Equivalence of Classical Statistics and Quantum Dynamics

for Bosonic Field Theories

Paul J. Werbos

National Science Foundation*

Room 675, Arlington, VA 22230, USA

Ludmilla D. Werbos

IntControl, Arlington, VA 22203, USA

 

Years ago, Einstein asserted that it is too soon to give up on modeling the universe as a system of continuous force fields governed by partial differential equations (PDE). He argued that the complex laws of quantum dynamics might be derived someday as the laws of evolution of the statistics of PDE.

љљљљљљљљљљљ Most physicists gave up this hope years ago. They often cite a famous book by J.S. Bell. Bell argues that experiments based on a theorem of Clauser et al (often called "Bell's Theorem") prove that there cannot exist any kind of objective reality at all, or else that the laws of evolution must be nonlocal, leading to elaborate speculations about parallel universes and so on.[1]. Physicists of the Einstein school were stymied by the problem of "closure of turbulence" or "infinite regress" which makes it difficult to compute the statistics predicted by PDE without introducing extraneous apriori assumptions. The usual stoichiometric equations of thermodynamics, which were the foundation of the early work of Prigogine, were originally derived from the approximation of point particles in a free space of near-zero particle density [2]. Prigogine and others encountered a stubborn problem of "infinite regress" from higher moments to lower moments when they tried to start out from the more general assumption of continuous fields and interactions.

љљљљљљљљљљљ A recent paper (published in the International Journal of Bifurcation and Chaos[3], archived at arXiv.org as quant-ph 0309031) suggested that Einstein may have been right after all. It showed how to encode the statistical moments ("correlations") of bosonic PDE or ODE systems into a new mathematical object, the classical density matrix r. It provided a tutorial on field operators, and extensive details for bosonic ODE, ODE derived from the Lagrangian:

where H is the Hamiltonian and j and p ÎRⁿ. It proved that the usual field operators F and P defined in the standard text of Weinberg, applied to the classical density matrix r as in quantum theory (Tr(rF)), yield the classical expected values of j and p, and obey the quantum dynamical law given by Weinberg. It explained how "Bell's Theorem" experiments and the like result from differences in assumptions about measurement rather than differences in dynamics, and discussed the PDE case. It provided an updated version of the Backwards Time Interpretation of quantum mechanics first published in 1973 [4]; according to the updated version[3,5], subtle interactions between time-symmetry at the microscopic level and time-forwards thermodynamic effects at the macroscopic level are essential to a complete understanding of quantum measurement.

љљљљљљљљљљљ Nevertheless, discrepancies between classical statistics and quantum dynamics do exist in the general case. This paper derives the free space master equation governing r. It calculates the classical/quantum discrepancies in predicting the expected time flux <_tg(j,p)> for a general observable g(j,p). It shows that we can easily reformulate the classical field theory (by rescaling the field components j_j and p_j without changing the content of the system) so as to zero out the major discrepancy terms. Similar reformulations mirroring the usual renormalization procedures of quantum theory may zero out the small higher-order discrepancies in the case where the quantum version is renormalizable or "finite" (requirements for mathematical validity of a quantum field theory). Proper scaling of j and p is analogous to proper scaling of data prior to statistical analysis. An alternative method given in the Appendix of this paper may provide a more general