Exotic probability: Difference between revisions
Conforming to Wikipedia's Manual of Style, which see. Got rid of gratuitous capital letters, highlighted the title phrase. |
Conforming to Wikipedia's Manual of Style, which see. Got rid of gratuitous capital letters, highlighted the title phrase. |
||
| Line 22: | Line 22: | ||
=== Complex-probability === |
=== 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 |
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. |
||
==See also== |
==See also== |
||
Revision as of 18:01, 25 May 2004
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
|
|
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.