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Einstein's critique of Quantum Mechanics in the
EPR paper was an attempt to show that Quantum Mechanics was
somehow not complete. The argument indicates that, eventually, physicists
will need to develope some new theory to describe nature. One way to escape
from this critique is suppose that the quantum mechanical approach to
describing nature is valid and the apparent incompleteness arises from
an incomplete understanding of the important quantities and constraints
inherent in nature. If this is true, then one can expect that new dynamical
variables will be discovered in the future which will eliminate the problems
associated with the EPR paradox.
For example, when one flips a coin it appears that the result (either heads or tails) is a completely random process. On the other hand, if we exactly how the air in the room was moving (or, if we did the toss in a place where there was no air) and we used a machine to flip the coin so we would know exactly HOW the coin was flipped, then we could predict precisely the outcome. Since we lack this knowledge, the results of tossing the coin appear random but only because there are variables (the air movement and how we flipped the coin) that we don't know about. For many years, physicists investigated what theories with hidden variables might look like. John Bell, in 1964, laid out a mathematical description of how these new variables might be used to overcome the incompleteness of Quantum Mechanics. His work was later used to construct experiments which would test for the existence of these "hidden" variables.1 First, lets denoted these hidden variables by the letter h. Now, h could be a single variable, a set of variables, a set of functions, etc. It merely represents all those things which we don't yet know but need to in order to understand nature. Some of properties of h can be determined right away. If you examine the arguments made by Einstein, you will find that he makes two rather simple assumptions. The first condition is that measurements which are made far apart (at spacelike distances) cannot affect each other. Secondly, if two particles, A and B, have previously interacted but are now separated, then the results of measurements on A should not affect B in any way. The first condition means that the probability for a measurement at A is only a function of A's variables. This condition is known as the "locality" condition. The second condition means that one can "separate" the probabilities for measurements on A and B (ie: the probabilities for measurements at A and B are two independent functions). This is the "separability" condition. 2 Now let us assume that the result of making measurements on two particles depends upon the hidden variable h such that, given h one can predict the exact outcome of the measurement. Let A stand for the outcome of a measurement on particle 1 and let a stand for the orientation of that measurement. For example, if we are measuring the polarization of light or the spin of a particle, then then a would define the projection axis of the spin or the polarization. One can also assume that these measurements are done in a system of units where A is always between -1 and 1. Again, if we are measuring electron spins, we would get results of ± 1 for spins of ± h/2. One can also define B and b as similar quantities for particle 2. How can we formulate these conditions mathematically? The functions A and B, which gives us the result of any measurement made on particle 1 and particle 2 respectively, will be of the form | A(a,h) | <= 1 and | B(b,h) | <= 1 (where |x| represents the absolute value of x. ). Notice that the result of a measurement on particle 1 is not affected by the orientation of detector "B" nor does it depend on any measurement which might have taken place on particle 2 (ie. A is A(a,h) and not A(a,b,h)). The fact that the functions for A and B can be written down independently of each other is a consequence of the separability condition. The fact that A is a function of only the orientation of detector A and the hidden variable h (and B is only a function of the orientation of detector B and the hidden variable h) is a result of the locality condition. The expectation value for a particular measurement of this system, with specific values of a and b is given by the formula:
In this formula, p(h) represents the probability density function for the unknown hidden variables 3. So, you can see that, even though we do not know the exact form that these hidden variables will take and even though we do not know the exact impact they will have on the functions A and B, we can still incorperate them into our theory as long as we properly average over their density of states, p(h). |
