|
The probability of the outcome of an experiment is never negative, but quasiprobability distributions can be defined that allow a negative probability for some events. These distributions may apply to unobservable events or conditional probabilities. ==Physics== In 1942, Paul Dirac wrote a paper "The Physical Interpretation of Quantum Mechanics" where he introduced the concept of negative energies and negative probabilities: : "Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money." The idea of negative probabilities later received increased attention in physics and particularly in quantum mechanics. Richard Feynman argued that no one objects to using negative numbers in calculations: although "minus three apples" is not a valid concept in real life, negative money is valid. Similarly he argued how negative probabilities as well as probabilities above unity possibly could be useful in probability calculations. Mark Burgin gives another example: Negative probabilities have later been suggested to solve several problems and paradoxes.〔Khrennikov, A. Y. (1997): ''Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models''. Kluwer Academic Publishers. ISBN 0-7923-4800-1〕 ''Half-coins'' provide simple examples for negative probabilities. These strange coins were introduced in 2005 by Gábor J. Székely.〔 Székely, G.J. (2005) (Half of a Coin: Negative Probabilities ), Wilmott Magazine July, pp 66–68.〕 ''Half-coins'' have infinitely many sides numbered with 0,1,2,... and the positive even numbers are taken with negative probabilities. Two half-coins make a complete coin in the sense that if we flip two half-coins then the sum of the outcomes is 0 or 1 with probability 1/2 as if we simply flipped a fair coin. In ''Convolution quotients of nonnegative definite functions'' and ''Algebraic Probability Theory'' 〔Ruzsa, I.Z. and Székely, G.J. (1988): ''Algebraic Probability Theory'', Wiley, New York ISBN 0-471-91803-2〕 Imre Z. Ruzsa and Gábor J. Székely proved that if a random variable X has a signed or quasi distribution where some of the probabilities are negative then one can always find two other independent random variables, Y, Z, with ordinary (not signed / not quasi) distributions such that X + Y = Z in distribution thus X can always be interpreted as the `difference' of two ordinary random variables, Z and Y. Another example known as the Wigner distribution in phase space, introduced by Eugene Wigner in 1932 to study quantum corrections, often leads to negative probabilities, or as some would say "quasiprobabilities". For this reason, it has later been better known as the Wigner quasiprobability distribution. In 1945, M. S. Bartlett worked out the mathematical and logical consistency of such negative valuedness. The Wigner distribution function is routinely used in physics nowadays, and provides the cornerstone of phase-space quantization. Its negative features are an asset to the formalism, and often indicate quantum interference. The negative regions of the distribution are shielded from direct observation by the quantum uncertainty principle: typically, the moments of such a non-positive-semidefinite quasiprobability distribution are highly constrained, and prevent ''direct measurability'' of the negative regions of the distribution. But these regions contribute negatively and crucially to the expected values of observable quantities computed through such distributions, nevertheless. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「negative probability」の詳細全文を読む スポンサード リンク
|