Dear Professor Youssef,

I'm reading your newest hep-th/0110253 [Ref. 1] with immense interest. I believe the question in the subject line, taken from [Ref. 1], is related to the main issue of how to cancel probabilities: "A modified probability theory must provide a way for probabilities to cancel each other and so an obvious first guess is to allow probabilities to be complex numbers" [Ref. 2].

Have you thought about introducing two modes of time, one in which "particle in  R^3  is somewhere in  R^3  at each time", interpreted in [Ref. 1] as the "proper time or path length parameter" (I call it 'global time'), and another mode of time (local time mode in which things "happen"), in which a student will observe in her/his past light cone that a state $(1/2,1/2)$  had*already* evolved into  $(1,0)$ [Ref. 2]?

I'm sure this is not new at all, these two modes of time can be traced back to Whitehead, but I couldn't find relevant references at 

http://physics.bu.edu/~youssef/quantum/quantum.html
http://physics.bu.edu/~youssef/quantum/quantum_refs.html

With kind regards,

Dimiter G. Chakalov
http://members.aon.at/chakalov
(last update 27 October 2001)
 

References

[Ref. 1] Saul Youssef. Physics with exotic probability theory. Mon, 29 Oct 2001 06:07:02 GMT,
http://xxx.lanl.gov/abs/hep-th/0110253

"With probability theory modified, there is no need for the usual "wave-particle duality" and one is free to assume, for example, that a particle in  R^3  is somewhere in  R^3  at each time.  Introducing such "state spaces" and assuming that probabilities have a square norm, exotic probabilities acquire the power to predict real non-negative frequencies and are limited to three algebras: reals, complex numbers and quaternions.
...

"Srinivasan has realized that one should expect even more interesting  results in field theory because exotic probability theory cannot produce the apparent divergences which are so common in quantum field theory. Indeed, he has shown that with his quaternionic probability version of canonical quantization, he gets the correct result for the Lamb shift without having to add a cutoff momentum [S.K. Srinivasan, J. Phys. A Math. Gen. 31 (1998)].
...

"Bell's result and two of the more well known variations are considered in reference 3 in some detail and are shown not to eliminate exotic  probabilities. There has also been an increasing tendency to refer to Bell and similar results as "non-local" effects because they cannot be explained by local correlations [S. Youssef, Phys. Lett. A204, 181 (1995)]. The point is, however, that if one has the wrong probability theory, one may also have the wrong notion of what is just a correlation. Within exotic probability theory, we expect that Bell's results are just correlations in the new probability theory.

... 

"What is the meaning of the "time" parameter?

"Although the time parameter in exotics seems essential once the state space axioms are introduced, this does not mean that exotics are non relativistic. "Time" in the complex   R^4  theory, for example, can be interpreted as the proper time or path length parameter.  One suspects however, that "time" is really the order in which one discovers facts about the system rather than anything more intrinsic. In this case, one might expect that automorphisms of the time parameter should result in equivalent theories with modified moments of $(x_t\right arrow x'_{t'})$.  Is this correct and, if so, what are the consequences of invariance under time automorphisms?
...

"It is clear from the exotic probability point of view that a naive picture of quantum computers doing computations "on all paths simultaneously" must not be correct. In some sense, the particle can only do so much because it only, in fact, follows one path through the system.
...

"\bibitem{nonmeasurablesets} Assuming that $X\times T$ is a sublattice of $L$ implies that probabilities such as $(a\rightarrow N_t)$ can occur where $N$ is a non-measurable set.  This does not cause problems because we only assume that  $(a\rightarrow b\vee c)=(a\rightarrow b)+(a\rightarrow c)$  for pairs $(b,c)$ and not for countable subsets $b_1,b_2,\dots$ with $b_i\wedge b_j=0$ for all $i\neq j$.  "Measure space" in this paper refers to a measure space with a finite measure.
 
 

[Ref. 2] Saul Youssef. Quantum Mechanics as an Exotic Probability Theory. Presented at the Workshop on Maximum Entropy and Bayesian Methods, St. John's College, Santa Fe, New Mexico, August, 1995.Mon, 11 Sep 1995 16:08:05 -0400,
http://xxx.lanl.gov/abs/quant-ph/9509004

"A modified probability theory must provide a way for probabilities to cancel each other and so an obvious first guess is to allow probabilities to be complex numbers.
...

"You might expect that if quantum mechanical phenomena can be described by complex probability theory, the Bayesian view might help in understanding some of the long standing semi-paradoxical measurement and observer questions in quantum mechanics. Here, it's helpful to  first think about a purely classical experiment where a single coin is flipped and then uncovered, revealing that it landed "heads."

"From the Bayesian point of view, of course, the situation before the
observation could be described by the distribution  $(1/2,1/2)$  and after observing heads our description would be adjusted to  $(1,0)$ . The problem is, what would you say to a student who then asks:

"Yes, but what causes  $(1/2,1/2)$  to evolve into  $(1,0)$ ? How does it happen?"