The way to determine the renormalized energy of inhomogeneous systems of a quantum field under an external potential is established for both equilibrium and nonequilibrium scenarios based on thermo field dynamics. The key step is to find an extension of the on-shell concept valid in homogeneous case. In the nonequilibrium case, we expand the field operator by time-dependent wavefunctions that are solutions of the appropriately chosen differential equation, synchronizing with temporal change of thermal situation, and the quantum transport equation is derived from the renormalization procedure. Through numerical calculations of a triple-well model with a reservoir, we show that the number distribution and the time-dependent wavefunctions are relaxed consistently to the correct equilibrium forms at the long-term limit.

© 2016 Elsevier Inc.The formulation for zero mode of a Bose–Einstein condensate beyond the Bogoliubov approximation at zero temperature [Y. Nakamura et al., Phys. Rev. A 89 (2014) 013613] is extended to finite temperature. Both thermal and quantum fluctuations are considered in a manner consistent with a concept of spontaneous symmetry breakdown for a finite-size system. Therefore, we need a proper treatment of the zero mode operators, which invoke non-trivial enhancements in depletion condensate and thermodynamical quantities such as the specific heat. The enhancements are visible in the weak interaction case. Our approach reproduces the results of a homogeneous system in the Bogoliubov approximation in a large particle number limit.

１２Cの3アルファ粒子構造のホイル状態がボーズ凝縮状態であり、南部ゴールドストーン、ヒッグスモードの状態が実在すること、２＋及び４＋状態は回転バンド状態ではなく、凝縮体の振動状態であることを示した。

© 2016 American Physical Society.An effective field theory of α-cluster condensation is formulated as a spontaneously broken symmetry in quantum field theory to understand the raison d'être and the nature of the Hoyle and α-cluster states in C12. The Nambu-Goldstone and Higgs mode operators in infinite systems are replaced with a pair of canonical operators whose Hamiltonian gives rise to discrete energy states in addition to the Bogoliubov-de Gennes excited states. The calculations reproduce well the experimental spectrum of the α-cluster states. The existence of the Nambu-Goldstone-Higgs states is demonstrated and crucial. The γ-decay transitions are also obtained.

The dynamical instability of weakly interacting two-component Bose-Einstein condensates with coaxial quantized vortices is analytically investigated in a two-dimensional isotopic harmonic potential. We examine whether complex eigenvalues appear on the Bogoliubov-de Gennes equation, implying dynamical instability. Rather than solving the Bogoliubov-de Gennes equation numerically, we rely on a perturbative expansion with respect to the coupling constant which enables a simple, analytic approach. For each pair of winding numbers and for each magnetic quantum number, the ranges of intercomponent coupling constant where the system is dynamically unstable are exhaustively obtained. Corotating and counter-rotating systems show distinctive behaviors. The latter is much more complicated than the former with respect to dynamical instability, particularly because radial excitations contribute to complex eigenvalues in counter-rotating systems.

© 2016 American Physical Society.The dynamical instability of weakly interacting two-component Bose-Einstein condensates with coaxial quantized vortices is analytically investigated in a two-dimensional isotopic harmonic potential. We examine whether complex eigenvalues appear on the Bogoliubov-de Gennes equation, implying dynamical instability. Rather than solving the Bogoliubov-de Gennes equation numerically, we rely on a perturbative expansion with respect to the coupling constant which enables a simple, analytic approach. For each pair of winding numbers and for each magnetic quantum number, the ranges of intercomponent coupling constant where the system is dynamically unstable are exhaustively obtained. Corotating and counter-rotating systems show distinctive behaviors. The latter is much more complicated than the former with respect to dynamical instability, particularly because radial excitations contribute to complex eigenvalues in counter-rotating systems.

<p>近年、光共振器内に冷却原子気体によるBose-Einstein凝縮体を入れた系が実験的に調べられている。特に、光共振器中のフォトンと凝縮体との相互作用による超流動相-超固体相転移が実現され、注目を集めている。本講演ではこの系の場の量子論に基づいた定式化と量子ゆらぎの解析について報告する。</p>

To formulate the zero modes in a finite-size system with spontaneous breakdown of symmetries in quantum field theory is not trivial, for in the naive Bogoliubov theory, one encounters difficulties such as phase diffusion, the absence of a definite criterion for determining the ground state, and infrared divergences. An interacting zero mode formulation that has been proposed for systems with a single zero mode to avoid these difficulties is extended to general systems with multiple zero modes. It naturally and definitely gives the interactions among the quantized zero modes, the consequences of which can be observed experimentally. In this paper, as a typical example, we consider an atomic Bose-Einstein condensed system with a dark soliton that contains two zero modes corresponding to the spontaneous breakdown of the U(1) gauge and translational symmetries. Then we evaluate the standard deviations of the zero mode operators and see how the mutual interaction between the two zero modes affects them.

A complex eigenvalue in the Bogoliubov-de Gennes equations for a stationary Bose-Einstein condensate in the ultracold atomic system indicates the dynamical instability of the system. We also have the modes with zero eigenvalues for the condensate, called the zero modes, which originate from the spontaneous breakdown of symmetries. Although the zero modes are suppressed in many theoretical analyses, we take account of them in this paper and argue that a zero mode can change into one with a pure imaginary eigenvalue by applying a symmetry breaking external perturbation potential. This emergence of a pure imaginary mode adds a new type of scenario of dynamical instability to that characterized by the complex eigenvalue of the usual excitation modes. For illustration, we deal with two one-dimensional homogeneous Bose-Einstein condensate systems with a single dark soliton under a respective perturbation potential, breaking the invariance under translation, to derive pure imaginary modes. (C) 2014 Elsevier Inc. All rights reserved.

We analyze an ultracold fermionic atom system in a three-dimensional optical lattice with a confinement harmonic potential, using the Hubbard model and a time-dependent Gutzwiller variational approach to numerical calculation. Our study is focused on the time evolution of particle transfer when the lattice potential is modulated by adding a periodic one. The choice of parameters, such as the modulation frequency, modulation amplitude, and number of particles, affects particle transfer. We calculate the time evolution of the variance in the particle distribution and show its dependences on the parameters. Lattice modulation is found to effectively control particle transfer, and is a useful method in experiments on fermionic atom systems.

The unperturbed Hamiltonian for the Bose-Einstein condensate, which includes not only the first and second powers of the zero mode operators but also the higher ones, is proposed to determine a unique and stationary vacuum at zero temperature. From the standpoint of quantum field theory, it is done in a consistent manner that the canonical commutation relation of the field operator is kept. In this formulation, the condensate phase does not diffuse and is robust against the quantum fluctuation of the zero mode. The standard deviation for the phase operator depends on the condensed atom number with the exponent of -1/3, which is universal for both homogeneous and inhomogeneous systems. © 2014 American Physical Society.

The 2 x 2-matrix structure of Green's functions is a common feature for the real-time formalisms of quantum field theory under thermal situations, such as the closed time path formalism and Thermo Field Dynamics (TFD). It has been believed to originate from quantum nature. Recently, Galley has proposed the Hamilton's principle with initial data for nonconservative classical systems, doubling each degree of freedom [1]. We show that the Galley's Hamilton formalism can be extended to quantum field and that the resulting theory is naturally identical with nonequilibrium TFD. (C) 2013 Elsevier B.V. All rights reserved.

Emphasizing that the specification of the representation space or the quasiparticle picture is essential in nonequilibrium quantum field system, we have constructed the unique unperturbed representation of the interaction picture in the superoperator formalism. To achieve it, we put the three basic requirements (the existence of the quasiparticle picture at each instant of time, the macroscopic causality and the relaxation to equilibrium). From the resultant representation follows the formulation of nonequilibrium Thermo Field Dynamics (TFD). The two parameters, the number distribution and excitation energy, characterizing the representation, are to be determined by the renormalization condition. While we point out that the diagonalization condition by Chu and Umezawa is inconsistent with the equilibrium theory, we propose a new renormalization condition as a generalization of the on-shell renormalization on the self-energy which derives the quantum transport equation and determines the renormalized excitation energy. © 2012 Elsevier Inc.

A relativistic neutral scalar held is investigated in non-equilibrium thermo field dynamics. The canonical quantization is applied to the fields out of equilibrium. Because the thermal Bogoliubov transformation becomes time-dependent, the equations of motion for the ordinary unperturbed creation and annihilation operators are modified. This forces us to introduce a thermal counter term in the interaction Hamiltonian which generates additional radiative corrections. Imposing the self-consistency renormalization condition on the total radiative corrections, we obtain the quantum Boltzmann equation for the relativistic scalar field.

We study the superfluid-Mott insulator (SF-MI) transition in a two-dimensional optical lattice system and employ the Bose-Hubbard model in three dimensions with a combined potential of an optical lattice in two directions and a confining harmonic trap in the other direction, which we refer to as the pseudo-two-dimensional Bose-Hubbard model. Some excited states with respect to the harmonic trap are taken into account in this paper. The Mott lobes shrink in the mu and J directions of the mu-J phase diagram. The shrinkage occurs because the interactions involving the excited states become weaker than that between particles in the ground state. The dispersion of the in-site particle number increases because the energy spacing between the eigenstates of the Hamiltonian decreases at finite temperature. The presence of the excited states significantly affects the robustness of the MI phase at finite temperature.

We present an extension of Nelson&apos;s stochastic quantum mechanics to finite temperature. Utilizing the formulation of Thermo Field Dynamics (TFD), we can show that Ito&apos;s stochastic equations for tilde and non-tilde particle positions reproduce the TFD-type Schrodinger equation which is equivalent to the Liouville-von Neumann equation. In our formalism, the drift terms in the Ito&apos;s stochastic equation have the temperature dependence and the thermal fluctuation is induced through the correlation of the non-tilde and tilde particles. We show that our formalism satisfies the position-momentum uncertainty relation at finite temperature. (C) 2011 Elsevier B.V. All rights reserved.

The coupled equations which describe the temporal evolution of the Bose-Einstein condensed system are derived in the framework of nonequilibrium Thermo Field Dynamics. The key element is that they are not the naive assemblages of assumed equations, but are the self-consistent ones derived by appropriate renormalization conditions. While the order parameter is time-dependent, an explicit quasiparticle picture is constructed by a time-dependent expansion. Our formulation is valid even for the system with a unstable condensate, and describes the condensate decay caused by the Landau instability as well as by the dynamical one. (C) 2010 Elsevier Inc. All rights reserved.

The non-Markovian transport equations for the systems of cold Bose atoms confined by a external potential both without and with a Bose-Einstein condensate are derived in the framework of non-equilibrium thermal field theory (Thermo Field Dynamics). Our key elements are an explicit particle representation and a self-consistent renormalization condition which are essential in thermal field theory. The non-Markovian transport equation for the non-condensed system, derived at the two-loop level, is reduced in the Markovian limit to the ordinary quantum Boltzmann equation derived in the other methods. For the condensed system, we derive a new transport equation with an additional collision term which becomes important in the Landau instability. (C) 2009 Elsevier Inc. All rights reserved.

We study the dynamics of a trapped Bose-Einstein condensate with a multiply-quantized vortex, and investigate the roles of the fluctuations in the dynamical evolution of the system. Using the perturbation theory of the external potential, and assuming the situation of the small coupling constant of self-interaction, we analytically solve the time-dependent Gross-Pitaevskii equation. We introduce the zero mode and its adjoint mode of the Bogoliubov-de Gennes equations. Those modes are known to be essential for the completeness condition. We confirm how the complex eigenvalues induce the vortex splitting. It is shown that the physical role of the adjoint zero mode is to ensure the conservation of the total condensate number. The contribution of the adjoint mode is exponentially enhanced in synchronism with the exponential growth of the complex mode, and is essential in the vortex splitting. (C) 2009 Elsevier Inc. All rights reserved.

We study the dynamics of Bose-Einstein condensates flowing in optical lattices on the basis of quantum field theory. For such a system, a Bose-Einstein condensate shows an unstable behavior which is called the dynamical instability. The unstable system is characterized by the appearance of modes with complex eigenvalues. Expanding the field operator in terms of excitation modes including complex ones, we attempt to diagonalize the unperturbative Hamiltonian and to find its eigenstates. It turns out that although the unperturbed Hamiltonian is not diagonalizable in the conventional bosonic representation the appropriate choice of physical states leads to a consistent formulation. Then we analyze the dynamics of the system in the regime of the linear response theory. Its numerical results are consistent with those given by the discrete nonlinear Schrodinger equation. (c) 2007 Elsevier Inc. All rights reserved.

The condition for the appearance of dynamical instability of the Bose-condensed system, characterized by the emergence of complex eigenvalues in the Bogoliubov-de Gennes equations, is studied analytically. It is concluded that the degeneracy between a positive-norm eigenmode and a negative-norm one is essential for the emergence of complex modes. Based on the conclusion, we justify the two-mode approximation applied in our previous work [E. Fukuyama , Phys. Rev. A 76, 043608 (2007)], in which we analytically studied the condition for the existence of complex modes when the condensate has a highly quantized vortex.

We consider a trapped Bose-Einstein condensate (BEC) with a highly quantized vortex. For the BEC with a doubly, triply, or quadruply quantized vortex, the numerical calculations have shown that the Bogoliubov-de Gennes equations, which describe the fluctuation of the condensate, have complex eigenvalues. In this paper, we obtain the analytic expression of the condition for the existence of complex modes, using the method developed by Rossignoli and Kowalski [R. Rossignoli and A. M. Kowalski, Phys. Rev. A 72, 032101 (2005)] for the small coupling constant. To derive it, we make the two-mode approximation. With the derived analytic formula, we can identify the quantum numbers of the complex modes for each winding number of the vortex. Our result is consistent with those obtained by the numerical calculation in the case that the winding number is two, three, or four. We prove that the complex modes always exist when the condensate has a highly quantized vortex.

7 The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo's linear. response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation. (c) 2007 Elsevier Inc. All rights reserved.

We consider a trapped Bose-Einstein condensate with a highly quantized vortex. Pu et al. found numerically the parameter region in which complex eigenvalues arise. Recently, the splitting of a highly quantized vortex into two singly quantized vortices is observed in the experiment. We derive analytically the condition for the existence of complex eigenvalues by using the small coupling constant expansion and the two-mode approximation. We check that our results agree with those by Pu et al.

The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. We consider the case in which these equations have complex eigenvalues. We give the complete set including a pair of complex modes whose eigenvalues are complex conjugate to each other. The expansion of the quantum fields which represent neutral atoms in terms of the complete set brings about the operators associated with the complex modes, which are simply neither bosonic nor fermionic ones. The eigenstate of the Hamiltonian is given. Introducing the notion of the physical states, we discuss the instability of the condensates in the context of Kubo's linear response theory.

The Bogoliubov-de Gennes equations are used for a number of theoretical works to describe quantum and thermal fluctuations of trapped Bose-Einstein condensates. We consider the case in which the condensate has a highly quantized vortex. It is known that these equations have complex eigenvalues in this case. We give the complete set including a pair of complex modes whose eigenvalues are complex conjugates to each other. The expansion of the quantum fields which represent neutral atoms in terms of the complete set brings the operators associated with the complex modes, which are simply neither bosonic nor fermionic ones. The eigenstate of the Hamiltonian is given. Introducing the notion of the physical states, we discuss the instability of the condensates in the context of Kubo's linear response theory.

We construct an approximate scheme based on the concept of the spontaneous symmetry breakdown, satisfying the Goldstone theorem, for finite volume Bose-Einstein condensed gases in both zero and finite temperature cases. In this paper, we discuss the Bose-Einstein condensation in a box with periodic boundary condition and do not assume the thermodynamic limit. When energy spectrum is discrete, we found that it is necessary to deal with the Nambu-Goldstone mode explicitly without the Bogoliubov's prescription, in which zero-mode creation- and annihilation-operators are replaced with a c-number by hand, for satisfying the Goldstone theorem. Furthermore, we confirm that the unitarily inequivalence of vacua in the spontaneous symmetry breakdown is true for the finite volume system. (c) 2006 Elsevier Inc. All rights reserved.

We approach the problem of the Bose-Einstein condensation of neutral atoms in an external trap, which is a finite system in size, from the viewpoint of quantum field theory. The phenomenon is considered as a spontaneous symmetry breakdown. The main result of this paper is that the vacua for this system with finite size and finite number of trapped neutral atoms, parametrized by different phases of the order parameter, are orthogonal to each other, namely the Fock spaces associated with each vacuum are unitarily inequivalent to each other. In the proof, the zero-energy mode which is the Nambu-Goldstone mode, and is discrete due to the presence of the trap, plays a crucial role. We introduce the discrete zero-energy mode in the formulation of the generalized Bogoliubov transformation and introduce an additional infinitesimal symmetry-breaking term in the Hamiltonian to control the zero-energy mode properly. (c) 2005 Elsevier B.V. All rights reserved.

The dynamical response of a trapped Bose-Einstein condensate (BEC) is formulated consistently with quantum field theory and is numerically evaluated. We regard the BEC as a manifestation of the breaking of the global phase symmetry. Then, the Goldstone theorem implies the existence of a zero energy excitation mode (the zero-mode). We calculate the effect of the zero-mode on the response frequency and show that the contribution of the zero-mode to the first excitation mode is not so important in the parameter set realized in the existing experiment. This is the reason that experimental results can be described using the Bogoliubov prescription, although it breaks the consistency of the description in quantum field theory.

The response of the trapped Bose-Einstein condensate (BEC) is investigated. We regard the BEC as a manifestation of the spontaneous breakdown of the global phase symmetry. Then, the Goldstone theorem leads to the existence of the zero energy excitation mode (zero-mode). We calculate the effect of the zero-mode to the response frequency and show that the contribution of the zero-mode to the first excitation mode becomes dominant as the temperature and/or the coupling constant are increased.

The two approaches of consistent quantum field theory for systems of the trapped Bose-Einstein condensates are known, one is the Bogoliubov-de Gennes approach and the other is the generalized Bogoliubov approach. In this paper, we investigate the relation between the two approaches and show that they are formally equivalent to each other. To do this one must carefully treat the Nambu-Goldstone mode which plays a crucial role in the condensation. It is emphasized that the choice of vacuum is physically relevant. (C) 2005 American Institute of Physics.

We prove that the Ward-Takahashi relations at finite temperature are satisfied at the tree level for a system of a trapped Bose-Einstein condensate when the zero mode is included in the quasi-particle picture property. This implies that the zero mode represents the Nambu-Goldstone mode and that it is introduced in a way consistent with the Goldstone theorem. In the proof, we employ thermo field dynamics which is a real-time operator formalism of quantum field theory at finite temperature. (C) 2004 Published by Elsevier B.V.

The effects of quantum coordinates associated with the Nambu-Goldstone mode in a trapped Bose-Einstein condensate are investigated at zero and finite temperatures. We develop a loop expansion and perform numerical calculations over a wide range of temperature, i.e., between zero and the critical temperatures. Our results are compared with those obtained in cases without the Nambu-Goldstone mode (the Bogoliubov approximation) and with the Nambu-Goldstone mode as a zero-energy particle mode.

The system of the trapped Bose-Einstein condensation is investigated from the viewpoint of Quantum Field Theory. It is pointed out that the Nambu-Goldstone mode, which is a discrete mode due to the trapping potential, creates observable effects through fluctuations. The way how the Nambu-Goldstone mode is treated is not unique. The introduction of quantum coordinates is a possible candidate.

So far much theoretical consideration of experiments on the Bose-Einstein condensation (BEC) of alkali-metal atoms in harmonic traps is based on the Gross-Pitaevskii (GP) equation. In this paper, we attempt to formulate the BEC in the language of quantum field theory and to estimate the quantum and thermal fluctuation effects, which are neglected in the approximation using the GP equation. First, the formulation at zero temperature is developed, and then it is extended to the finite-temperature case by means of thermofield dynamics. We treat the zero-energy mode with care, so that the canonical commutation relations hold. As a result, an infrared divergence appears, but it can be renormalized into the observed condensate number. Numerical calculations are performed. For illustration, the corrections at one-loop level to the original GP equation are given. We also calculate numerically the effects of quantum and thermal fluctuations on the distribution of condensed atoms.

So far much theoretical consideration of experiments on the Bose-Einstein condensation (BEC) of alkali-metal atoms in harmonic traps is based on the Gross-Pitaevskii (GP) equation. In this paper, we attempt to formulate the BEC in the language of quantum field theory and to estimate the quantum and thermal fluctuation effects, which are neglected in the approximation using the GP equation. First, the formulation at zero temperature is developed, and then it is extended to the finite-temperature case by means of thermofield dynamics. We treat the zero-energy mode with care, so that the canonical commutation relations hold. As a result, an infrared divergence appears, but it can be renormalized into the observed condensate number. Numerical calculations are performed. For illustration, the corrections at one-loop level to the original GP equation are given. We also calculate numerically the effects of quantum and thermal fluctuations on the distribution of condensed atoms. © 2003 The American Physical Society.

We analyze the Nambu-Goldstone (NG) mode for a system of trapped Bose-Einstein condensation (BEC). The NG mode is a discrete mode in our case, contrary to the homogeneous case in which the NG mode is in a continuum labeled by a momentum index. In order to control the NG mode, an infinitesimal symmetry-breaking term is introduced. This breaking term plays the role of infinitesimal magnetic field in spin system which fixes the macroscopic "direction" of spin. We calculate a tadpole diagram to evaluate fluctuation effects, taking account of the NG mode, then find that the coefficient of the wave function of condensation is infrared divergent while the remaining terms are finite.

Nonequilibrium Thermo Field Dynamics (TFD) is reviewed. The aspect that nonequilibrium TFD respects Quantum Field Theory is emphasized. A new type of renormalization condition, implying a self-consistent treatment, plays an important role. We also comment on the correspondence between TFD and the conventional density matrix formalism. The recent experiments of the Bose-Einstein condensation of neutral atoms are expected to offer a test field for thermal field theories.

The Bose-Einstein condensation in recent experiments is formulated as a problem in thermal quantum field theory. From this formulation, we carry out numerical analysis so as to investigate effects of quantum and thermal fluctuation to the condensate phase and properties of quasi-particles in this situation.

調和振動子型ポテンシャル中のアルカリ原子のBose-Einstein凝縮(BEC)の実験に対する多くの理論的考察はGross-Pitaevskii(GP)方程式に基づいている.この報告書では, GP方程式を超える試みとして,場の理論による定式化を行った.まず,零度で定式化し,それをThermo Field Dynamicsにより有限温度に拡張する.非摂動的な定式化のために必要な数値計算も行い,例として, GP方程式に対する1-ループレベルの補正を計算した.また,凝縮相に対する量子・温度補正の影響も評価した.

Experiments of Bose-Einstein condensation (BEC) of alkali atoms in harmonic traps which were achieved successfully have inspired various theoretical studies. So far many theoretical considerations are based on the Gross-Pitaevskii equation. In this report, we attempt to formulate the BEC in the language of quantum field theory, so that effects of quantum fluctuation can be estimated properly.

We analyze the effects of inelastic scattering on the tunneling time theoretically, using generalized Nelson's quantum mechanics. This generalization enables us to describe quantum system with channel couplings and optical potential in a real time stochastic approach, which seems to give us a new insight into quantum mechanics beyond Copenhagen interpretation.

We analyze the effects of inelastic scattering on the tunneling time theoretically, using generalized Nelson’s quantum mechanics. This generalization enables us to describe a quantum system with optical potential and channel couplings in a real-time stochastic approach, which seems to give us a new insight into quantum mechanics beyond Copenhagen interpretation. © 1997 The American Physical Society.

The concept of quasiparticle (representation particle) is a central issue when thermofield dynamics (TFD) for nonequilibrium systems of quantum fields is constructed. The development of nonequilibrium TFD, mainly clone by the Alberta group, is reviewed from the viewpoint of the quasiparticle.

We examine whether the chaotic behavior of classical systems with a limited number of degrees of freedom can produce quantum dephasing, against the conventional idea that dephasing takes place only in large systems with a huge number of constituents and complicated internal interactions. On the basis of this analysis, we briefly discuss the possibility of defining quantum chaos and of inventing a ''chaos detector''.

Conventional transport theory is not really applicable to nonequilibrium systems which exhibit strong quantum effects. We present two different approaches to overcome this problem. Firstly we point out how transport equations may be derived that incorporate a nontrivial spectral function as a typical quantum effect, and test this approach in a toy model of a strongly interacting degenerate plasma. Secondly we explore a path to include nonequilibrium effects into quantum field theory through momentum mixing transformations in Fock space. Although the two approaches are completely orthogonal, they lead to the same coherent conclusion.

We present a time-dependent description of tunneling phenomena, using Nelson's stochastic approach. It appears to be capable to describe individual experimental runs, and provides a new insight into quantum measurement processes. We propose a new way to evaluate the tunneling time in this approach and give the results of numerical simulations.

In the framework of thermofield dynamics for spatially inhomogeneous time-dependent nonequilibrium situations, we derive the kinetic equation, which is different from the Boltzmann equation due to quantum effects. It is shown that this kinetic equation leads to the entropy law. An expression for the entropy how is found.

Making use of the thermo field dynamics (TFD) we formulate a calculable method for time-dependent nonequilibrium systems in a time representation (t-representation) rather than in the k0-Fourier representation. The corrected one-body propagator in the t-representation has the form of B-1 (diagonal matrix) B (B being a thermal Bogoliubov matrix). The number parameter in B here is the observed number (the Heisenberg number) with a fluctuation. With the usual definition of the on-shell self-energy a self-consistent renormalization condition leads to a kinetic equation for the number parameter. This equation turns out to be the Boltzmann equation, from which the entropy law follows.

A new perturbative scheme for interacting nonequilibrium thermal quantum fields using thermo field dynamics is outlined by explicitly considering the temporal change of the thermal vacuum as it moves through many inequivalent state vector spaces. One is then naturally led to two sources of time dependence, one from the dynamics and the other from the change of thermal vacuum, which are taken care of by the Hamiltonian and the thermal generator, respectively. To obtain a practical scheme we restrict ourselves by the demand that a spectral representation for the full propagator exists. This leads to a time dependent temperature. The addition of a diagonalization condition for the quasi-particle Hamiltonian provides the master equation for the number density. We show that our formalism is equivalent to an extended form of the path-ordering method. This formalism is a first step towards the study of the origin of heat and temperature in high-energy heavy ion collisions.

A model with spontaneous breakdown of symmetry in quantum field theory is studied in the framework of the Parisi-Wu stochastic quantization (SQ). Looking at the conventional treatment of the model in the canonical and the path-integral quantizations including the problems of many inequivalent Hilbert spaces and measures, we explicitly construct the SQ formulation for the model in the conventional treatment of quantum field theory. The real power of SQ is found in extending quantum field theory beyond the existing frameworks; SQ easily avoids the problems imposed by choices of Hilbert space or measure. We demonstrate it by introducing c-numbers depending on the fictitious time into the starting formal Langevin equations for the model. This new formulation in SQ thus obtained is mathematically well-defined, leading to a definite calculational scheme. The result from this shows that the masses and number of excitation modes change from those in the conventional treatment.

By making use of time-dependent Bogoliubov transformations, we develop a calculation technique for time-dependent non-equilibrium systems of quantum fields in a time-representation (t-representation). The corrected one-body propagator in the t-representation turns out to have the form B-1 (diagonal matrix) B (B being a thermal Bogoliubov matrix). Applying the usual on-shell concept to the diagonal matrix part of the self-energy, we formulate a self-consistent renormalization scheme. This renormalization determines the vacuum and leads to a kinetic equation for the number density parameter, which reduces to the Boltzmann equation in the lowest approximation. This gives us the increasing entropy in time (the second law of thermodynamics).

The recently developed formulation of time-dependent non-equilibrium thermofield dynamics (TFD) is extended to spatially inhomogeneous field systems in this paper. This inhomogeneous formalism is constructed from introducing momentum mixing thermal Bogoliubov transformation. The momentum mixing number density in the Bogoliubov matrix is related to the number density distribution defined at each space-time point which is to be observed.

In this paper the so-called a degree of freedom appearing in thermal quantum field theory is discussed, using thermo field dynamics (TFD). This paper is confined to stationary thermal situations, both nonequilibrium and equilibrium. The main result is that when the stationary number distribution differs from the equilibrium one the use of time ordered and antitime ordered formalisms picks up alpha = 1 and alpha = 0, in contrast to the general belief that the Feynman diagram method is usable for any other alpha as well. This situation in TFD will be compared with the other approaches. The reason why the Feynman diagram method becomes available for any alpha in the case of the equilibrium distributions is also studied.

We use the formalism of thermo field dynamics and the classical Cramer's theorem to show that if two quantum systems are prepared independently and their centre of mass is found to be a thermal squeezed state, then both systems were prepared in thermal squeezed states. This constitutes an alternative derivation of the quantum version of Cramer's theorem. Further, we consider the case of two systems that are prepared independently and their centre of mass is in a pure state in the expanded space of thermo field dynamics. In this case we show that the system is also separable in the centre of mass and relative coordinates, and again all the states involved are thermal squeezed states.

The view of the thermal degree of freedom for the quantum field system, using the language of thermo field dynamics, is elaborated in this paper: the thermal degree of freedom is the degree of freedom to move through degenerate thermal vacua associated with spontaneous breakdown of a certain symmetry (G triple-over-dot-symmetry). Based on this view, we study a quasi-particle picture, in which the G triple-over-dot-transformation becomes the thermal Bogoliubov transformation.

This paper is aimed to clarify the appearance of non-quantum noises in quantum systems in various subjects from a unified viewpoint. Examples taken are the squeezed state in quantum optics, the Hawking radiation around the black hole and thermo field dynamics (TFD) in thermal quantum field theory. Under the use of mathematical formulation of TFD, the origin of noises is traced to the Bogoliubov transformation, i.e., the pair condensation of bare particles in the physical vacua. Detailed analysis of the noise energy reveals a crucial distinction among them: Calling the operator generating the noise energy the noise generator, we find two cases, dynamical and spontaneous, depending on whether the noise generator is part of the Hamiltonian or not. The squeezed state belongs to the former, TFD does to the latter.