Towards a Non-linear Extension of Stochastic

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Towards a Non-linear Extension of Stochastic Calculus(Quantum Stochastic Analysis and Related Fields)

ACCARDI, LUIGI; LU, YUN-GANG; OBATA, NOBUAKI

数理解析研究所講究録 (1996), 957: 1-15

1996-08

http://hdl.handle.net/2433/60449

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Textversion

Departmental Bulletin Paper

publisher

Kyoto University

数理解析研究所講究録 957 巻 1996 年 1-15

1

Towards a Non-linear Extension of Stochastic Calculus LUIGI 1)

2) 3)

$\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{A}\mathrm{R}\mathrm{D}\mathrm{I}^{1,2)},$

$\mathrm{Y}\mathrm{U}\mathrm{N}$

-GANG

$\mathrm{L}\mathrm{U}^{2,3)}$

AND

NOBUAKI

$\mathrm{o}_{\mathrm{B}\mathrm{A}\mathrm{T}\mathrm{A}^{1}}$

)

Graduate School of Polymathematics, Nagoya University, Nagoya, 464-01, Japan Centro Matematico Vito Volterra, Universit\‘a di Roma, Roma, 00133, Italia Dipartimento di Matematica, Universit\‘a di Bari, Bari, 70125, Italia

Introduction In a series of recent works [3], [4], [5], [6] a new approach to quantum stochastic calculus has been launched out through the study of stochastic limit of quantum theory. This new approach is not based upon a quantum It\^o theory but upon a new type of distribution theory on simplexes and reveals intrinsic features of quantum white noise. Meanwhile, independently of that, quantum stochastic processes have been discussed on the basis of white noise (Hida) distribution theory [11], [12], [19], [20], [21]. This approach seems useful for studying differential equations for Fock space operators involving higher powers of quantum white noises, at least for the existence and uniqueness of a solution in a distribution sense [22]. Thus it is quite natural to ask what can be said by combining the above two ideas together. In the present paper we discuss the simplest problem of a unitary equation driven by and , which is already beyond the reach of the It\^o or Hudson-Parthasarathy stochastic calculus. Although our argument stays at somehow formal level, it is most plausible that we are led along a new direction towards a non-linear stochastic calculus. $b_{t}^{\uparrow 2}$

1

$b_{t}^{2}$

Classsical and quantum stochastic calculi

Classsical stochastic calculus was initiated by It\^o [14]. It gives a meaning to equations of the form

$df(X_{t})=zf(x_{t})dt+Lf(x_{t})dW_{t}$ ,

(1.1)

.

are -valued Brownian motion starting at zero, and where $W=W_{t}$ is the standard . and respectively a second order and a first order differential operators on for space but function Here the Schwartz test function space can be replaced with a larger the moment we stick to this choice. Equation. (1.1) is interpreted as a symbolic notation for the integral equation: $L$

$\mathbb{R}^{d}$

$f(X_{t})=f(x_{0})+ \int_{0}^{t}zf(X)SdS+\int_{0}^{t}Lf(x_{s})dW_{s}$

$Z$

$f\in S(\mathbb{R}^{d})$

$\mathbb{R}^{d}$

(1.2)

,

where the second integral is a stochastic integral. Equation (1.2) can be shown to admit is a random a unique solution for any initial condition of the form $X(0)=X_{0}$ , where $W$ variable wich is measurable with respect to a -algebra independent of that of . $X_{0}$

$\sigma$

2

Quantum stochastic calculus was initiated by Hudson and Parthasarathy [13]. It gives a meaning to equations of the form $dU_{t}=(DdB^{\uparrow_{-}}tD \mathrm{t}dBt+(-\frac{\gamma}{2}D^{\uparrow D}+iH)dt)U_{t}$

,

$U_{0}=I$

,

(1.3)

where the pair is the Fock Brownian motion with variance $\gamma>0$ acting on the Boson Fock space , and $D,$ $H=H^{*}$ are operators on a Hilbert space . Thus will be an operator acing on . More generally, one can replace by , where is any Hilbert space, see e.g., [17], [23]. Equation (1.3) is interpreted as a symbolic notation for the integral equation $(B_{tt}^{\uparrow,B})$

$\Gamma(L^{2}(\mathbb{R}))$

$\mathcal{H}_{S}$

$U_{t}$

$\mathcal{H}_{S}\otimes\Gamma(L^{2}(\mathbb{R}))$

$L^{2}(\mathbb{R})\otimes \mathcal{K}$

$L^{2}(\mathbb{R})$

$\mathcal{K}$

$U_{t}=1+ \int_{0}^{t}(DdB^{1}-SD\dagger dB_{s})US+\int_{0}^{t}(-\frac{\gamma}{2}D^{\mathrm{t}}D+\dot{i}H)dsU_{s}$

,

(1.4)

where the first integral is a quantum stochastic integral. Equation (1.3) can be shown to admit a unique solution which is a unitary operator on the space whenever the operators $D$ and $H$ satisfy some conditions. Quantum stochastic calculus includes the classical one. Let $(\Omega, P, F)$ be the probability space of $W$ . Then each random variable of the process defines an homomorphism : by $\mathcal{H}_{S}\otimes\Gamma(L^{2}(\mathbb{R}))$

$X_{t}$

$j_{t}$

$S(\mathbb{R}^{d})arrow S(\mathbb{R}^{d})\otimes L^{\infty}(\Omega, P, \mathcal{F})$

it $(f)=f(Xt)$ .

(1.5)

The family of homomorphisms characterizes the process and constitutes a natural algebraic characterization of it. On the other hand, the classical Brownian motion $W$ can be realized as an operator process (i.e. ) on the Fock space is a Hilbert space corresponding to a space of possible initial conditions for the It\^o equation (1.1), the process $f(X_{t})$ can be thought as a multiplication operator on the space and the classical flow , defined by (1.5), is given by $j_{t}$

$W_{t}=B_{t}+B_{t}^{\mathrm{t}}$

$\Gamma(L^{2}(\mathbb{R})),$

$\mathcal{H}_{S}$

$\mathcal{H}_{S}\otimes\Gamma(L^{2}(\mathbb{R}))$

$j_{t}$

$j_{t}(f)=f(x_{t})=U_{t}^{*}f(X_{0})U_{t}$

,

where is the solution of a QSDE of the form (1.3) for an appropriate choice of the coefficients operator $D$ , see [9], [10], [23]. Thus, the flow plays a crucial role in the link between classical and quantum stochastic processes, see e.g., [7]. $U_{t}$

$j_{t}$

2

Stochastic limit

As is clear from the previous section, for a generalization of stochastic calculus fundamental object to start with is the flow or the associated one-parameter family of unitary operators . The stochastic limit of quantum theory developed recently in [6] suggests a new possibility of constructing unitary operators . The starting point of this theory is not a stochastic equation but a usual Schr\"odinger equation in interaction representation: $j_{t}$

$U_{t}$

$U_{t}$

$\partial_{t}U_{t}=-\dot{i}\lambda H_{I}(t)U_{t}$

,

$U_{0}=I$

,

(2.1)

3

where

$\lambda>0$

is a coupling constant. For example,

$H_{I}(t)$

$H_{I}(t)=e^{itH_{0}}(D\otimes A^{\dagger}(g)+D^{\mathrm{t}}\otimes A(g))e^{-itH_{0}}$

which is known as a Hamiltonian

is given as

,

$H_{0}=H_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}}\mathrm{m}+H_{\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{e}}\mathrm{r}\mathrm{v}\circ:\mathrm{r}$

’

of laser type. A simple case would be

$H_{I}(t)=D\otimes A_{t}^{\uparrow}+D\dagger\otimes A_{t}$

,

$A_{t}=A(S_{t}g)$

.

The (formal) solution of (2.1) is given by the iterated series: $U_{t}$

$=$

$I+(-i \lambda)\int_{0}^{t}dt_{1}HI(t1)Ut_{1}$

$=$

$I+ \sum_{n=1}^{\infty}(-i\lambda)^{n}\int^{t}\mathrm{o}dt_{1}\int_{0}^{t}1td2\ldots\int^{t_{n}}0)-1di_{n}H_{I}(t1)HI(t_{2})\cdots H_{I}(t_{n}$

.

, which Particularly interesting is the scaling limit of (2.1) according to the law and by physics mathematics limit theorem) (Friedrichs-van (central is motivated both by Hove rescaling). The rescaled solution is $tarrow t/\lambda^{2}$

$U_{t/\lambda}^{(\lambda)}2$

$=$

$I+ \sum_{n=1}^{\infty}(-i\lambda)^{n}\int^{t}0\int_{0}^{t_{1}}/\lambda 2\ldots t_{n-1}dt_{1}dt2\int \mathrm{o}(dt_{n}HI(t_{1})HIt_{2})\cdots HI(t)n$

$=$

$I- \dot{i}\lambda\int_{0}^{t/\lambda^{2}}dt_{1}HI(t_{1})-\lambda^{2}\int_{0}^{t/\lambda^{2}}dt1\int_{0}^{t_{1}}dt_{2}HI(t_{1})H_{I}(t_{2})+\cdots$

(2.2)

and the rescaled equation is of the form: $\partial_{t}U_{t}^{()}/\lambda^{2}\lambda=-i(D\otimes A_{t}^{(\lambda)\uparrow}+D^{\uparrow}\otimes A_{t}^{(\lambda)})U_{/\lambda^{2}}t(\lambda)$

,

$U_{0}^{(\lambda)}=I$

.

(2.3)

to the solution It was proved in [3] that the iterated series solution (2.2) converges as , in the sense of the quantum stochastic differential equation (1.3) with of quantum convergence in law. More recently, it has been proved in [5] that the iterated series solution (2.2) converges term by term to the same limit in the same sense as above to the iterated series solution of the distribution equation $\lambdaarrow 0$

$H–\kappa D^{\uparrow}D,$

$\partial_{t}U_{t}=-i(D\otimes b_{t}^{\uparrow}+D^{\uparrow}\otimes b_{t})U_{t}$

where variance

$b_{t}^{1},$

$b_{t}$

,

It is known that

$b_{t}^{\uparrow},$

$b_{t}$

(2.4)

.

are characterized (up to unitary equivalence) by the algebraic relations [

$b_{t},$

$b_{S}^{1]}$

$b_{t}\Phi$

$\Phi$

,

are the annihilation and creation operators of the Boson Fock white noise with $\gamma=\int_{-\infty}^{\infty}\langle g, S_{t}g\rangle dt>0$

where

$U_{0}=I$

$\kappa\in \mathbb{R}$

is the Fock vacuum.

$=$

$\gamma\delta(s-t)$

$=$

$0$

,

,

$t\in \mathbb{R}$

$t,$ $s\in \mathbb{R}$

,

,

(2.5)

4

3

The white noise approach to stochastic calculus

Equation (2.4) has a well-defined meaning in terms of matrix elements in the number vectors. Let us recall from [5] how to give an intrinsic meaning to equation (2.4). This shall be useful for comparison with the nonlinear extension of this equation to be discussed below. The number vectors are defined by the realtion $b_{t}^{\mathrm{t}}\cdots b_{t_{k}}^{\dagger}\Phi 1=|t_{1},$

$\cdots,$

$t_{k}\rangle$

(3.1)

,

which has to be interpreted in the distribution sense, i.e. for any test functions , one defines

$\phi_{1},$

$\cdots$

,

$\phi_{k}\in$

$S(\mathbb{R})$

$b \dagger(\phi_{1})\cdots b\uparrow(\phi_{k})\Phi=\int dt_{1}\cdots\int dt_{k}\phi_{1}(t_{1})\cdots\phi k(t_{k})|t1,$ $\cdots,$

$t_{k}\rangle$

.

(3.2)

In what follows we put $D=I$ for simplicity. Then (2.4) becomes $\partial_{t}U_{t}=-\dot{i}(b_{t}^{\dagger}+b_{t})U_{t}$

,

$U_{0}=I$

(3.3)

,

and its weak formulation is $\partial_{t}\langle s_{1}, \cdots, S_{h}|U_{t}|t_{1}, \cdots, tk\rangle=-i\langle S1, \cdots, sh|(b_{t}+b_{t}\uparrow)Ut|t1, \cdot i\cdot, tk\rangle$

$=-i \sum_{j=1}\gamma\delta(t-hS_{j})\langle s1, \cdots, sj\wedge, \cdot, .

, Sh|U_{t}|t_{1}, \cdots, t_{k}\rangle$

(3.4)

$-i\langle t, S_{1}, \cdots, S_{h}|Ut|t_{1}, \cdots, t_{k}\rangle$

The right hand side of equation (3.4) is still a distribution. To reduce everything to ordinary functions one interprets (3.3) as the integral equation $U_{t}=I-i \int_{0}^{t}(b_{s}\dagger+b_{s})U_{S}ds$

(3.5)

.

Then $\langle s_{1,h}\ldots, S|U_{t}|t1, \cdots,tk\rangle=$

$=\langle s1, \cdots, Sh|t1, \cdots, tk\rangle$

$-i \sum_{1j=}^{h}\gamma\int_{0}t|\delta(s-Sj)\langle_{S_{1},\cdots,s_{j}^{\wedge}}, \cdots, ShUS|t1, \cdots, tk\rangle d_{S}$

$- \dot{i}\int_{0}^{t}\langle_{S,S_{1}}, \cdots, sh|US|t1, \cdots, tk\rangle d_{S}$

$=\langle s1, \cdots, Sh|t1, \cdots, tk\rangle$

$-i. \gamma\sum_{j=1}^{h}\chi 10,t](S_{j})\langle_{S}1, \cdots, S^{\wedge}j, \cdots, sh|U_{S}|jt_{1}, \cdots, t_{k}\rangle$

.

$- \dot{i}\int_{\mathbb{R}}\chi[0,t](S)\langle_{S}, S1, \cdots, Sh|U_{s}|t_{1}, \cdots, tk\rangle d_{S}$

(3.6)

5

In view of

$U_{t}=\langle\partial_{S}U_{S}, x_{[0},t](S)\rangle$

, we replace

$\chi_{[0,t]}$

in (3.6) with

$\phi\in S(\mathbb{R})$

to obtain

$\langle s_{1}, \cdots, S_{h}|\langle\partial_{SS}U, \phi\rangle|t_{1,k}\ldots, t\rangle=$

$=\langle s_{1,,h}\ldots S|t1, \cdots, tk\rangle$

$-i \gamma\sum_{=j1}^{h}\phi(S_{j})\langle s1, \cdots, s_{j},\cdot, S_{h}\wedge..|U_{s}|jt_{1}, \cdots, t_{k}\rangle$

$- \dot{i}\int_{\mathbb{R}}\phi(_{S})\langle S, S_{1}, \cdots, Sh|Us|t_{1}, \cdots, tk\rangle d_{S}$

.

So the equation (3.3) has an meaning weakly on number vectors. A similar discussion can be done for the exponential vectors. In order to identify the distribution equation (3.3) with a quantum stochastic equation, it is convenient to write equation (3.3) in normal form. Since $\partial_{t}U_{t}=-\dot{i}(b_{tt}\uparrow U+U_{t}b_{t}+[b_{t}, U_{t}])$

,

in order to bring equation (2.4) into a normal form we have to calulate the commutator . This is done through the following result [1], [6]. $[b_{t}, U_{t}]$

Theorem 3.1

If

$U_{t}$

is a solution

of (1.3)

with

$[b_{t}, U_{t}]=\gamma-DU_{t}$

$H=\kappa D\dagger D$

,

, then

$\gamma-=\frac{\gamma}{2}-i\kappa$

.

where the identity is understood weakly on the exponential or number vectors. The proof of Theorem 3.1 is based upon two fundamental principles of the quantum white noise theory on the standard simplex established in [5]. $\bullet$

The causal commutator rule When the operators in the iterated series for are brought into a normal ordered form, we must not use the usual commutation relation (2.5) but the causal commutation relation defined by $b_{t},$

$b_{t}^{\dagger}$

$U_{t}$

[

$b_{\sigma},$

$b^{\uparrow]}\mathcal{T}=\gamma_{-}\delta+(\sigma-\tau)$

,

$\sigma\geq\tau$

,

(3.7)

where is a complex number and is the causal delta function on a simplex. Notice that the causal commutator is defined only for a (the time index of annihilator) greater than (the time index of creator). So the causal commutator rule should not be interpreted as a new commutation relation, but only as a rule to bring in normal order the creation and annihilation operator which appear in the iterated series. $\delta_{+}$

$\gamma_{-}$

$\prime r$

$\bullet$

The principle of consecutive times Any commutator of the form $\int_{0}^{t}$

[

$bt,$

Us] $dS$

vanishes identically. This is, in fact, a corollary of the principle of consecutive times but is all we shall need in the present paper.

6

From Theorem 3.1 it follows that the normal form of (3.3) is given by $\partial_{t}U_{t}=-i(b_{tt}^{\uparrow U}+U_{t}b_{t}+\gamma_{-}U_{t})$

.

(3.8)

Similarly, the normal form of (2.4) is given as $\partial_{t}U_{t}=-\dot{i}(Db_{t}^{\uparrow U_{t}}+D^{\mathrm{t}}Utbt+\gamma_{-D^{\mathrm{t}_{DU}}}t)$

,

(3.9)

where short hand notation such as $D=D\otimes I,$ is used. This result solves the above conjecture affirmatively. In fact, by taking matrix elements in the exponential vectors of both sides of the two equations (3.3) and (3.8) one easily obtains the same ordinary differential equation. The above argument suggests a new approach to stochastic calculus based directly on white noise and not through the intermediate Brownian motion. The basic idea of this approach can be formulated as follows: $b_{t}=I\otimes b_{t}$

1. Write an equation in normal form: $\partial_{t}U_{t}=Ab^{\dagger}tUt+BU_{t}b_{t}+CU_{t}$

where

$A,$ $B,$ $C$

,

$U_{0}=I$

,

(3.10)

are system operators.

2. Prove the existence and uniqueness of the solution in terms of operator symbols $q_{t}(f,g)=\langle\psi_{f}, U_{t}\psi_{g}\rangle$

where

$\psi_{f},$

$\psi_{g}$

,

are exponential vectors.

3. Prove a regularity condition showing that the solution of equation (3.10) is a bona fide operator and not only a distribution. 4. Formulate the unitarity condition for

$U_{t}$

, namely,

$U_{t}^{*}U_{t}=U_{t}U_{t}^{*}=I$

, and prove it.

In the paper [5] this program was realized for the usual (classical and quantum) stochastic calculi. The advantage of this formulation with respect to the traditional ones lies in the fact that it naturally suggests the problem of extending equation (3.3) to the case of an equation which is still normally ordered, but now depending on higher powers of the white noise . Such equations had already been studied in the theory of white noise and existence and uniqueness theorems in the space of Hida distributions are already available [22]. It is therefore a natural program to combine the result of [22] with the above mentioned ones and to investigate regularity and unitarity. $b_{t}^{\uparrow},$

4

$b_{t}$

Non-linear white noise stochastic calculus

In order to realize the program described at the end of the previous section we have to overcome two competing requirements. On the one hand, we would like to start with an equation in a normally ordered form. On the other hand, we come to a non-trivial task of recognizing the unitarity condition in an equation in normal form. The advantage of the singular Hamiltonian formulation of equation (2.4) is that the formal unitarity condition is

7

automatically satisfied. Namely, one can hope that, if one starts from a formally unitary equation and then puts it in normal order, one obtains an equation whose solution is effectively unitary. We are more interested in realizing this program in the non-linear case. Again we can apply the two basic principles formulated in the previous section, because they are universally valid in the Fock space and do not depend in any way on the linearity of the equation. In fact, they do not depend at all on any particular equation but are intrinsic properties of the Boson Fock white noise calculus. Thus our starting point shall be the equation $\partial_{t}U_{t}=-i(b_{t}\uparrow 2+b_{t}^{2})U_{t}$

,

$U_{0}=1$

,

(4.1)

or in the integral form: $U_{t}=1-i \int_{0}^{t}(b_{S}^{\mathrm{t}}2+b_{s}^{2})U_{S}ds$

.

(4.2)

Our first goal is to give a meaning to it. Contrary to the linear case, equation (4.2) has no meaning even weakly on the number vectors. In fact, if we calculate formally the matrix elements of the both sides of (4.2) with respect to two number vectors, we obtain the expression: $\langle s_{1,h}\ldots, S|U_{t}|t1, \cdots, tk\rangle=$

$=\langle s1, \cdots, Sh|t1, \cdots,tk\rangle$

$st\delta(s-S\alpha)\delta(s-s\beta)\langle s_{1}, \cdots,\overline{S}\cdots,\overline{s}\alpha’\beta,h|U_{S}|t_{1}, \cdots, t_{k}\rangle d_{S}$

$- \dot{i}\sum_{\alpha\neq\beta}\gamma^{2}\int 0\ldots,$

$- \dot{i}\int_{0}^{t}\langle s, s, S_{1,h}\ldots, S|Us|t1, \cdots, tk\rangle d_{S}$

,

which involves the ill-defined product of two -functions. In order to give a meaning to equation (4.1) we adopt a different approach, which can be considered as an elaborated version of the scheme of thought one applies when, in usual distribution theory, one wants to give a meaning to the derivative of a non-differentiable function. We use the commutation relations to write equation (4.1) in normal form. In doing –the -function evaluated at zero! so we shall meet some ill-defined quantities such as However, as we shall see below these ill-defined quantities appear only as formal additive constants in the Hamiltonian equation and therefore can be eliminated by a subtraction procedure which is well known in physics under the name of renormalization. Once written in this form and once the ill-defined quantities have been subtracted, the equation has a weak meaning on the number vectors. We take this normal form as the definition of the a priori meaningless equation (4.1). $\delta$

$\delta(0)$

Lemma 4.1 Any (formal) solution relations:

of

equation

$(\mathit{4}\cdot \mathit{1})$

$\delta$

satisfies

$[b_{t}, U_{t}]=-2i\gamma_{-b}tt\uparrow U$

$[b_{t}, U_{t}^{*}]=2_{\dot{i}}\gamma-U_{tt}*b\mathrm{t}$

$[U_{t}^{*}, b_{t}^{\mathrm{t}}]=2i\overline{\gamma_{-U}}t*b_{t}$

$[U_{t}, b_{t}^{\uparrow}]=-2i\overline{\gamma-}b_{tt}U$

the following commutation

(4.3) (4.4) (4.5) (4.6)

8

PROOF.

Writing equation (4.1) in the integral form: $U_{t}=I-i \int_{0}^{t}(b^{\uparrow 2}S+b_{s}^{2})U_{s}ds$

,

we obtain $[b_{t}, U_{t}]$

$=$

$[b_{t}, I]- \dot{i}\int_{0}^{t}[b_{t}, (b_{s}^{\uparrow 2}+b_{s}^{2})Us]ds$

$=$

$-i \int_{0}^{t}[b_{t}, (b_{s}^{|2}+b_{s}^{2})]Usds-\dot{i}\int_{0}^{t}(b_{S}^{\dagger}2+b_{S}^{2})[b_{t}, U_{S}]dS$

.

(4.7)

Using $[b_{t}, (b_{s}^{|2}+b_{s}^{2})]=[b_{t}, b_{S}^{\dagger 2}]=[b_{t}, b_{s}^{\uparrow}]b_{s}^{\uparrow}+b_{S}^{\mathrm{t}}[bt, b^{\dagger}]s=2\gamma_{-\delta_{+}}(t-S)bs\dagger$

and the time consecutive principle $\int_{0}^{t}[b_{t}, U_{S}]ds=0$

,

we see that the right hand side of (4.7) becomes $=-i \int_{0}^{t}2\gamma-\delta+(t-S)b_{S}|U_{s}d_{S}=-2i\gamma_{-}b_{t}^{\dagger}U_{t}$

.

qed

This proves (4.3). The remaining relations are proved similarly.

Lemma 4.2 We have formal identities: $[b_{t}^{2}, U_{t}]=-i4\gamma_{-b_{t}}tbt-\dagger_{U}2i\gamma_{-^{\delta_{+}(}}^{2}\mathrm{o})U_{t}-4\gamma_{-t}^{2}b^{\dagger}U_{t}2$

(4.8)

$[U_{t}^{*}, b_{t}\dagger 2]=i4\overline{\gamma}_{-}b_{t}^{1*}Utbt+2_{\dot{i}}\overline{\gamma}_{-+}^{2}\delta(\mathrm{o})U^{*}t-4\overline{\gamma}_{-t}U*b_{t}^{2}2$

(4.9)

$[b_{t}^{2}, U_{t}^{*}]=4(\dot{i}\gamma_{-}-\gamma-)2b^{\uparrow Ub_{t}-8(}tt*|\gamma-|^{2}+\dot{i}\gamma_{-})U_{t}*b_{t}^{2}+2i\gamma_{-}^{2}U_{t}*\delta_{+}(0)$

where

$\delta_{+}(0)=[b_{t}, b_{t}^{\uparrow}]$

PROOF.

is a formal quantity.

Note first that $[b_{t}^{2}, U_{t}]=b_{t}[b_{t}, U_{t}]+[b_{t}, U_{t}]b_{t}$

Since

$[b_{t}, U_{t}]=-2_{\dot{i}}\gamma_{-b}tU\dagger t$

.

by Lemma 4.1 (4.3), we obtain

$[b_{t}^{2}, U_{t}]$

$=$

$=$

$=$

$b_{t}(-2i\gamma-b^{\uparrow_{U)}}tt+(-2_{\dot{i}}\gamma_{-}b_{t}\uparrow U_{t})bt$

$-2_{\dot{i}}\gamma_{-^{b_{t}}}b_{t}^{\mathrm{t}}U_{t}-2\dot{i}\gamma_{-b_{tt}^{\uparrow_{U_{t}}}}b$

$-2i\gamma-([bt, b_{t}^{\uparrow}]+b_{t}^{\mathrm{t}}b_{t})U_{t}-2_{\dot{i}}\gamma-b^{\dagger}tU_{t}b_{t}$

Introducing a formal quantity $\delta_{+}(0)=[bt,$

$bt\mathrm{t}_{]}$

,

.

,

(4.10)

9

we continue the calculation: $=$

$-2_{\dot{i}}\gamma-(\gamma_{-}\delta(0)+b_{t}\uparrow b_{t})Ut-2i\gamma_{-}b_{t}^{\uparrow}U_{tt}b$

$=$

$-2_{\dot{i}}\gamma_{-^{\delta(}}^{2}\mathrm{o})U_{t}-2i\gamma-b_{t}\dagger_{b}Utt-2\dot{i}\gamma-b_{t}^{\dagger}U_{tt}b$

$=$

$-2i\gamma_{-^{\delta(0}}^{2})Ut-2_{\dot{i}}\gamma-b^{\dagger}t([b_{t}, U_{t}]+U_{t}b_{t})-2i\gamma_{-}b^{\mathrm{t}_{U}}ttbt$

$=$

$-2i\gamma_{-}^{2}\delta(0)U_{t}-2i\gamma-b_{t}\dagger(-2i\gamma_{-^{b^{\mathrm{t}}}}tU_{t}+U_{t}b_{t})-2i\gamma_{-b^{1}U_{t}b}tt$

$=$

$-2_{\dot{i}}\gamma^{2}-\delta(\mathrm{o})Ut-4\gamma_{-tt}^{2}b\mathrm{t}2-2_{\dot{i}}\gamma_{-b_{t}U_{t}}U\dagger bt-2i\gamma_{-}bt\dagger U_{tt}b$

$-2_{\dot{i}}\gamma_{-^{\delta(}}^{2}\mathrm{o})U_{t}-4\gamma_{-}^{2\dagger \mathrm{t}}b_{t}2Ut-4\dot{i}\gamma-btUtb_{t}$

.

This proves (4.8) whose adjoint is (4.9). We then prove (4.10). Note that $[b_{t}, U_{t}^{*}]$

$=$

$2_{\dot{i}}\gamma_{-U^{*}}tb_{t}^{1}$

$=$

$2i\gamma-([U_{t’ t}^{*}b^{\dagger}]+b_{t}^{\dagger}U_{t}^{*})$

$=$

$2i\gamma-(2i\overline{\gamma}-U^{*}b_{t}t+b_{t}^{\uparrow_{U_{t}^{*}})}$

$=$

$-4|\gamma-|^{2}U^{*}btt+2_{\dot{i}}\gamma_{-}b_{t}^{1U_{t}^{*}}$

.

(4.11)

Then $[b_{t}^{2}, U_{t}^{*}]$

$=$

$b_{t}[b_{t}, U_{t}*]+[b_{t}, U_{t}^{*}]b_{t}$

$=$

$b_{t}(-4|\gamma-|^{2}U_{tt}*b+2i\gamma-b_{t}^{\dagger*}U_{t})+(-4|\gamma-|^{2}U_{tt}*b+2i\gamma_{-}btU|*t)bt$

$=$

$-4|\gamma-|2b_{t}U*b_{t}t+2i\gamma_{-b_{t}b}tt-\dagger_{U^{*}}4|\gamma-|2U*b_{t}t2+2i\gamma-b_{t}^{\dagger}U_{t}*b_{t}$

.

(4.12)

The first two terms are to be computed. In view of (4.11) we have $-4|\gamma-|2b_{t}U*b_{t}t$

$=$

$-4|\gamma-|2([b_{t}, U*]t+U_{t}^{*}b_{t})b_{t}$

$=$

$-4|\gamma_{-}|^{2}(-4|\gamma-|^{2}U^{*}b_{t}t+2_{\dot{i}}\gamma-b^{\uparrow U^{*}}tt+U_{t}^{*}b_{t})b_{t}$

$=$

$-4|\gamma-|2(1-4|\gamma-|2)U_{tt}^{*}b^{2}-8\dot{i}|\gamma-|^{2}\gamma-btU^{*}\uparrow btt$

and $2i\gamma_{-}btb_{tt}\uparrow_{U^{*}}$

$=$

$2_{\dot{i}}\gamma-([bt, b_{t}^{\uparrow}]+b_{t}^{\uparrow}b_{t})U_{t}^{*}$

$=$

$2i\gamma-(\gamma_{-}\delta_{+}(\mathrm{o})+b_{t}^{\mathrm{t}}b_{t})U^{*}t$

$=$

$2i\gamma_{-^{\delta}+}^{2}(0)U_{t}*+2_{\dot{i}}\gamma_{-}b_{t}\uparrow_{b_{t}U_{t}}*$

$=$

$2i\gamma_{-}^{2}\delta+(\mathrm{o})U_{t}*+2_{\dot{i}}\gamma_{-b_{t}}(\dagger[b_{t,t}U^{*}]+U_{t}^{*}b_{t})$

.

Using (4.11) to obtain $2_{\dot{i}}\gamma-b_{t}b^{1}tU^{*}t$

$=$

$2i\gamma_{-}^{2}\delta+(\mathrm{o})U_{t}*+2_{\dot{i}}\gamma-b_{t}^{\dagger}(-4|\gamma_{-}|^{2*}Utb_{t}+2i\gamma-b_{tt}^{1}U*+U_{t}^{*}b_{t})$

$=$

$2i\gamma_{-^{\delta}+}^{2}(0)U_{t}*+2i\gamma-(1-4|\gamma-|2)b_{l}^{\dagger_{U_{t}^{*}b}}t-4\gamma_{-^{b_{t}U_{t}}}^{2\dagger}2*$

.

Then (4.12) becomes $[b_{t}^{2}, U_{t}^{*}]$

$=$

$-4|\gamma-|2(1-4|\gamma_{-}|2)U*b_{t}^{2}-8i|t\gamma-|^{2}\gamma-b_{t}|U_{t}*b_{t}$

$+2i\gamma_{-}^{2}\delta(0)U_{t}*+2i\gamma-(1-4|\gamma-|2)b_{tt\gamma^{2}}^{\uparrow U_{t}^{*}}b-4-b^{\dagger 2*}tU_{t}$

$-4|\gamma_{-}|^{2}U_{tt}^{*}b^{2}+2i\gamma_{-}b_{t}^{\uparrow}U^{*}btt$

$=$

$-8|\gamma-|2(1-2|\gamma_{-}|^{2})U_{t}^{*}b_{t}^{2}+4i\gamma-(1-4|\gamma-|2)b_{t}\dagger U*btt+2_{\dot{i}}\gamma_{-}^{2}\delta(0)U_{t}*$

,

10

qed

as desired.

Proposition 4.3 The normal form

of equation

is

$(\mathit{4}\cdot \mathit{1})$

$\partial_{t}U_{t}=i(4\gamma_{-}^{2}-1)b_{t}^{\dagger\uparrow_{U_{t}}(}2-b_{tt}b_{t}-iUUt4\gamma_{-}b2-t2\gamma^{2}-\delta_{+}0)U_{t}$

,

whose adjoint form is $\partial_{t}U_{t}^{*}=-i(4\overline{\gamma}_{-}2-1)U*b^{2}-tt\overline{\gamma}_{-}b_{t}^{\mathrm{t}*}U_{tt}bt+ib^{\dagger 2}tU^{*}-2\overline{\gamma}2-\delta+(40)U_{t}*$

PROOF.

.

This follows from Lemma 4.2 and the identity $\partial_{t}U_{t}=-i(b_{t}\uparrow 2+b_{t}^{2})U_{t}=-\dot{i}b_{tt}^{12}2U_{t}-iUb-ti[b_{t}2, Ut]$

. qed

We have thus put equation (4.1) in normal form. Then, as an application of a general theorem on normally ordered equations driven by white noise [22], we can guarantee the existence and uniqueness of its solution, at least in the space of white noise (Hida) distribube the Gelfand triple of white noise tions. To be more precise, let functions, for more details see [15], [18]. $(E)\subset\Gamma(L^{2}(\mathbb{R}))\subset(E)^{*}$

Lemma 4.4 Let $A,$ $B,$

$C$

be bounded operators on

$\mathcal{H}_{S}$

. Then the

$\partial_{t}U_{t}=Ab_{t}^{\uparrow 2}U_{t}+Bb_{tt}^{\dagger_{Ub_{t}}}+CU_{t}b_{t}^{2}$

admits a unique solution in the space

$U_{0}=I$

$\mathcal{L}(\mathcal{H}_{S}\otimes(E), \mathcal{H}_{S}\otimes(E)^{*})$

The proof follows by modifying the result [22, into account.

5

,

\S 5]

differential equation ,

.

where the initial space

$\mathcal{H}_{S}$

is not taken

The normal form of the flow equation

In order to study the isometricity condition for the solution of the equation (4.1), we start with the flow equation (5.1) $j_{t}(x)=U^{*}xUtt=x_{t}$ ,

where

$x$

commutes with

$b_{t}$

and

$b_{t}^{\uparrow}$

. We need some preliminary lemmas.

Lemma 5.1 It holds that (5.2)

,

$U_{t}*b_{t}\uparrow U_{t}b_{t}$

$=$

$\frac{1}{1-4|\gamma_{-1^{2}}}b_{ttt}^{\uparrow U_{t}Ub+\frac{2i\overline{\gamma}_{-}}{1-4|\gamma_{-1^{2}}}UU}*tt*tb^{2}$

$b_{tt}^{\dagger_{U_{t}^{*}b}}U_{t}$

$=$

$\frac{1}{1-4|\gamma_{-1^{2}}}b_{t}^{\dagger_{U_{t}^{*}U_{t}}|2*}b_{t}-\frac{2_{\dot{i}}\gamma_{-}}{1-4|\gamma_{-1^{2}}}b_{tt}UU_{t}$

.

(5.3)

11

PROOF.

By a direct calculation one obtains $U_{t}^{*}b_{tt}\uparrow Ub_{t}$

$=$

$b_{ttt}^{\mathrm{t}_{U}*}Ub_{t}+[U_{t}^{*}, b_{t}^{\dagger}]U_{t}b_{t}$

$=$

$b_{t}^{\mathrm{t}}U_{tt}^{*}Ubt+2i\overline{\gamma}_{-}U_{t}*b_{t}Utb_{t}$

$=$

$b_{t}^{\mathrm{t}_{U_{t}}*}U_{t}b_{t}+2i\overline{\gamma}_{-}U_{t}^{*}([b_{t}, U_{t}]+U_{t}b_{t})b_{t}$

$=$

$b_{t}^{\mathrm{t}}U_{t}^{*}U_{tt}b+2_{\dot{i}}\overline{\gamma}_{-}U_{t}*(-2i\gamma-b^{\uparrow U_{t}}t+U_{t}b_{t})b_{t}$

$=$

.

$b_{t}\dagger U_{t}*Utb_{t}+4|\gamma-|^{2}U_{t}^{*}btU\dagger btt+2i\overline{\gamma}_{-}U^{*}Ub^{2}ttt$

There.fore $(1-4|\gamma-|2)U_{t}^{*}b_{t}^{\mathrm{t}_{U_{t}}}b_{t}=b_{t}^{\mathrm{t}_{U_{t}}*}U_{t}b_{t}+2_{\dot{i}}\overline{\gamma}_{-}U*U_{t}b^{2}tt$

’

qed

which proves (5.2).

Lemma 5.2 Denoting $\alpha=\frac{1}{1-4|\gamma_{-1^{2}}}$

one has $U_{t}^{*}b_{t}^{2}U_{t}$

$=$

$\frac{-2i\gamma_{-}^{2}-8i|\gamma_{-}|^{4}}{1-16|\gamma-|^{4}}\delta_{+}(0)U^{*}tU_{t}+\alpha U_{t}^{*}2Utb^{2}t$

(5.4)

$-4i\gamma_{-^{\alpha bUU_{t}b}}ttt-2\dagger*4\gamma_{-^{\alpha^{2}}}^{2}b_{t}\uparrow 2U*tU_{t}$

PROOF.

For the proof we first compute $U_{t}^{*}b_{t}^{2}U_{t}=U_{t}^{*}[b_{t}^{2}, U_{t}]+U_{t}^{*}U_{t}b_{t}^{2}$

We know from (4.8) that $[b_{t}^{2}, U_{t}]=-i4\gamma_{-b}t\mathrm{t}_{Utbt-}2\dot{i}\gamma 2\delta-+(0)U_{t}-4\gamma-btU_{t}2\dagger 2$

.

Hence $U_{t}^{*}b_{t}^{2}U_{t}$

$=$

$=$

$=$

$U_{t}^{*}(-\dot{i}4\gamma_{-b}t-2\dot{i}\gamma b_{t}-^{\delta}+(\dagger_{U_{t}}20)Ut-4\gamma^{2}-b_{t}\mathrm{t}2U_{t})+U_{t}^{*}U_{t}b_{t}^{2}$

$-4i\gamma_{-}U_{t}*b^{\dagger*}tUtb_{t}-2i\gamma^{2}-^{\delta()U^{*}}+0U_{t}Ut-4\gamma^{2}-tb_{t}^{\mathrm{t}2}Ut+U_{t}^{*}U_{t}b_{t}^{2}$

.

$U^{*}Ub^{2}-ttt2\dot{i}\gamma^{2}-^{\delta}+(0)U^{*}Ut-4i\gamma-U*b^{\mathrm{t}_{U}2}b-4\gamma-^{U}ttUttttt*b^{\uparrow 2}t$

(5.5)

It follows from (5.2) that $-4i \gamma_{-U_{t}}b^{1U}tb*tt=\frac{-4_{\dot{i}}\gamma_{-}}{1-4|\gamma_{-1^{2}}}b\uparrow Utt*Utbt+\frac{8|\gamma_{-}|^{2}}{1-4|\gamma_{-1^{2}}}U_{t}*U_{t}b_{t}^{2}$

.

On the other hand, $-4\gamma_{-^{U_{t}b^{\dagger 2}}}^{2*}tUt$

$=-4\gamma_{-}^{2}([U_{t}^{*}, bt]\dagger 2\dagger 2U+btt*)Ut$

$=-4\gamma_{-((}^{2}i4\overline{\gamma}_{-t}b^{\mathrm{t}2}U_{t}*bt+2_{\dot{i}}\overline{\gamma}_{-+}\delta 0)U^{*}-t4\overline{\gamma}^{2*}-U_{tt}b^{2}+b_{t}^{\dagger}U*)2U_{t}t$

$=-4\gamma_{-}^{2}(_{\dot{i}}4\overline{\gamma}_{-^{bUb_{t}}}tt+2\mathrm{t}*\dot{i}\overline{\gamma}-^{\delta}+(20)U^{*}-t4\overline{\gamma}_{-}Utt+bUtt)2*b^{2\dagger 2*}U_{t}$

$=-16i|\gamma_{-}|2\dagger_{U_{t}}*b_{t}U_{t}\gamma-^{b}t-8i|\gamma-|^{4}\delta_{+}(\mathrm{o})U_{t}^{*}Ut+16|\gamma-|^{4*}Ub_{t}^{2}Ut-t4\gamma_{-^{b^{\dagger 2}U^{*}}}^{2}tttU$

.

12

Then (5.5) becomes $U_{t}^{*}b_{t}^{2}U_{t}$

$=$

$U^{*}U_{t}ttb^{2}-2i\gamma_{-}2\delta+(0)U^{*}ttU$

$+ \frac{-4i\gamma_{-}}{1-4|\gamma_{-1^{2}}}b_{ttt}^{\uparrow U_{t}}*Ub+\frac{8|\gamma_{-}|^{2}}{1-4|\gamma_{-1^{2}}}U_{t}^{*}U_{t}b_{t}^{2}$

$-16i|\gamma_{-}|2b_{t}\dagger U-tb_{t}*U_{t}-8i\gamma|\gamma_{-}|^{4}\delta_{+}(\mathrm{o})U*U_{t}t+16|\gamma-|4U*b^{2}Ut-tt4\gamma_{-tt}b^{\dagger}22U*U_{t}$

.

Hence $(1-16|\gamma_{-}|^{4})U*b^{2}Uttt$

$U_{t}^{*}U_{t}b_{t}^{2}-2\dot{i}\gamma-\delta 2(+\mathrm{o})U^{*}ttU$

$=$

$+ \frac{-4_{\dot{i}}\gamma_{-}}{1-4|\gamma_{-}|^{2}}b_{ttt}^{\uparrow U_{t}}*Ub+\frac{8|\gamma_{-}|^{2}}{1-4|\gamma_{-1^{2}}}U_{tt}^{*}U_{t}b^{2}$

$-16_{\dot{i}}|\gamma_{-}|2btttt-i\gamma_{-}|\uparrow U*bU8\gamma_{-}|^{4}\delta+(\mathrm{o})U^{*}U_{t}-4\gamma_{-}b_{tt}2\mathrm{t}2U*U_{t}t$

$=$

$\frac{1+4|\gamma_{-}|^{2}}{1-4|\gamma_{-1^{2}}}U_{t}^{*}U_{t}b_{t}^{2}+(-2_{\dot{i}}\gamma_{-}^{2}-8i|\gamma_{-}|^{4})\delta+(0)U^{*}ttU$

.

$+ \frac{-4_{\dot{i}}\gamma_{-}}{1-4|\gamma_{-1^{2}}}b_{t}^{\dagger}U_{tt}^{*}Utb-16_{\dot{i}||^{2}}\gamma_{-}\gamma-btUtt\dagger*b_{t}U-4\gamma_{-^{b}t}^{2}\dagger 2*U_{t}U_{t}$

In view of (5.3) one obtains .

$-16_{\dot{i}}| \gamma-|2\gamma_{-}b_{tt}^{\uparrow}U^{*}b_{tt}U=\frac{-16i|\gamma_{-}|^{2}\gamma_{-}}{1-4|\gamma_{-}|^{2}}b_{tt}^{\dagger_{U^{*}}\mathrm{t}2*}U_{t}bt-\frac{32|\gamma-|^{2}\gamma_{-}2}{1-4|\gamma_{-}|^{2}}btU_{t}Ut$

Hence $(1-16|\gamma_{-}|^{4})U*b^{2}Uttt=$

$= \frac{1+4|\gamma_{-}|^{2}}{1-4|\gamma_{-1^{2}}}U_{t}^{*}U_{t}b_{t}2+(-2\dot{i}\gamma_{-}^{2}-8i|\gamma_{-}|4)\delta+(\mathrm{o})U_{t}*U_{t}+\frac{-4i\gamma_{-}}{1-4|\gamma_{-}|^{2}}b_{tt}^{\mathrm{t}}U^{*}Ut$

bt

$+ \frac{-16i|\gamma_{-}|^{2}\gamma_{-}}{1-4|\gamma_{-1^{2}}}b_{tt}^{\dagger\dagger 2}U_{t}^{*}Ub_{t}-\frac{32|\gamma-|^{2}\gamma_{-}2}{1-4|\gamma_{-}|^{2}}b2Utt-4\gamma_{-}*U_{t}b^{\uparrow*}t2U_{t}U_{t}$

$= \frac{1+4|\gamma-|^{2}}{1-4|\gamma_{-1^{2}}}U_{t}^{*}U_{t}b_{t}2+(-2_{\dot{i}}\gamma_{-}^{2}-8\dot{i}|\gamma-|^{4})\delta_{+}(\mathrm{o})U_{t}*U_{t}$

$+ \frac{-4i\gamma_{-}-16i|\gamma_{-}|2\gamma-}{1-4|\gamma_{-1^{2}}}b_{tt}^{\dagger_{U_{t}}*\uparrow}Ub_{t}+\frac{-32|\gamma_{-}|22-4\gamma^{2}\gamma_{-}-(1-4|\gamma_{-1^{2})}}{1-4|\gamma_{-1^{2}}}btU_{t}*U_{t}2$

$= \frac{1+4|\gamma-|^{2}}{1-4|\gamma_{-1^{2}}}U_{t}^{*}U_{t}b_{t}2+(-2i\gamma_{-}^{2}-8\dot{i}|\gamma_{-}|^{4})\delta+(\mathrm{o})U_{t}*U_{t}$

$+ \frac{-4_{\dot{i}}\gamma-(1+4|\gamma-|^{2})}{1-4|\gamma_{-1^{2}}}b_{t}^{\mathrm{t}}U^{*\uparrow}tUtbt+\frac{-4\gamma_{-}^{2}(1+4|\gamma-|^{2})}{1-4|\gamma_{-1^{2}}}bU_{t}tU_{t}2*$

which completes the proof.

Lemma 5.3 The equation

,

qed

satisfied

by

$x_{t}=U_{t}^{*}xU_{t}$

is

$\partial_{t}x_{t}=4{\rm Re}(\gamma_{-}^{2})\delta+(0)X_{t}-ib^{\dagger 2}X_{t}+\dot{i}x_{t}b_{t}^{2}$

.

(5.6)

13

Differentiating by the Leibnitz rule equation (5.1), we find

PROOF.

$\partial_{t}x_{t}=ib_{t}\dagger 2xt-\dot{i}xtbt2$

$+\dot{i}(4\gamma_{-}^{2}-1)U*xb^{\uparrow}tU_{t}t2+(h.c.)-4\gamma_{-Ub}t^{X}tU_{tt}*\dagger b+(h.c.)$

$-4{\rm Re}(\gamma-2)\delta_{+}(\mathrm{o})xt$

.

(5.7)

The result then follows using Lemmas 5.1 and 5.2. In fact, we compute $\dot{i}(4\gamma_{-}^{2}-1)U_{t^{X}}*b^{\uparrow}tUt2+(h.c.)-4\gamma_{-}U^{*}t^{X}tb^{\dagger}U_{t}b_{t}+(h.c.)=$

$=\dot{i}(4\gamma_{-}^{2}-1)U*2-txb_{t}^{1}Ut\dot{i}(4\overline{\gamma}--21)U^{*}b^{2}Ut-4\gamma_{-^{U}}t^{X}tU_{t}bt-4\overline{\gamma}b\uparrow b_{tt^{X}}^{1}-U_{t}^{*}*btt^{X^{*}}U_{t}*$

With no loss of generality we can omit above we find

$x$

.

from the equation. Inserting the results obtained

$i(4\gamma_{-}^{2}-1)U^{*}b^{\mathrm{t}2}U_{t}=tt$

$=i(4 \gamma_{-}^{2}-1)(\alpha b_{tt}\uparrow 2U2*U_{t}+\frac{2i\overline{\gamma}_{-}^{2}+8\dot{i}|\gamma_{-}|4}{1-16|\gamma-|^{4}}\delta_{+}(0)U_{t}^{*}U_{t}$

$+4i\overline{\gamma}_{-^{\alpha}}b_{tt}\dagger U_{t}*Ubt-24\overline{\gamma}_{-}^{22}\alpha U^{*}Uttb^{2)}t$

$= \dot{i}(4\gamma_{-}^{2}-1)\alpha^{2}b^{\dagger 2}U^{*}ttUt+\frac{-2(4\gamma_{-}^{2}-1)(\overline{\gamma}-+4|2\gamma_{-1^{4})}}{1-16|\gamma-|^{4}}\delta+(\mathrm{o})U_{t}*U_{t}$

.

$-4(4\gamma^{2}--1)\overline{\gamma}-\alpha^{2\dagger_{U_{t}^{*}U}22*}bttbt-4i(4\gamma_{-}-12)\overline{\gamma}_{-}\alpha UUtb^{2}tt$

Similarly $-\dot{i}(4\overline{\gamma}_{-}-21)U^{*}b^{2_{X}*}Uttt=$

$=- \dot{i}(4\overline{\gamma}^{2}--1)\alpha U2*Utb_{t}^{2}t+\frac{-2(4\overline{\gamma}_{-}^{2}-1)(\gamma-+42|\gamma_{-}|^{4})}{1-16|\gamma-|^{4}}\delta_{+(\mathrm{o}})U^{*}ttU$

.

$-4(4\overline{\gamma}_{-}^{2}-1)\gamma_{-}\alpha btU2\uparrow_{U_{t}}*btt+4i(4\overline{\gamma}_{-}^{2}-1)\gamma^{22\uparrow}-\alpha btUU_{t}t2*$

On the other hand, $-4\gamma_{-}U^{*}t^{X}b^{\dagger}tUtbt=-4\gamma-(\alpha b\dagger Uttb*U_{tt}+2i\overline{\gamma}_{-^{\alpha U_{t}U_{t}}}b_{t}^{2}*)$

and $-4\overline{\gamma}_{-}b^{\mathrm{t}}tU_{t}*bXtUt=-4\overline{\gamma}_{-}(\alpha b_{tt}|U*Utbt-2_{\dot{i}}\gamma-\alpha b\dagger 2*)tUUtt$

.

Summing up the above four expressions one finds $\frac{(2-32|\gamma_{-}|^{4})(\gamma 2+\overline{\gamma}^{2}-)}{1-16|\gamma-|^{4}}\delta_{+}(\mathrm{o})U^{*}U$

$+\dot{i}\{-(1-16|\gamma-|4)\alpha^{2}+8\alpha|\gamma_{-}|^{2}\}b^{\uparrow 2}tU*U_{t}t$

$+\dot{i}\{(1-16|\gamma-|^{4})\alpha^{2}-8\alpha|\gamma-|^{2}\}U^{*}U_{t}ttb^{2}$

$+\{4\alpha^{2}(\gamma+\overline{\gamma})-16\alpha^{2}|\gamma-|^{2}(\gamma-+\overline{\gamma}_{-})-4\alpha(\gamma_{-}+\overline{\gamma}_{-})\}b^{\dagger}U_{t}tU*tbt$

$=$

$4{\rm Re}(\gamma_{-}^{2})\delta_{+}(\mathrm{o})U^{*}Ut-t\dot{i}b\uparrow 2U_{t}*Ut+\dot{i}U_{t}^{*}U_{t}b^{2}t$

This completes the proof.

.

qed

14

6

Renormalization and formal unitarity of the solution Starting with $\partial_{t}U_{t}=-i(b_{t}^{\dagger 2}+b_{t}^{2})U_{t}$

,

$U_{0}=I$

,

(6.1)

we have found the normal form: $\partial_{t}U_{t}=i(4\gamma_{-}^{2}-1)b^{\dagger 2}tU_{t}-4\gamma_{-}bttb_{t}-iUtb\dagger_{U}2-t2\gamma-\delta_{+}(20)Ut$

(6.2)

,

where is see Proposition 4.3. Note that (6.2) involves the ill-defined quantity , where is a complex number. However, the addition of a quantity of the form a real constant, to the Hamiltonian in (6.1) does not change the basic quantity of interest, . After this addition we obtain itself but the flow which is not the operator $\gamma_{-}^{2}\delta_{+}(0)Ut$

$\gamma_{-}$

$\rho_{0}=\rho\delta_{+}(0)$

$\rho$

$U_{t}^{*}xU_{t}$

$U_{t}$

$\partial_{t}U_{t}=-i(b_{t}^{|2}+b_{t}^{2}+\rho\delta_{+}(0))U_{t}$

,

$U_{0}=I$

,

(6.3)

and its normal form: $\partial_{t}U_{t}=i(4\gamma_{-}^{2}-1)b^{\uparrow}tUt-4\gamma_{-}b^{\uparrow 4}tU_{t}b_{t}2-\dot{i}U_{t}b2-t({\rm Re}(\gamma^{2}-)+i\{{\rm Im}(\gamma^{2}-)-\rho/4\})\delta_{+}(\mathrm{o})Ut$

.

we see that the ill-defined quantity , then by putting Therefore, if vanishes identically. On the other hand, by a direct computation we obtain $\rho=4{\rm Im}(\gamma_{-}^{2})$

${\rm Re}(\gamma_{-}^{2})=0$

Lemma 6.1 The right hand side

of equation

(5.1) is identically zero

for any

choice

of

$\gamma_{-}$

.

The above result shows that the normal form equation (6.2) satisfies the formal tarity condition. Therefore the right hand side of (5.7) is identically zero as we expected. which renormalizes the equation for also Notice that the same condition renormalizes the equation for the flow . Our next task is to investigate the regularity and unitarity condition for or obeying the renormalized equation. The research is now in progress. $un\dot{i}-$

$U$

$({\rm Re}(\gamma_{-}^{2})=0)$

$j_{t}$

$U_{t}$

$j_{t}$

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