Exotic probability: Difference between revisions
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This article has 2 versions, the accuracy of the newer version at bottom is of especially questionable accuracy, considering the [http://en.wikipedia.org/enwiki/w/wiki.phtml?title=Special:Contributions&target=131.111.250.45 author]. |
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[[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 number]]s, the [[complex number]]s and the [[quaternion]]s. |
[[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 number]]s, the [[complex number]]s and the [[quaternion]]s. |
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* [http://flux.aps.org/meetings/YR97/BAPSAPR97/vpr/layn18-4.html Measuring Negative Probabilities, Demystifying Schroedinger's Cat and Exploring Other Quantum Peculiarities With Trapped Atoms] |
* [http://flux.aps.org/meetings/YR97/BAPSAPR97/vpr/layn18-4.html Measuring Negative Probabilities, Demystifying Schroedinger's Cat and Exploring Other Quantum Peculiarities With Trapped Atoms] |
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* [http://www.mathpages.com/home/kmath309.htm The Complex Domain of Probability] |
* [http://www.mathpages.com/home/kmath309.htm The Complex Domain of Probability] |
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'''Exotic probability''' is a branch [[measure theory|advanced measure theory]] dealing with probabilities outside the normal range of [0,1]. |
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== History == |
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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. |
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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. |
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In [[1996]] [[William Timothy Gowers|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. |
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== Forms of exotic probability == |
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=== Super-unitary probability === |
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<table align="right"><tr><td> |
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[[Image:20-sided_dice_250.jpg]] |
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</td></tr></table> |
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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. |
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=== Negative-probability === |
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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]]. |
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=== Complex-probability === |
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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. |
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==See also== |
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* [[Spin (physics)]] |
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* [[List of mathematical topics]] |
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* [[Probability]] |
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* [[Lorentz group]] |
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* [[Churchill College, Cambridge]] |
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==External links== |
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* http://physics.bu.edu/~youssef/quantum/quantum_refs.html |
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Revision as of 10:51, 28 May 2004
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