Classical trajectories and quantum field theory

Abstract

The density matrix and the Wigner function formalism requires the doubling of the degrees of freedom in quantum mechanics (QM) and quantum field theory (QFT). The doubled degrees of freedom play the role of the thermal bath or environment degrees of freedom and are entangled with the system degrees of freedom. They also account for quantum noise in the fluctuating random forces in the system-environment coupling. The algebraic structure of QFT turns out to be the one of the deformed Hopf algebra. In such a frame, the trajectories in the space of the unitarily inequivalent representations of the canonical commutation relations turn out to be classical trajectories and, under convenient conditions, they may exhibit properties typical of classical chaotic trajectories in nonlinear dynamics. The quantum Brownian motion and the two-slit experiment in QM are discussed in connection with the doubling of the degrees of freedom.

Bohm Trajectories and Feynman Paths in Light of Quantum Entropy

“…In synthesis we can say that by Fisher it is possible to characterize the deformations of the geometry in the presence of quantum effects in the space of parameters and to express Bohm's quantum potential as the information channel indicating the modification of the geometry of the configuration space determined by the quantum entropy. Novello, Salim and Falciano [19] have recently proposed a geometrical approach in which the presence of quantum effects is linked with the Weyl length L W = 1 √ R , and thus with the curvature scalar; in analogous way, in the approach proposed by the authors and illustrated in this section (see also [2]), the quantum effects are owed to the microstates characterizing the system under consideration and thus to the quantum entropy (5). Both the Novello, Salim and Falciano's and our approach are realistic models that aim to provide a geometrical framework to quantum mechanics in a Bohmian picture, the one in the context of Weyl integrable space, the other in the context of an entropic background in the condition of Fisher information.…”

“…where Q is the Bohm quantum potential derived from Fisher information as extremal, that is a measure of distance in the entropy space [1]. On the basis of the formalism described by equations (5)- (10), it becomes permissible the following re-reading of the mathematical formalism in non-relativistic Bohmian quantum mechanics: the distribution probability of the wave function determines the functions W k defining the number of microstates of the physical system under consideration, a quantum entropy is fixed by these functions W k given by equations (5), by determining a change of the geometry -with respect to the Euclidean space of classical physics -expressed by the Weyl-like gauge potential (7) and characterized by the deformation of the moments (8). Moreover, on the basis of equation (10), one can interpret Bohm's quantum potential as an information channel determined by the functions W k given by equations (5): these functions W k , and therefore the quantum entropy determine the action of the quantum potential (in the extreme condition of the Fisher information) on the basis of equation 1…”

“…In other words, the concept of trajectory emerges only in appropriate conditions that can be strictly defined in terms of Quantum Field Theory. This is the conceptual limit of any "geometrodynamics" [5,6]. The link between Bohm's quantum potential and Feynman's path integrals is explored here within a new formalism based on quantum entropy as deformation of the geometry of the configuration space.…”

Unification of Quantum and Gravity by Non Classical Information Entropy Space

“…It has to be mentioned here another advantage of Bohm's theory. What is usually defined as "background space" in quantum mechanics is nothing but an "open door" on a typical level of quantum field theory; it is possible to show that the Feynman integral paths can be correlated with Bohm's quantum potential and the same classic concept of "trajectory" emerges from the dynamics of quantum fields [21][22][23][24].…”

Almost‐anywhere theories: Reductionism and universality of emergence

“…14; there are at least two theories deriving from Bohm: see-on the one hand-the ''classical limit'' by Allori and Zanghi [15]., and-on the other hand-the switching from nonlocal to local information in Hiley and Maroney [16]. A particularly interesting and effective approach, based upon Quantum Field Theory, is that of Vitiello [17], which describes the emerging of classical Physics as a dissipative quantum effect.…”