1 The Bohr-Einstein Debate. 2 In May 1935, Einstein published a paper in the journal Physical Review co-authored with Boris Podolsky and Nathan Rosen

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The Bohr-Einstein Debate

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In May 1935, Einstein published a paper in the journal Physical Review co-authored with Boris Podolsky and Nathan Rosen. This paper is entitled:

Can quantum-mechanical description of physical reality be considered complete?

The Third Challenge

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In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then, either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.

Abstract

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The EPR thought experiment involves measurements made of one of the two quantum particles that have somehow interacted and moved apart. We will denote these as particle A and particle B. The position and momentum of particle A are complementary observables and we cannot measure one without introducing an uncertainty in the other in accordance with Heisenberg’s uncertainty principle. Similar arguments can be made for the properties and of particle B.

Ap

Bp

Aq

Bq

BA qqQ

BA ppP AA qip

BB qip

Now consider the quantities and , where and

. The commutator is given by: ],[

PQ

))(())((,

BABABABA qqppppqqQPPQPQ

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],[

PQ

0,,,,

)()()()(

)(

BBABBAAA

BBBBBAABABBAAAAA

BBABBAAABBABBAAA

pqpqpqpq

qppqqppqqppqqppq

qpqpqpqppqpqpqpq

Here: and , since these operators

refer to different quantum particles.

ipqpq BBAA

,, 0,,

ABBA pqpq

Since operators and commute, there is no restriction on the precision with which we can measure the difference between the positions of the particles A and B and the sum of their momenta.

PQ

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Suppose we allow the two particles to interact and move a long distance apart. We perform an experiment on particle A to measure its position with certainty. We know that must be a physically real quantity and so we in principle deduce the position of particle B also with certainty. We therefore conclude that must be an element of physical reality according to the EPR definition. However, suppose instead that we choose to measure the momentum of particle A with certainty. We know that must be physically real, and so we can in principle deduce the momentum of particle B with certainty. We conclude that it too must be an element of physical reality. Thus, although we have not performed any measurement on particle B following its separation from A, we can, in principle, establish the reality of either position or its momentum from measurement we choose to perform on A which, by definition, do not disturb B.

BA qq

Bq

BA pp

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The Copenhagen interpretation denies that we can do this. We are forced to accept that if this interpretation of quantum theory is correct, the physical reality of either the position or momentum of particle B is determined by the nature of the measurement we choose to make on a completely different particle an arbitrarily long distance away. EPR argued that ‘no reasonable definition of reality could be expected to permit this.’

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Spooky action at a distance

The EPR thought experiment strikes right at the heart of the Copenhagen interpretation. If the uncertainty principle applies to an individual quantum particle, then it appears that we must invoke some kind of action at a distance if the reality of the position or momentum of particle B is to be determined by measurements we choose to perform on A.

Whether it involves a change in the physical state of the system or merely some kind of communication, the fact that this action at a distance must be exerted instantaneously on a particle an arbitrarily long distance away from our measuring device suggests that it violets the postulates of special relativity, which restricts any signal to be communicated no faster than the speed of light. EPR did not believe that such action at a distance is necessary: the position and momentum of particle B are defined all along and, as there is nothing in the wave function which tells us how these quantities are defined, quantum theory is incomplete.

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Bohr’s reply

Bohr rejects the argument that the EPR thought experiment creates serious difficulties for the Copenhagen interpretation and stresses once again the importance of taking into account the necessary interactions between the objects of study and the measuring devices.

His emphasis is once again on the important role of the measuring instrument in defining the elements of reality that we can observe. Thus, setting up an apparatus to measure the position of particle A with certainty, from which we can infer the position of particle B, excludes the possibility of measuring the momentum of A and hence inferring the momentum of B.

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In June 1935, Schrödinger wrote to congratulate Einstein on the EPR paper, He wrote:

I was very happy that in the paper just published in [Physical Review] youhave evidently caught dogmatic [quantum mechanics] by the coat-tails ... Myinterpretation is that we do not have a [quantum mechanics] that is consistentwith relativity theory, i.e, with a finite transmission speed of all influences. Wehave only the analogy of the old absolute mechanics ... The separation processis not at all encompassed by the orthodox scheme.

Einstein Separability

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Schrödinger's reference to the 'separation process' highlights the essential difficulty that the EPR argument creates for the Copenhagen interpretation. According to this interpretation, the wavefunction for the two-particle quantum state does not separate as the particles themselves separate in space-time. Instead of dissolving into two completely separate wavefunctions, one associated with each particle, the wavefunction is 'stretched' out and, when a measurement is made, collapses instantaneously despite the fact that it may be spread out over a large distance.


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EPR's definition of physical reality requires that the two panicles are considered to be isolated from each other, i.e. they are no longer described by a single wavefunction at the moment a measurement is made. The reality thus referred to is sometimes called 'local reality' and the ability of the particles to separate into two locally real independent physical entities is sometimes referred to as 'Einstein separability'. Under the circumstances of the EPR thought experiment, the Copenhagen interpretation denies that the two particles are Einstein separable and therefore denies that they can be considered to be locally real (at least,before a measurement is made on one or other of the particles, at which point they both become localized).


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You might have assumed that the collapse occurs at the moment the microscopic quantum system interacts with the macroscopic measuring device. But is this assumption justified? After all, a macroscopic measuring device is composed of microscopic entities—molecules, atoms, protons, neutrons and electrons. We could argue that the interaction takes place on a microscopic level and should, therefore, be treated using quantum mechanics.

Schrödinger’s Cat

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In other words, the problem is that the collapse is itself not contained in any of the mathematical apparatus of quantum theory: we simply cannot obtain a collapse of the state vector using Schrödinger continuous, deterministic equation of motion from which the time evolution of the interaction between a quantum system and a measuring device is deduced. It is impossible to separate the combined description of quantum system plus measuring device into its constitutents parts except by invoking an indeterministic collapse, or von Neumann’s ‘projection postulate’.

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One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.

Schrödinger’s cat: Schrödinger writes:

An illustration of both states, a dead and living cat. According to quantum theory, after an hour the cat is in a quantum superposition of coexisting alive and dead states. Yet when we look in the box we expect to only see one of the states, not a mixture of them.

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atomlive 0At t = 0

At time t decayeddeadatomlive 2

1

live

dead

atom

decayed

: State vector of the live cat

: State vector of the dead cat

: State vector of the non-decayed atom

: State vector of the decayed atom

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The argument would seem to suggest that we should express the state vector of the system-plus-cat as a linear superposition of the products of the state vectors describing a disintegrated atom and a dead cat and of the state vectors describing an intact atom and a live cat. Prior to measurement, the cat is neither alive nor dead but some peculiar combination of both states. We can perform measurement on the cat by opening the chamber and ascertaining its physical state.

The Copenhagen interpretation says that elements of an empirical reality are defined by the nature of the experimental apparatus we construct to perform measurements on a quantum system. It insists that we resist the temptation to ask what physical state a particle (or a cat) was actually in prior to measurement as such a question is quite without meaning.

Arguments such as EPR thought experiment and Schrödinger’s cat led some scientists to set about searching for an alternative theory. A hidden variable theory can be a good one.


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According to Gell-Mann the problem with the Schrödinger’s cat example is that of decoherence. A cat is a large, macroscopic system, not an element of the microscopic quantum world. As such, the cat interacts with its environment very extensively: it breathes air, it absorbs and emits heat radiation, it eats and drinks. Therefore, it is impossible for the cat to behave “both dead and live,” as an electron in a superposition of more than one state.

A Reply

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Hidden Variables

If we reject the 'spooky' action at a distance that seems to be required in the Copenhagen interpretation of quantum theory, and which is highlighted by the EPR thought experiment, then we must accept the EPR argument that the theory is somehow incomplete. In essence, this involves the rejection of the first postulate of quantum theory: the state of a quantum mechanical system is not completely described by the wavefunction.

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Hidden Variables

Those physicists who in the 1930s were uncomfortable with the Copenhagen interpretation were faced with two options. Either they could scrap quantum theory completely and start all over again or they could try to extend the theory to reintroduce strict causality and local reality. There was a general recognition that quantum theory was too good to be consigned to history's waste bin of scientific ideas. The theory did an excellent job of rationalizing the available experimental information on the physics of the microscopic world of quantum particles, and its predictions had been shown to be consistently correct. What was needed, therefore, was some means of adapting the theory to bring back those aspects of classical physics that it appeared to lack.

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Hidden Variables

Einstein had hinted at a statistical interpretation. In his opinion, the squares of the wave functions of quantum theory represented statistical probabilities obtained by averaging over a large number of real particles. The obvious analogy here is with Boltzmann's statistical mechanics, which allows the calculation of observable physical quantities (such as gas pressure and thermodynamic functions like entropy) using atomic ormolecular statistics. Although the theory deals with probabilities, these are derived from the behaviour of an ensemble of atoms or molecules which individually exist in predetermined physical states and which obey the laws of a deterministic classical mechanics.

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Hidden Variables

A completely deterministic, locally real version of quantum theory demands that the physical states of the particles are ‘set’ at the moment of their interaction, and that the particles separate as individually real entities in those physical states (separability).

Quantum theory in the form taught to undergraduate students of chemistry and physics tells us nothing about such physical states. This is either because they have no basis in reality (Copenhagen interpretation) or because the theory is incomplete (EPR argument). One way in which quantum theory can be made ‘complete’ in this sense is to introduce a new set of variables. These variables determine which physical states will be preferred as a result of a quantum process. As these variables are not revealed in laboratory experiments, they are necessarily ‘hidden’ from us.

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Hidden Variables

Hidden variable theories of one form or another are not without precedent in the history of science. Any theory which rationalizes the behaviour of a system in terms of parameters that are for some reason inaccessible to experiment is a hidden variable theory. These variableshave often later become 'unhidden' through the application of new experimental technologies. The obvious example is again Boltzmann's use of the ‘hidden’ motions of real atoms and molecules to construct a statistical theory of mechanics. Mach's opposition to Boltzmann's ideaswas based on the extreme view that introducing such hidden variables unnecessarily complicates a theory and takes science no further forward. History has shown Mach's views to have been untenable.

Documents

1 The Bohr-Einstein Debate. 2 In May 1935, Einstein published a paper in the journal Physical Review co-authored with Boris Podolsky and Nathan Rosen