Realistic Derivation of Heisenberg Dynamics

Paul J. Werbos

National Science Foundation

Room 675, Arlington, VA 22203, USA

e-mail: уpwerbos@nsf.gov

Einstein conjectured long ago that much of quantum mechanics might be derived as a statistical formalism describing the dynamics of classical systems. Bell’s Theorem experiments have ruled out complete equivalence between quantum field theory (QFT) and classical field theory (CFT), but an equivalence between dynamics is not only possible but provable in simple bosonic systems. Future extensions of these results might possibly be useful in developing provably finite variations of the standard model of physics.

 

The goal of this paper is to present a new formalism for analyzing the statistical dynamics of “classical fields” – of ODE or PDE derived from a classical Lagrangian density of the usual Hamiltonian form. To keep the notation simple, I will focus on the ODE case with:

                                                                                            (1)

where the state of the system at time  t  is defined by the two classical vectors j , p Î Rⁿ. The generalization to PDE is straightforward.

            This paper will show how a probability distribution or statistical ensemble of possible states, Pr(j, p) may be “encoded” or “mapped” into a “classical density matrix” r, which is dual to the usual field operators of the Heisenberg formulation of QFT. I will then define simple field operators F_j and P_j which have the properties that Tr(F_jr)=<j_j>, Tr(P_jr)=<p_j> and both obey the usual Heisenberg/Liouville dynamics as described by Weinberg. Note that I use angle brackets to denote the expectation value of classical stochastic variables.

            The possible significance and future extension of these results are discussed in paper conclusion.