PHL 356 Philosophy of Physics Week XI EPR & the Bell results 1

The material in this lecture is by far the most interesting and important in the QM part of this course. It stems from Einstein’s rejection of the Copenhagen interpretation. Einstein was a realist and thought that QM needed to be modified by adding so-called hidden variables. The EPR thought experiment supported this view. J.S. Bell was very sympathetic to Einstein; he generalized the EPR line of thought and showed that it had empirical, testable consequences. The outcome was very counter-intuitive. Some of this has led to the current work on quantum computation. Because of the intimate mix of philosophy and physics, these issues are sometimes called “experimental metaphysics.” 2

Background (again) Realism: –A quantum system has all its properties all the time. (Eg, an electron has position AND momentum all the time; a photon has spin in every direction all the time, etc. They have their properties independent from measurement.) –Superpositions merely reflect ignorance of what objectively exists, independently from us. –Measurements do not make reality; they reveal it. Copenhagen (anti-realism): –A quantum system has a property only when it is in an eigenstate of that property; otherwise it is indeterminate. –Superpositions reflect the “fuzzy” nature or non-existence of objective reality –Measurements somehow create reality. Reality is not independent from us. 3

EPR Einstein, Podolsky, and Rosen produced a famous thought experiment in The aim was to refute Bohr & Copenhagen interpretation 4

Solvay Conference (Belgium, 1927) 5

EPR: Three Philosophical Principles Completeness: Every element of reality must have a counterpart in the theory. –Eg, if an electron simultaneously has both position and momentum, then a complete theory must represent both of them simultaneously. 6

Criterion of Reality: If, without disturbing the system, we can predict with probability one the value of a property, then there is an element of reality corresponding to that value. –If my prediction is always right and I’m not somehow interfering, then I must be discovering something objectively real, not making it. 7

Locality: A measurement that is space-like separated from a remote system does not disturb that remote system. –Otherwise the effect of a measurement would travel faster than light, which is taken to be impossible, thanks to SR. 8

Dirac’s notation (review) Dirac’s notation is often useful; many articles and books use it, so you should know what it is. State: |ψ> Eigenstates: |a 1 >, |a 2 >,… Eigenvalues: a 1, a 2,… Innerproduct: (These are called “bra” and “ket” vectors.) 9

EPR set up 10

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An experiment Suppose we are at L (ie, the left side) and we measure spin in the y-direction. Suppose the result is spin up By the projection postulate: Thus, the whole system is now: Thus, the right photon is: From this we can be sure (ie, probability one) that a measurement will get spin-down on the R photon. 12

The EPR Argument 1.By locality, the result at R was not disturbed by the measurement at L. 2.We could predict the R outcome with probability one, so by the criterion of reality the spin component in the y- direction is an element of reality; it is not (contra Bohr) indeterminate. 3.The initial description with |ψ LR > being a superposition does not account for this element of reality. Therefore, by the completeness criterion, QM is not complete. 13

Aftermath The EPR argument impressed many. Those who liked it looked for a hidden variable theory to replace or modify QM. But most physicists were not persuaded; they stayed with Bohr. Often the claim was: “It’s just metaphysics” (meaning: leave it to philosophers; it has no empirical content; it’s just a waist of time; shut up and calculate; etc.). This is how things stood from 1935 until Bell’s work in

John Stewart Bell 1928 – 1990, British Worked at CERN in high-energy physics Philosophically close to Einstein (ie, a realist) Thought Bohr obscure & fuzzy Foundations work was just a “hobby” (advised students to make it only a sideline – no job prospects. This is no longer true.) Bell results made him extremely famous 15

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The Bell Results We need to distinguish “Bell’s inequality” (also known as “Bell’s theorem,” which we will now prove) from the “Bell results,” which refers to the inequality and the subsequent experiments by Aspect and others that show that nature violates the inequality. The fact that nature does not obey the inequality does not show that Bell made a mistake. The proof is fine. It shows that (at least one of) the premisses are false (something Bell had suspected). That is the stunning outcome, since the premisses are Einstein’s and very commonsensical. How to understand all of this remains an open question. 17

Bell’s inequality Consider an EPR-type setup, with possible measurements in different directions. There are four possible measurements: (ie, we could measure for spin in the a-direction on left, b-direction on right, and so on): (a,b), (a ',b), (a,b ' ), (a ',b ' ) Let a spin-up result be +1 and a spin-down result be -1. (a=1 means that the spin in the a-direction is up; b ’ =-1 means that the spin is down in the b ' -direction, and so on.) 18

Next, we form the correlation function: If a=1 and b=1, then c(a,b) = 1 x 1 = 1 If a=1 and b=-1, then c(a,b) = 1 x -1 = -1 And so on. After N tests (ie, we send out pairs of particles N times), with a i, etc. being the i th result, we have: 19

The Argument Realist assumption (following Einstein): The particle has spin-up or spin-down in each direction, whether it is measured or not. In the math we represent this assumption by letting a i (or b i, etc) be the result (either +1 or -1) of the i th measurement made in the a or b (or a ' or b ’ ) direction. The value of +1 or -1 exists whether we make the measurement in that direction or not. (Measurements discover; they do not create.) 20

Locality Assumption (ie, assume Special Relativity): The measurement result at one side does not depend on or influence what happens at the other side. Next, define the formula F as follows: Note that (b i ± b i ' ) must equal 0 or +2 or -2, and a i and a i ' must each equal +1 or

Thus, the absolute value is, Therefore, after N pairs of particles are measured, In terms of the correlation function, we have, This is Bell’s Inequality 22

The two assumptions that lead to this are together called “local realism” or “local hidden variables.” (The spin component in each direction is called “hidden” because it is thought to exist, even though it isn’t measured.) As a derivation, Bell’s inequality is simple and correct. (This much is not controversial.) What does it mean? (This is controversial.) Assuming local realism is true, if we ran an experiment with many pairs measured in the various directions (eg, a i and b i ' ), then when we plug the particular outcomes (+1 or -1) into the equation, we get the result that the overall value must be equal to or less than 2. 23

QM Prediction Local realism implies Bell’s inequality. (Bell proved this.) QM, however, implies something different. We let the possible setting on the left and right be as follows. 24

QM predicts the following correlations: C(a,b) = -cos 0 = -1 C(a,b ' ) = -cos 45 = -1/√2 C(a ',b) = -cos 45 = -1/√2 C(a ',b ' ) = -cos 90 = 0 Using these values in the same form as Bell’s formula above, we have: |-1 -1/√2 -1/√2 - 0| = 1 + √2 > 2 Thus, QM predicts a violation of Bell’s inequality. 25

The Aspect experiment (1982) 26

Outcome Alain Aspect (Paris) carried out a series of very meticulous experiments in the 1980s. The results conformed with QM but not with the Bell inequality. Conclusions: –The EPR argument is faulty. –Local realism is false. –Nature is non-local. 27

What’s going on? What are some options? Some possibilities: Brute fact (“pre-established harmony”). There is no explanation for the pattern of outcomes for the distant events. It is just a remarkable coincidence. Realism is true, but locality false (ie, special relativity is false; signals travel from one side of the apparatus to the other faster than light). Bohm’s version of QM (which Bell liked) makes this assumption (it’s a non-local hidden variable theory). 28

Superdeterminism Everything is set from the Big Bang, including the decision by experimenter of what to measure. This is implausible. 29

Abner Shimony called the Bell results “experimental metaphysics,” meaning a mix of physics and philosophy, where metaphysical issues are – contrary to all expectation – settled (sort of) by empirical means. The issue is up in the air. No one knows what’s going on. Summer project: Solve this problem (and win a Nobel Prize). 30

The Bell results are (arguably) the most important thing to happen in physics in the 20 th century. (Even more important that QM or SR & GR.) It is independent from QM, in that it establishes the fact that reality is nonlocal, regardless of the truth of QM. 31

Further Reading There are lots of good books on the topic. Here are a few. Rae, Quantum Theory: Appearance or Reality? Maudlin, Quantum Non-locality and Relativity Albert, Quantum Mechanics and Experience Redhead, Incompleteness, Non-locality, and Realism Hughes, The Structure and Interpretation of Quantum Mechanics 32

Final Week Next week will be a final quiz: 25% Similar to the first quiz. 30 mins. Before the quiz there will a short class on some aspects of QM that we didn’t have time to cover, eg, “many worlds.” 33