Stochastic variational method as quantization scheme: Field quantization of the complex Klein–Gordon equation

Abstract

Quantization of electromagnetic fields is investigated in the framework of stochastic variational method (SVM). Differently from the canonical quantization, this method does not require canonical form and quantization can be performed directly from the gauge invariant Lagrangian. The gauge condition is used to choose dynamically independent variables. We verify that, in the Coulomb gauge condition, SVM result is completely equivalent to the traditional result. On the other hand, in the Lorentz gauge condition, SVM quantization can be performed without introducing the indefinite metric. The temporal and longitudinal components of the gauge filed, then, behave as c-number functionals affected by quantum fluctuation through the interaction with charged matter fields. To see further the relation between SVM and the canonical quantization, we quantize the usual gauge Lagrangian with the Fermi term and argue a stochastic process with a negative second order correlation is introduced to reproduce the indefinite metric.

The Hydrodynamic Representation of the Klein-Gordon Equation with Self- Interacting Field

“…As shown by Weiner et al [6], the outputs of the quantum hydrodynamic model agree with the outputs of the Schrödinger problem and, more recently, as shown by Koide and Kodama [7], it agrees with the outputs of the stochastic variational method.…”

The Non-Euclidean Hydrodynamic Klein–Gordon Equation with Perturbative Self-Interacting Field

“…The generalized model to non-Euclidean space-time allows for the determination of the quantum energy impulse tensor density of mesons, for the gravitational equation of quantum mechanical systems.Keywords: quantum hydrodynamic representation; Bhom-Madelung approach; self-interaction; Einstein equation for quantum mechanical system     .As shown by Weiner et al [71], the outputs of the quantum hydrodynamic model agree with the outputs of the Schrödinger problem, but not only for the semi-classical limit or for a single particle [62,63,69]. More recently, Koide and Kodama [72], showed that it agrees with the outputs of the stochastic variational method.…”

“…as a function of the two real variables, | | and S [60,62,63,[69][70][71][72][73]. The model gives rise to classical-like analogy describing the motion of particles' density…”

Theoretical Derivation of the Cosmological Constant in the Framework of the Hydrodynamic Model of Quantum Gravity: Can the Quantum Vacuum Singularity Be Overcome?

“…q,tq , only one form (that minimizes the action) of Λ p . q, tq is possiblein Equation (28) . In order to determine Λ p .…”

“…More recently, Koide and Kodama [28] showed that the hydrodynamic quantum field model can also be obtained by means of the stochastic variational method.…”