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Physical Review Letters

New Interpretation of the Euclidean-Markov Field in the Framework of Physical Minkowski Space-Time

Franesco Guerra, Patrizia Ruggiero

Published Oct 15, 1973• 159 citations
Publisher

Abstract

The Euclidean-Markov field, extensively used in constructive quantum field theory, is shown to be the same as the lowest energy generalized stochastic process associated with classical field theory through the procedure of Nelson's stochastic mechanics. As a result, the underlying four-dimensional manifold, on which the Markov field is defined, can be considered as the physical space-time.

DOI
10.1103/physrevlett.31.1022

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The Euclidean-Markov field, extensively used in constructive quantum field theory, is shown to be the same as the lowest energy generalized stochastic process associated with classical field theory through the procedure of Nelson's stochastic mechanics. As a result, the underlying four-dimensional manifold, on which the Markov field is defined, can be considered as the physical space-time.
A new idea for the quantization of dynamic systems, as well as space time itself, using a stochastic metric is proposed. The quantum mechanics of a mass point is constructed on a space time manifold using a stochastic metric. A stochastic metric space is, in brief, a metric space whose metric tensor is given stochastically according to some appropriate distribution function. A mathematically consistent model of a space time manifold equipping a stochastic metric is proposed in this work. The quantum theory in the local Minkowski space can be recognized as a classical theory on the stochastic Lorentz-metric-space. A stochastic calculus on the space time manifold is performed using white noise functional analysis. A path-integral quantization is introduced as a stochastic integration of a function of the action integral, and it is shown that path-integrals on the stochastic metric space are mathematically well-defined for large variety of potential functions. The Newton–Nelson equation of motion can also be obtained from the Newtonian equation of motion on the stochastic metric space. It is also shown that the commutation relation required under the canonical quantization is consistent with the stochastic quantization introduced in this work. The quantum effects of general relativity are also analyzed through natural use of the stochastic metrics. Some example of quantum effects on the Universe is discussed.
The state-of-the-art physics consists of two irreconcilable branches, i.e., the quantum theory and the general relativity, which work well in their own territories, independently. However, what are quantum and spacetime after all? The key question was never addressed, satisfactorily. In this work, we describe a possibility to reformulate the quantum theory in the Minkowski spacetime from the viewpoint of classic physics in the Euclidean spacetime, i.e., classic mechanics and stochastic process theory. We show that quantum theory and Minkowski spacetime may connect with each other and emerge from a single fundamental entity.
Markov diffusions corresponding to solutions of the Klein-Gordon equation are introduced in a similar way as in Nelson's stochastic mechanics. A relativistic version of the Nelson-Newton law for Markov diffusions is established. Stochastic generalizations of the proper time and of the relativistic action functional are proposed and a stochastic version of the relativistic principle of least action is stated. The formalism can be regarded as a quantization procedure leading from classical special relativity to the Klein-Gordon equation.
The relation between quantum electrodynamics and (classical) stochastic electrodynamics is elucidated by means of a general construction which associates with every Gaussian quantum field (for example, vacuum-free fields or coherent states at zero or nonzero temperature) a classical random field, which in the case of quantum electrodynamics yields stochastic electrodynamics. The associated classical and quantum theories yield the same results for all nondisturbing (causally separated) field measurements. This is an additional classical characteristic of coherent quantum states which implies that Bell's inequality is satisfied by quantum field measurements. Furthermore, stochastic electrodynamics is identified as the Wigner distribution of quantum electrodynamics. We point out that in these Gaussian states stochastic electrodynamics is more suitable as a classical model of the quantum field than is the Euclidean field, which results from applying Nelson's stochastic mechanics to each field mode.