Heisenberg's uncertainty relation as a case of stochastic dependence

Hennadii Kovalov

I. Fenyes

Timur E. Gureyev, Alexander Kozlov, David M. Paganin +2 more

We show that the width of an arbitrary function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. In the case of ergodic stochastic processes, an ensuing uncertainty relationship is then demonstrated for the product of correlation length and variance. A closely related uncertainty principle is also established for the average degree of fourth-order coherence and the spatial width of modes of bosonic quantum fields. However, it is shown that, in the case of stochastic and quantum observables, certain non-classical states with sub-Poissonian statistics, such as for example photon number squeezed states in quantum optics, can overcome the "classical" noise-resolution uncertainty limit. This uncertainty relationship, which is fundamentally different from the Heisenberg and related uncertainty principles, can define an upper limit for the information capacity of communication and imaging systems. It is expected to be useful in a variety of problems in classical and quantum optics and imaging.

Hennadii Kovalov

Stephan Völlinger

The Schrödinger equation is deduced from a geometric principle. Lagrange, Hamilton-Jacobi, and Hamilton formalisms are defined almost analogously to the deterministic case, which can be identified as geometric optic. The form of these formalisms is identical to the deterministic formalisms. Furthermore, it will be shown how the deterministic case is “superficial” to the stochastic one if Planck’s constant is very small. An elementary proof of Heisenberg’s uncertainty relation finishes the paper.

Paul Busch, Brigitte Falkenburg