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Can Chance Be Less Than Zero?

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Can Chance Be Less Than Zero? _NA_T_U_R_E_V_O_L._3_20_1_0 _AP_R_IL_19_86_______ NEWS AND VIEWS---------------48-1 Can chance be less than zero? An intriguing review ofphysical problems yielding negative probabilities may fail to demonstrate their reality, but should not be overlooked on that account. By simple definition, as everybody knows, which describe it and of the conjugate do not commute with each other cannot in the values assigned to quantities repre­ momenta. Although the uncertainty prin­ any circumstances be measured simultan­ senting the probability of events must be ciple disallows the simultaneous exact eously, and from which the more recent positive real numbers. How could it be measurement of these quantities, it is discussions of Bell's inequality and otherwise? The chance that a flipped coin clear that there must be conjugately link­ Aspect's experimental tests of quantum will come down showing heads, not tails, ed probability distribution functions for correlations have flowed. Muckenheim is roughly 0.5, the probability of the only position and momentum. proves, by demonstration, his belief that other possible outcome is roughly the For a system with only one degree of negative probabilities signal connections same, and the two numbers added togeth­ freedom with a position coordinate x and with the important problems of physics. er make exactly 1.0, which is the probabil­ momentum p, for example, is is a simple But what do negative probabilities ity that there will be an outcome of one matter to calculate the properties of the mean? Bartlett (who wrote a paper on kind or the other. In such a context, the momentum distribution function for any negative probability in 1945) provides one notion that probabilities might have nega­ stationary state ofthe system for which the simple way of fixing ideas: a convenient tive values is entirely devoid of meaning. wavefunction is known, say as u(x). For way of handling the probability that a That is what everybody knows. then the successive moments of the biased coin will fall on one side or the None of this has deterred Dr W. momentum distribution function can be other is to represent the chance that it will Muckenheim of Gottingen from compil­ calculated directly from the wavefunction tum up heads as 0.50 + p, in which case ing a substantial and fascinating review of as the integral over the range of x of the the extra variable is plainly capable of be­ negative probability, called "extended function u*(x)pu(x) where * denotes the ing negative as well as positive. Even so, probability" no doubt so as to avoid the complex conjugate of the wavefunction Bartlett was able to show (in 1945) that charge of sensation-mongering (Physics and p is the momentum operator (.wi) quantities representing probabilities Reports 166, 337; 1986). Part of Mucken­ (dldx). which can take negative values can be heim's case is that if it has been useful to So why not calculate the distribution manipulated by most of the usual rules of extend the set of natural numbers to in­ function that gives the joint probability the probability calculus. His view then has clude negative and irrational and imagin­ that x is within a certain interval and p the not changed much over the years; most of ary numbers, to allow systems to have scalar value of the momentum within the difficulties with negative probabilities negative energy (as in some solutions of another? This is what Wigner did in 1932. appear to arise from people's determina­ Dirac's equations) and even negative The result was a distribution function (for tion to follow Max Born in the interpreta­ temperatures (as in laser and maser mat­ one degree of freedom) P(x,p) which, tion of the square of the wave function as a erials with artificial populations of excited apart from a factor lI(pi.h), is the integral probability density, for which he sees no states), there can be no good reason why over the dummy variable y of the quantity physical justification. probabilities should be sacrosanctly posi­ u(x+y).u(x-y).exp(2ipyth). The result, Running through this commentary on tive. His more tangible starting point is a which is obviously generalized to systems negative probability is the opinion that, in puzzle which has been about in quantum with more than one degree of freedom, is practice, sensible people will ensure that mechanics since it was first identified by a distribution function with some of the they do not interpret probability distribu­ Eugene Wigner in 1932. right properties, but which also has the tions physically unless they have somehow Muckenheim comes to no particular disconcerting property of being negative manipulated them so as to be everywhere conclusion about the meaning of negative over some parts of the range of x and p. positive. probability. Properly, he notes that the Muckenheim gives an interesting account Other ways of squaring this circle are question is almost certainly linked with of the circumstances in which the Wigner more ingenious. Dewdney, Holland, Kip­ other current imponderables, such as the function and its many generalizations mis­ rianidis and Vigier from the Poincare In­ question whether quantum mechanics behave in this way. stitute in Paris suggest that it might be requires a revision of classical causality This is not the only connection in which worthwhile to consider negative probabil­ or even of the principles of logic (as first negative probabilities are known to occur. ity distributions as the difference of two suggested by von Neumann); tinkering Relativistic generalizations of classical separate and presumably positive den­ with the meaning of probability may be a quantum mechanics habitually yield nega­ sities which may, for example, represent simpler task. tive energies and, with them, functions the distributions of particles and antipar­ But instead of a conclusion, Mucken­ which it is tempting to interpret as proba­ ticles respectively. heim has assembled the opinions of sever­ bility distributions that are capable of Gallantly for one who clearly wishes al of those who have contributed to the being negative, so much so that Dirac is that somebody would give meaning to field, including refreshingly those of E.T. quoted as believing them to be insepar­ negative probability, Muckenheim gives Jaynes of Washington University, St able. They also crop up in field theories, as almost the last word to Jaynes who, while Louis, and M.S. Bartlett, long-since retir­ for example is chemes for accounting for sharing the general doubt about the valid­ ed from the University of Manchester but the interaction between real particles by ity of Born's interpretation of the square plainly doing more than merely growing the emission and absorption of "virtual" of the wave equation as a probability den­ roses at his private address in Devon. field particles. Similar difficulties arise in sity, makes the simple point that the onus Wigner's conundrum arises from his the discussion of the Einstein-Rosen­ of proof rests on those who talk about attempt (with Szilard) to calculate for a Podolski paradox, the Gedankenexperi­ negative probability. That, unfortunately general quantum system the joint prob­ ment designed in 1935 as a challenge to perhaps, will be the general view. ability distribution of the coordinates Bohr's dictum that physical variables that John Maddox © 1986 Nature Publishing Group.
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