Exotic probability
This article has 2 versions, the accuracy of the newer version at bottom is of especially questionable accuracy, considering the author.
Exotic probability is a branch of probability theory that deals with probabilities which are outside the normal range of [0, 1]. The most common author of papers on exotic probability theory is Saul Youssef. According to Youssef, the valid possible alternatives for probability values are the real numbers, the complex numbers and the quaternions.
Youssef also cites the work of Richard Feynman, P. A. M. Dirac, Stanley Gudder and S. K. Srinivasan as relevant to exotic probability theories.
Of the application of such theories to quantum mechanics, Bill Jefferys has said: Such approaches are also not necessary and in my opinion they confuse more than they illuminate. [1]
External links
- http://physics.bu.edu/~youssef/quantum/quantum_refs.html
- Physics with exotic probability theory - paper by Youssef on arXiv.
- http://fnalpubs.fnal.gov/library/colloq/colloqyoussef.html
- Measuring Negative Probabilities, Demystifying Schroedinger's Cat and Exploring Other Quantum Peculiarities With Trapped Atoms
- The Complex Domain of Probability
Exotic probability is a branch advanced measure theory dealing with probabilities outside the normal range of [0,1].
History
The original exotic probablity theories (EPTs) were studied in papers by such mathematicians as Henri Lebesgue and Paul Erdös in the early 1920s, but aroused little interest.
In the 1970s, researchers in quantum electrodynamics, especially Richard Feynman and Lawrence M. Krauss, realised that by using a renormalized form of exotic probability theory, many problems involving quark interactions could be reformulated into simpler soluble mathematical problems. Krauss won the Nobel Prize for his work in this field.
In 1996 Timothy Gowers of the University of Cambridge published a ground-breaking paper linking EPT to advanced combinatorics, bringing Exotic Probability Theory to the forefront of modern mathematical research.
Forms of exotic probability
Super-unitary probability
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This is when the probability of an event (naturally we use the Lebesgue definition of probability), is strictly greater than 1. Lebesgue illustrated these examples with the famous hypothetical Lebesgue dice. If the probability of rolling a five on the Lebesgue dice were 2, then the we expect on average after 10 rolls, five to appear 20 times. Super-unitary probability can be associated with the SU(3) group of matrices.
Negative-probability
This is when the Lebesgue-probability of an event is strictly less than 0. If the probability of rolling a five on the Lebesgue Dice were −1, then the we expect on average after 10 rolls, five never to appear by a factor of 1. negative probability can be associated with the GL(4) group of matrices over the field of quaternions.
Complex-probability
This is when the real part of the Lebesgue-probability of an event is not equal to zero. If the probability of rolling a five on the Lebesgue dice were 0.5+i, then the we expect on average after 10 rolls, five never to appear 5 times and the disappear through another dimension. No isomorphism has been found between complex probability and a standard group, and the Manning-McArdle conjecture states that complex-probability does not lead to consistent answers unless the imaginary part is equal to some integer times the square root of 7.