It was bound to happen, sooner or later. You give 'em an inch and they'll take
several more dimensions. Once probabilities were allowed to be complex
numbers, and we had to speak of probability *amplitudes* to make sense of
anything, it was a matter of time before probabilities became quaternions.
Colloquium: Exotic Probabilities & Quantum Mechanics
Saul Youssef, Florida State University
Fermi National Accelerator Laboratory
1 West, 4:00 PM, July 21, 1999
The Bayesian development of probability theory can be followed without
assuming that probabilities are real and non-negative. The result is
three copies of probability theory with real, complex or quaternion
values. From the complex copy, all of standard quantum mechanics can be
derived with ordinary probability theory restored in the classical
limit. This type of theory escapes Bell's result and other similar
limitations on local realistic theories and also nicely resolves the
long standing semi-paradoxical observer and measurement problems
associated with quantum theory. Recently S.K. Srinivasan and G.
Sudarshan have also shown that quaternion probabilities lead to the
Dirac equation and Srinivasan has further intriguing results in field
theory.
Cheers,
Wayne
I spoke with a Hamiltonian operator and several Hamiltonian mechanics (on the
Greens outside LaGrange) about this, and they assure me that things are easier
when time is momentous http://www.ugcs.caltech.edu/~toby/papers/time.ps and
imaginary to boot. But then old Alexander himself distrusted the wisdom and
political capacity of the common man and other strong dispersed correlations.
http://www-oss.fnal.gov/~baisley