The Albert's Socks Illustration Of Negative Probabilities
A recent blog post by Johannes Koelman on negative probabilities in quantum mechanics is the most lucid explanation of the concept aimed at a layman that I have ever seen. An academic paper made available in preprint form the same week covers much of the same territory in a manner that is more mathematically rigorous and that explores two subtypes of the concept.
The post uses the analogy of "Albert's Socks", involving a mysterious three drawer by three drawer sock chest in which only one row, or one column of drawers may be opened on any given day (with socks redistributed between days). Any given drawer may have either one sock or no sock. Any row will always have either zero or two socks with equal probability. Any column will always have one sock.
This seems to be contradictory. If every column has an odd number of socks, there must always be a odd number of socks in the chest. If every row has an even number of socks, there must always be an even number of socks in the chest. But, a set of rules that involve negative probabilities can fit the data.
A probability function that produces this outcome has ten possible states. Nine states (each of which is assigned a probability of 1/6) have every possible way to have two socks in a row. One more state has no socks in the chest at all (which is assigned a probability of negative 1/2). By adding up all ten states for any given row, or for any given column, one can get the probability that is observed when any given row or column is measured.
The state with negative probability suppresses a particular subset of outcomes of the other "realistic" outcomes (whose combined probabilities exceed one in isolation whenever negative probabilities come into play) in a particular, well defined way.
Since it is impossible to view all of the drawers at once, an observer can never know which of the ten possible states came up when a particular column or row was observed, and the negative probability state itself is never observed, for example, when an observe chooses to look at a column.
In general, negative probabilities are only possible in systems (1) where only part of the system is observed at once (Heisenberg's uncertainty principle decrees that this cosmic censorship principle is necessarily true in all real world quantum mechanical systems), (2) where the sum of all probabilities of given states, positive and negative sum to one, and (3) all observable probabilities are non-negative.
This basically parallels the notion of complex numbers, which often are used in intermediate steps in applied applications, but in the real world, arising in formulas where the observable outcomes are always real valued rather than complex.
Probability Amplitudes Methods Compared
The alternative way of reaching the same result without the radical concept of negative probabilities involves probability amplitudes, in which probabilities are represented as vector quantities (i.e. arrows in space), rather than as scalar quantities (i.e. simple numbers), and the final probability is the square of the magnitude of the the sum of the probability amplitude vectors for all of the possibilities combined. The best layman oriented explanation of that mathematically equivalent alternative to using negative probabilities is by Richard Feynman as a part of his transcribed series of four lectures entitled "QED" (for quantum electrodynamics, which he was pivotal in formulating in terms of probability amplitudes).
Thinking of negative probabilities that contribute to an outcome as probabilities oriented in different directions in some sort of phase space is easier to get one's head around than trying to understand the meaning of individual negative probabilities in isolation (and in much the same way, conceptualizing complex numbers as vector rather than scalar quantities that are opposed in different directions on a complex plane is much easier than simply trying to understand the square root of negative one in isolation).
The probability amplitude method makes the condition that all observed probabilities are non-negative be non-negative when the formulas are applied appear obviously and trivally true. This is reassuring, because it is not easy to state the conditions that must be present once negative probabilities are introduced, for this condition to hold when an algebraic as opposed to a geometric description of the concept is used.
But, it is far less obvious when using probability amplitudes that the sum of all of the probabilities for every possible outcome in a given application really do sum to exactly one, which is often trivial in an algebraic approach using negative probabilities.
Also, an algebraic approach using negative probabilties makes the point that there is a deep and fundamental connection between the fact that not of all a system is observable at once (even in theory) and the phenomena (like quantum tunnelling) that are possible in quantum electrodyamics (and quantum physics generally) which would be impossible using Maxwell's equations of electromagnetism (and in purely classical physics generally). This deep and fundamental connection is not nearly so self-evident when probability amplitudes are used.
The way that probability amplitudes dovetail nicely with the notion of complex numbers likewise being vector rather than scalar quantities that are manipulated accordingly, also provides a natural segue between complex numbers and related notions like quaternions, which have a very natural vector representation that generalizes transparently from the complex number concept even though the abstract algebra generalization of complex numbers is rather daunting.
Because the concepts of probability amplitudes and negative probabilities are so fundamentally necessary in every day, routine quantum mechanics, the mathematics involved are quite well developed and the issues are quite well understood.
Of course, like any other kind of mathematical idea, there is no good reason that a concept like this that has applications in one context might not have applications in another totally unrelated field.
To my knowledge, this hasn't been done in any field outside of the physical sciences in any serious way. But, the Albert's Socks analogy suggests one place where it might have an application, which is in describing public opinion, for example, in politics.
One of the odd features of public opinion that comes up a lot in the political theory of voting is that political preferences are not necessarily transitive. The fact that a majority of people prefer Quinn to Erin, and that a majority of people prefer Erin to Dave, does not in fact necessarily imply that a majority of people prefer Quinn to Dave.
This reality is known as Arrow's Paradox (after economist Kenneth Arrow), and is often stated as follows:
[W]hen voters have three or more distinct alternatives (options), no rank order voting system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specific set of criteria. These criteria are called unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. . . . Although Arrow's theorem is a mathematical result, it is often expressed in a non-mathematical way with a statement such as "No voting method is fair," "Every ranked voting method is flawed," . . . These statements are simplifications of Arrow's result which are not universally considered to be true. . . .what Arrow's theorem really shows is that any majority-wins voting system is a non-trivial game, and that game theory should be used to predict the outcome of most voting mechanisms. This could be seen as a discouraging result, because a game need not have efficient equilibria, e.g., a ballot could result in an alternative nobody really wanted in the first place, yet everybody voted for.
(Note that Arrow's paradox merely says that no voting method is fair for all possible distributions of preferences - the fortunate reality is that most of the time there is a single consistent ranking but that there are gray area cases where ranking preferences cannot be done consistently in any particular system.)
Given that Arrow's paradox involve circumstances where incomplete information gathering about the total data set at any given time can produce seemingly inconsistent outcomes, a negative probability framework might have application to these kinds of problems in the social sciences and provide insight into their deeper structure.
The subdiscipline of economics where Arrow's paradox and related theories come into play are mostly related to how to maximize utility for a group of people even though one person's utility can't strictly and rigorously be equated with another person's utility, is called "welfare economics", not because it is about welfare programs, but because it is about the general welfare of the members of the community at large.
One also might imagine that the use of negative probability outcomes might be helpful in game theory to identify circumstances and conditions in which using seemingly inconsistent strategies makes sense, depending upon the manner in which one part of a larger problem presents itself to a decision maker. Negative probabilities, also known as quasi-probabilities are also used in mathematical finance and derivative pricing.