Two-time correlation functions: stochastic and conventional quantum mechanics

An investigation of two-time correlation functions is reported within the framework of (i) stochastic quantum mechanics and (ii) conventional Heisenberg-Schr?dinger quantum mechanics. The spectral functions associated with the two-time electric dipole moment correlation functions are worked out in detail for the case of the hydrogen atom. While the single time averages are identical for stochastic and conventional quantum mechanics, differences arise in the two approaches for multiple time correlation functions.     

Lévy's stochastic area in stochastic mechanics

Using the methods of Hida's white noise calculus, we study Lévy's stochastic area spanned by a two-dimensional stochastic process of relevance in Nelson's stochastic mechanics. An intriguing partial relationship is found with quantum mechanical angular momentum for the associated Schrödinger wave-function.     

Momentum in stochastic quantum mechanics

Momentum is analyzed as a random variable in stochastic quantum mechanics. Arbitrary potential energy functions are considered. The oscillator is presented as an example.     

Doubly stochastic matrices in quantum mechanics

The general set of doubly stochastic matrices of order n corresponding to ordinary nonrelativistic quantum mechanical transition probability matrices is given. Landé's discussion of the nonquantal origin of such matrices is noted. Several concrete examples are presented for elementary and composite angular momentum systems with the focus on the unitary symmetry associated with such systems in the spirit of the recent work of Bohr and Ulfbeck. Birkhoff's theorem on doubly stochastic matrices of order n is reformulated in a geometrical language suitable for application to the subset of quantum mechanical doubly stochastic matrices. Specifically, it is shown that the set of points on the unit sphere in cartesian n!-space is surjective with the set of doubly stochastic matrices of order n. The question is raised, but not answered, as to what is the subset of points of this unit sphere that correspond to the quantum mechanical transition probability matrices, and what is the symmetry group of this subset of matrices.     

Spin correlation in dilute magnetic alloys

The correlation of a magnetic impurity spin with the spin density of the conduction electrons in a dilute magnetic alloy is calculated non-perturbationally on the basis of the Nagaoka theory. It is shown that there are anomalies due to the Kondo effect in the long range behaviour of this correlation which contradicts the bound state interpretation of the Kondo effect. The most interesting detail is the appearance of a non-oscillating contribution to the correlation.     

Spin & statistics in nonrelativistic quantum mechanics,II

Recently a sufficient and necessary condition for Pauli's spin‐statistics connection in nonrelativistic quantum mechanics has been established [1]. The two‐dimensional part of this result is extended to n‐particle systems and reformulated and further simplified in a more geometric language.     

The transitions among classical mechanics,quantum mechanics,and stochastic quantum mechanics

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.     

Quantum mechanics derived from stochastic electrodynamics

The connection between stochastic electrodynamics (SED) and the quantum theory of matter is further explored. The main result is that the Fokker-Planck-like equation of SED can be recast into the form of a Schrödinger equation with radiative corrections, when the system is close to a state of equilibrium. The phase-space distribution can be written as Wigner's pseudo-distribution plus corrections due to the nonlinearity of the external force and to radiative effects. The radiative corrections predicted by the theory, namely the Lamb shift and the decay of excited atomic states, coincide with those predicted by QED. Moreover, the theory offers a clear physical interpretation of these phenomena as due to the coupling of the electric dipole of the system with the zero-point radiation field and to radiation reaction, respectively.     

Finite temperature field from stochastic mechanics

Stochastic mechanics of Nelson when generalized to positive temperature for a scalar field, gives rise to a stochastic field which appears to be a hybrid of euclidean and minkowskian field if the usual value of the diffusion parameter is taken. The stochastic process associated with it is a gaussian non-Markov process. The thermal expectations of this stochastic field fails to satisfy the KMS periodic condition. If the diffusion parameter is allowed to continue analytically to a purely imaginary value, the resulting field can be identified with the usual finite temperature quantum field in minkowskian space-time. The relation of this field with that of thermo field dynamics is discussed.     

Generalized stochastic mechanics on Riemannian manifolds

Dankel's stochastic mechanics on Riemannian manifolds is generalized such that the diffusion constant becomes a free parameter in the theory.     

Spin echo formation in the presence of stochastic dynamics

Spin echo formation in magnetic field gradients in the presence of fast stochastic motion is studied for hyperpolarized 3He gas at different diffusivities. The fast translational motion leads to frequency shifts already during echo formation, which can be described analytically for a linear gradient. Despite complete signal loss at the position of the spin echo itself, considerable intensity can be preserved at an earlier time (sqrt[2]tau rather than 2tau, where tau is the pulse delay). Hence, the phenomenon is designated as a pseudo spin echo.     

Fuzzy amplitude densities and stochastic quantum mechanics

Fuzzy amplitude densities are employed to obtain probability distributions for measurements that are not perfectly accurate. The resulting quantum probability theory is motivated by the path integral formalism for quantum mechanics. Measurements that are covariant relative to a symmetry group are considered. It is shown that the theory includes traditional as well as stochastic quantum mechanics.     

Quantum statistical mechanics on stochastic phase space

Within the context of the theory of stochastic phase spaces, introduced in some earlier papers, a systematic mathematical procedure is developed for expressing quantum mechanical observables as generalized functions on a stochastic phase space. The states in such a theory are normalized, positive semidefinite, continuous functions of the phase space variables, satisfying marginality conditions appropriate to the stochastic nature of the underlying phase space. The action of a general quantum mechanical observable on the state space is then shown to lead in general to formal differential operators of finite or infinite order. Explicit computations of some typical operators are made to illustrate the theory. As a useful practical application, the theory is employed to derive a Bloch equation from which the Husimi transform of the canonical equilibrium state is then computed, after expressing it as an infinite series in powers of .Supported in part by a research grant from the National Research Council of Canada.     

Statistical mechanics of a nonlinear stochastic model

A multivariable Fokker-Planck equation (FPE) is used to investigate the equilibrium and dynamical properties of a nonlinear stochastic model. The model displays a phase transition. The equilibrium distributions are found to be non-Gaussian; the deviation from Gaussian is especially significant near the transition point. To study the nonequilibrium behavior of the model, a self-consistent dynamic mean field (SCDMF) theory is derived and used to transform the FPE to a systematic hierarchy of equations for the cumulant moments of the time-dependent distribution function. These equations are numerically solved for a variety of initial conditions. During the time evolution of the system from an initial unstable equilibrium state to the final equilibrium state, three distinct time stages are found.Supported by a grant from the National Research Council of Canada (to RCD) and by the Sherman Fairchild Foundation (to RZ).Also Sherman Fairchild Distinguished Scholar, 1974–75, at the California Institute of Technology, where the early part of this research was done.     

On reformulating quantum mechanics and stochastic theory

A stochastic theory approach is used to formulate the theory of quantum mechanical motion. Apart from giving a unifying point of view to quantum mechanics and stochastic theory, the new formulation is not limited to forces derivable from a potential. A nonlinear dynamical law is deduced in contradistinction to previous works in whichad hoc linear laws are postulated.     

Relativistic quantum mechanics over stochastic phase space

Stochastic quantum mechanics is a quantum theory in which the basic limitations of real-world measuring instruments, due to their intrinsically quantum nature, are taken into account. Among other things this leads to a new operational definition of space-time, called quantum space-time. Fundamental to this approach is the formulation of quantum mechanics over phase space rather than just over position or momentum space. A concept of extended particle is a natural outgrowth of this development. Gauge and internal symmetry have a natural place within the theory, and preliminary computations combining some old ideas due to Born with more recent ideas on symmetry breaking suggest that the theory could lead to a mass formula compatible with known data on the low-lying baryons.Supported in part by NSERC Grant, No. A8403.     

Spin and quark states in the probability representation of quantum mechanics

A short review of the general principles of constructing tomograms of spin and quark states is presented.     

Spin correlation coefficients for neutron-proton radiative capture

We use a theoretical model for deuteron photodisintegration below the pion production threshold, which has previously been used to calculate the observables for which experimental data is available, in order to calculate the spin correlation coefficients for radiative capture of a polarized neutron beam by a polarized proton target. We give results for the coefficients whose measurement is possible in the near future and explain how such experimental results could improve our understanding of the reaction mechanism. We also comment on exciting measurements of the neutron analyzing power at very low energies.