We give a pedagogical introduction of the stochastic variational method and show that this generalized variational principle describes classical and quantum mechanics in a unified way.
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 t
he 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.
Stochastic Variational Method (SVM) is the generalization of the variation method to the case with stochastic variables. In the series of papers, we investigate the applicability of SVM as an alternative field quantization scheme. Here, we discuss th
e complex Klein-Gordon equation. In this scheme, the Euler-Lagrangian equation for the stochastic fields leads to the functional Schroedinger equation, which in turn can be interpreted as the ideal fluid equation in the functional space. We show that the Fock state vector is given by the stationary solution of these differential equations and various results in the usual canonical quantization can be reproduced, including the effect of anti-particles. The present formulation is a quantization scheme based on commutable variables, so that there appears no ambiguity associated with the ordering of operators, for example, in the definition of Noether charges.
The microscopic formulae of the bulk viscosity $zeta $ and the corresponding relaxation time $tau_{Pi}$ in causal dissipative relativistic fluid dynamics are derived by using the projection operator method. In applying these formulae to the pionic fl
uid, we find that the renormalizable energy-momentum tensor should be employed to obtain consistent results. In the leading order approximation in the chiral perturbation theory, the relaxation time is enhanced near the QCD phase transition and $tau_{Pi}$ and $zeta $ are related as $tau_{Pi}=zeta /[beta {(1/3-c_{s}^{2})(epsilon +P)-2(epsilon -3P)/9}]$, where $epsilon $, $P$ and $c_{s}$ are the energy density, pressure and velocity of sound, respectively. The predicted $zeta $ and $% tau_{Pi}$ should satisfy the so-called causality condition. We compare our result with the results of the kinetic calculation by Israel and Stewart and the string theory, and confirm that all the three approaches are consistent with the causality condition.
We show that relativistic fluids behave as non-Newtonian fluids. First, we discuss the problem of acausal propagation in the diffusion equation and introduce the modified Maxwell-Cattaneo-Vernotte (MCV) equation. By using the modified MCV equation, w
e obtain the causal dissipative relativistic (CDR) fluid dynamics, where unphysical propagation with infinite velocity does not exist. We further show that the problems of the violation of causality and instability are intimately related, and the relativistic Navier-Stokes equation is inadequate as the theory of relativistic fluids. Finally, the new microscopic formula to calculate the transport coefficients of the CDR fluid dynamics is discussed. The result of the microscopic formula is consistent with that of the Boltzmann equation, i.e., Grads moment method.
The transport coefficients of causal relativistic dissipative fluid dynamics are calculated both in a field-theoretical and a kinetic approach. We find that the results from the traditional kinetic calculation by Israel and Stewart are modified. The
new expressions for the viscous transport coefficients agree with the results obtained in the field-theoretical approach when the contributions from pair creation and annihilation are neglected.
We extended our formulation of causal dissipative hydrodynamics [T. Koide textit{et al.}, Phys. Rev. textbf{C75}, 034909 (2007)] to be applicable to the ultra-relativistic regime by considering the extensiveness of irreversible currents. The new equa
tion has a non-linear term which suppresses the effect of viscosity. We found that such a term is necessary to guarantee the positive definiteness of the inertia term and stabilize numerical calculations in ultra-relativistic initial conditions. Because of the suppression of the viscosity, the behavior of the fluid is more close to that of the ideal fluid. Our result is essentially same as that from the extended irreversible thermodynamics, but is different from the Israel-Stewart theory. A possible origin of the difference is discussed.
A new formula to calculate the transport coefficients of the causal dissipative hydrodynamics is derived by using the projection operator method (Mori-Zwanzig formalism) in [T. Koide, Phys. Rev. E75, 060103(R) (2007)]. This is an extension of the Gre
en-Kubo-Nakano (GKN) formula to the case of non-Newtonian fluids, which is the essential factor to preserve the relativistic causality in relativistic dissipative hydrodynamics. This formula is the generalization of the GKN formula in the sense that it can reproduce the GKN formula in a certain limit. In this work, we extend the previous work so as to apply to more general situations.
The stability and causality of the Landau-Lifshitz theory and the Israel-Stewart type causal dissipative hydrodynamics are discussed. We show that the problem of acausality and instability are correlated in relativistic dissipative hydrodynamics and
instability is induced by acausality. We further discuss the stability of the scaling solution. The scaling solution of the causal dissipative hydrodynamics can be unstable against inhomogeneous perturbations.
We studied the shock propagation and its stability with the causal dissipative hydrodynamics in 1+1 dimensional systems. We show that the presence of the usual viscosity is not enough to stabilize the solution. This problem is solved by introducing a
n additional viscosity which is related to the coarse-graining scale of the theory.
