Variations on a Wignerian theme

The Wigner distribution and its equation of motion in the scalar potential case are arrived at in an unusual way. This in turn suggests (a) a departure from the standard Wigner distribution treatment for a charged particle in a magnetic field and (b) a new approach to quantization of nonconservative systems. Suggestion (a) is found to be, like the standard treatment, in agreement with Schrödinger's equation but, unlike it, also satisfies local classical-type conservation laws and employs a distribution which is gauge-invariant rather than merely gauge-covariant. Suggestion (b) gives a clear result only in the case of resistance proportional to velocity, when it agrees with the Schrödinger-Langevin equation; for other dissipative systems a fresh assumption is required, and a proposal in that direction is put forward.     

Prediction for Two Processes and the Nehari Problem

We exploit an analogy between the trigonometric moment problem and prediction theory for a stationary stochastic process. Extending this theory, we show how to use correlations between two processes to predict one from the other. In turn, this gives rise to a simple and unified treatment of the Caratheodory and Nehari moment problems.     

Empirical two-point correlation functions

Let A ₁ , A ₂ , A ₃ A ₄ be four observables, the compatible observables among them being (A ₁ , A ₃ ), (A ₁ , A ₄ ), (A ₂ , A ₃ ), (A ₂ , A ₄ ). In order that the empirical data be reproducible by a quantum or a classical theory, the two-point correlation functions $$\{ C_{ij} = \left\langle {A_i A_j } \right\rangle :i,j a compatible pair\} $$ must necessarily satisfy $$|X_{13} X_{14} - X_{23} X_{24} | \leqslant \left( {1 - X_{13} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - X_{14} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \left( {1 - X_{23} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - X_{24} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} (*)$$ where X_ij=C_ijC _ii ^?1/2 C _jj ^?1/2 . In the case ofGaussian data, this inequality is alsosufficient; If (*) holds, there is a Gaussian joint distribution for A ₁ , A ₂ , A ₃ , A ₄ which reproduces the Gaussian data for compatible pairs. It follows that Bell's inequality is satisfied by all true-false propositions about the Gaussian data. A further consequence of the analysis is thatquantum Gaussian fields satisfy Bell's inequality for all true-false propositions aboutfield measurements. The maximum violation of (*) corresponds to Rastall's example in the case of two-valued observables.