Negative probability
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In 1942 Paul Dirac wrote a paper: "The Physical Interpretation of Quantum Mechanics"[1] 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 has later got increased attention in physics and particular in quantum mechanics. Another famous physicist, Richard Feynman[2] 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.[3]
Negative probabilities have more recently been tried 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 Haug (2007) speculates that this can give certain finance models more flexibility without being inconsistent with real observed probabilities.
[edit] See also
- 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#General ghosts in theoretical physics.
[edit] Notes
- ^ 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.
Haug, E. G. (2007): Derivatives Models on Models, John Wiley & Sons, New York
