quantum stochastic diﬀerential equations for boson and fermion … · 2005-04-17 · quantum...

Annals of Physics 308 (2003) 395–446

www.elsevier.com/locate/aop

Quantum stochastic differential equationsfor boson and fermion systems—methodof non-equilibrium thermo field dynamics

A.E. Kobryn,1 T. Hayashi, and T. Arimitsu*

Institute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan

Received 15 April 2003

Abstract

A unified canonical operator formalism for quantum stochastic differential equations, in-

cluding the quantum stochastic Liouville equation and the quantum Langevin equation both

of the Itoo and the Stratonovich types, is presented within the framework of non-equilibrium

thermo field dynamics (NETFD). It is performed by introducing an appropriate martingale

operator in the Schr€oodinger and the Heisenberg representations with fermionic and bosonic

Brownian motions. In order to decide the double tilde conjugation rule and the thermal state

conditions for fermions, a generalization of the system consisting of a vector field and Fad-

deev–Popov ghosts to dissipative open situations is carried out within NETFD.

� 2003 Published by Elsevier Inc.

Keywords:Non-equilibrium thermo field dynamics; Stochastic differential equations; Martingale operator;

Fermionic Brownian motion; Bosonic Brownian motion

1. Introduction

In this paper we study time-dependent behavior of non-equilibrium quantum

systems involving stochastic forces which can be boson or fermion type and are

called quantum Brownian motion. Present consideration is an extension of previous

* Corresponding author. Fax: +81298534492.

E-mail address: [email protected] (T. Arimitsu).1 Present address: Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan.

0003-4916/$ - see front matter � 2003 Published by Elsevier Inc.

doi:10.1016/S0003-4916(03)00178-7

mail to: [email protected]

396 A.E. Kobryn et al. / Annals of Physics 308 (2003) 395–446

analysis reported comprehensively by one of the authors [1] and is given in terms of

non-equilibrium thermo field dynamics (NETFD) [2–4]. NETFD is a unified formal-

ism, which enables us to treat dissipative quantum systems by the method similar to

usual quantum mechanics or quantum field theory, which accommodates the con-

cept of the dual structure in the interpretation of nature, i.e., in terms of the operatoralgebra and the representation space. The representation space in NETFD is com-

posed of a direct product of two Hilbert spaces: one is for non-tilde fields and the

other for tilde fields. Within the statistical operator (density operator) formalism

there is entanglement between operators and statistical operator due to their non-

commutativity. Introduction of two kinds of operators, without tilde and with tilde,

made it possible to resolve the entanglement between relevant operators and the sta-

tistical operator.

We are deriving a unified system of quantum stochastic differential equations(QSDEs) under the influence of quantum Brownian motion, including the quantum

stochastic Liouville equation and the quantum Langevin equation. The quantum

Fokker–Planck equation is derived by taking the random average of the correspond-

ing stochastic Liouville equation. The relation between the Langevin equation and

the stochastic Liouville equation, as well as between the Heisenberg equation for op-

erators of gross variables and the quantum Fokker–Planck equation obtained here,

is similar to the one between the Heisenberg equation and the Schr€oodinger equationin quantum mechanics and field theory. Our extension of analysis [1] consists of es-sentially three items. Two of them include definition of fermionic Brownian motion

and treatment of fermions in NETFD, i.e., the tilde conjugation rule and the thermal

state conditions in the case of fermion systems. Third item is the simultaneous con-

sideration of hermitian and non-hermitian interaction Hamiltonians.

To begin with, we first remind briefly some standard steps that people usually take

in order to obtain the irreversible evolution of macroscopic systems starting from the

microscopic level. At present, there are many viewpoints giving us tools how to de-

scribe N -body systems out of equilibrium. At the same time, one usually follows oneof several basic approaches: (i) the behavior of the systems is expressed in terms of

not the total (N -particle) distribution function but s-particle ones (with s being usu-

ally 1 and/or 2), (ii) the dynamics of the systems is characterized by the evolution of a

‘‘coarse grained’’ phase-space distribution function or statistical operator, and (iii)

the evolution of the systems is described by the equations of motion for the dynam-

ical gross variables.

The approach (i) is intimately related to the Bogoliubov method of a reduced de-

scription of many-particle systems [5], which is widely used for construction of ki-netic equations based on the Liouville or the Liouville–von-Neumann equation.

Bogoliubov�s hypothesis that the time dependence of higher-particle distribution

functions enter through the one-particle distribution provides a fundamental impor-

tance in various schemes of truncation of the BBGKY hierarchy.

In the approach (ii), the most frequently used tools are projection operators intro-

duced by Nakajima [6] and Zwanzig [7,8]. The basic idea underlying the application

of their techniques to complex systems is to regard the operation of tracing over

the environment as a formal projection in the space of the total system. It became

A.E. Kobryn et al. / Annals of Physics 308 (2003) 395–446 397

especially popular in quantum optics where the so-called quantum master equation

for reduced statistical operator of a relevant system now bears their names and is

called the Nakajima-Zwanzig equation [9].

The general framework, called sub-dynamics, at the Brussels school is also related

to the approach (ii) but the underlying concept is different from the one by Nakajimaand Zwanzig. The main point here is the notion of the increase of the number of cor-

relations within a system in time. It has been expounded in detail by Prigogine and

coauthors, see, e.g., [10,11].

Regarding to the approach (iii), we should mention projection operator by Mori

[12] and the one by Kawasaki and Gunton [13]. The former is used to derive linear

equations of motion for gross variables out of non-linear equations. Originally, one

of the intentions to introduce such an operator was to obtain expressions for phys-

ical (measurable) kinetic coefficients. The latter is an improved version of the time-dependent projection operator by Robertson [14].

Zubarev introduced the concept of non-equilibrium ensemble as a generalization of

Green�s works on the statistical mechanics of linear dissipation processes [15,16] and

Kubo�s theory of linear response of systems to mechanical [17] and thermodynamical

[18] external perturbations. This generalization is known as the method of non-equi-

librium statistical operator [19]. It is shown that this has a close relationship to the

projection operator methods [20].

The above-mentioned methods do not exhaust the entire list, but they may be themost generic ones. However, in this paper we do not follow them. In the case of pres-

ence of additional degree(s) of freedom, e.g., stochastic force(s), description may be

given also in some optional way (in a sense that consideration does not start from the

very microscopic level). The theory of Brownian motion is an example. The funda-

mental equation here is the Langevin equation and it is the stochastic differential

equation for dynamical variables [21,22]. Random forces in Langevin equation are

usually described by Gaussian white stochastic processes. Stochastic integral with re-

spect to such processes is defined as a kind of a Riemann-Stieltjes one [23] wheremultiplication between the stochastic increment and integrand is commonly consid-

ered in the form of Itoo [24] or Stratonovich [25] (for Itoo and Stratonovich multipli-

cations see Appendix A).

The Langevin equation can be used to calculate various time correlation func-

tions. Now it is radically extended to solve numerous problems in different areas

[26–30]. In particular, the theory of Brownian motion itself has been extended to sit-

uations where the ‘‘Brownian particle’’ is not a real particle anymore, but instead

some collective properties of a macroscopic system. Corresponding equation in thephase space or the Liouville space of statistical operators can be considered as a sort

of stochastic differential equation too. In order to investigate classical stochastic sys-

tems, the stochastic Liouville equation was introduced first by Anderson [31] and

Kubo [32–34].

There were several attempts to extend the classical theory (both the Langevin and

the stochastic Liouville equations) for quantum cases. Study of the Langevin equa-

tion for quantum systems has its origin in papers by Senitzky [35–37], Schwinger

[38], Haken [39–42], and Lax [43], where they investigated a quantum mechanical


damped harmonic oscillator in connection with laser systems. In particular, it was

shown that the quantum noise, i.e., the spontaneous emission, can be treated in a

way similar to the thermal fluctuations, and that the noise source has non-zero sec-

ond moments proportional to a quantity which can be associated with a quantum

analog of a diffusion coefficient. As it was noticed by Kubo [44] in his discussion withvan Kampen, the random force must be an operator defined in its own Hilbert space,

which does not happen in classical case since there is no consideration of space for

the random force.

Mathematical study of the quantum stochastic processes was initiated by Davies

[45,46], Hudson [47–52], Accardi [53,54], Parthasarathy [55–57], and their co-authors.

Quantum mechanical analogs of Wiener processes [47] and quantum Itoo formula for

boson systems [48–51] were defined first by Hudson et al. The classical Brownian mo-

tion is replaced here by the pair of one-parameter unitary group authomorphisms,namely by the annihilation and creation boson random force operators with time in-

dices in the boson Fock space, named quantum Brownian motion. Fermion stochas-

tic calculus were defined by Applebaum, Hudson and Parthasarathy [58–63]. In these

papers, they developed the fermion analog of the corresponding boson theory [49,50]

in which the annihilation and creation processes are fermion field operators in the fer-

mion Fock space. Within the frame of this formalism, the Itoo-Clifford integral [64–

67]—fermion analog of the classical Brownian motion—is contained as a special case.

It should be noted, however, that in both boson and fermion theories of quantum sto-chastic calculus mathematicians were debating unitary processes only. For readers�convenience, clue of mathematicians� theory of quantum Brownian motion and the

extension with allowance for thermal degree of freedom are put into Appendix B.

Contrary to expectations, attempts to extend the classical stochastic Liouville

equation for quantum case were not very successful so far. In the present work we

construct our consideration using the formalism of NETFD. It is an alternative

way to the above-mentioned general methods of non-equilibrium statistical mechan-

ics in the sense that it provides us with a general structure of the canonical operatorformalism for dissipative non-equilibrium quantum systems without starting from

the microscopic description, and turns out to be especially successful in the inclusion

of quantum stochastic forces. In particular, a unified canonical operator formalism

of QSDEs for boson systems was constructed first within NETFD [1,68–73] on the

basis of the quantum stochastic Liouville equation.

The paper is organized as follows. First, in Section 2, we remind a brief essence of

the formalism of NETFD by giving its technical basics and some fundamentals. In

Section 3 we derive the semi-free time evolution generator for systems in non-station-ary case. The semi-free generator is bi-linear and globally gauge invariant. The an-

nihilation and creation operators are introduced by means of a time-dependent

Bogoliubov transformation. We close the section by calculating the two-point func-

tion. The generating functional method, which gives us the relation between the

method of NETFD and the one of the Schwinger closed-time path, is introduced

in Section 4. Interaction with external fields is considered in Section 5. Here we

study two cases: hermitian and non-hermitian interaction hat-Hamiltonians.

To make possible their simultaneous consideration we introduce an auxiliary param-


eter k which plays the role of a switch between the cases. In Section 6, the general

expression of the stochastic semi-free time evolution generator is derived for a

non-stationary Gaussian white quantum stochastic process by means of the interac-

tion hat-Hamiltonian with arbitrary k. Correlations of the random force operators

are also derived generally. With the generator, quantum stochastic Liouville equa-tions and quantum stochastic Langevin equations of both Itoo and Stratonovich types

of the system are investigated in a unified manner. We conclude the section by deriv-

ing the equation of motion for the expectation value of an arbitrary operator of the

relevant system. In Section 7, we consider a semi-free system with a stationary

process and check explicitly the irreversibility of such a process in terms of its Boltz-

mann entropy. In Section 8 we investigate relation to the Monte Carlo wave-function

method. Summary and open questions are put into Section 9. Auxiliary material is

put into Appendices A, B, C, D.

2. Basics of NETFD

Information about the general method of NETFD can be found in many papers

and we refer first of all to the original source [2–4] and the review article [1]. To make

our paper self-contained, we include some standard steps which are necessary at least

to fix the notations. The formalism of NETFD is constructed upon the followingfundamental requirements.

An arbitrary operator A in NETFD is accompanied by its tilde conjugated partner~AA, called tilde operator, according to the rule

2 W

system

ðA1A2Þ� ¼ ~AA1~AA2; ð2:1Þ

ðc1A1 þ c2A2Þ� ¼ c�1 ~AA1 þ c�2 ~AA2; ð2:2Þ

ð ~AAÞ� ¼ A; ð2:3Þ
where c1 and c2 are c-numbers. It should be noted that in the present paper the
double tilde conjugation rule (2.3) is of the same form for both bosonic and ferm-

ionic operators and leaves them unchanged.

To indicate commutation or anti-commutation of two operators, say A1 and A2,

we will use the notation ½A1;A2g and call it (anti-)commutator, which should be un-

derstood as

½A1;A2g ¼ ½A1;A2�þ ¼ fA1;A2g ¼ A1A2 þ A2A1; ð2:4Þ

when both operators are fermionic, or

½A1;A2g ¼ ½A1;A2�� ¼ ½A1;A2� ¼ A1A2 � A2A1 ð2:5Þ
otherwise.2
hen one operator is bosonic and another one is fermionic, the rule of commutation depends on the

. In this paper for such combinations we assume (2.5).


Tilde and non-tilde operators, say A1 and ~AA2, are supposed to be mutually (anti-)

commutative at equal time, i.e.,

½A1; ~AA2g ¼ 0: ð2:6Þ
Tilde and non-tilde operators are related with each other through the thermal state
condition (TSC)

hhj ~AAy ¼ s�hhjA; ð2:7Þ
where hhj represents the thermal bra-vacuum; s is the complex parameter which takes
two values:

s ¼ 1 for bosonic operators;i for fermionic operators:

�ð2:8Þ

Derivation of the double tilde conjugation rule and TSC for fermionic operatorsused in this paper is given in Appendix C.

Within the framework of NETFD, the dynamical evolution of a system is

described by the Schr€oodinger equation (here we use the system with �h ¼ 1)

o

otj0ðtÞi ¼ �iHH j0ðtÞi; ð2:9Þ

where j0ðtÞi represents the thermal ket-vacuum. It can be also called the quantum

master equation or the quantum Fokker–Planck equation in this paper. The thermal

vacuums are tilde invariant, i.e., hhj� ¼ hhj and j0ðtÞi� ¼ j0ðtÞi, and are normalized

as hhj0ðtÞi ¼ 1. The hat-Hamiltonian HH , an infinitesimal time-evolution generator,

satisfies the tildian condition:

ðiHHÞ� ¼ iHH : ð2:10Þ

The tildian hat-Hamiltonian is not necessarily hermitian operator. It has zero eigen-

values for the thermal bra-vacuum

hhjHH ¼ 0; ð2:11Þ

which is nothing but manifestations of conservation of probability.

Introducing the time-evolution operator VV ðtÞ by
d
dtVV ðtÞ ¼ �iHHVV ðtÞ; ð2:12Þ

with the initial condition VV ð0Þ ¼ 1, we can define the Heisenberg operator

AðtÞ ¼ VV �1ðtÞAVV ðtÞ ð2:13Þ
satisfying the Heisenberg equation for dissipative systems
d

dtAðtÞ ¼ i½HHðtÞ;AðtÞ�; ð2:14Þ

where HHðtÞ is the hat-Hamiltonian in the Heisenberg representation. The existence

of the Heisenberg equation of motion for coarse-grained operators enables us to


construct a canonical formalism of the dissipative quantum field theory. Note that

with the help of TSC we have an equation of motion for a vector hhjAðtÞ

3 W

inclusi

operat

d

dthhjAðtÞ ¼ ihhj½HHðtÞ;AðtÞ� ð2:15Þ

in terms of only non-tilde operators. The expectation value of an observable oper-

ator A at time t is given by

hAðtÞi ¼ hhjAj0ðtÞi ¼ hhjAðtÞj0i; ð2:16Þ
where j0i ¼ j0ðt ¼ 0Þi. We define that observable operators consist only of non-tilde
operators.3

3. Semi-free hat-Hamiltonian

Let us consider a system specified by the total hat-Hamiltonian

HH tott ¼ HHt þ HH1 þ HHI;t; ð3:1Þ

where HHt is a semi-free hat-Hamiltonian, whereas HH1 and HHI;t are, respectively, the

interaction hat-Hamiltonian within the relevant system and the one representing the

coupling with external fields. The system itself is supposed to be consistent with allthe requirements of NETFD given in the previous section. Some general remark

about treatment of interaction within the relevant system is given in Section 4. Ex-

plicit treatment of interaction with external fields is given in Section 5. Here we

concentrate on derivation and study of properties of the semi-free hat-Hamiltonian,

i.e., renormalized unperturbed hat-Hamiltonian.

3.1. Derivation of the semi-free hat-Hamiltonian

The semi-free hat-Hamiltonian is bilinear in operators a, ay, ~aa, and ~aay, and is

invariant under the phase transformation a ! aei/:

HHt ¼ h1ðtÞayaþ h2ðtÞ~aay~aaþ h3ðtÞa~aaþ h4ðtÞay~aay þ h0ðtÞ; ð3:2Þ
where hjðtÞ are time-dependent complex c-number functions. Operators a, ay, ~aa, and~aay satisfy the canonical (anti-)commutation relations
½ak; ayk0 ��r ¼ dk;k0 ; ½~aak; ~aayk0 ��r ¼ dk;k0 ; ð3:3Þ
where we use r ¼ 1 for bosonic systems and r ¼ �1 for fermionic ones. According to
(2.6), tilde and non-tilde operators are mutually (anti-)commutative. In the following

account, a subscript k for specifying a momentum and/or other degrees of freedom

will be dropped unless it is necessary. Number of unknown functions hjðtÞ can be

e can include tilde operators in addition to non-tilde ones in the definition of observable. However,

on of tilde operators may give us a set of different but equivalent definitions for one observable

or.


reduced by the use of (2.11) and tildian (2.10) for the semi-free hat-Hamiltonian.

It results in

HHt ¼ HHS;t þ iPPt; ð3:4Þ
where
HHS;t ¼ xðtÞðaya� ~aay~aaÞ; ð3:5Þ

PPt ¼ c1ðtÞðayaþ ~aay~aaÞ � s 2c1ðtÞ½ þ c2ðtÞ�ay~aay þ rsc2ðtÞa~aaþ r 2c1ðtÞ½ þ c2ðtÞ�;ð3:6Þ

with

xðtÞ ¼ Reh1ðtÞ; ð3:7Þ

c1ðtÞ ¼ Imh1ðtÞ; ð3:8Þ

c2ðtÞ ¼Imh3ðtÞ for bosonic systems;Reh3ðtÞ for fermionic systems:

�ð3:9Þ

Let us introduce operators aðtÞ and ayyðtÞ in the interaction representation definedby

aðtÞ ¼ VV �1ðtÞaVV ðtÞ; ayyðtÞ ¼ VV �1ðtÞayVV ðtÞ; ð3:10Þ
where
d

dtVV ðtÞ ¼ �iHHtVV ðtÞ; ð3:11Þ

with the initial condition VV ð0Þ ¼ 1. They satisfy the equal-time (anti-)commutation

relations

½aðtÞ; ayyðtÞ��r ¼ 1: ð3:12Þ
The Heisenberg equation (2.14) for aðtÞ and ayyðtÞ with
HHðtÞ ¼ VV �1ðtÞHHtVV ðtÞ ð3:13Þ
are explicitly given by
daðtÞdt

¼ c1ðtÞ½ � ixðtÞ�aðtÞ � s 2c1ðtÞ½ þ c2ðtÞ�~aayyðtÞ; ð3:14Þ

dayyðtÞdt

¼ ixðtÞ½ � c1ðtÞ�ayyðtÞ � sc2ðtÞ~aaðtÞ: ð3:15Þ

In these formulae we used a symbol yy instead of usual dagger because the semi-free

hat-Hamiltonian HHt is not necessarily hermitian.Since the semi-free hat-Hamiltonian HHt satisfies (2.11), we have TSC for the bra-

vacuum at time t

hhj~aayyðtÞ ¼ s�hhjaðtÞ: ð3:16Þ


By making use of the Heisenberg equations (3.14) and (3.15), and of TSC (3.16),

one obtains the equation of motion for a vector hhjayyðtÞaðtÞ in the form

d

dthhjayyðtÞaðtÞ ¼ �2jðtÞhhjayyðtÞaðtÞ þ iR<ðtÞhhj; ð3:17Þ

where jðtÞ and iR<ðtÞ are defined by

jðtÞ ¼ c1ðtÞ þ c2ðtÞ; ð3:18Þ

iR<ðtÞ ¼ �r½2c1ðtÞ þ c2ðtÞ�: ð3:19Þ

Substituting (3.18) and (3.19) into (3.14) and (3.15), one gets equations of motion for

operators aðtÞ and ayyðtÞ in the form

daðtÞdt

¼ � ixðtÞ½ þ jðtÞ�aðtÞ � riR<ðtÞ½aðtÞ � s~aayyðtÞ�; ð3:20Þ

dayyðtÞdt

¼ ixðtÞ½ þ jðtÞ�ayyðtÞ þ riR<ðtÞ½ayyðtÞ � s~aaðtÞ� � 2sjðtÞ~aaðtÞ: ð3:21Þ

Applying the thermal ket-vacuum j0i at the initial time to (3.17), we obtain theequation of motion for the one-particle distribution function

nðtÞ ¼ hhjayyðtÞaðtÞj0i ð3:22Þ
in the form
d

dtnðtÞ ¼ �2jðtÞnðtÞ þ iR<ðtÞ: ð3:23Þ

Equation (3.23) can be identified as the generalized Boltzmann equation of the

system. The function iR<ðtÞ is given when the interaction hat-Hamiltonian HH1 is

defined.

The initial ket-vacuum j0i is specified by TSC

~aaj0i ¼ sfayj0i; ð3:24Þ
with f 2 ½0; 1Þ when r ¼ 1, f 2 ½0;1Þ when r ¼ �1. The initial value for the one-
particle distribution function n ¼ nðt ¼ 0Þ is determined by f . Since

n ¼ n� ¼ hhjayaj0i� ¼ hhj~aay~aaj0i ¼ s�hhja~aaj0i ¼ jsj2f hhjaayj0i ¼ f ½1þ rn�;ð3:25Þ

we have

n ¼ f ½1� rf ��1: ð3:26Þ

In the first equality of (3.25) we used the fact that n is a real number; in the third

equality we used the tilde invariance of the thermal vacuums hhj and j0i; finally, inthe fourth and fifth equalities we used TSCs (2.7) and (3.24), respectively.

Solving the Heisenberg equations for aðtÞ, ayyðtÞ, and their tilde conjugates and

using TSC at initial time (3.24), we find TSC for the ket-vacuum at time t


~aaðtÞj0i ¼ snðtÞ1þ rnðtÞ a

yyðtÞj0i; ð3:27Þ

where nðtÞ satisfies the Boltzmann equation (3.23).Substituting (3.18), (3.19), and (3.23) into (3.6) one gets the most general form of

PPt in the interaction representation:

PPt ¼� fjðtÞ 1½ þ 2rnðtÞ� þ r _nnðtÞgðayaþ ~aay~aaÞ þ rsf2jðtÞ 1½ þ rnðtÞ� þ r _nnðtÞga~aaþ rsf2jðtÞnðtÞ þ _nnðtÞgay~aay � f2jðtÞnðtÞ þ _nnðtÞg: ð3:28Þ

Here, we used the abbreviation _nnðtÞ ¼ dnðtÞ=dt.By introducing thermal doublet notations

�aam ¼ ay;�

� s~aa�; al ¼ collon a; s~aay

� �; ð3:29Þ

canonical (anti-)commutation relations are written as

½al; �aam��r ¼ dlm: ð3:30Þ
The resulting semi-free hat-Hamiltonian (3.4) can be presented in a compact
form as

HHt ¼ xðtÞ�aalal þ i�aalAðtÞlmam þ r½xðtÞ þ ijðtÞ�; ð3:31Þ
where matrix AðtÞlm has the following structure:
AðtÞlm ¼ r�jðtÞ 2nðtÞ þ r½ � � _nnðtÞ; 2jðtÞnðtÞ þ _nnðtÞ�2jðtÞ nðtÞ þ r½ � � _nnðtÞ; jðtÞ 2nðtÞ þ r½ � þ _nnðtÞ

� �: ð3:32Þ

3.2. Annihilation and creation operators

Let us introduce annihilation and creation operators by

cðtÞ ¼ 1½ þ rnðtÞ�aðtÞ � rsnðtÞ~aayyðtÞ; ð3:33Þ

~cc$ðtÞ ¼ ~aayyðtÞ � rsaðtÞ: ð3:34Þ
From TSCs (3.16) and (3.27) at time t, we see that they annihilate the vacuums:
hhjc$ðtÞ ¼ 0; cðtÞj0i ¼ 0; ð3:35Þ

hhj~cc$ðtÞ ¼ 0; ~ccðtÞj0i ¼ 0: ð3:36Þ
With the thermal doublet notations
�aaðtÞl ¼ ðayyðtÞ;�s~aaðtÞÞ; aðtÞm ¼ collonðaðtÞ; s~aayyðtÞÞ; ð3:37Þ

�ccðtÞm ¼ ðc$ðtÞ;�s~ccðtÞÞ; cðtÞl ¼ collonðcðtÞ; s~cc$ðtÞÞ; ð3:38Þ
(3.33), (3.34), and their tilde conjugates can be written as
�ccðtÞm ¼ �aaðtÞl½B�1ðtÞ�lm; cðtÞl ¼ BðtÞlmaðtÞm; ð3:39Þ


where BðtÞlm is a matrix of the time-dependent Bogoliubov transformation:

BðtÞlm ¼ 1þ rnðtÞ �rnðtÞ�1 1

� �: ð3:40Þ

This transformation is the canonical one since it leaves the canonical (anti-)-com-

mutation relations unchanged

½cðtÞl; �ccðtÞm��r ¼ dlm: ð3:41Þ
The equation of motion for the thermal doublet cðtÞl is derived as
d

dtcðtÞl ¼ ½ � ixðtÞdlm � jðtÞslm3 �cðtÞ

m; ð3:42Þ

where

slm3 ¼ 1 0

0 �1

� �: ð3:43Þ

The solution of (3.42) then is obtained in the form

cðtÞl ¼ exp

Z t

0

dt0�� ixðt0Þdlm � jðt0Þslm3

��cð0Þm: ð3:44Þ

3.3. Schr€oodinger representation

Annihilation and creation operators in the Schr€oodinger representation are intro-

duced by the relations

�ccðtÞm ¼ VV �1ðtÞ�ccmt VV ðtÞ; cðtÞl ¼ VV �1ðtÞclt VV ðtÞ; ð3:45Þ
with VV ðtÞ being specified by (3.11) and the thermal doublet notations
�ccmt ¼ c$;�

� s~cct�; clt ¼ collon ct; s~cc

$� �

: ð3:46Þ

Using (3.29), one can write

�ccmt ¼ �aal½B�1ðtÞ�lm; clt ¼ BðtÞlmam; ð3:47Þ
where matrix BðtÞlm is given by (3.40).
We see that the annihilation and creation operators in the Schr€oodinger represen-tation annihilate the vacuums at time t:

hhjc$ ¼ 0; ctj0ðtÞi ¼ 0; ð3:48Þ

hhj~cc$ ¼ 0; ~cctj0ðtÞi ¼ 0: ð3:49Þ
Note that creation operator c$ and its tilde conjugated partner ~cc$ do not depend on
time. It is consistent with the fact that the vacuum hhj does not depend on time due to

the property hhjHHt ¼ 0.The terms (3.5) and (3.28) of the semi-free hat-Hamiltonian HHt, (3.4), now read

HHS;t ¼ xðtÞðc$ct � ~cc$~cctÞ; ð3:50Þ


PPt ¼ �jðtÞðc$ct þ ~cc$~cctÞ þ rs _nnðtÞc$~cc$; ð3:51Þ
in the normal ordering with respect to the annihilation and creation operators in the
Schr€oodinger representation. When the system is semi-free, putting (3.50) and (3.51)

into (3.4) and substituting HHt for HH into the Schr€oodinger equation (2.9), one has

o

otj0ðtÞi ¼ rs _nnðtÞc$~cc$j0ðtÞi: ð3:52Þ

It is solved to give

j0ðtÞi ¼ expf�hhj~cctctj0ic$~cc$gj0i: ð3:53Þ
Here, we introduced a kind of the order parameter
hhj~cctctj0i ¼ rs½nð0Þ � nðtÞ�; ð3:54Þ
which gives a measure of difference of the system from the initial state. From (3.53)
we see that the evolution of the ket-vacuum is realized by a condensation of tilde and

non-tilde particle pairs into initial ket-vacuum. The ket-vacuum itself is the func-

tional of the one-particle distribution function nðtÞ. The dependence of the thermal

ket-vacuum on nðtÞ is given by

ddnðtÞ j0ðtÞi ¼ rsc$~cc$j0ðtÞi: ð3:55Þ

Then the Schr€oodinger equation can be written in an alternative way:

o

ot

�� _nnðtÞ d

dnðtÞ

�j0ðtÞi ¼ 0: ð3:56Þ

This shows that the vacuum j0ðtÞi is migrating in the super-representation space

spanned by the one-particle distribution function fnkðtÞg with the velocity f _nnkðtÞg as

a conserved quantity [74,75].

3.4. Two-point function of the semi-free field

A time-ordered two-point function Gðt; t0Þlm (propagator), defined by

Gðt; t0Þlm ¼ �ihhjT ½aðtÞl�aaðt0Þm�j0i; ð3:57Þ
is given by
Gðt; t0Þlm ¼ ½B�1ðtÞ�lkGðt; t0ÞkqBðt0Þqm; ð3:58Þ
with
Gðt; t0Þkq ¼ �ihhjT ½cðtÞk�ccðt0Þq�j0i ¼ GRðt; t0Þ 0

0 GAðt; t0Þ

� �; ð3:59Þ

where non-zero matrix elements are:

GRðt; t0Þ ¼ �ihhjT ½cðtÞc$ðt0Þ�j0i

¼ �ihðt � t0Þ expZ t

t0dt00�� ixðt00Þ � jðt00Þ

��; ð3:60Þ


GAðt; t0Þ ¼ irhhjT ½~cc$ðtÞ~ccðt0Þ�j0i

¼ ihðt0 � tÞ expZ t0

tdt00 ixðt00Þ�(

� jðt00Þ�)

: ð3:61Þ

Here, T is the time ordering operator.

4. Generating functional method

Let us define the generating functional for the semi-free field by

Z½K; ~KK� ¼ hhjT exp

(� i

Z �tt

0

dtSðtÞ)j0i; ð4:1Þ

where the source function SðtÞ reads
SðtÞ ¼ �KKðtÞlaðtÞl þ �aaðtÞlKðtÞl ¼ �KKcðtÞlcðtÞl þ �ccðtÞlKcðtÞl; ð4:2Þ
with

�KKðtÞl ¼ ðKðtÞ�;�s ~KKðtÞÞ; KðtÞm ¼ collonðKðtÞ; s ~KKðtÞ�Þ; ð4:3Þ
and similar notations for �KKcðtÞm and KcðtÞl. The K �s are related by the Bogoliubov
transformation

�KKcðtÞm ¼ �KKðtÞl½B�1ðtÞ�lm; KcðtÞl ¼ BðtÞlmKðtÞm: ð4:4Þ
Matrix BðtÞlm here is the one given by (3.40). External fictitious fields KðtÞl and �KKðtÞmare c-numbers or Grassmann numbers corresponding to r ¼ 1 or r ¼ �1, and
satisfy

½KðtÞl; �KKðtÞm��r ¼ 0: ð4:5Þ
Operators aðtÞl, �aaðtÞm, cðtÞl, and �ccðtÞm are those in the interaction representation
introduced in Section 3.

Taking the functional derivative of the generating functional (4.1), one has

d ln Z½K; ~KK� ¼ �i

Z �tt

0

dt½d �KKcðtÞlhcðtÞliK þ h�ccðtÞliKdKcðtÞl�; ð4:6Þ

where hcðtÞliKand h�ccðtÞli

Kare defined by

hcðtÞliK¼ i

d

d �KKcðtÞlln Z½K; ~KK�

¼ 1

Z½K; ~KK�hhjT cðtÞl exp

("� i

Z �tt

0

dt0Sðt0Þ)#

j0i; ð4:7Þ

h�ccðtÞliK¼ ri

ddKcðtÞl

ln Z½K; ~KK�

¼ 1

Z½K; ~KK�hhjT �ccðtÞl exp

("� i

Z �tt

0

dt0Sðt0Þ)#

j0i: ð4:8Þ


The equation of motion for hcðtÞliKis obtained in the form

d

dthcðtÞli

K¼ ½�ixðtÞdlm � jðtÞslm3 �hcðtÞ

miK� iKcðtÞl: ð4:9Þ

With the boundary conditions

hcð0Þl¼1iK¼ hcð0Þi

K¼ 0; ð4:10Þ

hcð�ttÞl¼2iK¼ sh~cc$ð�ttÞi

K¼ 0; ð4:11Þ

h�ccð�ttÞl¼1iK¼ hc$ð�ttÞi

K¼ 0; ð4:12Þ

h�ccð0Þl¼2iK¼ �sh~ccð0Þi

K¼ 0; ð4:13Þ

it can be solved as

hcðtÞliK¼Z �tt

0

dt0Gðt; t0ÞlmKcðt0Þm; ð4:14Þ

where Gðt; t0Þlm is given by (3.59). The boundary conditions in (4.10)–(4.13) are de-

rived by TSCs (3.35) and (3.36).

Substituting (4.14) into (4.6), one finally obtains [76]

Z½K; ~KK� ¼ exp

(� i

Z �tt

0

dtZ �tt

0

dt0 �KKcðtÞlGðt; t0ÞlmKcðt0Þm)

¼ exp

(� i

Z �tt

0

dtZ �tt

0

dt0 �KKðtÞlGðt; t0ÞlmKðt0Þm): ð4:15Þ

This expression has been derived first by Schwinger for a boson system within the

closed-time path method [38]. Derivation of this result shown in the present section

reveals the relation between the quantum operator formalism of dissipative fields

(realized for the first time within NETFD) and their path integral formalism [38].

The effect of the interaction HH1 � HH1ðal; �aamÞ within the system, which induces thedynamical correlations, can be taken into account by the generating functional

Z1½K; ~KK� ¼ exp

(� i

Z �tt

0

dtHH1 id

d �KKðtÞl ; rid

dKðtÞm

!)Z½K; ~KK�: ð4:16Þ

Note that HH1 should satisfy hhjHH1 ¼ 0.

5. Interaction with external fields

5.1. Hermitian interaction hat-Hamiltonian

The simplest hat-Hamiltonian representing an interaction with an external fieldmay be given by


HH 0t ¼ H 0

t � ~HH 0t ; ð5:1Þ

with a hermitian interaction Hamiltonian

H 0t ¼ i aybt

� byt a

; ð5:2Þ

where bt, byt , and their tilde conjugates are operators of the external system and are

assumed to be (anti-)commutative with operators a, ay, and their tilde conjugates of

the relevant system. The subscript t indicates that these operators may depend on

time. Note that the hat-Hamiltonian (5.1) is tildian, i.e.,

iHH 0t

� ��¼ iHH 0

t : ð5:3Þ

The tilde and non-tilde operators of the external system are related with each

other by

hj~bbyt ¼ s�hjbt; ð5:4Þ
where hj is the bra-vacuum for the external system. Applying the bra-vacuum hhj forthe relevant system on (5.1), one has
hhjHH 0t ¼ �ihhj b$

t a�

þ rs~bb$

t ay�: ð5:5Þ

Here we introduced a new operator

b$

t ¼ byt � s~bbt; ð5:6Þ
which annihilates the bra-vacuum hj for the external system:
hjb$

t ¼ 0: ð5:7Þ
As it is seen from (5.6), the subscript t of the new operator b$
t has been inherited from

the original operators of the external system. By applying the bra-vacuum hj on HH 0t in

addition to hhj, we observe that

hhhjHH 0t ¼ 0; ð5:8Þ

where the bra-vacuum of a total system is introduced by

hhhj ¼ hj � hhj: ð5:9Þ
The dynamics of the system is described by the Schr€oodinger equation for the ket-
vacuum j0ðtÞii of the whole system:

o

otj0ðtÞii ¼ �iHH tot

t j0ðtÞii; ð5:10Þ

where HHI;t in HH tott is replaced by HH 0

t . Conservation of the probability is guaranteed by

hhhjHH tott ¼ 0 for the total system, i.e., the relevant system and the external system.

5.2. Non-Hermitian interaction hat-Hamiltonian

Let us consider if we can have an interaction hat-Hamiltonian which satisfies the

conservation of probability within the relevant system. This feature is consistent


with the one we have in the case of stochastic differential equations for classical

systems.

We assume that the interaction hat-Hamiltonian is globally gauge invariant and

bilinear:

HH 00t ¼ ifh1aybt:þ h2aybyt þ h3~aabt þ h4~aa~bb

yt þ h5~aay~bbt þ h6~aaybyt þ h7a~bbt þ h8abyt g;

ð5:11Þ
where quantities hj (j ¼ 1; . . . ; 8) are time-independent complex c-numbers. The tildian
iHH 00t

� ��¼ iHH 00

t ð5:12Þ

gives us

h�1 ¼ h5; h�2 ¼ h6; h�3 ¼ h7; h�4 ¼ h8: ð5:13Þ
By applying hhj from the left to the Schr€oodinger equation
o

otj0ðtÞii ¼ �iHH tot

t j0ðtÞii; ð5:14Þ

with HHI;t in HH tott being replaced by HH 00

t , we see that the requirement of the conser-

vation of probability within the relevant system leads to

hhjHH 00t ¼ 0: ð5:15Þ

HHt in HH tott is the semi-free hat-Hamiltonian of the relevant system satisfying (2.11).

From (5.15) we obtain

h1 þ rsh3 ¼ 0; h7 þ rsh5 ¼ 0; ð5:16Þ

h2 þ rsh4 ¼ 0; h8 þ rsh6 ¼ 0; ð5:17Þ
which are solved as
h3 ¼ �sh1; h7 ¼ �rsh�1; ð5:18Þ

h4 ¼ �sh2; h8 ¼ �rsh�2: ð5:19Þ

Then the structure of HH 00t can be expressed in terms of only h1, h2, and their complex

conjugates as

HH 00t ¼ ifa$bt þ ~aa$ ~bbtg; ð5:20Þ

where we introduced new operators

a$ ¼ ay � s~aa; ð5:21Þ

bt ¼ h1bt þ h2~bbyt ; ð5:22Þ

and their tilde conjugates. Note that the creation operator a$ annihilates the bra-

vacuum hhj:
hhja$ ¼ 0: ð5:23Þ


In order to investigate parameters h1 and h2 we consider the moments

hbt~bbti ¼ ðh1 þ rsh2Þ sh�1hbyt bti

�þ h�2hbtbyt i

�; ð5:24Þ

h~bbtbti ¼ ðh�1 þ sh�2Þ rsh1hbyt bti�

þ h2hbtbyt i�; ð5:25Þ

where we are using the symbol h� � �i ¼ hj � � � jti without specifying the dynamics

which determines the ket-vacuum jti of the external system. For the present purpose,

the details of its dynamics are not required. Here we assume, however, that the

external ket-vacuum may evolve in time. The further use of the property of the (anti-)

commutativity, i.e., hbt~bbti ¼ rh~bbtbti, gives the necessary two relations to define h1

and h2:

ðsh1 þ h2Þh�1 ¼ ðsh�1 þ rh�2Þh1; ð5:26Þ

rðsh1 þ h2Þh�2 ¼ ðsh�1 þ rh�2Þh2; ð5:27Þ

which reduce to

h�1h2 ¼ rh1h�2: ð5:28Þ
We can express h1 and h2 as
h1 ¼ leih1 ; h2 ¼ meih2 ; ð5:29Þ
where l; m 2 Rþ, namely l ¼ jh1j, m ¼ jh2j. From the requirement (5.28), one has
h2 ¼ h1 for r ¼ 1 and h2 ¼ h1 � p=2 for r ¼ �1. Substituting (5.29) into (5.22) and

putting the phase factor eih1 into bt and ~bbyt , we have

bt ¼ lbt þ rsm~bbyt : ð5:30Þ
Thus, the vector hjbt is calculated as
hjbt ¼ hjðlbt þ rsm~bbyt Þ ¼ ðlþ rmÞhjbt: ð5:31Þ
The further requirement that the norm of hjbt should be equal to that of hjbt, i.e.,khjbtk ¼ khjbtk, leads one to the relation
lþ rm ¼ 1: ð5:32Þ

5.3. Relation between the two interaction hat-Hamiltonians

Note that the hermitian interaction hat-Hamiltonian HH 0t and the non-hermitian

one HH 00t are related to each other by

HH 0t ¼ HH 00

t � ifb$

t ðlaþ rsm~aayÞ þ ~bb$

t ðl~aaþ rs�mayÞg: ð5:33Þ

With an auxiliary parameter 06 k6 1, it is possible to make a simultaneous con-

sideration of both hermitian and non-hermitian interaction hat-Hamiltonians by

introducing

HHI;t ¼ ifa$bt þ ~aa$ ~bbtg � ikfb$

t ðlaþ rsm~aayÞ þ ~bb$

t ðl~aaþ rs�mayÞg: ð5:34Þ


It is easy to see that this expression is reduced to HH 0t in the case k ¼ 1 and to HH 00

t in the

case k ¼ 0, respectively. The dynamics of the system is now described by the

Shr€oodinger equation (5.10) or (5.14) with HHI;t in HH tott being given by (5.34).

6. Quantum stochastic differential equations

6.1. Quantum stochastic Liouville equations

6.1.1. Itoo type

Let us derive the general form of the semi-free hat-Hamiltonian HHF ;t dt for a sto-

chastic Liouville equation of the Itoo type

dj0F ðtÞi ¼ �iHHF ;t dtj0F ðtÞi; ð6:1Þ
where a subscript F is added to indicate that we are considering a system under the
influence of a random force. We assume that the hat-Hamiltonian HHF ;t dt for the

stochastic semi-free field is bilinear in a, ay, dFt, dFyt , and their tilde conjugates, and

that it is invariant under the phase transformation a ! aei/ and dFt ! dFt ei/. Here,

a, ay, and their tilde conjugates are operators of a relevant system satisfying the

canonical (anti-)commutation relation

½a; ay��r ¼ 1; ð6:2Þ
whereas dFt, dF
yt , and their tilde conjugates are random force operators. The tilde

and non-tilde operators are related with each other by the TSC

hhj~aay ¼ s�hhja; ð6:3Þ

hjd ~FF yt ¼ s�hjdFt; ð6:4Þ

where hhj and hj are, respectively, the thermal bra-vacuum of the relevant system and

of the random force.From the investigation in Section 5, we can propose that the required form of the

hat-Hamiltonian should be

HHF ;t dt ¼ HHt dt þ dMMt; ð6:5Þ
where HHt is specified by (3.4) with PPt having the same structure as (3.28) or (3.51).
For later convenience, we rewrite PPt as

PPt ¼ PPR þ PPD; ð6:6Þ
with
PPR ¼ �jðtÞfa$aþ ~aa$~aag; ð6:7Þ

PPD ¼ rsf2jðtÞ½nðtÞ þ g� þ _nnðtÞga$~aa$; ð6:8Þ
where we introduced
a ¼ naþ rsg~aay; nþ rg ¼ 1; ð6:9Þ


which forms a canonical set with a$ defined by (5.21), i.e.,

½a; a$��r ¼ 1: ð6:10Þ
The one-particle distribution function nðtÞ is defined by
nðtÞ ¼ hhjayaj0F ðtÞi �

; ð6:11Þ

and satisfies the Boltzmann equation (D.11) (see Appendix D). Here, h� � �i means to

take the random average, i.e., the vacuum expectation value with respect to the

thermal bra- and ket-vacuums of random force: h� � �i ¼ hj � � � ji. Terms PPR and PPD

are, respectively, the relaxational and diffusive parts of the damping operator PPt.

The martingale dMMt is the term containing operators representing quantum

Brownian motion and satisfies

hdMMti ¼ 0: ð6:12Þ
Associating dFt and dF y
t with bt dt and byt dt in (5.34), respectively, we have

dMMt ¼ ifa$ dWt þ ~aa$ d ~WWtg � ikfdW $t ðlaþ rsm~aayÞ þ d ~WW $

t ðl~aaþ rs�mayÞg; ð6:13Þ
where we introduced new operators
dWt ¼ ldFt þ rsmd ~FF yt ; ð6:14Þ

dW $t ¼ dF y

t � sd ~FFt: ð6:15Þ
Note that dW $
t and d ~WW $t annihilate the bra-vacuum for random force hj:

hjdW $t ¼ 0: ð6:16Þ

In the Itoo multiplication, the random force operators dWt , dW $t and their tilde

conjugates do not correlate with quantities at time t, e.g., j0F ðtÞi:

hdMMtj0F ðtÞii ¼ 0: ð6:17Þ
Thus, taking the random average of the stochastic Liouville equation (6.1), we arrive
at the Fokker–Planck equation

o

otj0ðtÞi ¼ �iHHtj0ðtÞi; ð6:18Þ

where j0ðtÞi ¼ j0F ðtÞih i.The formal solution of (6.1) can be written as

j0F ðtÞi ¼ VVF ðtÞj0F ð0Þi; ð6:19Þ
where the time-evolution generator is defined through
dVVF ðtÞ ¼ �iHHF ;t dtVVF ðtÞ ð6:20Þ
with the initial condition VVF ð0Þ ¼ 1.
6.1.2. Fluctuation–dissipation theorem of the second kind

By making use of the relation between the Itoo and the Stratonovich stochastic

multiplications (see Appendix A), we can rewrite the Itoo type stochastic Liouville


equation into the Stratonovich type as follows. Relation (A.6) makes the term con-

taining random force operators in the r.h.s of (6.1) be

dMMtj0F ðtÞi ¼ dMMt � j0F ðtÞi � 12dMMt dj0F ðtÞi; ð6:21Þ

where the symbol � has been introduced to indicate the Stratonovich stochastic

multiplication (see Appendix A). Substituting (6.1) into the last term for dj0F ðtÞi andneglecting terms of the higher order than dt, we arrive at the quantum stochastic

Liouville equation of the Stratonovich type

dj0F ðtÞi ¼ �iHHF ;t dt � j0F ðtÞi; ð6:22Þ
with
HHF ;t dt ¼ HHS;t dt þ iPPt dt þ dMMt þ i2dMMt dMMt ð6:23Þ

¼ HHS;t dt þ ið1� kÞPPR dt þ dMMt: ð6:24Þ

In order to obtain expression (6.24) we used the generalized fluctuation–dissipation

theorem of the second kind, which can be written as

dMMt dMMt ¼ �2ðkPPR þ PPDÞdt ð6:25Þ
(refer to Appendix D for derivation). Note that in the Stratonovich multiplicationrandom force operators dWt , dW $
t and their tilde conjugates correlate with quantities

at time t, i.e.,

hdMMt � j0F ðtÞii 6¼ 0: ð6:26Þ
The formal solution of (6.22) has the form (6.19), where the stochastic time-evo-
lution generator VVF ðtÞ is defined through

dVVF ðtÞ ¼ �iHHF ;t dt � VVF ðtÞ ð6:27Þ
with the initial condition VVF ð0Þ ¼ 1.
6.1.3. Correlations of the random force operators

Operators dWt , dW $t , and their tilde conjugates are of the quantum stochastic

Wiener process satisfying (for derivation of the results see Appendix D)

hdWti ¼ hd ~WWti ¼ 0; ð6:28Þ

hdWt dWsi ¼ hd ~WWt d ~WWsi ¼ 0; ð6:29Þ

hdWt d ~WWsi ¼ rhd ~WWs dWti ¼ sf2jðtÞ½nðtÞ þ m� þ _nnðtÞgdðt � sÞdtds; ð6:30Þ
and
hdW $t i ¼ hd ~WW $

t i ¼ 0; ð6:31Þ

hdW $t dW $

s i ¼ hdW $t d ~WW $

s i ¼ 0; ð6:32Þ

hdW $t dWsi ¼ hd ~WW $

t d ~WWsi ¼ 0; ð6:33Þ


hdWt dW $s i ¼ hd ~WWt d ~WW $

s i ¼ 2jðtÞdðt � sÞdtds: ð6:34Þ
Due to the argument of Appendix D we also have n ¼ l and g ¼ m, which leads to
a ¼ laþ rsm~aay: ð6:35Þ
In the following, we will use this definition in both PPt and dMMt. Especially, the latterbecomes
dMMt ¼ ifa$ dWt þ ~aa$ d ~WWtg � ikfdW $t aþ d ~WW $

t ~aag: ð6:36Þ

It is important to note here that the martingale dMMt is introduced in the normalordering with respect to all operators a$, a, dW $

t , dWt , and their tilde conjugates.

Within the weak relations, the correlations (6.29), (6.30), and (6.32)–(6.34) reduce,

respectively, to

dWt dWs ¼ d ~WWt d ~WWs ¼ 0; ð6:37Þ

dWt d ~WWs ¼ rd ~WWsdWt ¼ sf2jðtÞ½nðtÞ þ m� þ _nnðtÞgdðt � sÞdsdt; ð6:38Þ

and

dW $t dW $

s ¼ dW $t d ~WW $

s ¼ 0; ð6:39Þ

dW $t dWs ¼ d ~WW $

t d ~WWs ¼ 0; ð6:40Þ

dWtdW $s ¼ d ~WWt d ~WW $

s ¼ 2jðtÞdðt � sÞdsdt: ð6:41Þ

6.2. Stochastic semi-free operators

The stochastic semi-free operators are defined by

AðtÞ ¼ VV �1F ðtÞAVVF ðtÞ; ð6:42Þ

whereas the random force operators in the Heisenberg representation by

W ðtÞ ¼ VV �1F ðtÞWtVVF ðtÞ; W $ðtÞ ¼ VV �1

F ðtÞW $t VVF ðtÞ; ð6:43Þ

and their tilde conjugates. We also use the convenient operators introduced by

�dW ðtÞ ¼ VV �1F ðtÞdWtVVF ðtÞ; �dW $ðtÞ ¼ VV �1

F ðtÞdW $t VVF ðtÞ; ð6:44Þ

and their tilde conjugates. Here,

dVV �1F ðtÞ ¼ iVV �1

F ðtÞHH�F ;t dt; ð6:45Þ

with VV �1F ð0Þ ¼ 1. HH�

F ;t dt is specified by

HH�F ;t dt ¼ HHF ;t dt þ i dMMt dMMt ¼ HHS;t dt þ iPP�

t dt þ dMMt ð6:46Þ

with

PP�t ¼ ð1� 2kÞPPR � PPD: ð6:47Þ


In particular cases when A represents a or ay, we have

aðtÞ ¼ VV �1F ðtÞaVVF ðtÞ; ~aayyðtÞ ¼ VV �1

F ðtÞ~aayVVF ðtÞ: ð6:48Þ

Since the stochastic tildian hat-Hamiltonian HHF ;t dt is not necessarily hermitian, we

introduced the symbol yy in order to distinguish it from the hermite conjugation y. Itis assumed that, at initial time t ¼ 0, the relevant system starts to contact with the

irrelevant system representing the stochastic process described by the random force

operators dFt, dFyt and their tilde conjugates. Within the formalism, the random

force operators dFt and dF yt are assumed to (anti-)commute with any relevant system

operator A in the Schr€oodinger representation, i.e.,

½A; dFtg ¼ 0; ½A; dF yt g ¼ 0: ð6:49Þ

The semi-free operators aðtÞ, ayyðtÞ, and their tilde conjugates keep the equal-time

canonical (anti-)commutation relations

½aðtÞ; ayyðtÞ��r ¼ 1; ð6:50Þ

and satisfy TSC

hhhj~aayyðtÞ ¼ s�hhhjaðtÞ: ð6:51Þ
Calculating the time derivatives of Heisenberg operators of the quantum Brown-
ian motion (6.43) within the Itoo calculus (A.15), and taking into account (6.20) and

(6.45) with the characteristics of the Itoo multiplication

½dW $t ;Wt ��r ¼ ½dWt;W $

t ��r ¼ 0; ð6:52Þ

½dWt ; VVF ðtÞ� ¼ ½dW $t ; VVF ðtÞ� ¼ 0; ð6:53Þ

and their tilde conjugates, one has

dW ðtÞ ¼ dWt � iVV �1F ðtÞ½dWt ; dMMt�VVF ðtÞ; ð6:54Þ

dW $ðtÞ ¼ dW $t � iVV �1

F ðtÞ½dW $t ; dMMt�VVF ðtÞ; ð6:55Þ

and their tilde conjugates. With the help of (6.37)–(6.41), the expressions (6.54) and

(6.55) reduce, respectively, to

dW ðtÞ ¼ dWt � 2kjðtÞaðtÞdt; ð6:56Þ

dW $ðtÞ ¼ dW $t � 2jðtÞa$ðtÞdt ð6:57Þ

(and their tilde conjugates), while (6.53) gives

�dW ðtÞ ¼ dWt ; �dW $ðtÞ ¼ dW $t ð6:58Þ

(and their tilde conjugates).

We see that the martingale operator �dMMðtÞ � VV �1F ðtÞdMMtVVF ðtÞ being written in

terms of the Heisenberg operators reads


�dMMðtÞ ¼ ifa$ðtÞ �dW ðtÞ þ ~aa$ðtÞ �d ~WW ðtÞg � ikf �dW $ðtÞaðtÞ þ �d ~WW $ðtÞ~aaðtÞg¼ ifa$ðtÞdW ðtÞ þ ~aa$ðtÞd ~WW ðtÞg � ikfdW $ðtÞaðtÞ þ d ~WW $ðtÞ~aaðtÞg� dMMðtÞ; ð6:59Þ

and keeps the property

h �dMMðtÞi ¼ hdMMðtÞi ¼ 0 ð6:60Þ
for arbitrary k. The beautiful relation (6.59) is manifestations of the normal ordered
definition (6.36). Note that the increments in the martingale (6.36) are introduced

just through the random force operators dW $t , dWt , and their tilde conjugates.

Therefore, dMMðtÞ is different from the operator calculated by d i a$ðtÞW ðtÞþfðt:c:g � ik W $ðtÞaðtÞ þ t:c:f gÞ. Here, t.c. indicates the tilde conjugation.

6.3. Quantum Langevin equations

6.3.1. Itoo type

Substituting dVVF ðtÞ, (6.20), and dVV �1F ðtÞ, (6.45), into the time derivative of the dy-

namical quantity AðtÞ within the Itoo calculus (A.15), we obtain the quantum Lange-

vin equation of the Itoo type in the form

dAðtÞ ¼ i½HHF ðtÞdt;AðtÞ� � dMMðtÞ½dMMðtÞ;AðtÞ� ð6:61Þ¼ i½HHSðtÞ;AðtÞ�dt þ jðtÞða$ðtÞ½aðtÞ;AðtÞg þ ~aa$ðtÞ½~aaðtÞ;AðtÞg

þ ð2k� 1Þð½AðtÞ; a$ðtÞgaðtÞ þ ½AðtÞ; ~aa$ðtÞg~aaðtÞÞÞdtþ sð2jðtÞ½nðtÞ þ m� þ _nnðtÞÞ½~aa$ðtÞ; ½a$ðtÞ;AðtÞggdtþ ½AðtÞ; a$ðtÞgdW ðtÞ þ ½AðtÞ; ~aa$ðtÞgd ~WW ðtÞþ kðdW $ðtÞ½aðtÞ;AðtÞg þ d ~WW $ðtÞ½~aaðtÞ;AðtÞgÞ ð6:62Þ

¼ i½HHSðtÞ;AðtÞ�dt � jðtÞð½AðtÞ; a$ðtÞgaðtÞ þ ½AðtÞ; ~aa$ðtÞg~aaðtÞþ ð2k� 1Þða$ðtÞ½aðtÞ;AðtÞg þ ~aa$ðtÞ½~aaðtÞ;AðtÞgÞÞdtþ sð2jðtÞ½nðtÞ þ m� þ _nnðtÞÞ½~aa$ðtÞ; ½a$ðtÞ;AðtÞggdtþ ½AðtÞ; a$ðtÞgdWt þ ½AðtÞ; ~aa$ðtÞgd ~WWt

þ kðdW $t ½aðtÞ;AðtÞg þ d ~WW $

t ½~aaðtÞ;AðtÞgÞ; ð6:63Þ

where (6.37)–(6.41) for multiplications among the random force operators are em-ployed. To derive (6.63) from (6.62), we used (6.56) and (6.57). Note that the

Langevin equations (6.62) and (6.63) written, respectively, by means of the quantum

Brownian motion in the Heisenberg representation and by means of that in the

Schr€oodinger representation may be related with the ‘‘out’’ and ‘‘in’’ fields introduced

by Gardiner et al. [77,78].

With the help of (6.62) one can verify that the calculus rule for the product of ar-

bitrary relevant stochastic operators, say AðtÞ and BðtÞ, satisfies the Itoo calculus

(A.15). This proves that QSDE (6.62) is of the Itoo type indeed. Furthermore, since


(6.62) is the time-evolution equation for any relevant stochastic operator AðtÞ, it isItoo�s formula for quantum systems.

6.3.2. Stratonovich type

The quantum Langevin equation of the Stratonovich type can be derived similarlyif one starts from the expression for a dynamical quantity AðtÞ, (6.42), and considers

its derivative in the Stratonovich calculus (A.16) with the help of

dVV �1F ðtÞ ¼ iVV �1

F ðtÞ � HHF ;t dt: ð6:64Þ
Substituting (6.27) for dVVF ðtÞ and (6.64) for dVV �1
F ðtÞ into dAðtÞ, we have for the

stochastic Heisenberg equation of the Stratonovich type

dAðtÞ ¼ i½HHF ðtÞdt�;AðtÞ� ð6:65Þ¼ i½HHSðtÞ;AðtÞ�dt þ jðtÞða$ðtÞ½aðtÞ;AðtÞg þ ~aa$ðtÞ½~aaðtÞ;AðtÞg

þ ð2k� 1Þð½AðtÞ; a$ðtÞgaðtÞ þ ½AðtÞ; ~aa$ðtÞg~aaðtÞÞÞdtþ ½AðtÞ; a$ðtÞg � dW ðtÞ þ ½AðtÞ; ~aa$ðtÞg � d ~WW ðtÞþ kðdW $ðtÞ � ½aðtÞ;AðtÞg þ d ~WW $ðtÞ � ½~aaðtÞ;AðtÞgÞ ð6:66Þ

¼ i½HHSðtÞ;AðtÞ�dt � jðtÞð½AðtÞ; a$ðtÞgaðtÞ þ ½AðtÞ; ~aa$ðtÞg~aaðtÞþ ð2k� 1Þða$ðtÞ½aðtÞ;AðtÞg þ ~aa$ðtÞ½~aaðtÞ;AðtÞgÞÞdtþ ½AðtÞ; a$ðtÞg � dWt þ ½AðtÞ; ~aa$ðtÞg � d ~WWt

þ kðdW $t � ½aðtÞ;AðtÞg þ d ~WW $

t � ½~aaðtÞ;AðtÞgÞ: ð6:67Þ

Here, we defined

½X ðtÞ�; Y ðtÞ� ¼ X ðtÞ � Y ðtÞ � Y ðtÞ � X ðtÞ ð6:68Þ
for arbitrary operators X ðtÞ and Y ðtÞ, and
HHF ðtÞdt ¼ VV �1F ðtÞ � HHF ;t dt � VVF ðtÞ: ð6:69Þ

Note that

VV �1F ðtÞ � dMMt � VVF ðtÞ ¼ VV �1

F ðtÞdMMtVVF ðtÞ þ 12VV �1F ðtÞdMMt dVVF ðtÞ þ 1

2dVV �1

F ðtÞdMMtVVF ðtÞ¼ dMMðtÞ: ð6:70Þ

Using expression (6.66), one can readily verify that the calculus rule for the

product of arbitrary relevant system operators, say AðtÞ and BðtÞ, satisfies the

Stratonovich type calculus (A.16). This fact proves that QSDE (6.66) is indeed of

the Stratonovich type, and provides us with the reason why the stochastic Hei-

senberg equation (6.65) has the same structure as the one (2.14) for non-stochastic

operators.

The quantum Langevin equation of the Stratonovich type can be also derived

from that of the Itoo type by making use the connection formulae (A.13) and(A.14). When dY ðtÞ is dW ðtÞ or d ~WW ðtÞ, and X ðtÞ is constituted by the relevant


operator, say AðtÞ, satisfying the quantum Langevin equation of the Itoo type, the

connection formula (A.13) reduces, respectively, to

AðtÞ � dW ðtÞ ¼ AðtÞ � dW ðtÞ � 12rsð2jðtÞ½nðtÞ þ m� þ _nnðtÞÞ½AðtÞ; ~aa$ðtÞgdt; ð6:71Þ

AðtÞ � d ~WW ðtÞ ¼ AðtÞ � d ~WW ðtÞ � 12sð2jðtÞ½nðtÞ þ m� þ _nnðtÞÞ½AðtÞ; a$ðtÞgdt: ð6:72Þ

Similarly, when dX ðtÞ is dW $ðtÞ or d ~WW $ðtÞ, and Y ðtÞ is AðtÞ, the connection formula(A.14) reduces, respectively, to

dW $ðtÞ � AðtÞ ¼ dW $ðtÞ � AðtÞ; ð6:73Þ

d ~WW $ðtÞ � AðtÞ ¼ d ~WW $ðtÞ � AðtÞ: ð6:74Þ
Using these relations in (6.62), the quantum Langevin equation of the Stratonovich
type is obtained in the form (6.66).

Substituting a and ~aa$ for A as an example, we see that both (6.61) and (6.65) result in

daðtÞ ¼ �ixðtÞaðtÞdt � ð1� 2kÞjðtÞaðtÞdt þ dW ðtÞ¼ � ixðtÞ½ þ jðtÞ�aðtÞdt þ dWt ; ð6:75Þ

d~aa$ðtÞ ¼ �ixðtÞ~aa$ðtÞdt þ jðtÞ~aa$ðtÞdt þ kd ~WW $ðtÞ¼ � ixðtÞ½ � ð1� 2kÞjðtÞ�~aa$ðtÞdt þ kd ~WW $

t ; ð6:76Þ

which are written in terms of the original operators as

daðtÞ ¼ � ixðtÞ½ � kjðtÞ�aðtÞdt � ð1� kÞjðtÞ½ðl� rmÞaðtÞ þ 2rsm~aayyðtÞ�dtþ dW ðtÞ � krsmd ~WW $ðtÞ

¼ � ixðtÞ½ þ kjðtÞ�aðtÞdt � ð1� kÞjðtÞ½ðl� rmÞaðtÞ þ 2rsm~aayyðtÞ�dtþ dWt � krsmd ~WW $

t ; ð6:77Þ

d~aayyðtÞ ¼ � ixðtÞ½ � kjðtÞ�~aayyðtÞdt þ ð1� kÞjðtÞ½ðl� rmÞ~aayyðtÞ � 2rslaðtÞ�dtþ rsdW ðtÞ þ kld ~WW $ðtÞ

¼ � ixðtÞ½ þ kjðtÞ�~aayyðtÞdt þ ð1� kÞjðtÞ½ðl� rmÞ~aayyðtÞ � 2rslaðtÞ�dtþ rsdWt þ kld ~WW $

t : ð6:78Þ

These equations are the same in both Itoo and Stratonovich multiplications as theyshould be with the martingale (6.36).

6.4. Averaged equation of motion

Applying the total bra-vacuum hhhj to the Itoo type quantum Langevin equations

(6.61)–(6.63), one can derive the stochastic equation of motion of the Itoo type for the

bra-vector state hhhjAðtÞ in the form


dhhhjAðtÞ ¼ ihhhj½HHSðtÞ;AðtÞ�dtþ ð2k� 1ÞjðtÞhhhjðAðtÞ½a$ðtÞaðtÞ þ ~aa$ðtÞ~aaðtÞ�Þdtþ rs 2jðtÞ½nðtÞð þ m� þ _nnðtÞÞhhhjAðtÞa$ðtÞ~aa$ðtÞdtþ hhhjðAðtÞ½a$ðtÞdW ðtÞ þ ~aa$ðtÞd ~WW ðtÞ�Þ ð6:79Þ

¼ ihhhj½HHSðtÞ;AðtÞ�dt� jðtÞhhhjðAðtÞ½a$ðtÞaðtÞ þ ~aa$ðtÞ~aaðtÞ�Þdtþ rs 2jðtÞ½nðtÞð þ m� þ _nnðtÞÞhhhjAðtÞa$ðtÞ~aa$ðtÞdtþ hhhjðAðtÞ½a$ðtÞdWt þ ~aa$ðtÞd ~WWt �Þ: ð6:80Þ

In terms of operators aðtÞ and ayyðtÞ it becomes

dhhhjAðtÞ ¼ ihhhj½HHSðtÞ;AðtÞ�dt � jðtÞhhhj ½AðtÞ; ayyðtÞgaðtÞ

þ ayyðtÞ½aðtÞ;AðtÞgdt � rð2jðtÞnðtÞ þ _nnðtÞÞ

� hhhj½½AðtÞ; ayyðtÞg; aðtÞgdt þ hhhj ½AðtÞ; ayyðtÞgdFt

� r½AðtÞ; aðtÞgdF yt

; ð6:81Þ

where we used

hjdWt ¼ hjdFt; hjd ~WWt ¼ s�hjdF yt : ð6:82Þ

The stochastic equation of motion of the Stratonovich type for the bra-vector

state hhhjAðtÞ is derived similarly in the form

dhhhjAðtÞ ¼ ihhhj½HHSðtÞ;AðtÞ�dt � jðtÞhhhj ayyðtÞ½aðtÞ;AðtÞg

� raðtÞ½ayyðtÞ;AðtÞgdt þ hhhj ½AðtÞ; ayyðtÞg � dFt

� r½AðtÞ; aðtÞg � dF y

t

: ð6:83Þ

Applying to (6.81) the random force ket-vacuum ji and the ket-vacuum j0i of therelevant system, one obtains the equation of motion for the expectation value of anarbitrary operator AðtÞ of the relevant system as

d

dthhAðtÞii ¼ihh½HHSðtÞ;AðtÞ�ii � jðtÞhh ½AðtÞ; ayyðtÞgaðtÞ

þ ayyðtÞ½aðtÞ;AðtÞg

ii

� rð2jðtÞnðtÞ þ _nnðtÞÞhh½½AðtÞ; ayyðtÞg; aðtÞgii: ð6:84Þ

Here, hh� � �ii ¼ hjhhj � � � j0iji means to take both random average and vacuum ex-pectation. This is the exact equation of motion for systems with linear-dissipative

coupling to reservoir, which can be also derived by means of the Fokker–Planck

equation (6.18). Here, we used the property

h½AðtÞ; ayyðtÞgdFti ¼ h½AðtÞ; aðtÞgdF yt i ¼ 0; ð6:85Þ

which is the characteristics of the Itoo multiplication. Note that equation of

motion for expectation value of an arbitrary operator AðtÞ does not depend on the

parameter k.


7. Semi-free system with a stationary process

One possible way to specify a model is to give the Boltzmann equation (3.23). For

the cases of a semi-free system corresponding to the stationary quantum stochastic

processes, one needs to make substitutions

iR<ðtÞ ¼ 2j�nn; xðtÞ ¼ x; jðtÞ ¼ j; ð7:1Þ

where �nn is an average quantum number in equilibrium given by

�nn ¼ ex=T

� r�1

; ð7:2Þ

and T is the temperature of environment (here we use the system with the Boltzmann

constant kB ¼ 1). Then, the Boltzmann equation (3.23) becomes

_nnðtÞ ¼ �2jðnðtÞ � �nnÞ: ð7:3Þ

It describes the system of a damped harmonic oscillator.

Substituting the Boltzmann equation (7.3) into the semi-free hat-Hamiltonian

(3.4) with (3.5) and (3.28) or with (3.50) and (3.51), one obtains

HH ¼ xðaya� ~aay~aaÞ þ 2rsijð1þ r�nnÞa~aaþ 2rsij�nnay~aay

� ijð1þ 2r�nnÞðayaþ ~aay~aaÞ � 2ij�nn ð7:4Þ¼ x�aalal � ij�aalAlmam þ rðxþ ijÞ ð7:5Þ¼ xðc$ct � ~cc$~cctÞ � ijðc$ct þ ~cc$~cctÞ � 2rsijðnðtÞ � �nnÞc$~cc$; ð7:6Þ

where

Alm ¼ 1þ 2r�nn �2r�nn2ð1þ r�nnÞ �ð1þ 2r�nnÞ

� �: ð7:7Þ

The Fokker–Planck equation of the model is given by

o

otj0ðtÞi ¼ �iHH j0ðtÞi; ð7:8Þ

with (7.6). It is solved as (3.53) with the order parameter

hhj~cctctj0i ¼ rsðnð0Þ � �nnÞ 1

� e�2jt; ð7:9Þ

where ct, c$, and their tilde conjugates are defined by (3.47) and (3.40) with nðtÞ being

replaced by the solution of (7.3). The expression (3.53) with the order parameter (7.9)

led us to the notion of a mechanism named the spontaneous creation of dissipation[79–84].

Introducing a set of new operators

�ddm ¼ ðdy;�s~ddÞ; dl ¼ collonðd; s~ddyÞ; ð7:10Þ
defined by
�ddm ¼ �aal Blm½ ��1; dl ¼ Blmam; ð7:11Þ


with

Blm ¼ 1þ r�nn �r�nn�1 1

� �; ð7:12Þ

the hat-Hamiltonian HH can be also written in the form

HH ¼ xðdyd � ~ddy~ddÞ � ijðdyd þ ~ddy~ddÞ: ð7:13Þ

We see that the new operators satisfy the canonical ðanti-Þcommutation relation

½dl; �ddm��r ¼ dlm; ð7:14Þ
and that TSC (3.24) for the thermal ket-vacuum j0i can be expressed as
~ddj0i ¼ sðnð0Þ � �nnÞdyj0i: ð7:15Þ
It is easy to see from the diagonalized form (7.13) of HH that
dðtÞ ¼ VV �1ðtÞ~ddyVV ðtÞ ¼ ~ddy e�ðixþjÞt; ð7:16Þ

~ddyyðtÞ ¼ VV �1ðtÞ~ddyVV ðtÞ ¼ ~ddy e�ðix�jÞt: ð7:17Þ
On the other hand, it is easy to see from the normal ordered form (7.6) that HHsatisfies hhjHH ¼ 0, since the annihilation and creation operators satisfy (3.48) and
(3.49). The difference between the operators which diagonalize HH and the ones which

make HH in the form of normal product is one of the features of NETFD, and shows

the point that the formalism is different from usual quantum mechanics and quan-

tum field theory. This is manifestations of the fact that the hat-Hamiltonian is a

time-evolution generator for irreversible processes.The second law of thermodynamics tells us that for a closed system the entropy

increment dS of the relevant system should be given by [85]

dS ¼ dSi þ dSe; ð7:18Þ

dSi P 0; ð7:19Þ
where dSi is the change of intrinsic entropy of the system and dSe the change due to
the heat flow �dQ into the system from the thermal reservoir with temperature T :

dSe ¼�dQT

: ð7:20Þ

We can check this for the present model [1]. The entropy of the relevant system isgiven by [86]

SðtÞ ¼ � nðtÞ ln nðtÞf � r 1½ þ rnðtÞ� ln 1½ þ rnðtÞ�g; ð7:21Þ
whereas the heat change of the system can be identified with
�dQðtÞ ¼ xdnðtÞ ð7:22Þ
leading to
dSe ¼xTdnðtÞ: ð7:23Þ


Putting (7.21) and (7.23) into (7.18) for dS and dSe, respectively, we have a relation

for the entropy production rate [1]

dSi

dt¼ dS

dt� dSe

dt¼ 2jðnðtÞ � �nnÞ ln nðtÞð1þ r�nnÞ

�nnð1þ rnðtÞÞ P 0; ð7:24Þ

giving the inequality (7.19). It is easy to check that inequality holds for both cases,

i.e., nðtÞ > �nn and nðtÞ < �nn, and that the equality realizes for the thermal equilibrium

state, nðtÞ ¼ �nn, or for the quasi-static process with j ! 0.

8. Relation to the Monte Carlo wave-function method

In this section, we will investigate the Fokker–Planck equation (7.8) in order toreveal the relation of NETFD to the Monte Carlo wave-function method, i.e., the

quantum jump simulation [9,87–94] in which evolution with a non-hermitian hat-

Hamiltonian is described in terms of randomly decided quantum jumps followed

by the wave-function normalization.

Let us decompose the hat-Hamiltonian (7.4) as

HH ¼ HH ð0Þ þ HH ð1Þ; ð8:1Þ

with

HH ð0Þ ¼ xðaya� ~aay~aaÞ � ijð1þ 2r�nnÞðayaþ ~aay~aaÞ; ð8:2Þ

HH ð1Þ ¼ 2irsjðð1þ r�nnÞa~aaþ �nnay~aayÞ � 2ij�nn; ð8:3Þ
and consider an equation
o

otj00ðtÞi0 ¼ �iHH ð0Þj00ðtÞi0: ð8:4Þ

Note that HH ð1Þ contains cross terms among tilde and non-tilde operators. We see that

HH ð0Þ and HH ð1Þ have the properties

hhjHH ð0Þ ¼ �2ijð1þ 2r�nnÞhhjaya; ð8:5Þ

hhjHH ð1Þ ¼ 2ijð1þ 2r�nnÞhhjaya: ð8:6Þ
Introducing the wave-functions jwðtÞi and j ~wwðtÞi through the relation
j00ðtÞi0 ¼ jwðtÞij ~wwðtÞi; ð8:7Þ
we have from (8.4) the Schr€oodinger equations of the form
o

otjwðtÞi ¼ �iH ð0ÞjwðtÞi; ð8:8Þ

and its tilde conjugate, where

H ð0Þ ¼ xaya� ijð1þ 2r�nnÞaya: ð8:9Þ


This procedure is possible because HH ð0Þ does not contain cross terms among tilde and

non-tilde operators. The Monte Carlo simulations for quantum systems are per-

formed for the Schr€oodinger equation (8.8) [87–90].

The time evolution generated by the hat-Hamiltonian HH ð0Þ does not preserve the

normalization of the ket-vacuum, i.e., the inner product hh j0ðtÞi evolves for timeincrement dt as

hhj00ðt þ dtÞi0 ¼ hhjð1� iHH ð0Þ dtÞj0ðtÞi ¼ 1� dpðtÞ; ð8:10Þ
with
dpðtÞ ¼ ihhjHH ð0Þj0ðtÞidt ¼ 2jð1þ 2r�nnÞnðtÞdt: ð8:11Þ
The recipe of the quantum jump simulation is that, for a time increment dt,
1) when dpðtÞ < e with a given positive constant e, the normalized ket-vacuum

evolves as

j0ðtÞi ! j00ðt þ dtÞi ¼ j00ðt þ dtÞi0

1� dpðtÞ ¼ jwðt þ dtÞiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� dpðtÞ

p j ~wwðt þ dtÞiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� dpðtÞ

p ; ð8:12Þ

2) in the case dpðtÞ > e, a quantum jump comes in

j01ðt þ dtÞi ¼ �iHH ð1Þ dtj0ðtÞidpðtÞ : ð8:13Þ

The time increment dt should be chosen as the condition dpðtÞ 1 being satisfied.

Averaging the processes j00ðtÞi and j01ðtÞi with respective probabilities 1� dpðtÞand dpðtÞ:

j0ðt þ dtÞi ¼ ½1� dpðtÞ�j00ðt þ dtÞi þ dpðtÞj01ðt þ dtÞi; ð8:14Þ
we can obtain the Fokker–Planck equation (7.8). Note that the ket-vacuums j00ðtÞiand j01ðtÞi look like satisfying a certain kind of stochastic Liouville equation.
9. Summary

The aim of this paper has been to study the system of QSDEs from a physical ba-

sis. We have formulated everything from the starting point using the method of

NETFD. In the presented approach, boson and fermion systems are considered si-

multaneously. The obtained results have two fixed parameters: the real parameter

r specifying different commutation rules for boson and fermion operators, and the

complex parameter s, (2.8), specifying different thermal state conditions for bosonand fermion systems. Such a combined consideration was made possible due to

the unification of fermion and boson stochastic calculus (Appendix B), where fer-

mion annihilation and creation processes are realized in a boson Fock space by

means of a simple stochastic integral prescription leading to similar multiplication

rules for stochastic differentials.

The dissipation mechanism is considered through the concept of a quantum noise,

i.e., as a quantum field interacting with the relevant system. In our paper we consid-


ered two types of interaction with external fields: hermitian (06 k < 1) and non-her-

mitian (06 k < 1). With the latter, conservation of the probability is satisfied within

the relevant system. With the former, information about only relevant system is not

enough and instead of that we can speak about conservation of the probability with-

in the total system: relevant system plus environment system.As we are concentrated on the stochastic equations, there are two types of sto-

chastic calculus: Itoo and Stratonovich. Correspondingly, equations used one or an-

other type of stochastic calculus are classified as QSDE of the Itoo or Stratonovich

types. The Langevin equation of the Stratonovich type (6.65) has structure similar

to one of the Heisenberg equation of motion for a dynamical quantity in quantum

mechanics and quantum field theory. As a result of different stochastic multiplication

rule, the Langevin equation of the Itoo type (6.61) contains an extra term propor-

tional to a product of random forces dWt d ~WWt . The corresponding Fokker–Planckequation is then obtained most easily from the quantum stochastic Liouville equa-

tion of the Itoo type by taking the random average. Though in fermion case the con-

nection with the classical Brownian motion is only formal, the Itoo/Stratonovich

product formula is the same as in boson case (relations (A.15) and (A.16)). The av-

eraged equation of motion for a dynamical quantity can be obtained in two ways.

From the Langevin equation by taking both random average and the relevant vac-

uum expectation, or from the Fokker–Planck equation by taking the vacuum expec-

tation of operators corresponding to the dynamical quantity. In our study weshowed that QSDEs constructed upon hermitian and non-hermitian interaction

hat-Hamiltonians lead to the same averaged equation of motion (6.84) for an arbi-

trary operator of the relevant system. In the case of stationary semi-free quantum

stochastic process, its irreversibility is checked in terms of the Boltzmann entropy.

We also demonstrated the relationship between the presented formulation and the

method of quantum jump simulations.

The approach we followed in this paper is rather formal and we are looking now

for some demonstrative examples of its application for particular problems. An in-teresting result is obtained, for instance, for the model of a continuous quantum

non-demolition measurement—continuous observation of a particle track in the

cloud chamber [95,96] and for the system corresponding to the quantum Kramers

equation [97,98]. More detailed report about them will be presented elsewhere.

Acknowledgments

Authors thank Mr. Y. Fukuda and Mr. Y. Kaburaki for their fruitful discussions.

Appendix A. Ito and Stratonovich calculus

Definitions of the Itoo [24] and Stratonovich [25] multiplications for arbitrarystochastic operators Xt and Yt in the Schr€oodinger representation are given, respec-

tively, by


Xt � dYt ¼ XtðYtþdt � YtÞ; ðA:1Þ

dXt � Yt ¼ ðXtþdt � XtÞYt; ðA:2Þ
and
Xt � dYt ¼ 12ðXtþdt þ XtÞðYtþdt � YtÞ; ðA:3Þ

dXt � Yt ¼ ðXtþdt � XtÞ12ðYtþdt þ YtÞ: ðA:4Þ

From these relations we have the connection formulae between the Itoo and Stra-

tonovich products in the differential form as

Xt � dYt ¼ Xt � dYt þ 12dXt � dYt; ðA:5Þ

dXt � Yt ¼ dXt � Yt þ 12dXt � dYt: ðA:6Þ

Note that random average of the stochastic multiplication (A.1) or (A.2) of the Itoo

type is equal to zero.Definitions of the Itoo and Stratonovich multiplications for stochastic operators

X ðtÞ and Y ðtÞ in the Heisenberg representation are given in the same form by

X ðtÞ � dY ðtÞ ¼ X ðtÞ Y ðt½ þ dtÞ � Y ðtÞ�; ðA:7Þ

dX ðtÞ � Y ðtÞ ¼ X ðt½ þ dtÞ � X ðtÞ�Y ðtÞ; ðA:8Þ

and

X ðtÞ � dY ðtÞ ¼ 12X ðt½ þ dtÞ þ X ðtÞ� Y ðt½ þ dtÞ � Y ðtÞ�; ðA:9Þ

dX ðtÞ � Y ðtÞ ¼ X ðt½ þ dtÞ � X ðtÞ�12Y ðt½ þ dtÞ þ Y ðtÞ�; ðA:10Þ

where operators X ðtÞ and dX ðtÞ are introduced, respectively, through relations

X ðtÞ ¼ VV �1F ðtÞXtVVF ðtÞ; ðA:11Þ

dX ðtÞ ¼ dðVV �1F ðtÞXtVVF ðtÞÞ; ðA:12Þ

with VVF ðtÞ being a stochastic time evolution operator.

From (A.7)–(A.10), we have the connection formulae between the Itoo and Stra-

tonovich products in the differential form as

X ðtÞ � dY ðtÞ ¼ X ðtÞ � dY ðtÞ þ 12dX ðtÞ � dY ðtÞ; ðA:13Þ

dX ðtÞ � Y ðtÞ ¼ dX ðtÞ � Y ðtÞ þ 12dX ðtÞ � dY ðtÞ: ðA:14Þ

Stochastic multiplications (A.7)–(A.10) are consistent with corresponding types of

differential calculus for products of stochastic operators, which for the case of the Itootype calculus and the Stratonovich type calculus read, respectively, as

d½X ðtÞY ðtÞ� ¼ dX ðtÞ � Y ðtÞ þ X ðtÞ � dY ðtÞ þ dX ðtÞ � dY ðtÞ ðA:15Þ


and

d½X ðtÞY ðtÞ� ¼ dX ðtÞ � Y ðtÞ þ X ðtÞ � dY ðtÞ: ðA:16Þ

Appendix B. Boson and fermion Brownian motion

Let C0s denotes the boson Fock space (the symmetric Fock space) over the Hilbert

space H ¼ L2ðRþÞ of square integrable functions, and bt and byt denote, respectively,boson annihilation and creation operators at time t 2 ½0;1Þ satisfying the canonical

commutation relations

½bt; bys � ¼ dðt � sÞ; ½bt; bs� ¼ ½byt ; bys � ¼ 0: ðB:1Þ
The bra- and ket-vacuums (j and j) are defined, respectively, by
ðjbyt ¼ 0; btjÞ ¼ 0: ðB:2Þ
Note that ðj ¼ jÞy since here we are considering the unitary representation of bt andbyt . The space C0
s is equipped with a total family of exponential vectors

ðeðf Þj ¼ ðjexpZ 1

0

dtf �ðtÞbt� �

; ðB:3Þ

jeðgÞÞ ¼ exp

Z 1

0

dtgðtÞbyt� �

jÞ; ðB:4Þ

whose overlapping is

ðeðf ÞjeðgÞÞ ¼ exp

Z 1

0

dtf �ðtÞgðtÞ� �

: ðB:5Þ

Here, f , g 2 H. The dense span of exponential vectors is denoted by E. Operators bt,byt and exponential vectors are characterized by the relations

ðeðf Þjbyt ¼ ðeðf Þjf �ðtÞ; btjeðgÞÞ ¼ gðtÞjeðgÞÞ: ðB:6Þ
Let us introduce operator Ut defined as
Ut ¼ r<P½0;t� þ r>Pðt;1Þ; ðB:7Þ

where r< and r> are two independent parameters taking values 1, and P½a;b� (a6 b)is an operator on H of multiplication by the indicator function whose action reads

P½a;b�

Z 1

0

dtgðtÞ ¼Z b

adtgðtÞ ¼

Z 1

0

dthðt � aÞhðb� tÞgðtÞ: ðB:8Þ

Here, hðtÞ is the step function specified by

hðtÞ ¼ 1 for tP 0;0 for t < 0:

�ðB:9Þ

The operator P½a;b� has the following properties:

P 2½a;b� ¼ P½a;b�; P y

½a;b� ¼ P½a;b�; P½a;b�P½c;d� ¼ P½c;d�P½a;b�; ðB:10Þ


which are easily verified using the definition (B.8). Then, we see that operator Ut is

unitary, and satisfies

U 2t ¼ I ; U y

t ¼ Ut; UtUs ¼ UsUt; ðB:11Þ
where I is the identity operator on H.
The so-called reflection process Jt � JtðUtÞ, t 2 Rþ, whose action on E is given

by [62]

JtjeðgÞÞ ¼ jeðUtgÞÞ ¼ exp Ut

Z 1

0

dt0gðt0Þbyt0� �

jÞ; ðB:12Þ

inherits properties of the operator Ut, (B.11), i.e.,

J 2t ¼ 1; J y

t ¼ Jt; JtJs ¼ JsJt; ðB:13Þ
and does not change the vacuum
ðjJt ¼ ðj; JtjÞ ¼ jÞ: ðB:14Þ
Here, 1 is the unit operator defined in C0
s .

Let us now consider new operators

bt ¼ Jtbt; byt ¼ byt Jt: ðB:15Þ
Apparently, they annihilate vacuums
ðjbyt ¼ 0; btjÞ ¼ 0: ðB:16Þ
The following matrix elements:
ðeðf Þj½Jt; bs��rjeðgÞÞ ¼ ðeðf Þjf1� r½r> þ ðr< � r>Þhðt � sÞ�ggðsÞJtjeðgÞÞ;ðB:17Þ

ðeðf Þj½bt;bys ��rjeðgÞÞ ¼ ðeðf Þjdðt� sÞjeðgÞÞþ ðeðf ÞjJtJsðr>r< � rÞf �ðsÞgðtÞjeðgÞÞ ðB:18Þ

are valid for f , g 2 H. Then the requirement of equal-time (anti-)commutativity

between Jt and bt

½Jt; bt��r ¼ 0 ðB:19Þ
gives
1� rr< ¼ 0; ðB:20Þ
while the requirement of canonical (anti-)commutation relation
½bt; bys ��r ¼ dðt � sÞ ðB:21Þ
leads to
r>r< � r ¼ 0: ðB:22Þ

All those conditions are satisfied when r< ¼ r and r> ¼ þ1. Then the operator Ut

turns out to be


Ut ¼ rP½0;t� þ Pðt;1Þ: ðB:23Þ

Note that for a boson system, i.e., r ¼ 1, Ut ¼ I and the operators bt and byt reduce,respectively, to bt and byt .

We see that the generalized quantum Brownian motion, defined by

Bt ¼Z t

0

dt0bt0 ; Byt ¼

Z t

0

dt0byt0 ; ðB:24Þ

with B0 ¼ 0, By0 ¼ 0, satisfies

½Bt;Bys ��r ¼ minðt; sÞ: ðB:25Þ

The case r ¼ 1 represents the boson Brownian motion [49,55], whereas the case

r ¼ �1 the fermion Brownian motion [62]. Their increments

dBt ¼ Btþdt � ~BByt ¼ bt dt; ðB:26Þ

dByt ¼ By

tþdt � Byt ¼ byt dt; ðB:27Þ

annihilate the vacuum, i.e.,

ðjdByt ¼ 0; dBtjÞ ¼ 0; ðB:28Þ

and their matrix elements read

ðeðf ÞjdBtjeðgÞÞ ¼ ðeðf ÞjJtgðtÞdtjeðgÞÞ; ðB:29Þ

ðeðf ÞjdByt jeðgÞÞ ¼ ðeðf Þjf �ðtÞdtJtjeðgÞÞ; ðB:30Þ

ðeðf ÞjdBt dBtjeðgÞÞ ¼ 0; ðB:31Þ

ðeðf ÞjdByt dBtjeðgÞÞ ¼ 0; ðB:32Þ

ðeðf ÞjdBt dByt jeðgÞÞ ¼ dtðeðf ÞjeðgÞÞ: ðB:33Þ

Here, we neglected terms of the higher order than dt. The latter equations are

summarized in the following table of multiplication rules for increments dBt and dByt :

ðB:34Þ

Now we consider a tensor product space CC ¼ C0s � ~CC0

s . Its vacuum states jÞÞ and ex-

ponential vectors jeðf ; gÞÞÞ are defined through the ‘‘principle of correspondence’’ [3]

jÞÞ $ jÞðj; ðB:35Þ

jeðf ; gÞÞÞ $ jeðf ÞÞðeðgÞj: ðB:36Þ


Annihilation and creation operators acting on CC are defined through

btjeðf ; gÞÞÞ $ btjeðf ÞÞðeðgÞj; ðB:37Þ

byt jeðf ; gÞÞÞ $ byt jeðf ÞÞðeðgÞj; ðB:38Þ

~bbtjeðf ; gÞÞÞ $ jeðf ÞÞðeðgÞjbyt ; ðB:39Þ

~bbyt jeðf ; gÞÞÞ $ jeðf ÞÞðeðgÞjbt; ðB:40Þ
and similarly for Jt and ~JJt, i.e.,
Jtjeðf ; gÞÞÞ $ Jtjeðf ÞÞðeðgÞj; ðB:41Þ

~JJtjeðf ; gÞÞÞ $ jeðf ÞÞðeðgÞjJt: ðB:42Þ
Algebra of commutation relations between these operators reads
½bt; bys � ¼ ½~bbt; ~bbys� ¼ dðt � sÞ; ðB:43Þ

½bt; ~bbs� ¼ ½bt; ~bbys � ¼ 0; ðB:44Þ

½Jt; ~bbs� ¼ ½ ~JJt; bs� ¼ 0; ðB:45Þ

½Jt; bt��r ¼ ½ ~JJt; ~bbt��r ¼ 0: ðB:46Þ
Let us now consider new operators defined by
bt ¼ Jtbt; byt ¼ byt Jt; ðB:47Þ

~bbt ¼ ss ~JJt~bbt; ~bbyt ¼ ss~bbyt ~JJt; ðB:48Þ
where ss is an operator satisfying the following (anti-)commutation relations:
½ss; Jt� ¼ ½ss; ~JJt� ¼ 0; ðB:49Þ

½ss; bt��r ¼ ½ss; byt ��r ¼ 0; ðB:50Þ

½ss; ~bbt��r ¼ ½ss; ~bbyt ��r ¼ 0; ðB:51Þ
and the condition
ss2 ¼ r: ðB:52Þ
Operators bt, b
yt and their tilde conjugates annihilate vacuums

ððjbyt ¼ ððj~bbyt ¼ 0; btjÞÞ ¼ ~bbtjÞÞ ¼ 0; ðB:53Þ
and satisfy canonical (anti-)commutation relations
½bt; bys ��r ¼ ½~bbt; ~bbys ��r ¼ dðt � sÞ; ðB:54Þ

½bt; ~bbs��r ¼ ½bt; ~bbys��r ¼ 0: ðB:55Þ


Since ð~bbtÞy and ðbyt Þ�are calculated as

ð~bbtÞy ¼ ~bbyt ~JJt ssy ¼ rssy~bbyt ~JJt; ðB:56Þ

ðbyt Þ� ¼ ssðbyt JtÞ

� ¼ ss~bbyt ~JJt; ðB:57Þ
the commutativity of tilde conjugation and hermitian conjugation for operators
(B.47) and (B.48) implies

ssy ¼ rss: ðB:58Þ
In order to fulfill the requirement that double tilde conjugation applied to operators
bt�s leaves them unchanged one needs to put

ðssÞ� ¼ ssy; ðB:59Þ
since
ð~bbtÞ� ¼ ssðssÞ�Jtbt ¼ ssðssÞ�bt: ðB:60Þ
Because of
~bbtjÞÞ ¼ ~bbtssjÞÞ ¼ 0; ðB:61Þ
one can conclude that ssjÞÞ / jÞÞ. The proportionality factor is a phase factor since
the norm of ssjÞÞ is unity:
ððjssyssjÞÞ ¼ ðjrssssjÞ ¼ r2 ¼ 1: ðB:62Þ
Hence one can write

ssjÞÞ ¼ ei/=2jÞÞ: ðB:63Þ
Multiplying both sides by ss, one has
ss2jÞÞ ¼ ei/jÞÞ ¼ rjÞÞ; ðB:64Þ
which gives ei/ ¼ r, or
ssjÞÞ ¼ffiffiffir

pjÞÞ: ðB:65Þ

Thermal degree of freedom can be introduced by the Bogoliubov transformation in

CC. For this purpose we require that the expectation value of bytbs should be

hbytbsi ¼ �nndðt � sÞ; ðB:66Þ
with �nn 2 Rþ, where h� � �i ¼ hj � � � ji indicates the expectation with respect to tilde
invariant thermal vacuums hj and ji. The requirement (B.66) is consistent with TSCfor states hj and ji such that

hj~bbyt ¼ s�hjbt; ~bbtji ¼s�nn

1þ r�nnbyt ji: ðB:67Þ

Let us introduce annihilation and creation operators

ct ¼ ½1þ r�nn�bt � rs�nn~bbyt ; ðB:68Þ

~cc$ ¼ ~bby � rsbt; ðB:69Þ
t t


and their tilde conjugates. From the TSC (B.67) one has

hjc$t ¼ hj~cc$t ¼ 0; ctji ¼ ~cctji ¼ 0: ðB:70Þ

With the thermal doublet notations

�bblt ¼ byt ;�

� s~bbt�; bmt ¼ collon bt; s~bb

yt

� �; ðB:71Þ

and

�cclt ¼ c$t ;�

� s~cct�; cmt ¼ collon ct; s~cc

$t

� �; ðB:72Þ

(B.68), (B.69), and their tilde conjugates can be written in form of the Bogoliubov

transformation

clt ¼ Blmbmt ; �ccmt ¼ �bblt ½B�1�lm; ðB:73Þ
with (7.12). This new operators satisfy the canonical (anti-)commutation rela-tions
½ct; c$s ��r ¼ dðt � sÞ: ðB:74Þ
In the following, we will use the representation space constructed on vacuums hj
and ji. Note that hj 6¼ jiy, i.e., it is not a unitary representation. Let CCb denotes theFock space spanned by the basic bra- and ket-vectors introduced by a cyclic opera-

tions of ct, ~cct on the thermal bra-vacuum hj, and of c$t , ~cc$t on the thermal ket-vacuum

ji. Quantum Brownian motion at finite temperature is defined in the Fock space CCb

by operators

B]t ¼

Z t

0

dsb]s; ~BB]t ¼

Z t

0

ds~bb]s; ðB:75Þ

with B]0 ¼ 0 and ~BB]

0 ¼ 0, where ] stands for null or dagger. The explicit representa-

tion of processes B]t and

~BB]t can be performed in terms of the Bogoliubov transfor-

mation. The couple Bt and Byt , for example, is calculated as

Bt ¼Z t

0

dsðcs þ rs�nn~cc$s Þ ¼ Ct þ rs�nn~CC$

t ; ðB:76Þ

Byt ¼

Z t

0

dsð½1þ r�nn�c$s þ s~ccsÞ ¼ ½1þ r�nn�C$

t þ s~CCt; ðB:77Þ

where we defined new operators

C]t ¼

Z t

0

dsc]s; ~CC]t ¼

Z t

0

ds~cc]s; ðB:78Þ

with C]0 ¼ 0 and ~CC]

0 ¼ 0, and ] standing for null or the Venus-mark. Since vacuum

expectations of dC]t and d~CC]

t in thermal space CCb read

~ $ ~$
hdCti ¼ hdCCti ¼ hdCt i ¼ hdCCt i ¼ 0; ðB:79Þ


hdC$

t dCti ¼ hd~CC$

t d~CCti ¼ 0; ðB:80Þ

hdCt dC$

t i ¼ hd~CCt d~CC$

t i ¼ dt; ðB:81Þ

calculation of moments of quantum Brownian motion in the thermal space CCb can be

performed, for instance, as

hdBt dByt i ¼ hðdCt þ rs�nnd~CC$

t Þð½1þ r�nn�dC$

t þ sd~CCtÞi¼ ½1þ r�nn�hdCt dC

$

t i ¼ ½1þ r�nn�dt: ðB:82Þ

Repeating this for other pair products of dB]t , d

~BB]t , and dt, multiplication rules for

these increments can be summarized in the following table:

ðB:83Þ

Appendix C. Treatment of fermions in thermo field dynamics

We are deciding the double tilde conjugation rule and the thermal state conditions

for fermions [99,100] by considering the system consisting of a vector field and Fad-

deev–Popov ghosts [101].

In the case of pure Abelian gauge field within the Feynman gauge, the system is

specified by the Hamiltonian Hvfþgh ¼ Hvf þ Hgh defined on the total state vectorspace V ¼ Vvf � Vgh. Hvf and Hgh are, respectively, Hamiltonians for the vector

field and ghosts defined on the vector field sector Vvf and the ghost sector Vgh

given by

Hvf ¼ �Z

d3keð~kkÞglmaylð~kkÞamð~kkÞ; ðC:1Þ

with eð~kkÞ ¼ j~kkj being the energy spectrum and glm ¼ diagð1;�1;�1;�1Þ, and by

Hgh ¼ �i

Zd3keð~kkÞ �ccyð~kkÞcð~kkÞ

h� cyð~kkÞ�ccð~kkÞ

i: ðC:2Þ

Here, aylð~kkÞ and alð~kkÞ are, respectively, creation and annihilation operators of the

gauge field of the mode ~kk satisfying the canonical commutation relations

½alð~kkÞ; aymð~qqÞ� ¼ �glmd3ð~kk �~qqÞ; ðC:3Þ


while cyð~kkÞ and cð~kkÞ [�ccyð~kkÞ, and �ccð~kkÞ] are, respectively, creation and annihilation

operators of ghosts [anti-ghosts] satisfying the following canonical anti-commuta-

tion relations:

½cð~kkÞ; �ccyð~qqÞ�þ ¼ �½�ccð~kkÞ; cyð~qqÞ�þ ¼ id3ð~kk �~qqÞ: ðC:4Þ

Other combinations of ghost/anti-ghost operators anti-commute with each other.

The BRS charge—generator of the BRS transformation [102] and the ghost charge

[103,104] acting on the total space V are, respectively, given by

QB ¼ �Z

d3kkl alð~kkÞcyð~kkÞh

þ aylð~kkÞcð~kkÞi; ðC:5Þ

Qc ¼Z

d3k cyð~kkÞ�ccð~kkÞh

þ �ccyð~kkÞcð~kkÞi; ðC:6Þ

which satisfy

½iQc;QB� ¼ QB: ðC:7Þ

Let us introduce a set of new operators faðrÞð~kkÞ j r ¼ þ;�;L; Sg through the

relation

alð~kkÞ ¼ aðrÞð~kkÞ�ðrÞl ð~kkÞ; ðC:8Þ

where �ðrÞl ð~kkÞ are polarization vectors defined by

�ðÞl ð~kkÞ ¼ ð0;~eeÞ; ðC:9Þ

�ðLÞl ð~kkÞ ¼ �ikl ¼ �iðj~kkj;~kkÞ; ðC:10Þ

�ðSÞl ð~kkÞ ¼ �i�kkl=2j~kkj2 ¼ �iðj~kkj;�~kkÞ=2j~kkj2; ðC:11Þ

with ~ee satisfying ~ee �~kk ¼ 0, ~ee� �~ee� ¼ 0 and ~ee� �~ee ¼ 1. The polarization vectors

�ðÞl ð~kkÞ correspond, respectively, to the transverse modes with helicity 1, while

�ðLÞl ð~kkÞ and �ðSÞl ð~kkÞ indicate, respectively, the longitudinal mode and the scalar mode.

With the definition (C.10) and (C.11), we see that �ðLÞ�l ð~kkÞ � �ðLÞ;lð~kkÞ ¼ �ðSÞ�l ð~kkÞ� �ðSÞ;lð~kkÞ ¼ 0, and �ðLÞ�l ð~kkÞ � �ðSÞ;lð~kkÞ ¼ 1. Introducing a ‘‘metric’’

gðrsÞ ¼ gðrsÞ ¼

�1 0 0 0

0 �1 0 00 0 0 1

0 0 1 0

0BB@

1CCA; ðC:12Þ

we can define ‘‘contravariant’’ polarization vectors through �lðrÞð~kkÞ ¼ gðrsÞ�ðsÞ;lð~kkÞ,and see that

�l�ðrÞð~kkÞ � �ðrÞ;mð~kkÞ ¼ glm; ðC:13Þ

�ðrÞ�l ð~kkÞ � �ðsÞ;lð~kkÞ ¼ gðrsÞ: ðC:14Þ


The commutation relations (C.3) being rewritten in terms of operators aðrÞð~kkÞ and

ayðrÞð~kkÞ become

½aðrÞð~kkÞ; ayðsÞð~qqÞ� ¼ �gðrsÞdð~kk �~qqÞ: ðC:15Þ

Also the Hamiltonian for the vector field and generator of the BRS transformation

read

Hvf ¼ �Z

d3keð~kkÞgðrsÞayðrÞð~kkÞaðsÞð~kkÞ; ðC:16Þ

QB ¼ �i

Zd3k ayðSÞð~kkÞcð~kkÞh

� aðSÞð~kkÞcyð~kkÞi: ðC:17Þ

In the local covariant operator formalism [103,104] of gauge theories, the space Vof state vectors has inevitably an indefinite metric as can be seen by (C.15) with

(C.12). The physical subspace Vphys of V, defined by [104]

QBVphys ¼ 0; ðC:18Þ
can be shown to have a positive semi-definite metric [104]. Dividing Vphys by its
subspace V0 consisting of normless states, we have, as a quotient space, the physical

Hilbert space Hphys (¼ Vphys=V0) with positive definite metric in which the probabi-

listic interpretation of quantum theory works. Hphys is isomorphic to the Hilbert

space Hphys spanned by the Fock states created by the cyclic operation of ayðÞð~kkÞ on a

certain vacuum. The space spanned by the Fock states created by ayðSÞð~kkÞ is classifiedin V0, while the space spanned by the Fock states created by ayðLÞð~kkÞ is classified in a

space complemented to Vphys. This reflects the fact that physical modes for photonsare two transverse modes only.

As it is sufficient to pay attention to one mode in the following manipulation, we

will pick up a mode, say~kk, from each type of particles, and drop the index~kk, for sim-

plicity. Let us span the state vector space V by means of a set of the bases

fjfnðrÞgÞ � jnc; n�ccÞg whose elements, being the bases of the vector field sector Vvf

and the ghost sector Vgh, respectively, are defined by

jfnðrÞgÞ ¼Y

r¼;L;S

1ffiffiffiffiffiffiffiffiffinðrÞ!

p ayðrÞ� �nðrÞ

jf0gÞ; ðC:19Þ

jnc; n�ccÞ ¼ cy ncð�ccyÞn�cc j0; 0Þ: ðC:20Þ

They constitute the eigenstates of Hvf , Hgh, and iQc:

Hvf jfnðrÞgÞ ¼ EvfðfnðrÞgÞjfnðrÞgÞ; ðC:21Þ

Hghjnc; n�ccÞ ¼ Eghðnc; n�ccÞjnc; n�ccÞ; ðC:22Þ

iQcjnc; n�ccÞ ¼ Nghðnc; n�ccÞjnc; n�ccÞ; ðC:23Þ

with EvfðfnðrÞgÞ ¼ eP

r nðrÞ, Eghðnc; n�ccÞ ¼ e nc þ n�ccð Þ, and Nghðnc; n�ccÞ ¼ nc � n�ccð Þ,where nðrÞ are non-negative integers, and nc and n�cc take values of 0 or 1. We will


denote the basis vectors fjfnðrÞgÞ � jnc; n�ccÞg by fjnÞg for brevity. Then, the metric

tensor of V is

gn;m ¼ ðnjmÞ: ðC:24Þ

At the finite temperature, the statistical average of an observable quantity A,satisfying

½QB;A� ¼ 0; ðC:25Þ
is given by [105]
hAi ¼ TrAqP ð0Þ ¼ TrAqepQc ðC:26Þ
with P ð0Þ being a projection operator onto Hphys and q ¼ Z�1 e�bHvfþgh being the sta-
tistical operator acting on V with the partition function Z ¼ Tr e�bHvfþghþpQc ; the traceoperation is taken in the space V. Here, for the second equality in (C.26), we used the

BRS-invariance of the statistical operator

½QB; q� ¼ 0: ðC:27Þ
Let us express the statistical average (C.26) as the vacuum expectation in the dou-
bled state space (thermal space) VV ¼ V � ~V which is introduced as follows. If A is an

operator on V so that

A ¼Xn;m

AnmjnÞðmj; ðC:28Þ

the corresponding vector jAi in VV is obtained as

jAi ¼ Anmjn; ~mmi; ðC:29Þ
where fjn; ~mmi � jfnðrÞg; f ~mmðrÞgi � jnc; n�cc; ~mmc; ~mm�ccig is the set of the bases spanning VV and
defined through the ‘‘principle of correspondence’’ [3]:

jfnðrÞg; f ~mmðrÞgi $ jfnðrÞgÞðfmðrÞgj; ðC:30Þ

jnc; n�cc; ~mmc; ~mm�cci $ jnc; n�ccÞðmc;m�ccj: ðC:31Þ
The inner product in VV is given by
hAjBi ¼ TrAyB: ðC:32Þ
Annihilation and creation operators acting on VV are defined through
aðsÞayðsÞ

� �jfnðrÞg; f ~mmðrÞgi $

aðsÞayðsÞ

� �jfnðrÞgÞðfmðrÞgj; ðC:33Þ

~aaðsÞ~aayðsÞ

� �jfnðrÞg; f ~mmðrÞgi $ jfnðrÞgÞðfmðrÞgj ayðsÞ

aðsÞ

� �; ðC:34Þ

for vector field, and through

c�cc

� �jnc; n�cc; ~mmc; ~mm�cci $

c�cc

� �jnc; n�ccÞðmc;m�ccj; ðC:35Þ


cy

�ccy

� �jnc; n�cc; ~mmc; ~mm�cci $ cy

�ccy

� �jnc; n�ccÞðmc;m�ccj; ðC:36Þ

~cc~�cc�cc

� �jnc; n�cc; ~mmc; ~mm�cci $ ð�1Þuþ1jnc; n�ccÞðmc;m�ccj

cy

�ccy

� �; ðC:37Þ

~ccy~�cc�ccy

!jnc; n�cc; ~mmc; ~mm�cci $ ð�1Þujnc; n�ccÞðmc;m�ccj

c�cc

� �; ðC:38Þ

for ghosts, where u ¼ Nghðnc; n�ccÞ � Nghðmc;m�ccÞ. Also bases jfnðrÞg; f ~mmðrÞgi and jnc; n�cc;~mmc; ~mm�cci are generated from the vacuums jf0g; f~00gi and j0; 0; ~00; ~00i, respectively, as

jfnðrÞg; f ~mmðrÞgi ¼Yr

ðayðrÞÞnðrÞ ð~aayðrÞÞ

mðrÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinðrÞ!mðrÞ!

p jf0g; f~00gi; ðC:39Þ

jnc; n�cc; ~mmc; ~mm�cci ¼ ð�1ÞvðcyÞncð�ccyÞn�ccð~ccyÞmcð~�cc�ccyÞm�cc j0; 0; ~00; ~00i; ðC:40Þ

where v ¼ mcm�cc. For the total vacuum, we will use a collective designation j0; ~00i.Let us introduce thermal vacuums hhj and j0ðbÞi 2V such that

hAi ¼ hhjAj0ðbÞi: ðC:41Þ
We require them to satisfy
hhjQQ�B ¼ 0; QQ�

B j0ðbÞi ¼ 0; ðC:42Þ
and
hhjQQc ¼ 0; QQcj0ðbÞi ¼ 0; ðC:43Þ
where QQ�
B and QQc are the generator of the BRS transformation and the ghost hat-

charge, respectively, in V [99]

QQ�B ¼ QB � ~QQB ¼ �i ayðSÞc

h� aðSÞcy þ ~aayðSÞ~cc� ~aaðSÞ~ccy

i; ðC:44Þ

QQc ¼ Qc � ~QQc ¼ cy�ccþ �ccyc� ~ccy~�cc�cc� ~�cc�ccy~cc: ðC:45Þ
To satisfy (C.42), we need a trick. Namely, by rewriting (C.26) as
hAi ¼ TrhAqepQch�1; ðC:46Þ
we introduce an operator h with the basic requirement that its inverse exists. Then we
settle the correspondence

hhj $ h; j0ðbÞi $ qepQch�1: ðC:47Þ
It gives
hhjQQc ¼ hhjðcy�ccþ �ccyc� ~ccy~�cc�cc� ~�cc�ccy~ccÞ $ hðcy�ccþ �ccycÞ � ðcy�ccþ �ccycÞh ¼ ½h;Qc�:ðC:48Þ

With the requirement given by the first equality in (C.43), the expression in (C.48) isequal to zero and tells us that h and Qc commute with each other, i.e. ½h;Qc� ¼ 0.

Then


QQcj0ðbÞi ¼ cy�cc�

þ �ccyc� ~ccy~�cc�cc� ~�cc�ccy~cc�j0ðbÞi

$ ðcy�ccþ �ccycÞqepQch�1 � qepQch�1ðcy�ccþ �ccycÞ ¼ ½Qc; q�epQch�1; ðC:49Þ

and the second equality in (C.43) is automatically satisfied as far as ½Qc; q� ¼ 0.

Based upon ½h;Qc� ¼ 0 and existence of h�1, let us try the following form of h:

h ¼ expfi/1ðicy�ccÞ þ i/2ð�i�ccycÞ þ i/3ðicy�ccÞð�i�ccycÞg; ðC:50Þ
where /1, /2, and /3 are real numbers which should be determined. Then, taking
into account the first correspondence in (C.47), the definition (C.44) and (C.50), the

calculation of the first equality in (C.42) goes as

hhjQQ�B ¼ �ihhjðayðSÞc� aðSÞcy þ ~aayðSÞ~cc� ~aaðSÞ~ccyÞ

$ �ifhðayðSÞc� aðSÞcyÞ þ ðaðSÞcy þ ayðSÞcÞhg

¼ �ihfayðSÞc� aðSÞcy þ e�i/1�i/3ð�i�ccycÞaðSÞcy þ ei/2þi/3ðicy�ccÞayðSÞcg: ðC:51Þ
If we take /1 ¼ 0, /2 ¼ p, and /3 ¼ 0, we have hhjQQ�
B ¼ 0 with the choice

h ¼ expfipð�i�ccycÞg: ðC:52Þ
This structure for h allows us to calculate the second equality in (C.42) as
QQ�B j0ðbÞi ¼ �iðayðSÞc� aðSÞcy þ ~aayðSÞ~cc� ~aaðSÞ~ccyÞj0ðbÞi

$ �ifðayðSÞc� aðSÞcyÞqepQch�1 þ qepQch�1ð�aðSÞcy � ayðSÞcÞg

¼ QBqepQch�1 þ qepQcQBh

�1 ¼ ½QB; q�epQch�1; ðC:53Þ

where we also used ½epQc ;QB�þ ¼ 0 which is obtained from (C.7). Taking into accountthe BRS-invariance of the statistical operator q (C.27), the expression (C.53) is equal

to zero, and both requirements (C.42) are fulfilled with the choice (C.52). We see that

introduction of factors h and h�1 indeed is necessary to satisfy the BRS-invariance of

the thermal vacuums hhj and j0ðbÞi. Expression of the unit operator in V

1 ¼Xn;m

jnÞg�1n;mðmj; ðC:54Þ

and correspondences (C.47) with (C.52) enable us to see the structure of thermal

vacuums as

hhj ¼Xn;m

ðg�1n;mÞ

�hn; ~mmjh

¼X

nðrÞ;mðrÞ

ðg�1nðrÞ;mðrÞ

Þ�hfnðrÞg; f ~mmðrÞgjXnc;n�ccmc;m�cc

ðg�1ðnc;n�ccÞ;ðmc;m�ccÞÞ

�hnc; n�cc; ~mmc; ~mm�ccjh

¼ hf0g; f~00gj expn� gðrsÞ~aaðrÞaðsÞ

oh0; 0; ~00; ~00j½1þ i~cc�cc�½1þ i~�cc�ccc�

¼ h0; ~00j exp i~cc�ccn

þ i~�cc�ccc� gðrsÞ~aaðrÞaðsÞo; ðC:55Þ

and


j0ðbÞi ¼ Z�1 expf�e�begðrsÞayðrÞ~aayðsÞ � ie�beðcy~�cc�ccy þ �ccy~ccyÞgj0; ~00i: ðC:56Þ

It may be instructive to note that considering

hhjQQþB ¼ 0; QQþ

B j0ðbÞi ¼ 0; ðC:57Þ

with QQþB ¼ QB þ ~QQB, instead of (C.42) with (C.44), leads to the choice

h ¼ expfipðicy�ccÞg.After determination of parameters /i the thermal state conditions with the

ghost operators can be derived through the following steps. First, for the bra-vac-

uum we see

hhj cy

�ccy

� �$ h

cy

�ccy

� �k

hhj~cc

�~�cc�cc

� �$ cy

��ccy

� �h;

ðC:58Þ

therefore

hhj cy

�ccy

� �¼ hhj ~cc

~��cc��cc

� �: ðC:59Þ

Similarly, taking into account structures of the statistical operator q, the ghostcharge Qc, (C.6), and h, (C.52), for the ket-vacuum we have

c

�cc

� �j0ðbÞi $

c

�cc

� �qepQch�1

ke�be~ccy

�e�be~�cc�ccy

!j0ðbÞi $ qepQch�1 e�bec

�e�be�cc

� �;

ðC:60Þ

which gives

c�cc

!j0ðbÞi ¼ e�be ~ccy

�~�cc�ccy� �

j0ðbÞi: ðC:61Þ

The double tilde conjugation rule must be defined so that it leaves thermal vacu-

ums unchanged. To this end, we put

~aaðrÞ� ��

¼ aðrÞ; ðC:62Þ

~cc~�cc�cc

� ��

¼ nc�nn�cc

� �; ðC:63Þ


and determine parameters n and �nn so that hhj� ¼ hhj and j0ðbÞi� ¼ j0ðbÞi. Taking thetilde conjugation of (C.55) we have

hhj� ¼ h0; ~00j� expf�inc~�cc�cc� i�nn�cc~cc� gðrsÞaðrÞ~aaðsÞg¼ h0; ~00j expfin~�cc�cccþ i�nn~cc�cc� gðrsÞ~aaðrÞaðsÞg; ðC:64Þ

where we assumed h0; ~00j� ¼ h0; ~00j. The requirement hhj� ¼ hhj gives n ¼ 1 and �nn ¼ 1,which leads to

~cc~�cc�cc

� ��

¼ c�cc

� �: ðC:65Þ

As a consequence, we obtain the tilde-invariance of the thermal ket-vacuum too:

j0ðbÞi� ¼ ðZ�1 expf�e�begðrsÞayðrÞ~aayðsÞ � ie�beðcy~�cc�ccy þ �ccy~ccyÞgj0; ~00iÞ�

¼ Z�1 expf�e�begðrsÞ~aayðrÞayðsÞ þ ie�beð~ccy�ccy þ ~�cc�ccycyÞgj0; ~00i ¼ j0ðbÞi: ðC:66Þ

A similar line of reasoning can be used to derive the tilde conjugation rule and the

thermal state conditions for a system consisting of physical fermions. Let us consider

the system specified by the Hamiltonian

Hf ¼Z

d3keð~kkÞayð~kkÞað~kkÞ ðC:67Þ

with að~kkÞ and ayð~kkÞ being, respectively, fermion annihilation and creation operators

satisfying the canonical anti-commutation relation

½að~kkÞ; ayð~qqÞ�þ ¼ dð~kk �~qqÞ: ðC:68Þ

As in previous consideration, in the following manipulation we will pay attention to

one mode, say ~kk, and drop the index ~kk for simplicity. In that case the bases of the

state vector space Vf will be denoted as j0Þ and j1Þ defined by aj0Þ ¼ 0 and

j1Þ ¼ ayj0Þ.At the finite temperature, the statistical average of an observable quantity A is

given by

hAi ¼ TrAqf ðC:69Þ
with qf being the statistical operator of the system
qf ¼ Z�1f e�bHf ¼ Z�1

f j0Þð0j�

þ e�bej1Þð1j�; ðC:70Þ

where Zf is the partition function.

Within the doubled state space Vf ¼ Vf � ~Vf , the statistical average (C.69) can beexpressed in terms of the vacuum expectation with respect to the thermal bra- and

ket-vacuums for physical fermions. The bases of Vf are defined through the principle

of correspondence:

jn; ~mmi $ jnÞðmj; ðC:71Þ
where n and m take values of 0 or 1. Annihilation and creation operators acting on VVf
are defined through


aay

� �jn; ~mmi $ a

ay

� �jnÞðmj; ðC:72Þ

~aajn; ~mmi $ ð�1Þn�mþ1jnÞðmjay; ðC:73Þ

~aayjn; ~mmi $ ð�1Þn�mjnÞðmja: ðC:74Þ
The bases jn; ~mmi are generated from the vacuum j0; ~00i:
jn; ~mmi ¼ ðayÞnð~aayÞmj0; ~00i: ðC:75Þ

This time we do not have peculiar symmetries for thermal vacuums like in the case

of gauge theories with the BRS symmetry. However, in the derivation of thermal

vacuums, let us use a trick similar to the one in (C.46)

hAi ¼ Tr ei/ayaAqe�i/aya ¼ hhjAj0ðbÞi; ðC:76Þ

where / is a real number which should be decided. We settle the correspondence for

hhj and j0ðbÞi as

hhj $ ei/aya; j0ðbÞi $ qf e

�i/aya: ðC:77Þ

They are normalized, hhj0ðbÞi ¼ 1, and generated from j0; ~00i as

hhj ¼ h0; ~00j½1þ ei/~aaa�; ðC:78Þ

j0ðbÞi ¼ Z�1f ½1þ e�i/ e�beay~aay�j0; ~00i: ðC:79Þ

A requirement of the tilde invariance for the thermal bra-vacuum

hhj� ¼ hhj ðC:80Þ
determines the tilde conjugation rule for physical fermion operators up to the phasefactor
ðaÞ� ¼ ~aa; ð~aaÞ� ¼ �e2i/a: ðC:81Þ

We have seen in (C.65) that the ghost operators, which are fermion operators, were

unchanged under the double tilde conjugation. Let us adopt the same rule forphysical fermion operators, i.e., put / ¼ p=2 to obtain

ðaÞ� ¼ ~aa; ð~aaÞ� ¼ a: ðC:82Þ
With this choice of /, the thermal vacuums for physical fermions read
hhj ¼ h0; ~00j½1þ i~aaa�; ðC:83Þ

j0ðbÞi ¼ Z�1f ½1� ie�beay~aay�j0; ~00i ðC:84Þ

and satisfy the following thermal state conditions

hhj~aay ¼ �ihhja; ~aaj0ðbÞi ¼ ie�beayj0ðbÞi: ðC:85Þ


Appendix D. Correlation of random force operators

The random force operators are of the Wiener process whose first and second mo-

ments are given by real c-numbers:

4 In

hdFti ¼ hdF yt i ¼ 0; ðD:1Þ

hdFt dFti ¼ hdF yt dF

yt i ¼ 0; ðD:2Þ

hdFt dF yt i ¼ a real c-number; ðD:3Þ

hdF yt dFti ¼ a real c-number; ðD:4Þ

where h� � �i ¼ hj � � � ji represents the random average referring to the random force

operators dFt. From (D.1), (D.2), and TSC (6.4) we have for operators (6.14) and

(6.15)

hdWti ¼ hd ~WWti ¼ hdW $t i ¼ hd ~WW $

t i ¼ 0; ðD:5Þ

hdWt dWsi ¼ hd ~WWt d ~WWsi ¼ 0; ðD:6Þ
while from (6.16) it follows
hdW $t dWsi ¼ hdW $

t d ~WWsi ¼ 0; ðD:7Þ

hdW $t dW $

s i ¼ hdW $t d ~WW $

s i ¼ 0; ðD:8Þ
and their tilde conjugates. Using (D.6)–(D.8) the explicit structure of dMMtdMMt in
(6.23) is written as

dMMt dMMt ¼ �2rdWt d ~WWta$~aa$ þ kðdWt dW $

t a$aþ d ~WWt d ~WW $

t ~aa$~aaÞ ðD:9Þ

in a ‘‘weak sense.’’4 We demand that the Stratonovich type time evolution generator

should not contain a diffusion term, i.e., the term proportional to a$~aa$. Then the

correlation hdWt d ~WWti is determined to be

dWt d ~WWt ¼ sf2jðtÞ½nðtÞ þ g� þ _nnðtÞgdt ¼ hdWt d ~WWti ðD:10Þ
so that PPD in (6.23) is cancelled by the first term in the r.h.s. of (D.9). Here, the first
equality in (D.10) should be understood in a weak sense as well. Expression (D.10) is

compatible with the assumption that the process is white. Let us put the subscript Fto R<ðtÞ in the Boltzmann equation in order to remember that it is due to the in-

teraction with the random force dFt:

_nnðtÞ ¼ �2jðtÞnðtÞ þ iR<F ðtÞ: ðD:11Þ

Making use of two previous equations, we have

the case of classical systems it corresponds to the stochastic convergence.


iR<F ðtÞdt ¼ 2jðtÞnðtÞdt þ _nnðtÞdt ¼ �2jðtÞgdt þ rshdWt d ~WWti

¼ �2jðtÞgdt þ hdF yt dFti þ m hdFt dF y

t i�

� rhdF yt dFti

�; ðD:12Þ

where (6.14) has been used and l has been erased with the help of (5.32).

We can assume that the quantity g may depend on m, i.e., g ¼ gðmÞ, and that thephysical quantities jðtÞ, R<

F ðtÞ, hdFyt dFti, and hdFtdF y

t i may not depend on m. Then,differentiating equation (D.12) with respect to m, one has

0 ¼ �2jðtÞ ogom

dt þ hdFt dF yt i � rhdF y

t dFti: ðD:13Þ

This leads to

ogom

¼ kðtÞ; ðD:14Þ

which is solved as

g ¼ kðtÞmþ lðtÞ; ðD:15Þ
where kðtÞ and lðtÞ are real numbers independent of m. With this solution one
has

hdFt dF yt i � rhdF y

t dFti ¼ 2jðtÞkðtÞdt; ðD:16Þ
and
iR<F ðtÞdt ¼ �2jðtÞlðtÞdt þ hdF y

t dFti; ðD:17Þ
which leads to
hdF yt dFti ¼ f2jðtÞ½lðtÞ þ nðtÞ� þ _nnðtÞgdt; ðD:18Þ

where we have used (D.11). The substitution of (D.18) into (D.16) gives us

hdFt dF yt i ¼ f2jðtÞ½kðtÞ þ rlðtÞ þ rnðtÞ� þ r _nnðtÞgdt: ðD:19Þ

For the case of stationary quantum stochastic process, the Boltzmann equation

(D.11) reduces to

_nnðtÞ ¼ �2j½nðtÞ � �nn�; ðD:20Þ
where �nn is the average quantum number in equilibrium. Therefore, (D.18) and (D.19)
reduce, respectively, to

hdF yt dFti ¼ 2j½�nnþ lðtÞ�dt; ðD:21Þ

hdFt dF yt i ¼ 2j½kðtÞ þ rlðtÞ þ r�nn�dt: ðD:22Þ

Since in the white noise assumption the Boltzmann equation (D.20) is compatible

with the stationary process specified by [41]

hdF yt dFti ¼ 2j�nndt; ðD:23Þ

hdFt dF yt i ¼ 2j½1þ r�nn�dt; ðD:24Þ


one concludes now that

lðtÞ ¼ 0; kðtÞ ¼ 1; ðD:25Þ
which leads to
g ¼ m; n ¼ l: ðD:26Þ
Note that the result (D.24) can be obtained using the Bogoliubov transformation as
it is described in Appendix B.

Substituting (D.25) into (D.18) and (D.19), one obtains

hdF yt dFti ¼ ½2jðtÞnðtÞ þ _nnðtÞ�dt; ðD:27Þ

hdFt dF yt i ¼ f2jðtÞ½1þ rnðtÞ� þ r _nnðtÞ�gdt; ðD:28Þ

which leads to

hdWt dW $t i ¼ hdFtdF y

t i � rhdF yt dFti ¼ 2jðtÞdt: ðD:29Þ

Assembling (D.9), (D.10), (D.26), and (D.29), one obtains expression (6.25).

References

[1] T. Arimitsu, Cond. Matter Phys. Issue 4 (1994) 26, Available from http://www.px.tsukuba.ac.jp/

home/tcm/arimitsu/cmp4.pdf.

[2] T. Arimitsu, H. Umezawa, Progr. Theoret. Phys. 74 (1985) 429.



[5] N.N. Bogoliubov, in: J. de Boer, G.E. Uhlenbeck (Eds.), Studies in Statistical Mechanics, vol. 1,

North-Holland, Amsterdam, 1962, pp. 1–118.

[6] S. Nakajima, Progr. Theoret. Phys. 20 (1958) 948.

[7] R. Zwanzig, J. Chem. Phys. 33 (1960) 1338.

[8] R. Zwanzig, in: W.E. Brittin, B.W. Downs, J. Downs (Eds.), Lectures in Theoretical Physics, vol.

III, Interscience Publishers, New York, 1961, pp. 106–141.

[9] H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press,

Oxford, 2002.

[10] I. Prigogine, C. George, F. Henin, Physica A 45 (1969) 418.

[11] I. Prigogine, Physica A 263 (1999) 528.

[12] H. Mori, Progr. Theoret. Phys. 33 (1965) 423.

[13] K. Kawasaki, J.D. Gunton, Phys. Rev. A 8 (1973) 2048.

[14] B. Robertson, Phys. Rev. 144 (1966) 151.

[15] M.S. Green, J. Chem. Phys. 20 (1952) 1281.

[16] M.S. Green, J. Chem. Phys. 22 (1954) 398.

[17] R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570.

[18] R. Kubo, M. Yokota, S. Nakajima, J. Phys. Soc. Jpn. 12 (1957) 1203.

[19] D.N. Zubarev, Nonequilibrium Statistical Thermodynamics, Consultant Bureau, New York, 1974.

[20] D.N. Zubarev, J. Soviet Math. 16 (1981) 1509.

[21] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1.

[22] M.C. Wang, G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323.

[23] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences,

second ed., Springer-Verlag, Berlin, 1985.

[24] K. Itoo, Proc. Imp. Acad. Tokyo 20 (1944) 519.

http://www.px.tsukuba.ac.jp/home/tcm/arimitsu/cmp4.pdf

http://www.px.tsukuba.ac.jp/home/tcm/arimitsu/cmp4.pdf


[25] R. Stratonovich, J. SIAM Control 4 (1966) 362.

[26] K.E. Shuler (Ed.), Stochastic Processes in Chemical Physics, in: I. Prigogine, S. Rice (Eds.),

Advances in Chemical Physics, vol. XV, Interscience Publishers, New York, 1969.

[27] W. Coffey, in: M.W. Evans (Ed.), Dynamical Processes in Condensed Matter, pp. 69–252, in: I.

Prigogine, S. Rice (Eds.), Advances in Chemical Physics, vol. LXIII, Interscience Publishers, New

York, 1985.

[28] K. Sobczyk, Stochastic Differential Equations: With Application to Physics and Engineering,

Kluwer Academic, Dordrecht, 1991.

[29] N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam,

1992.

[30] W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation. With Applications in

Physics, Chemistry and Electrical Engineering, World Scientific, Singapore, 1996.

[31] P.W. Anderson, J. Phys. Soc. Jpn. 9 (1954) 316.

[32] R. Kubo, J. Phys. Soc. Jpn. 9 (1954) 935.

[33] R. Kubo, in: D. ter Haar (Ed.), Fluctuation, Relaxation and Resonance in Magnetic Systems, Oliver

& Boyd, Edinburgh-London, 1962, pp. 23–68.

[34] R. Kubo, J. Math. Phys. 4 (1963) 174.

[35] I.R. Senitzky, Phys. Rev. 119 (1960) 670.



[38] J. Schwinger, J. Math. Phys. 2 (1961) 407.

[39] H. Haken, Z. Phys. 181 (1964) 96.

[40] H. Haken, Z. Phys. 182 (1965) 346.

[41] H. Haken, Optik. Handbuch der Physik, vol. XXV/2c, Springer-Verlag, Berlin, 1970 (Reprinted as

H. Haken, Laser Theory, Berlin, Springer-Verlag, 1984).

[42] H. Haken, Rev. Mod. Phys. 47 (1975) 67.

[43] M. Lax, Phys. Rev. 145 (1966) 110.

[44] R. Kubo, J. Phys. Soc. Jpn. Suppl. 26 (1969) 1.

[45] E.B. Davies, Commun. Math. Phys. 15 (1969) 277.

[46] E.B. Davies, Quantum Theory of Open Systems, Academic Press, London, 1976.

[47] A.M. Cockroft, R.L. Hudson, J. Multivar. Anal. 7 (1977) 107.

[48] R.L. Hudson, R.F. Streater, Phys. Lett. A 86 (1981) 277.

[49] R.L. Hudson, K.R. Parthasarathy, Commun. Math. Phys. 93 (1984) 301.

[50] R.L. Hudson, K.R. Parthasarathy, Acta Appl. Math. 2 (1984) 353.

[51] R.L. Hudson, J.M. Lindsay, in: L. Accardi, W. von Waldenfels (Eds.), Quantum Probability and

Applications II, Lecture Notes in Mathematics, vol. 1136, Berlin, Springer, 1984, pp. 276–305.

[52] R.L. Hudson, J.M. Lindsay, J. Func. Anal. 61 (1985) 202.

[53] L. Accardi, A. Frigerio, J.T. Lewis, Publ. RIMS Kyoto Univ. 18 (1982) 97.

[54] L. Accardi, Rev. Math. Phys. 2 (1990) 127.

[55] K.R. Parthasarathy, Pram~aana—J. Phys. 25 (1985) 457.

[56] K.R. Parthasarathy, Rev. Math. Phys. 1 (1989) 89.

[57] K.R. Parthasarathy, in: An Introduction to Quantum Stochastic Calculus, Monographs in

Mathematics, vol. 85, Basel, Birkh€aauser, 1992.

[58] D. Applebaum, R.L. Hudson, Commun. Math. Phys. 96 (1984) 473.

[59] D. Applebaum, R.L. Hudson, J. Math. Phys. 25 (1984) 858.

[60] D. Applebaum, J. Phys. A 19 (1986) 937.

[61] D. Applebaum, J. Phys. A 28 (1995) 257.

[62] R.L. Hudson, K.R. Parthasarathy, Commun. Math. Phys. 104 (1986) 457.

[63] K.R. Parthasarathy, K.B. Sinha, Pram~aana—J. Phys. 27 (1986) 105.

[64] C. Barnett, R.F. Streater, I.F. Wilde, J. Funct. Anal. 48 (1982) 172.

[65] C. Barnett, R.F. Streater, I.F. Wilde, J. London Math. Soc. 27 (1983) 373.

[66] C. Barnett, R.F. Streater, I.F. Wilde, Commun. Math. Phys. 89 (1983) 13.

[67] C. Barnett, R.F. Streater, I.F. Wilde, J. Func. Anal. 52 (1983) 19.


[68] T. Arimitsu, Phys. Lett. A 153 (1991) 163.

[69] T. Arimitsu, M. Ban, T. Saito, Physica A 177 (1991) 329.

[70] T. Saito, T. Arimitsu, Mod. Phys. Lett. B 6 (1992) 1319.



[73] T. Saito, T. Arimitsu, J. Phys. A 30 (1997) 7573.

[74] T. Arimitsu, Phys. Essays 9 (1996) 591.

[75] T. Arimitsu, in: L. Accardi, S. Tasaki (Eds.), Non-Commutativity, Infinite-Dimensionality and

Probability at the Crossroads, Quantum Probability and White Noise Analysis, vol. 17, World

Scientific, Singapore, 2003, pp. 24–27.

[76] T. Arimitsu, J. Pradko, H. Umezawa, Physica A 135 (1986) 487.

[77] C.W. Gardiner, M.J. Collet, Phys. Rev. A 31 (1985) 3761.

[78] C.W. Gardiner, P. Zoller, in: Quantum Noise: A Handbook of Markovian and Non-Markovian

Quantum Stochastic Methods with Applications to Quantum Optics, second ed., Springer Series in

Synergetics, vol. 56, Springer, Berlin, 2000.

[79] T. Arimitsu, Y. Sudo, H. Umezawa, Physica A 146 (1987) 433.

[80] T. Arimitsu, M. Guida, H. Umezawa, Europhys. Lett. 3 (1987) 277.

[81] T. Arimitsu, M. Guida, H. Umezawa, Physica A 148 (1988) 1.

[82] T. Arimitsu, H. Umezawa, in: G. Busiello, L. De Cesare, F. Mancini, M. Marinaro (Eds.), Advances

on Phase Transitions and Disordered Phenomena, World Scientific, Singapore, 1987, pp. 483–504.

[83] H. Umezawa, T. Arimitsu, in: M. Namiki, Y. Ohnuki, Y. Murayama, S. Nomura (Eds.),

Foundation of Quantum Mechanics—In the Light of New Technology, Physical Society of Japan,

Tokyo, 1987, pp. 79–90.

[84] T. Arimitsu, H. Umezawa, Y. Yamanaka, P. Papastamatiou, Physica A 148 (1988) 27.

[85] R. Kubo, H. Ichimura, T. Usui, N. Hashitsume, Statistical Mechanics. An Advanced Course with

Problems and Solutions, North-Holland, Amsterdam, 1965.

[86] R. Balescu, Equilibrium and Non-Equilibrium Statistical Mechanics, Wiley Interscience, New York,

1975.

[87] J. Dalibard, Y. Castin, K. Mølmer, Phys. Rev. Lett. 68 (1992) 580.

[88] K. Mølmer, Y. Castin, J. Dalibard, J. Opt. Soc. Am. B 10 (1993) 524.

[89] K. Mølmer, Y. Castin, Quant. Semiclass. Opt. 8 (1996) 49.

[90] C.W. Gardiner, A.S. Parkins, P. Zoller, Phys. Rev. A 46 (1992) 4363.

[91] R. Dum, A.S. Parkins, P. Zoller, C.W. Gardiner, Phys. Rev. A 46 (1992) 4382.

[92] H.J. Carmichael, in: An Open Systems Approach to Quantum Optics, Lecture Notes in Physics, vol.

m18, Springer-Verlag, Berlin, 1993.

[93] B.M. Garraway, P.L. Knight, Phys. Rev. A 49 (1994) 1266.

[94] M.B. Plenio, P.L. Knight, Rev. Mod. Phys. 70 (1998) 101.

[95] T. Arimitsu, in: A. Vijayakumar, M. Sreenivasan (Eds.), Stochastic Processes and their Applications,

Narosa Publ. House, New Delhi, 1999, pp. 279–294.

[96] T. Arimitsu, Y. Kaburaki, private communication.

[97] T. Saito, T. Arimitsu, in: A. Vijayakumar, M. Sreenivasan (Eds.), Stochastic Processes and their

Applications, Narosa Publ. House, New Delhi, 1999, pp. 323–333.

[98] T. Arimitsu, in: N. Obata, T. Matsui, A. Hora (Eds.), Non-Commutativity, Infinite-Dimensionality

and Probability at the Crossroads, Quantum Probability and White Noise Analysis, vol. 16, World

Scientific, Singapore, 2003, pp. 206–224.

[99] I. Ojima, Ann. Phys. (N. Y.) 137 (1981) 1.

[100] T. Hayashi, Master Thesis, University of Tsukuba, Japan, 2002 (in Japanese).

[101] L.D. Faddeev, V. Popov, Phys. Lett. B 25 (1967) 29.

[102] C. Becchi, A. Rouet, R. Stora, Ann. Phys. (N. Y.) 98 (1976) 287.

[103] T. Kugo, I. Ojima, Phys. Lett. B 73 (1978) 459.

[104] T. Kugo, I. Ojima, Progr. Theor. Phys. Suppl. 66 (1979) 1.

[105] H. Hata, T. Kugo, Phys. Rev. D 21 (1980) 3333.

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