On Negative Probabilities and Path Integrals

Alexios P. Polychronakos

Don Weingarten

We establish a necessary and sufficient condition for averages over complex-valued weight functions on R(N) to be represented as statistical averages over real, non-negative probability weights on C(N). Using this result, we show that many path integrals for time-ordered expectation values of bosonic degrees of freedom in real-valued time can be expressed as statistical averages over ensembles of paths with complex-valued coordinates, and then speculate on possible consequences of this result for the relation between quantum and classical mechanics.

Eugene P. Gross

Marco Patriarca

By using path integrals, the stochastic process associated to the time evolution of the quantum probability density is formally rewritten in terms of a stochastic differential equation, given by Newton's equation of motion with an additional multiplicative stochastic force. However, the term playing the role of the stochastic force is defined by a non-positive-definite probability functional, providing a clear example of the negative (or "extended") probabilities characteristic of quantum mechanics.

Telesforo López-Ciudad, Rafael Ramı́rez

Interesting approaches to study statical and dynamic properties of quantum systems, e.g., the quantum transition state theory and the centroid molecular dynamics, have been previously derived using fixed centroid path integrals. We show that these constrained propagators can be alternatively defined using an operator formalism. An interesting result is the finding of the differential equations that determine the temperature dependence of these propagators. One equation applies to path integrals with fixed-centroid position (i.e., those used in quantum transition state theory), and the other one to path integrals with fixed-centroid position and momentum (i.e., those used in centroid molecular dynamics). Both equations are solved for a harmonic oscillator, so that the spectral decomposition of the operators represented by fixed-centroid path integrals is derived for this particular case. Their eigenvalues build an alternating geometric series, showing explicitly the impossibility of considering such operators as true density operators, i.e., some eigenfunctions are associated to “negative probabilities.” The eigenfunctions are shown to be a generalization of the coherent and squeezed states of the harmonic oscillator. The physical meaning of centroid molecular dynamics, an approximation to study the time evolution of these mixed states, is clarified by considering the time evolution of the corresponding eigenfunctions. The mixed states constructed with “negative probabilities” display vanishing small position and momentum dispersion in the high temperature limit.

James Clerk