2024年5月5日发(作者：)

The probability of the outcome of an experiment is never negative, but quasi-probability

distributions can be defined that allow a negative probability for some events. These

distributions may apply to unobservable events or conditional probabilities.

经验结果的概率不可能为负，但准概率分布允许将某些事件定义为负概率。这种分

布适用于不可观察的事件或条件概率。

Physics

物理性质

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."

在1942年，保罗·狄拉克发表了论文“量子力学的物理解释”，引入了负能量和负概

率的概念：

“负能量和负概率不应该被认为是无稽之谈。他们是被明确数学化定义的概念，就

如同负的货币一样。”

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.

负概率后来在物理学中受到越来越多的关注，特别是量子力学。理查德·费曼认为，

没有人反对在计算中使用负数，尽管“负三个苹果”在现实生活中不是有效的概念。同样，

他认为负概率以及统一体上的概率（probabilities above unity）对概率计算是有用的。

Negative probabilities have later been suggested to solve several problems and

-coins provide simple examples for negative probabilities. These strange

coins were introduced in 2005 by Gábor J. Szé-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.

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后来有人用负概率解决了几个问题和悖论。不完全硬币（half-coins）是简单的负概

率的例子。这种奇怪的硬币是格贝尔·J·塞克利2005年提出的。不完全硬币有无限多的

侧面，编号分别为0，1，2，...，正偶数表示负概率。如果我们投掷两个不完全硬币，

概率为1/2，结果的合计为0或1，就如同我们投掷一个完整的硬币。从这个意义来说，

两个不完全硬币是一个完整的硬币。

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.

在“非负正定函数的卷积商数”和“代数概率论”中，伊姆雷·Z·鲁兹萨和格贝尔·J·塞克

利证明，如果一个随机变量X为负或概率为负的准分布，那么人们总是可以找到另外

两个独立的随机变量Y和Z，普通（非负或非准）分布等，在分布上有X + Y = Z，因

此X总是被解释为“两个普通的随机变量Z和Y的区别”。

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 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.

另一个著名的例子是相空间中的维格纳(Wigner)分布，1932年由尤金·维格纳引入，

研究往往形成负概率或者说“准概率”的量子修正。由于这个原因，后来被称为维格纳

准概率分布。1945年，M·S·巴特利特发现了这种负多值性（valuedness）的数学和逻

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辑的一致性。维格纳分布函数在当今物理学中被经常使用，提供相空间量化的基础。

它的负特性对形式主义来说是个优点，往往表明量子干涉。分布的负区免于量子测不

准原理的直接观察：通常情况下，这种非正半定的准概率分布的时刻被高度限制，防

止分布负区的直接测量。但是，这些区域对通过这样的分布计算的可观察的数量的预

期值，造成了消极和关键影响。

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.

最近负概率被应用于金融数学。在定量金融学中，大多数的概率并不是现实的概率，

而是往往被称为风险中性概率的伪概率。它们不是真正的概率，而是在一系列假设下

的理论“概率”，允许这种伪概率在某些情况下为负来简化计算，这是由豪格2004年首

先提出。

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.

最近马克•布尔金和冈特·迈斯纳（2011）提出了负概率及其属性的严格数学定义。

他们还说明了如何应用于金融期权定价运用负概率。

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