# Cushing, Chapter 22 The EPR Paper and Bell’s Theorem Mikey Pilger.

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Cushing, Chapter 22 The EPR Paper and Bell’s Theorem Mikey Pilger

Albert Einstein: Man or Ghost??

EPR Albert Einstein (1879 – 1955) Boris Podolsky (1896 – 1966) Nathan Rosen (1909 – 1995)

Copenhagen Quantum Physics (CQP) A few tenets: 1. The physical reality of a wave/particle system can be completely described by a wave function Ψ 2. Uncertainty Relation – Determining a definite measurement of one property of the system (e.g. momentum) limits our knowledge of other properties (e.g. position) to mere probabilities

Einstein on the Copenhagen View “Uncomfortable” elements of CQP: No single, objective reality until we observe the system directly The observation renders merely probabilistic all information other than the element of the system directly observed To Einstein, CQP seems more like a general statistical summary of the properties of systems than a complete theory

EPR’s Plan of Attack The EPR Paper is titled “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” (1935) EPR attempt to show that CQP is, if not illogical, then at least incomplete

Describing a Complete Theory What is a complete theory? It must contain a formal element corresponding to every relevant phenomenon in the physical world Which criterion should we use to distinguish complete and incomplete theories? If an observer can use a theory to predict a physical quantity (e.g. momentum, position) without disturbing the system and without uncertainty, then he or she establishes the quantity’s physical reality

EPR’s Assumptions Locality There is no spooky action at a distance Realism The physical world really exists, and not just as a wave function, even when it is unobserved

Introduction Imagine that a particle composed of two joined photons, at rest, decays into two physically separatde photons, A and B, which fly off in opposite directions Wave function Ψ can now describe the properties of both photons as members of a larger single system For example: the angular momenta of the photons must add up to zero, because they started from rest as one system Observing the spin of photon A to be +½ instantly forces photon B to conform to a -½ spin

Hidden Variables EPR believe that rather than an instantaneous force working on particle B at a distance, there are hidden variables in the system which explain the particles’ behavior There was information latent in the particles before they separated, which later influences their behavior

Hidden Variables Imagine a space traveler who takes one glove, out of a pair of gloves, with him on an interstellar voyage. He doesn’t look at which glove he takes, and before he can check, he seals it in a box. He leaves the other (also unknown) glove with his brother. When he gets to Alpha Centauri, many light-years away, and opens the box to finally look at the glove for the first time, he suddenly knows whether he has taken the left or the right glove, and he knows that the glove he left with his brother is the opposite glove.

The EPR Paradox: Thought Experiment If an element of reality exists, it has a definite value When the two photons separate, observing the position of photon A gives a definite value to that part of wave function Ψ, and puts the system in eigenstate 1, which also determines a definite position for photon B Simultaneously, observe the momentum of photon B, and arrive at a definite value for the momenta of both photon A and B But, the Heisenberg uncertainty relationship between position and momentum will not allow this Either CQP is incomplete, or position and momentum cannot have simultaneous reality

Conclusions from EPR CQP is incomplete, because we will not sacrifice the definite reality of the properties of a system to save CQP Inexplicable particle behavior is attributable to hidden variables (Locality is safe and it was never in trouble)

Criticism of EPR Did EPR really understand the indeterminacy relationship between position and momentum?

Locality EPR strongly influenced the thinking of prominent physicists, though the “paradox” itself is wanting Classical notions of determinism and locality BOTH survive in the minds of some physicists

Bell’s Theorem “On the Einstein, Rosen, Podolsky Paradox” (1964)

Bell’s Theorem: Simplified From the work of N. David Mermin

The Particles The central signal box sends one particle each to the detector boxes This is intended to be the only transfer of information between the signal and detection boxes The particles are equivalent E.g. two +1/2 spin particles The information conveyed is identical Assume that the particles are “programmed” with “instructions” – information that does not change between leaving the signal box and arriving at the detector box

Switch Settings Box A and Box B have three switches each (1, 2, and 3). For each round of the experiment, a switch on each box is randomly selected and engaged. There are nine possible AB switch combinations. If Box A is set to switch 2, and Box B is set to switch 3, this is represented as “23.” Switch1B2B3B 1A111213 2A212223 3A313233

Color Outputs (Information conveyed by particles) There should be at least a one-third chance that the lights will flash the same color after receiving their respective signal particles CaseS1S2S3Same Color? 1RRR1.000 2RRG0.333 3RGR 4RGG 5GGG1.000 6GGR0.333 7GRG 8GRR

Will the Light Color Match? Expected Results: Switch Setting Casexxx111213212223313233 1RRR ✓✓✓✓✓✓✓✓✓ 2RRG ✓✓✖✓✓✖✖✖✓ 3RGR ✓✖✓✖✓✖✓✖✓ 4RGG ✓✖✖✖✓✓✖✓✓ 5GGG ✓✓✓✓✓✓✓✓✓ 6GGR ✓✓✖✓✓✖✖✖✓ 7GRG ✓✖✓✖✓✖✓✖✓ 8GRR ✓✖✖✖✓✓✖✓✓

Expected Results Switch Setting Casexxx121321233132 2RRG ✓✖✓✖✖✖ 3RGR ✖✓✖✖✓✖ 4RGG ✖✖✖✓✖✓ 6GGR ✓✖✓✖✖✖ 7GRG ✖✓✖✖✓✖ 8GRR ✖✖✖✓✖✓ Excluding Cases 1 and 5, which will always yield matching light flashes

The Competing Predictions CQP: 25% Classical physics: >33%

How, CQP? First, a description of the experiment as Bell described it The particles emerging from the box are equivalent spin ½ particles Detector magnets are arranged at perpendicular to the vertical, or ±120° from perpendicular to the vertical in the path of the particle The orientation of the magnet depends on one of three “switch” positions If the magnets A and B have the same switch position, then they have the same magnet orientation

How, CQP? The detector indicates the particle’s orientation with respect to the field’s orientation, and activates a green or red light accordingly Green = particle spin is along the field Red = particle spin is against the field “It is a well-known elementary result that, when the orientations of the magnets differ by angle Θ, then the probability of spin measurements on each particle yielding opposite values is cos 2 ( Θ /2). This probability is unity when Θ = 0 and ¼ when Θ = ±120°” (Mermin 408).

R.I.P. Locality The two particles seem to interact instantaneously when one or the other is observed Though the detection system is, effectively, the cause of this change in the particles, it does not interfere with the particles as they leave the box, though it does measure the spin of each before reaching the magnet No hidden variable can explain the behavior of the experimental particles without sacrificing locality

(E)PR? “I was very pleased with your detailed letter, which speaks about the little essay. For reasons of language, this was written by Podolsky after many discussions. But still it has not come out as well as I really wanted; on the contrary, the main point was, so to speak, buried by the erudition” (Howard 175). - Einstein’s letter to Schrödinger, 19 June 1935

Competing Scientific Purposes Hesinberg said: “Einstein agreed... That the mathematical formulation of quantum mechanics, developed in Göttingen and consolidated further in Cambridge and Copenhagen, correctly described the phenomena within the atom. He may also have been willing to admit, for the time being at least, that the statistical interpretation of Schrödinger’s wave function, as formulated by Born, would have to be accepted as a working hypothesis...”

Competing Scientific Purposes “... But Einstein did not want to acknowledge that quantum mechanics represented a final, and even less a complete, description of these phenomena. The conclusion that the world could be completely divided into an objective and a subjective sphere, and the hypothesis that one should be able to make precise statements about the objective side of it, formed a part of his basic philosophical attitude. But quantum mechanics could not satisfy these claims, and it does not seem likely that science will ever find its way back to Einstein’s postulates” (Cushing 320).

Einstein: Realist, or Instrumentalist? The engineer of an unparalleled upheaval in physics, now defending a realist interpretation of the world against quantum revolutionaries

Discuss Why did Einstein propose his theory of relativity, but resist CPQ? Was it a mathematical, or an emotional or moral question? What was the role of the larger scientific community in establishing the dominance of quantum theory, despite the leading mind in the world’s adamant opposition?

Works Cited Mermin, N. David. "Quantum Mysteries for Anyone." The Journal of Philosophy 78.7 (1981): 397-408. 27 Feb. 2007. Web. 20 Apr. 2013. Howard, Don. "EINSTEIN ON LOCALITY AND SEPARABILITY." Studies in History and Philosophy of Science 16.3 (1985): 171-201. Print

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