Is Quantum Mechanics an Exotic Probability Theory?

Abstract

Quantum mechanics is often introduced with a discussion of the two-slit experiment where the observed interference pattern forces us to conclude that the particle goes through both slits at once and is "both a particle and a wave". This most-basic argument has a loophole, however. The conclusion rests on probability theory and, in particular, on the fact that probabilities are nonnegative so that, when the second slit is opened, P(x) = P(x via slit 1) + P(x via slit 2) 2 P(x via slit l),where P(x) is the probability for the particle to arrive at position x on the screen. Perhaps, then, it is possible that the particle does go through either one slit or the other after all and the interference effects can be explained by modifying probability theory itself. Here, we give a brief overview of this program, which is put forward in references 1-3. For other approaches to exotic probability theory, see references 4-7.Probability theory is most often presented with the "frequentist" approach where probabilities are defined as limits of experimental frequencies. For example, if n successes occur in N trials of an experiment, the large N limit of n/N is called the "probability of success". Such probabilities are then assumed to follow Kolmogorov's axioms8 From this point of view, probability theory is a theory of "random phenomena" that are well described by these axioms and, of course, probabilities are necessarily nonnegative. For our purposes, then, it is necessary to adopt the more general "Bayesian" view where probabilities are not a priori defined as frequencies, but where, instead, a frequency interpretation is derived as a consequence of the fundamental axiom^.^ With the Bayesian view, probability is introduced as a measure of "likelihood" where, if proposition a is known, the likelihood that proposition b is true is denoted1° "(a + b)". For "+" to be a useful likelihood measure, one expects it to have a few properties such as (i) if (a + b ) is known, this should determine (a + 7 b ) and (ii) the procedure to get from (a + b) to (a + T b ) should be independent of a and b. As shown by COX,^^ such considerations are enough to entirely fix probability theory, which then takes the form (a + b A c ) = (a + b)(a A b + c ) (a + b ) + (a + T b ) = 1 (a + -a ) = 0 (2a) (2b) (2c)for all propositions a, b, and c. 904

On Noncontextual, Non-Kolmogorovian Hidden Variable Theories

“…In the second approach to extending the range of the probability measure, one goes even further, allowing it to take on complex values, as proposed by Ivanović (1978); Youssef (1991Youssef ( , 1994Youssef ( , 1995Youssef ( , 1996Miller (1996); Srinivasan and Sudarshan (1994); and Srinivasan (1995Srinivasan ( , 1998Srinivasan ( , 2006. 10 Definition 11.…”

Toward a More Natural Expression of Quantum Logic with Boolean Fractions

“…On the other hand, accepting the Copenhagen interpretation of the two-slit experiment, S. Youssef [You91], [You94], You96], and [You01], showed that complex-valued probabilities can produce the interference results associated with that experiment. However, one must ask whether the probability function should be changed so much that we can no longer interpret it as a ratio?…”

Quantum phenomena via complex measure: Holomorphic extension

“…In a parallel contribution Yousseff [31][32][33] has advocated the use of 'exotic' probability theory wherein conditional complex probabilities are introduced thus accommodating violation of Bell's inequalities in a natural manner. The idea that a modified version of probability theory may be the appropriate tool to arrive at a theory of quantum phenomena is not entirely new and perhaps goes back to Dirac (see [34]). Our approach based on complex measures/measurable processes and their extension is direct and simple in the sense that it accommodates interference and internal motion ensuring at the same time (consistency with ) the violation of Bell's inequalities.…”