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Scaling Limit for the System of Dirac Fields Coupled to Quantized Radiation Fields with Cutoffs. (Duality and Scales in Quantum-Theoretical Sciences)

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Scaling

Limit

for the

System of Dirac Fields

Coupled

to

Quantized Radiation Fields

with Cutoffs.

Toshimitsu TAKAESU

FacultyofMathematics, Kyushu University,

Fukuoka, 812-8581, Japan

Abstract. In this paperthe system of the Dirac field interacting with the

quan-tized radiation fieldis investigated. By introducing ultraviolet cutoffs and spatial

cutoffs, itis

seen

that the total Hamiltonianis aself-adjoint operator

on

a

boson-fermion Fockspace. The scaledtotalHamiltonian is defined,andits asymptotic behavioris investigated. In the main theorem, it is shownthat the effective po-tential emerges.

This article is devoted to a short review on the obtained results in [13]. We consider the

system of the Quantum electrodynamics (QED), which describes the Dirac field coupled

to quantized radiation field in the Coulomb gauge. We analyze this system from purely

mathematical view point. The state space is defined by the boson-fermion Fock space

$\mathcal{F}_{QED}=\mathcal{F}_{Dirac}\otimes 9_{rad}^{\mathscr{J}}$ , where$\mathcal{F}_{Dirac}$ isthefermion Fock space on$L^{2}(R^{3};C^{4})$ and$\mathcal{F}_{rad}$ isthe

boson Fockspaceon$L^{2}(R^{3};C^{2})$. The free Hamiltonian ofthe Diracfield$H_{Dirac}$ is defined by

the secondquantizationof$aDir(p)=\sqrt{p^{2}+M^{2}}$withthe restmass$M>0$. Similarlythe free

Hamiltonian ofthe radiation field$H_{rad}$ is definedby the secondquantizationof akad$(k)=|k|$.

The field operators ofthe Dirac field and the radiation field are denoted by $\psi(x)$ andA(x),

respectively. Here we impose ultraviolet cutoffs on both $\psi(x)$ and A(x). The interaction

Hamiltonians are given by

$H_{I}’= \int_{R^{3}}\chi(x)\psi^{*}(x)\alpha\psi(x)\cdot A(x)dx$,

$H_{II}’= \frac{1}{8\pi}\int_{R^{3}\cross R^{3}}\frac{\chi(x)\chi(y)}{|x-y|}\psi^{*}(x)\psi(x)\psi^{*}(y)\psi(y)dxdy$,

where $\chi(x)$ denotes the spatial cutoff, and $\psi^{*}(x)\alpha\psi(x)\cdot A(x)=\sum_{j=1}^{3}\psi^{*}(x)\alpha^{j}\psi(x)A^{j}(x)$.

Then the total Hamiltonian is definedby

$H=H_{Dirac}+H_{rad}+eH_{I}’+e^{2}H_{II}’$ (1)

数理解析研究所講究録

(2)

Let

us

considerthe self-adjointness of$H$

.

Undersufficient conditions ofthe ultraviolet cutoff

andthe spatial cutoff, itis

seen

that$H_{I}$ is relatively bounded with respect to$H_{rad}^{1/2}$ and$H_{II}’$ is

aboundedoperator. Hence theinteractionsare infinitely small with respect to$H_{Dirac}+H_{rad}$.

Thenthe Kato-Rellich theorem shows that$H$ is self-adjoint and essentially self-adjoint any

coreof$H_{0}[12]$. The spectralproperties of$H$also have been investigatedin [2, 12].

Nowwe introducethe scaled QEDHamiltonian definedby

$H(\Lambda)=H_{Dirac}+\Lambda^{2}H_{rad}+e\Lambda H_{I}’+e^{2}H_{II}’$

.

(2)

We

are

interested in the asymptoticbehavior of$H(\Lambda)$

as

$\Lambdaarrow\infty$

.

Historilally scaling limits

of the Hamiltonians of the form (2) is introduced by E. B. Davies [3]. He investigates the

system of paricles coupled to a scalar bose field, and consider the scaled total Hamitonian

$H_{p}+\Lambda\kappa\phi(x)+\Lambda^{2}H_{b}$ where$H_{p}=A^{2}2M^{-}$ is a shcr\"odinger operator, $\phi(x)$ is the field operator

of the scalar bose field, and $H_{b}$ is the free Hamiltonian. Then

an

effective Hamiltonian $H_{p}+\kappa^{2}V_{eff}(x)$ is obtained Then ourresult canbe regarded as a extended $mode[$ of[3]. In

[1], ageneral theoryon scaling limits, whichcanbe applied toa spin-boson modeland

non-relativistic QED models, isinvestigated. In [6],byremovingultraviolet cutoffs and taking

a

scaling limit of the Nelson model simultaneously,

a

Schr\"odinger operator with the Yukawa

potential is derived. Refer to see also [4, 10, 11, 9, 12, 14]. It is noted that the unitary

evolutionof$H(\Lambda)$ is given by

$e^{-itH(\Lambda)}$ $=$

$e^{-it\Lambda^{2}(_{\Lambda}\tau^{H_{Dirac}}}-1+H_{rad}+( \frac{e}{\Lambda})H_{1}^{f}+(\frac{e}{\Lambda})^{2}H_{I\mathfrak{l}}’)$

,

(3)

and we see that $t\Lambda^{2}$ is the scaled

time and $\frac{e}{\Lambda}$ is the scaled coupling constant. The main

theorem is as follows:

Theorem 1 It

follows

that

for

$z\in C\backslash R$

$s- \lim_{\Lambdaarrow\infty}(H(\Lambda)-z)^{-1}=(H_{Dirac}+e^{2}H_{II}’+e^{2}V_{eff}-z)^{-1}P_{\Omega_{rad}}$, (4)

where

$V_{eff}=- \frac{1}{4}\sum_{j,l}\int_{R^{3}\cross R^{3}}\chi(x)\chi(y)\psi^{*}(x)\alpha\psi(x)\cdot\triangle(x-y)\psi^{*}(y)\alpha\psi(y)dxdy$, (5)

and $\triangle(z)=(\lambda^{j,l}(z)+\lambda^{j,l}(-z))_{j,i=1}^{3}$ is the $3\cross 3$ matrix with

afunction

$\lambda^{j,/}(z)$

defined

by $\lambda^{j,l}(z)=\int_{R^{3}}\frac{|\chi_{rad}(k)|^{2}}{(2\pi)^{3}|k|^{2}}(\delta_{j,l}-\frac{k^{j}k^{1}}{|k|^{2}})e^{-ik\cdot z}dk$. (6)

(3)

By thegeneral theorem([11], Lemma2.7),the following corollaryimmediately follows.

$Co$rollary2 It

follows

that

$s- \lim_{\Lambdaarrow\infty}e^{-itH(\Lambda)}P_{\Omega_{rad}}=e^{-}it(H_{Dirac}+e^{2_{H/ong}}+e^{2}V_{eff})_{P_{\Omega_{rad}}}$. (7)

The outline ofthe proof of themaintheorem isasfollows. Weconsidertheunitary

transfor-mation, called the dressing transformation, definedby

$U( \frac{e}{\Lambda})=e^{-i(_{X}^{e})T}$,

where

$T= \int_{R^{3}}\chi(x)\psi^{*}(x)\alpha\psi(x)\cdot\Pi(x)dx$,

with the conjugate operator$\Pi(x)=(\Pi(x)^{j}),$ $j=1,2,3$ satisfying $[\Pi(x)^{j}, H_{rad}]=-iA^{j}(x)$

and $[\Pi(x)^{j}, A^{l}(y)]=i\lambda^{j,l}$(x-y). Then the Hamiltonianis transformed by

$U( \frac{e}{\Lambda})^{-1}H(\Lambda)U(\frac{e}{\Lambda}I$ $=\tilde{H}_{0}(\Lambda)+K(\Lambda)$,

where $\tilde{H}_{0}(\Lambda)=H_{Dirac}+e^{2}H_{I,.I}’+\Lambda^{2}H_{rad}$, and$K(\Lambda)$ is an operator satisfying the following

properties :

Proposition3

(1) For$\epsilon>0$, thereexists $\Lambda(\epsilon)\geq 0$such that

for

all $\Lambda>\Lambda(\epsilon)$,

$\Vert K(\Lambda)\Psi\Vert\leq\epsilon\Vert\tilde{H}_{0}(\Lambda)\Psi\Vert+v(\epsilon)\Vert\Psi\Vert$. (8)

holds, where $v(\epsilon)$ isaconstantindependent$of\Lambda\geq\Lambda(\epsilon)$.

(2) For all$z\in C\backslash R$ it

follows

that

$s- \lim_{\Lambdaarrow\infty}K(\Lambda)(\tilde{H}_{0}(\Lambda)-z)^{-1}=K(H_{Dirac}+e^{2}H_{II}’-z)^{-1}P_{\Omega_{rad}}$ , (9)

where $K=- \frac{ie^{2}}{2}[T, H_{J}’]$.

By Proposition 3 andthe generaltheory([1] ; Thorem 2.1), it is seenthat

$s- \lim_{\Lambdaarrow\infty}(H(\Lambda)-z)^{-1}=(H_{Dirac}+e^{2}H_{II}’+K_{rad}-z)^{-1}P_{\Omega_{rad}}$,

where

$K_{rad}= \frac{-ie^{2}}{2}P_{\Omega_{rad}}[T,H_{I}’]P_{\Omega_{rad}}$.

By the simple computation of$K_{rad}$, themain theorem follows.

(4)

References

[1] A.Arai,Anasymptoticanalysis anditsapplicationtothenonrelativisticlimit ofthe Pauli-Fierz

andaspin-bosonmodel,J.Math. Phys. 32(1990)2653-2663.

[2] J.Barbaroux, M.Dimassi, and J.Guillot, Quantum electrodynamics ofrelativistic bound states withcutoffs, J.Hyper. Differ.Equa. 1 (2004) 271-314.

[3] E.B. Davies, Particle interactionsand the weak coupling limit, J. Math. Phys. 20 (1979) 345-351.

[4] F.Hiroshima, Scaling limit ofamodel ofquantum electrodynamics, J. Math. Phys. 34 (1993)

4478-4518.

[5] F.Hiroshima, Scaling limit ofa model ofquantum electrodynamics with many nonrelativistic

particles, Rev. Math. Phys. 9(1997) 201-225.

[6] F.Hiroshima, Weak coupling limit with a removal ofan ultraviolet cutoff for a Hamiltonian

ofparticles interacting with a massive scalar field, Inf. Dim. Ana. Quantum Prob. Rel. Top. 1

(1998)407-423.

[7] F.Hiroshima,Weakcoupling limitandaremoving ultraviolet cutoffforaHamiltonianof

parti-cles interactingwith aquantized scalar field. J. Math. Phys. 40 (1999) 1215-1236.

[8] F.Hiroshima, Observable effects andparametrized scaling limits ofa model innon-relativistic quantumelectrodynamics, J. Math. Phys. 43 (2002) 1755-1795.

[9] A.Ohkubo, Scaling limit for the Derezinski-G\’erard Model, to appear in Hokkaido Math. J.

(2009).

[10] A.Suzuki, Scaling limit for a general class of quantum field models and its applications to nuclear physics and condensed matter physics, Inf. Dim. Ana. Quantum Prob. Rel. Top. 10

(2007)43-65.

[11] A.Suzuki, Scaling limitfor ageneralization oftheNelson model and itsapplicationtonuclear

physics, Rev. Math. Phys. 19 (2007) 131-155.

[12] T.Takaesu,On the spectral analysis of quantum electrodynamics with spatial cutoffs.I.,J. Math. Phys, 50 (2009)06230.

[13] T.Takaesu, Scaling limit of quantum electrodynamics with spatial cutoffs. (arxiv :

0908.2080vl).

[14] T.Takaesu, Scaling limits for the system ofsemi-relativistic particles coupled to a scalar bose

field. (preprint)

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