Quantum Properties of a Field with Time as a Dynamical Variable

The distributed system $\mathcal{S}_D$ described by the Dirac equation is investigated simply as a dynamic system, i.e. without usage of quantum principles. The Dirac equation is described in terms of hydrodynamic variables: 4-flux $j^{i}$ , pseudo-vector of the spin $S^{i}$ , an action $\hbar ϕ$ and a pseudo-scalar $κ$ . In the quasi-uniform approximation, when all transversal derivatives (orthogonal to the flux vector $j^i$ ) are small, the system $\mathcal{S}_D$ turns to a statistical ensemble of classical concentrated systems $\mathcal{S}_{dc}$ . Under some conditions the classical system $\mathcal{S}_{dc}$ describes a classical pointlike particle moving in a given electromagnetic field. In general, the world line of the particle is a helix, even if the electromagnetic field is absent. Both dynamic systems $\mathcal{S}_D$ and $\mathcal{S}_{dc}$ appear to be non-relativistic in the sense that the dynamic equations written in terms of hydrodynamic variables are not relativistically covariant with respect to them, although all dynamic variables are tensors or pseudo-tensors. They becomes relativistically covariant only after addition of a constant unit timelike vector $f^{i}$ which should be considered as a dynamic variable describing a space-time property. This "constant" variable arises instead of $γ$ -matrices which are removed by means of zero divizors in the course of the transformation to hydrodynamic variables. It is possible to separate out dynamic variables $κ$ , $κ^i$ responsible for quantum effects. It means that, setting $κ,κ^i\equiv 0$ , the dynamic system $\mathcal{S}_D$ described by the Dirac equation turns to a statistical ensemble $\mathcal{E}_{Dqu}$ of classical dynamic systems $\mathcal{S}_{dc}$ .The distributed system $\mathcal{S}_D$ described by the Dirac equation is investigated simply as a dynamic system, i.e. without usage of quantum principles. The Dirac equation is described in terms of hydrodynamic variables: 4-flux $j^{i}$, pseudo-vector of the spin $S^{i}$, an action $\hbar ϕ$ and a pseudo-scalar $κ$. In the quasi-uniform approximation, when all transversal derivatives (orthogonal to the flux vector $j^i$) are small, the system $\mathcal{S}_D$ turns to a statistical ensemble of classical concentrated systems $\mathcal{S}_{dc}$. Under some conditions the classical system $\mathcal{S}_{dc}$ describes a classical pointlike particle moving in a given electromagnetic field. In general, the world line of the particle is a helix, even if the electromagnetic field is absent. Both dynamic systems $\mathcal{S}_D$ and $\mathcal{S}_{dc}$ appear to be non-relativistic in the sense that the dynamic equations written in terms of hydrodynamic variables are not relativistically covariant with respect to them, although all dynamic variables are tensors or pseudo-tensors. They becomes relativistically covariant only after addition of a constant unit timelike vector $f^{i}$ which should be considered as a dynamic variable describing a space-time property. This "constant" variable arises instead of $γ$-matrices which are removed by means of zero divizors in the course of the transformation to hydrodynamic variables. It is possible to separate out dynamic variables $κ$, $κ^i$ responsible for quantum effects. It means that, setting $κ,κ^i\equiv 0$, the dynamic system $\mathcal{S}_D$ described by the Dirac equation turns to a statistical ensemble $\mathcal{E}_{Dqu}$ of classical dynamic systems $\mathcal{S}_{dc}$.

arXiv:1101.5868Jan 31, 20117 citations

Dynamics of quantum fields

Jan Dereziński, Christian Gérard

Cambridge University Press eBooksMar 6, 20130 citations

Dynamic Probability Field Quantum Variable Analysis

Jincheng Zhang

This paper presents a novel approach to quantum variable analysis within dynamic probability fields. The core aim is to develop a method that accurately determines the distribution of quantum variables based on the inherent properties of dynamic probability fields. This approach offers a significant advancement over traditional methods, particularly in the context of quantum computation, by leveraging the dynamic nature of these fields to provide a more comprehensive and nuanced understanding of quantum variable behavior. The paper details the methodology, outlining the key steps involved in establishing the framework for this analysis. We demonstrate the effectiveness of this method through a focused analysis of a simplified quantum system, illustrating its potential to uncover unexpected patterns and provide valuable insights into the quantum world. The resulting analysis reveals a refined understanding of the interplay between quantum variables and the dynamic probability field.This paper presents a novel approach to quantum variable analysis within dynamic probability fields. The core aim is to develop a method that accurately determines the distribution of quantum variables based on the inherent properties of dynamic probability fields. This approach offers a significant advancement over traditional methods, particularly in the context of quantum computation, by leveraging the dynamic nature of these fields to provide a more comprehensive and nuanced understanding of quantum variable behavior. The paper details the methodology, outlining the key steps involved in establishing the framework for this analysis. We demonstrate the effectiveness of this method through a focused analysis of a simplified quantum system, illustrating its potential to uncover unexpected patterns and provide valuable insights into the quantum world. The resulting analysis reveals a refined understanding of the interplay between quantum variables and the dynamic probability field.

Aug 31, 20260 citations

Exact time-evolution of a generalized two-dimensional quantum parametric oscillator in the presence of time-variable magnetic and electric fields

Şirin A. Büyükaşık, Zehra Çayiç

The time-dependent Schrödinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei–Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin–Man’ko–Trifonov and the Lewis–Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy–Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.The time-dependent Schrödinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei–Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin–Man’ko–Trifonov and the Lewis–Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy–Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.

Journal of Mathematical PhysicsJul 1, 20221 citation

QUANTUM PROPERTIES OF THE FIELD

WORLD SCIENTIFIC eBooksMar 1, 20070 citations

The distributed system

\mathcal{S}_D

described by the Dirac equation is investigated simply as a dynamic system, i.e. without usage of quantum principles. The Dirac equation is described in terms of hydrodynamic variables: 4-flux

j^{i}

, pseudo-vector of the spin

S^{i}

, an action

\hbar ϕ

and a pseudo-scalar

κ

. In the quasi-uniform approximation, when all transversal derivatives (orthogonal to the flux vector

j^i

) are small, the system

\mathcal{S}_D

turns to a statistical ensemble of classical concentrated systems

\mathcal{S}_{dc}

. Under some conditions the classical system

\mathcal{S}_{dc}

describes a classical pointlike particle moving in a given electromagnetic field. In general, the world line of the particle is a helix, even if the electromagnetic field is absent. Both dynamic systems

\mathcal{S}_D

and

\mathcal{S}_{dc}

appear to be non-relativistic in the sense that the dynamic equations written in terms of hydrodynamic variables are not relativistically covariant with respect to them, although all dynamic variables are tensors or pseudo-tensors. They becomes relativistically covariant only after addition of a constant unit timelike vector

f^{i}

which should be considered as a dynamic variable describing a space-time property. This "constant" variable arises instead of

γ

-matrices which are removed by means of zero divizors in the course of the transformation to hydrodynamic variables. It is possible to separate out dynamic variables

κ

κ^i

responsible for quantum effects. It means that, setting

κ,κ^i\equiv 0

, the dynamic system

\mathcal{S}_D

described by the Dirac equation turns to a statistical ensemble

\mathcal{E}_{Dqu}

of classical dynamic systems

\mathcal{S}_{dc}