On the ground states of quantum electrodynamics with cutoffs (Mathematical Aspects of Quantum Fields and Related Topics)

On

the

_ground

states

of

_quantum

_{electrodynamics}

with cutoffs

Faculty

of Science and

_Engineering

Gunma

_University

Toshimitsu Takaesu

This article isashort review ofa

ground

stateofamodel of_quantum

electrodynamics

in

I13].

We

_investigate

a_systemofa

quantized

Dirac field

coupled

to a

_quantized

radiation filed in the

Coulomb gauge: The classical

Lagrangian density

is

_{given by}

\displaystyle \mathrm{C}_{\mathrm{Q}\mathrm{E}\mathrm{D}}=-\frac{1}{4}F^{ $\mu$ v}F_{ $\mu$ v}+\overline{ $\psi$}(i$\gamma$^{ $\mu$}D_{ $\mu$}-M) $\psi$,

where

_{F^{ $\mu$ v}=\partial^{ $\mu$}\mathrm{A}^{v}-\partial^{v}A^{ $\mu$},}

_{D_{ $\mu$}=\partial_{ $\mu$}-eA_{ $\mu$}}

and

_{\overline{ $\psi$}=$\psi$^{ $\dagger$}$\gamma$^{0}}

. Wedefine the Hilbert space for the

system

by

\mathcal{F}_{\mathrm{Q}\mathrm{E}\mathrm{D}}=\mathcal{F}_{\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{c}}\otimes T_{\mathrm{r}\mathrm{a}\mathrm{d}}

where

_yDirac

isafermion Fock spaceover

L^{2}(\mathbb{R}^{3};\mathbb{C}^{4})

and

_{3_{\mathrm{r}\mathrm{a}\mathrm{d}}^{\mathrm{i}}}

isa

boson Fock spaceover

L^{2}(\mathbb{R}^{3}\times\{1,2\})

.The total Hamiltonian for thesystemis defined

by

H_{\mathrm{Q}\mathrm{E}\mathrm{D}}=H_{\mathrm{D}}\displaystyle \otimes \mathrm{I}+\mathrm{I}\otimes H_{r\mathrm{a}\mathrm{d}}+$\kappa$_{\mathrm{I}}\sum_{j=1}^{3}\int_{\mathrm{R}^{3}}$\chi$_{\mathrm{I}}(\mathrm{x})($\psi$^{ $\dagger$}(\mathrm{x})$\alpha$^{j} $\psi$(\mathrm{x})\otimes A_{j}(\mathrm{x}))d\mathrm{x}

+$\kappa$_{\mathrm{I}\mathrm{I}}\displaystyle \int_{\mathrm{R}^{3}\mathrm{x}\mathrm{R}^{3}}\frac{$\chi$_{\mathrm{I}\mathrm{I}}(\mathrm{x})$\chi$_{1\mathrm{I}}(\mathrm{y})}{|\mathrm{x}-\mathrm{y}|}($\psi$^{ $\dagger$}(\mathrm{x}) $\psi$(\mathrm{x})$\psi$^{ $\dagger$}(\mathrm{y}) $\psi$(\mathrm{y})\otimes \mathrm{I})

dxdy,

where

H_{\mathrm{D}^{=\sum_{s=\pm 1/2}\int_{\mathrm{R}^{3}} $\dagger \dagger$}}$\omega$_{M}(\mathrm{p})(b_{s}(\mathrm{p})b_{s}(\mathrm{p})+d_{s}(\mathrm{p})d_{s}(\mathrm{p}))d\mathrm{p},

H_{\mathrm{r}\mathrm{a}\mathrm{d}}=\displaystyle \sum_{r=12}, $\omega$(\mathrm{k})a_{r}^{ $\dagger$}(\mathrm{k})a_{r}(\mathrm{k})d\mathrm{k},

with

_{$\omega$_{M}(\mathrm{p})=\sqrt{\mathrm{p}^{2}+M^{2}},M>0}

,and

to(k) =|\mathrm{k}|

. Themomentum

expansions

ofthe fields opera‐

torsare

$\psi$_{T}(\displaystyle \mathrm{x})=\sum_{\mathrm{s}=\pm 1/2}\frac{}{\frac{2$\pi$^{3}1}{()}}\int_{\mathrm{R}^{3}}$\chi$_{\mathrm{D}}(\mathrm{p})(u_{s}^{T}(\mathrm{p})b_{s}(\mathrm{p})e^{i\mathrm{p}\cdot \mathrm{x}}+\sqrt{s}(-\mathrm{p})d_{s}^{\uparrow}(\mathrm{p})e^{-i\mathrm{p}\cdot \mathrm{x}})

d\mathrm{p}

,

A_{j}(\displaystyle \mathrm{x})=\sum_{r=12},\frac{}{\frac{2$\pi$^{3}1}{()}}\int_{\mathrm{R}^{3}}\frac{$\chi$_{\mathrm{r}\mathrm{a}\mathrm{d}}(\mathrm{k})\dot{d}_{r}(\mathrm{k})}{\frac{2 $\omega$ \mathrm{k}}{()}}(a_{r}(\mathrm{k})e^{- $\iota$ \mathrm{k}\cdot \mathrm{x}}\wedge+a_{r}^{\uparrow}(\mathrm{k})e^{-i\mathrm{k}\cdot \mathrm{x}})

d\mathrm{k},

respectively.

数理解析研究所講究録

Thecanonicalanti‐commutation relationsare

\{b_{s}(\mathrm{p}),b_{s}^{ $\dagger$},(\mathrm{p}').\}=\{d_{s}(\mathrm{p}),d_{s}^{ $\dagger$},(\mathrm{p}')\}=$\delta$_{s,s'} $\delta$(\mathrm{p}-\mathrm{p}')

,

\{b_{s}(\mathrm{p}),b_{s'}(\mathrm{p}')\}=\{d_{s}(\mathrm{p}),

ゐ

_{(\mathrm{p}')\}_{\backslash }=0,}

\{b_{s}^{\uparrow}(\mathrm{p}),b_{s}^{ $\dagger$},(\mathrm{p}')\}=\{ffi_{s}(\mathrm{p}),d_{s}^{\mathrm{t}},(\mathrm{p}')\}.=\{b_{s}(\mathrm{p}),d_{s}^{$\dagger$_{l}}(\mathrm{p}')\}=0,

where

_{\{X, Y\}=X\mathrm{Y}+YX}

.The canonical commutation relationsare

[\^{a}_{r}(k),a_{f}^{ $\dagger$}(\mathrm{W})]=$\delta$_{r,\text{〆}} $\delta$(\mathrm{k}-\mathrm{k}'),\cdot

[a_{r}(\mathrm{k}),a

〆

(1d)]=[a_{r}^{ $\dagger$}(\mathrm{k}),a_{r}^{ $\dagger$},(\mathrm{W})]=0,

where

_{[X, \mathrm{Y}]=X\mathrm{Y}-\mathrm{Y}X.}

Assume the

following

conditions:

(A.l

;Ultraviolet cutoffsforDirac

fleld)

\displaystyle \int_{\mathrm{R}^{3}}|$\chi$_{\mathrm{D}}(\mathrm{p})|^{2}d\mathrm{p}<\infty.

(A.2

:Ultraviolet cutoffs forradiation

_field)

\displaystyle \int_{\mathrm{R}^{3}}\frac{|$\chi$_{\mathrm{r}\mathrm{a}\mathrm{d}}(\mathrm{k})|^{2}}{ $\omega$(\mathrm{k})^{l}}d\mathrm{k}<\infty, l=1,2.

(A.3

:

Spatial cutoffs)

\displaystyle \int_{\mathrm{R}^{3}}|$\chi$_{\mathrm{I}}(\mathrm{x})|d\mathrm{x}<\infty,

_{\displaystyle \int_{\mathrm{R}^{3}\mathrm{x}\mathrm{R}^{3}}\frac{|$\chi$_{\mathrm{I}\mathrm{I}}(\mathrm{x})$\chi$_{1\mathrm{I}}(\mathrm{y})|}{|\mathrm{x}-\mathrm{y}|}}

dxdy

<\infty.

Then

_{H_{\mathrm{Q}\mathrm{E}\mathrm{D}}}

isa

_{self‐adjoin}

on

\mathcal{D}(H_{0})

. We areinterested in the existence of the

ground

stateof

H_{\mathrm{Q}\mathrm{E}\mathrm{D}}

. LetHbea

self‐adjoint

operatoron aHilbert space. We say that H hasa

ground

stateif

the bottom of the_spectrum ofHis

_eigenvalue,

_i.e.,

_{E_{0}(H)=\dot{\mathrm{m}}\mathrm{f} $\sigma$(H)\in$\sigma$_{\mathrm{p}}(H)}

. It is seenthat

H_{0}(H)=H_{\mathrm{D}}\otimes \mathrm{I}+\mathrm{I}\otimes H_{\mathrm{r}\mathrm{a}\mathrm{d}}

hasa

ground

state. Since themass of the

photon

iszero,

E_{0}(H_{0})

is

embedded inacontinuous_spectrum.

Dimassi‐Guillot

[6]

andBrouxbaroux‐Dimassi‐Giollot

[3]

considera

_QED

model with gener‐

alized

_{perturbations,}

and

_proved

the existence of the

_ground

statefor

_sufficiently

small values of

couplin

\mathrm{g} constants. In

[11],

the existenceofthe

ground

state

_{H_{\mathrm{Q}\mathrm{E}\mathrm{D}}}

was_{proven for}

sufficiently

small

valuesof

_coupling

constants. The main purpose in the paper

[13]

isto_{prove the existence of the}

ground

state

_{H_{\mathrm{Q}\mathrm{E}\mathrm{D}}}

for all values of

_coupling

constants.

(A.4:

Momentum

_{regularization}

ofDirac

_fleld)

\displaystyle \int_{\mathrm{R}^{3}}|\partial_{P^{v}}$\chi$_{\mathrm{D}}(\mathrm{p})|^{2}d\mathrm{p}<\infty, \int_{\mathrm{R}^{3}}|$\chi$_{\mathrm{D}}(\mathrm{p})\partial_{p^{ $\nu$}}u_{s}^{l}(\mathrm{p})|^{2}d\mathrm{p}<\infty, \int_{\mathrm{R}^{3}}|\dot{ $\chi$}_{\mathrm{D}}(\mathrm{p})\partial_{p^{v}}v_{s}^{l}(-\mathrm{p})|^{2}d\mathrm{p}<\infty.

(A.5:

Momentum

_{regularization}

of radiation

_field)

\displaystyle \int_{\mathrm{R}^{3}}\frac{|$\chi$_{\mathrm{m}\mathrm{d}}(\mathrm{k})|^{2}}{|\mathrm{k}|^{5}}d\mathrm{k}<\infty, \int_{\mathrm{R}^{3}}\frac{|\partial_{k^{v}}$\chi$_{\mathrm{r}\mathrm{a}\mathrm{d}}(\mathrm{k})|^{2}}{|\mathrm{k}|^{3}}d\mathrm{k}<\infty, \int_{\mathrm{R}^{3}}\frac{|$\chi$_{\mathrm{r}\mathrm{a}\mathrm{d}}(\mathrm{k})\partial_{k^{v}}e_{r}^{j}(\mathrm{k})|^{2}}{|\mathrm{k}|^{3}}d\mathrm{k}<\infty.

(A.6 :Spatial localization)

\displaystyle \int_{\mathrm{R}^{3}}|\mathrm{x}||$\chi$_{\mathrm{I}}(\mathrm{x})|d\mathrm{x}<\infty,

_{\displaystyle \int_{\mathrm{R}^{3}\times \mathrm{R}^{3}}\frac{|$\chi$_{\mathrm{I}\mathrm{I}}(\mathrm{x})$\chi$_{\mathrm{I}\mathrm{I}}(\mathrm{y})|}{|\mathrm{x}-\mathrm{y}|}|\mathrm{x}|}

dxdy

<\infty.

The main theorem in

_[13]

isasfollows:

Theorem

_{([13];Theorem}

_2.1)

Assume

_{(\mathrm{A}.1)-\langle \mathrm{A}.6).}

Then

_{H_{\mathrm{Q}\mathrm{E}\mathrm{D}}}

hasa

ground

state. In

_particular,

its

_multiplicity

is

finite.

[Remaining Problemsl

(i)

Self‐adjointness

without cutoffs

Itisseenthat undermomentumcutoffs

_(A.I),(A.2)

and

_spatial

cutoffs

_(A.3),

_{H_{\mathrm{Q}\mathrm{E}\mathrm{D}}}

is

_{self‐adjoint.}

Forthe Nelson

_model,

which describes the non‐relativistic

_{particles coupled}

toascalar

_field,

itwas

proven that

by

subtracting

momentum

_divergence

termsfrom the

_Hamiltonian,

thereisan

unique

self‐adjoint

Hamiltonian

_[10].

(ii)

Infrared

_divergence

The condition

\displaystyle \int_{\mathrm{R}^{3}}\frac{$\chi$_{ $\alpha$ \mathrm{d}}(\mathrm{k})|^{2}}{|\mathrm{k}|^{5}}d\mathrm{k}<\infty

in

_(A.5)

is_strongerthan the standard infrared

_regularity

condition.

Non‐relativistic

_QED

model

_{[2, 7]}

and

_spin‐boson

model

_[8]

have the

_ground

statewithout infrared

cutoffs. On the other

_hand,

the non‐existence of the

_ground

statefor massless Nelson‐model

(see

e.g.,[5])

and the Generalized

_spin‐boson

model

_[1]

were

_{investigated.}

(iii)

Multiplicity

Weseethat

_multiplicity

of the

_ground

stateof

_{H_{\mathrm{Q}\mathrm{E}\mathrm{D}}}

is finite for all values of

_coupling

constants. In

_[9],

the

_multiplicity

of the

_ground

statefor various_quantumfiled modelswas

_investigated

for

sufficiently

small values of

_coupling

constants.

(1v) Asymptotic Completeness

In

[11],

the existence of

_asymptotic

field forthe Dirac field and the radiation fieldswas_proven,

however its

_{asymptotic completeness}

hasnotbeen shown

_(see

_e.g.

_[4])

References

[1] A.Arai,M.Hirokawa andH.Hiroshima,Onthe absence ofeigenvectorsofHamiltonians inaclass of

massless_quantumfield models without infraredcutoff,J. Funct. Anal. 168

_(1999),

470‐497.

[2] V_Bach,J.Fröhlich and \mathrm{L} M.

_Sigal,

_{Spectral analysis}for_systemsofatomsand molecules

_coupled

to

the

_quantized

radiationfield,Comm. Math._Phys.207

₍₁₉₉₉₎

249‐290.

[3] J. ‐M._Barbaroux,M._Dimassi,and J._{‐C.Guillot, Quantum electrodynamics}ofrelativisticboundstates

with_{cutoffs, JHyper}

_Differ.

_Equa.1

₍₂₀₀₄₎

271‐314.

[4] J.Derezin’ski and C._{Gérard, Asymptotic completeness} in_quantumfield

_theory.

MassivePauli‐Fierz

Hamiltonian,Rev. Math.

_Phys.

11₍₁₉₉₉₎383‐450.

[5] J.Derezi’ski and C._{Gérard, Scattering}

_theory

ofinfrared

_divergent

Pauli‐FierzHamiltonians,Ann. H

Poincaré,5₍₂₀₀₄₎523‐577.

[6] M.Dimassi and J. ‐C.Guillot,The_{quantumelectrodynamics}ofrelativistic boundstateswithcutoffs.I,

Appl.Math. Lett. 16₍₂₀₀₃₎551‐555.

[7] M._Griesemer,E. Lieband M.Loss,Groundstatesin non‐relativistic_quantum

_{electrodynamics,}

Invent.

Math. 145_{(2001)557-595.}

[8] D.Haslerand I.Herbst,Groundstatesin the

_spin

boson_model,Ann. HenriPoincaré 12(2011)621‐

677.

[9] F.Hiroshima,

Multiplicity ofground

statesinquantumfieldmodels;

applications

of

asymptotic

fields,

J. Funct.Anal. 224(2005),431‐470.

[10] E._Nelson,Interaction ofnonrelativistic

_particles

witha

quantized

scalarfield,J.Math.Phys.5(1964),

1190‐1197

[11] T.Takaesu,On the_spectral

_analysis

of_quantum

_{electrodynamics}

with

_spatial

cutoffs._\mathrm{L}JMath.Phys.

50₍₂₀₀₉₎06230.

[12] T._Takaesu,Existence ofa

_groumd

statesofa model ofrelativisticquantum

electrodynamics

withcutoffs

for all values

_ofcoupling

constants,RIMS_Kokyuroku1975₍₂₀₁₅₎23‐30.

[13] T._Takaesu,Groundstateof_quantum

_{electrodynamics}

withcutoffs,

(arxiv:

1509.0613)