cannot be if the particle is a point object; internal rotation requires spatial extension.
Singlet states
A singlet state of two electrons (or other fermions) is a state in which the total spin is zero, no matter how far from each other they are.
Since an electron (or any other fermion) always show, upon measurement, a spin value equal to±1/2 (in units ofh/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 when performing the measurement. According to quantum mechanics, the alignment of spin values happens
instantaneously, or at least faster than any signal between them. And experiments confirm the theory.
Is it possible to explain?
How does an electron look like?
We usually assume that electrons are very small particles having a diameter less than 10−14m or so.
This is certainly wrong!
Interference experiments show convincingly that electrons sometimes can have macroscopic extension in space.
This is so because interference pattern requires that each electron passes both slits.
Double slit experiment
Double slit experiment
How does an electron look like?
The interference pattern for light and for electrons are
structurally similar. The only difference is that one need much narrower slit distance with electrons.
The explanation in both cases is interference of coherent waves emerging from the two slits.
The conclusion is that both light and electronspropagateas (more or less) extended waves.
Further tests show convincingly that the interference pattern occurs even if only one electron is present at each point of time; so the interference is not the result of interaction between electrons.
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 in directions perpendicular to direction of propagation.
How does it look like in the direction of propagation?
Take the wave model seriously!
Theindeterminacyrelations,∆vx∆x≥ ~, tell us that if a quantum object has a well defined velocity in one direction, it has long extension in this direction, and vice versa.
A mathematical model of the object fitting this fact is a wave packet composed of an infinite number of waves.
If the bulk of the wave packet in real space is narrow, its component waves have all possible momenta.
Each component wave has a well defined momentum, hence it is 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 ofy=exp(−x2/2)
Fourier expansion
The fourier expansion of a function of a spatial coordinate gives us its resolution in component functions in momentum space.
f(x) = q2
π
R F(k)cos(kx)dk exp(−x22) =
q2 π
R∞
−∞exp(−k22)cos(kx)dk
Thus, we can view a wave packet of gaussian form as an infinite sum of infinitely extended cosine waves with different wave 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 ofy =exp(−k22)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 limited extension.
This means that the wave has a fairly well defined position in the direction of propagation.
Fourier analysis of this wave packet tells us that it is composed of an infinite number of partial waves with all possible wave numbers, i.e., momenta (p=hk)
Each suchpartial wavehas a definite wave number; hence it is infinitely extended in space.
It follows that two wave packets at different places has overlapping partial waves.
If the overlapping partial waves of two wave packets are coherent, 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 represents the probability for detection.
It is not possible to interprete the probability fordetectionat a certain place as the probability that the particlewasthere before the detection! See interference experiments!
The intensity distribution must be interpreted as a description of the distribution of an extended object.
This extended object however interacts at one point (narrowly defined region) in space.
Quantisation of interaction
The foundational postulate of quantum mechanics is:
Exchange of conserved quantities between matter and radiation field is quantized!
Quantisation means that state changes are discrete jumps, no matter how fine grained analysis we make.
Quantisation also means that interaction processes are independent of each other.
If a quantum system is described by a function which cannot be separated into terms with incoherent phases, then it will interact as one object.
Two electron wave packets
The wave function for a singlet state consists of two coherent one-electron functions.
The coherence of the two wave functions means that the system behaves as one indivisible unit during interactions with other objects, such as measurement devices.
This means that the two electrons in the singlet state is one object independent of its spatial extension.
But how can two particles act as one indivisible object?
Individuals in the quantum domain
Quantum particles arenotindividual objects.
The fact that we sometimes can count the number of particles in a system doesn’t mean that they areindividual objects.
Counting is possible if we can determine the total quantity of e.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 of quantities.
This is not just a matter of metaphysics; from Fermi-Dirac statistics (fermions) and Bose-Einstein statistics (bosons) we can infer that quantum particles are not individual objects.
If two such portions make up a singlet state they jointly interact with the environment as one unit.
Causation and interaction with singlet states
We cannot perform a measurement on one electron in a singlet state and by this interaction send a signal to the other electron!
Why?
Sending a signal from one place to another requires that we can identiy the state of one object at the first place and
independently of this identification identify the state of another object 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 of momentum hitting one part of the body spreadsinstantlyto the entire body, no matter its spatial extension.
Suppose two observers could observe its momentum by looking at differnt portions of the body without in any way affecting it.
They would always observe exactly the same state at the same time of the body.
From their observations one would conclude that they have observed the same body in the same state at all times.
They could never see parts of the body being in different states.
A thought experiment: the rigid body
Conclusion 1: Independently of the dimensions of the rigid body, we would be forced to conclude that it has no parts.
Conclusion 2: A singlet state consisting of two electrons behaves as a rigid body when it exchanges spin!
A singlet system does not behave as one object in some other interactions,viz., those represented by operators that commute with the spin operator.
Conclusion 3: individuation of objects depends on type of interactions done.
Ontology
The conclusion that individuation of objects, and hence the ontology, depends on which type of interactions a system undergoes may seem ad hoc. But it is not!
The same conclusion, albeit completely general, was arrived at by 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 the general terms.
The core argument is that any theory can be translated into another theory that assumes entirely different objects, mapping truths onto truths.
Causation
Causes are relations between events.
An event is a state change of an object.
individuation of events depends, among other things, on individuation 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 that occur simultaneously at the same object, they are different descriptions of thesameevent.
The descriptions ’Measuring the spin of the left electron of singlet state S at time t’ and ’Measuring the spin of the right electron of singlet state S at time t’ are descriptions of events that occur at the same time with the same object.
Therefore there cannot be any causal relation between the referents of these two event descriptions.
Compare coin tossing: the events ’landing head up’ and landing tail down’ are not two events, but one, if the descriptions refer to the same toss.
Summary
There is no reason to give up Reichenbachs principle.
If A has a causal effect on B, then a portion of some conserved quantity must have been transmitted from A to B.
This cannot go faster than light.
Nonlocal correlations between the states of two electrons in a singlet 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 pair is in fact at the same time a measurement of the other
electron’s spin.
A causal relation betweeen two events requires that there really are two different events!
General holism does not follow; It is false that everything is connected to everything else!