- Negative probability
- "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. Similarly he argued how negative probabilities as well as probabilities above unity possibly could be useful in probability calculations.
Negative probabilities have later been suggested to solve several problems and paradoxes. Half-coins provide simple examples for negative probabilities. These strange coins were introduced in 2005 by Gábor J. Székely.  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  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 "quasi-probabilities". For this reason, it has later been better known as the Wigner quasi-probability 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 quasi-probability 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.
Negative probabilities have more recently been applied to mathematical finance. In quantitative finance most probabilities are not real probabilities but pseudo probabilities, often what is known as risk neutral probabilities. These are not real probabilities, but theoretical "probabilities" under a series of assumptions that helps simplify calculations by allowing such pseudo probabilities to be negative in certain cases as first pointed out by Haug in 2004 .
A rigorous mathematical definition of negative probabilities and their properties was recently derived by Mark Burgin and Gunter Meissner (2011). The authors also show how negative probabilities can be applied to financial option pricing.
- Existence of states of negative norm—or fields with the wrong sign of the kinetic term, such as Pauli–Villars ghosts—allows the probabilities to be negative. See: Faddeev–Popov ghost
- Signed measure
- Wigner quasi-probability distribution
- ^ Dirac, P. (1942): "The Physical Interpretation of Quantum Mechanics," Proc. Roy. Soc. London, (A 180), 1–39.
- ^ Feynman, R. P. (1987): "Negative Probability," First published in the book Quantum Implications : Essays in Honour of David Bohm, by F. David Peat (Editor), Basil Hiley (Editor) Routledge & Kegan Paul Ltd, London & New York, pp. 235–248
- ^ Khrennikov, A. Y. (1997): Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers. ISBN 0792348001
- ^ Székely, G.J. (2005) Half of a Coin: Negative Probabilities, Wilmott Magazine July, pp 66–68.
- ^ Ruzsa, I.Z. and Székely, G.J. (1983) "Convolution quotients of nonnegative definite functions", Monatshefte fuer Matematik, 95, pp 235–239. doi:10.1007/BF01352002
- ^ Ruzsa, I.Z. and Székely, G.J. (1988): Algebraic Probability Theory, Wiley, New York ISBN 0471918032
- ^ Wigner, E. P. (1932): "On the quantum correction for thermodynamic equilibrium", Phys. Rev. 40 749–759. doi:10.1103/PhysRev.40.749
- ^ Bartlett, M. S. (1945). "Negative Probability". Math Proc Camb Phil Soc 41: 71–73. doi:10.1017/S0305004100022398.
- ^ Haug, E. G. (2004): Why so Negative to Negative Probabilities, Wilmott Magazine, Re-printed in the book (2007); Derivatives Models on Models, John Wiley & Sons, New York
- ^ Negative Probabilities in Financial Modeling Forthcoming Wilmott Magazine 2011
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