Negative probability
Encyclopedia
In 1942, Paul Dirac
wrote a paper "The Physical Interpretation of Quantum Mechanics" where he introduced the concept of negative
energies and negative probabilities:
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 value
s 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.
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
wrote a paper "The Physical Interpretation of Quantum Mechanics" where he introduced the concept of negative
Negative
- Science and mathematics :* Negative number* Negative mass* Negative energy * Electrical polarity* Negative result * Negative lenses, uses to describe diverging optics, see lens - Photography :...
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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
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
Gábor J. Székely
Gábor J. Székely is a Hungarian-American statistician/mathematician best known for introducing E-statistics or energy statistics [see E-statistics or Package energy in R ], e.g...
. 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
Imre Z. Ruzsa
Imre Z. Ruzsa is a Hungarian mathematician specializing in number theory.Ruzsa participated in the International Mathematical Olympiad for Hungary, winning a silver medal in 1969, and two consecutive gold medals with perfect scores in 1970 and 1971. He graduated from the Eötvös Loránd University...
and Gábor J. Székely
Gábor J. Székely
Gábor J. Székely is a Hungarian-American statistician/mathematician best known for introducing E-statistics or energy statistics [see E-statistics or Package energy in R ], e.g...
proved that if a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
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
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the 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
Wigner quasi-probability distribution
The Wigner quasi-probability distribution is a quasi-probability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics...
. In 1945, M. S. Bartlett
M. S. Bartlett
Maurice Stevenson Bartlett FRS was an English statistician who made particular contributions to the analysis of data with spatial and temporal patterns...
worked out the mathematical and logical
consistency of such negative valuedness. The Wigner distribution function is routinely used in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
nowadays, and provides the cornerstone of phase-space quantization
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
. 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
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
: 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 value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
s of observable quantities computed through such distributions, nevertheless.
Negative probabilities have more recently been applied to mathematical finance
Mathematical finance
Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
. In quantitative finance most probabilities are not real probabilities but pseudo probabilities, often what is known as risk neutral
Risk neutral
In economics and finance, risk neutral behavior is between risk aversion and risk seeking. If offered either €50 or a 50% chance of each of €100 and nothing, a risk neutral person would have no preference between the two options...
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.
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
- Signed measureSigned measureIn mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.-Definition:There are two slightly...
- Wigner quasi-probability distributionWigner quasi-probability distributionThe Wigner quasi-probability distribution is a quasi-probability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics...