Nonlocal correlations and causation: conflict or peaceful - - PowerPoint PPT Presentation

Nonlocal correlations and causation: conflict

• r peaceful coexistence?

Lars-G¨

• ran Johansson

Uppsala University

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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 Bertie always choose complementary colours. Their tie colours are strictly correlated. How could this be?

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Nonlocal correlations

Reichenbach’s principle: A true correlation between to types

• f 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

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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?

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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!

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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 randomly chosen 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’.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Nonlocal correlations in quantum theory

Since the choice is random and a definite spin is possible in

• nly 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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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

• f any conserved quantity is transmitted from the one to the
• ther.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Quantum spin-half objects

z √3/2 1/√2 1/2

Figure : A model of spin-half

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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 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 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?

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

How does an electron look like?

We usually assume that electrons are very small particles having a diameter less than 10−14 m 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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Double slit experiment

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Double slit experiment

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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 electrons propagate as (more or less) extended waves. Further tests show convincingly that the interference pattern

• ccurs even if only one electron is present at each point of

time; so the interference is not the result of interaction between electrons.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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?

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Take the wave model seriously!

The indeterminacy relations, ∆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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

A wave packet

• 10
• 7,5
• 5
• 2,5

2,5 5 7,5 10

• 0,5

0,5 1 1,5

Figure : Plot of y = exp(−x2/2)

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Fourier expansion

The fourier expansion of a function of a spatial coordinate gives us its resolution in component functions in momentum space. f(x) =

• 2

π

• F(k)cos(kx)dk

exp(− x2

2 ) =

• 2

π

∞

−∞ exp(− k 2 2 )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

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

A wave packet in momentum space

• 4,8
• 4
• 3,2
• 2,4
• 1,6
• 0,8

0,8 1,6 2,4 3,2 4 4,8

• 3
• 2
• 1

1 2 3

Figure : Plot of y = exp(− k 2

2 )cos(50k)

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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

• f an infinite number of partial waves with all possible wave

numbers, i.e., momenta (p=hk) Each such partial wave has a definite wave number; hence it is infinitely extended in space. It follows that two wave packets at different places has

• verlapping partial waves.

If the overlapping partial waves of two wave packets are coherent, the two wave packets will behave as one object!

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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 for detection at a certain place as the probability that the particle was there before the detection! See interference experiments! The intensity distribution must be interpreted as a description

• f the distribution of an extended object.

This extended object however interacts at one point (narrowly defined region) in space.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Two electron wave packets

The wave function for a singlet state consists of two coherent

• ne-electron functions.

The coherence of the two wave functions means that the system behaves as one indivisible unit during interactions with

• ther objects, such as measurement devices.

This means that the two electrons in the singlet state is one

• bject independent of its spatial extension.

But how can two particles act as one indivisible object?

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Individuals in the quantum domain

Quantum particles are not individual objects. The fact that we sometimes can count the number of particles in a system doesn’t mean that they are individual 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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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

• bject 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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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 spreads instantly to 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

• bserved the same body in the same state at all times.

They could never see parts of the body being in different states.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Ontology

The conclusion that individuation of objects, and hence the

• ntology, 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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

Causation

If A and B are event descriptions that describe events that

• ccur simultaneously at the same object, they are different

descriptions of the same event. 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.

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?

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!

Lars-G¨

• ran Johansson Uppsala University

Nonlocal correlations and causation: conflict or peaceful coexistence?