Infrared problem for the Nelson model on static space-times

0.33 MB·English

by  Christian Gérard

INFRARED PROBLEM
FOR THE NELSON MODEL ON STATIC SPACE-TIMES
C.GÉRARD,F.HIROSHIMA,A.PANATI,ANDA.SUZUKI
Abstract. We consider the Nelson model on some static space-times and
investigate the problem of existence of a ground state. Nelson models with
variablecoefficientsarisewhenonereplacesintheusualNelsonmodeltheflat
1
Minkowskimetricbyastatic metric, allowingalsothebosonmasstodepend
1
onposition. WeinvestigatetheexistenceofagroundstateoftheHamiltonian
0
in the presence of the infrared problem, i.e. assuming that the boson mass
2
m(x)tendsto0atspatialinfinity. Weshowthatifm(x)≥C|x|−1 atinfinity

n forsomeC>0thentheNelsonHamiltonianhasagroundstate.
a
J

3

] 1. Introduction
h
p The study of Quantum Field Theory on curved space-times has seen important
- developments since the seventies. Probably the mostspectacular prediction in this
h
domain is the Hawking effect [Ha, FH, Ba], predicting that a star collapsing to a
t
a
black hole asymptotically emits a thermal radiation. A related effect is the Unruh
m
effect [Un, Un-W, dB-M], where an accelerating observer in Minkowski space-time
[ sees the vacuum state as a thermal state.

3 Anotherimportantdevelopmentistheuseofmicrolocal analysis tostudyfreeor
v quasi-freestatesongloballyhyperbolicspace-times,whichstartedwiththeseminal
4 work by Radzikowski [Ra1, Ra2] , who proved that Hadamard states (the natural
0
substitutes forvacuumstatesoncurvedspace-times)canbe characterizedinterms
2
ofmicrolocalpropertiesoftheirtwo-pointfunctions. Theuseofmicrolocalanalysis
5
. in this domain was further developed for example in [BFK], [Sa].
4
Most of these works deal with free or quasi-free states, because of the well-
0
knowndifficulty to constructaninteracting,relativistic quantumfield theory,even
0
1 on Minkowski space-time.
: However in recent years a lot of effort was devoted to the rigorous study of
v
i interacting non-relativistic models on Minkowskispace-time, typically obtained by
X
coupling a relativistic quantum field to non-relativistic particles. The two main
r examplesarenon-relativistic QED,wherethe quantizedMaxwellfieldis minimally
a
coupled to a non-relativistic particle and the Nelson model, where a scalar bosonic
fieldislinearlycoupledtoanon-relativisticparticle. Forbothmodelsitisnecessary
to add an ultraviolet cutoff in the interaction term to rigorously construct the
associated Hamiltonian.
In both cases the models can be constructed on a Fock space with relatively
little efforts, and several properties of the quantum Hamiltonian H can be rigor-
ously studied. One of them, which will also be our main interest in this paper, is
the question of the existence of a ground state. Obviously the fact that H has a
ground state is an important physical property of the Nelson model. For example
a consequence of the existence of a ground state is that scattering states can quite
Date:November2010.
2010 Mathematics Subject Classification. 81T10,81T20,81Q10,58C40.
Keywordsandphrases. Quantumfieldtheory,Nelsonmodel,staticspace-times,groundstate.
1
2 C.GÉRARD,F.HIROSHIMA,A.PANATI,ANDA.SUZUKI
easily be constructed. These states describe the ground state of H with a finite
number of additional asymptotically free bosons.
When H has no ground state one usually speaks of the infrared problem or
infrared divergence. The infrared problem arises when the emission probability of
bosonsbecomesinfinitewithincreasingwavelength. Iftheinfraredproblemoccurs,
the scattering theory has to be modified: all scattering states contain an infinite
numberoflowenergy(soft)bosons(seeeg[DG3]). Amongmanypapersdevotedto
this question, let us mention [AHH, BFS, BHLMS, G, H, LMS, Sp] for the Nelson
model, and [GLL] for non-relativistic QED.
Our goalin this paper is to study the existence of a groundstate for the Nelson
model on a static space-time, allowing also for a position-dependent mass. This
model is obtained by linearly coupling the Lagrangians of a Klein-Gordon field
and of a non-relativistic particle on a static space-time (see Subsect. 2.2). We
believe that this model, although non-relativistic, is an interesting testing ground
for the generalizationof results for free or quasi-free models on curved space-times
to some interacting situations. Let us also mention that for the Nelson model on
Minkowskispace-timetheremovaloftheultravioletcutoffcanbedonebyrelatively
easy arguments. After removalof the ultraviolet cutoff, the Nelson model becomes
a local (although non-relativistic) QFT model. In a subsequent paper [GHPS3],
we will show that the ultraviolet cutoff can be removed for the Nelson model on a
static space-time.
Most of our discussion will be focused on the role of the variable mass term on
the ground state existence. Note that when one considers a massive Klein-Gordon
field in the Schwarzschild metric, the effective mass tends to 0 at the black hole
horizon(seeeg[Ba]). WebelievethatthestudyoftheNelsonmodelwithavariable
mass vanishing at spatial infinity will be a first step towards the extension of the
rigorous justification of the Hawking effect in [Ba] to some interacting models.
1.1. The Nelson model on Minkowski space-time. In this subsection we
quickly describe the usual Nelson model on Minkowski space-time. The Nelson
model describes a scalar bosonic field linearly coupled to a quantum mechanical
particle. It is formally defined by the Hamiltonian
1 1
H = p2+W(q)+ π2(x)+( ϕ(x))2+m2ϕ2(x)dx+ ϕ(x)ρ(x q)dx,
2 2 R3 ∇ R3 −
Z Z
where ρ denotes a cutoff function, p, q denote the position and momentum of the
particle,W(q)isanexternalpotentialandϕ(x),π(x)arethecanonicalfieldposition
and momentum.
The Nelson model arises from the quantization of the following coupled Klein-
Gordon and Newton system:
(2+m2)ϕ(t,x)= ρ(x q ),
t
(1.1) − −
( q¨t = qW(qt) ϕ(t,x) xρ(x qt)dx,
−∇ − ∇ −
were 2 denotes the d’Alembertian on thRe Minkowski space-time R1+3. The cutoff
function ρ plays the role of an ultraviolet cutoff and amounts to replacing the
quantum mechanical point particle by a charge density.
To distinguish the Nelson model on Minkowski space-time from its generaliza-
tions that will be described later in the introduction, we will call it the usual (or
constant coefficients) Nelson model.
For the usual Nelson model the situation is as follows: one assumes a stability
condition(seeSubsect. 4.5),implyingthatstateswithenergyclosetothebottomof
the spectrum are localized in the particle position. Then if the bosons are massive
INFRARED PROBLEM FOR THE NELSON MODEL ON STATIC SPACE-TIMES 3
i.e. if m > 0 H has a ground state (see eg [G]). On the contrary if m = 0 and
ρ(x)dx=0 then H has no ground state (see [DG3]).
6
R1.2. The Nelson model with variable coefficients. We now describe a gener-
alization of the usual Nelson model, obtained by replacing the free Laplacian ∆
x
−
by a generalsecond order differential operator and the constant mass term m by a
function m(x). We set:
h:= c(x)−1∂jajk(x)∂kc(x)−1+m2(x),
−
1 j,k d
≤X≤
foraRiemannianmetrica dxjdxk andtwofunctionsc(x),m(x)>0,andconsider
jk
the generalizationof (1.1):
∂2φ(t,x)+hφ(t,x)+ρ(x q )=0,
(1.2) t − t
( q¨t =−∇xW(qt)− R3φ(t,x)∇xρ(x−qt)|g|21d3x.
Quantizing the field equations (1.2)R, we obtain a Hamiltonian H acting on the
HilbertspaceL2(R3) Γ (L2(R3))(seeSect. 3),whichwecallaNelsonHamiltonian
s
⊗
with variable coefficients. Formally H is defined by the following expression:
(1.3)
1
H = p2+W(q)
2
1
+ π2(x)+ ∂ c(x) 1ϕ(x) ajk(x) ∂ c(x) 1ϕ(x) +m2(x)ϕ2(x)dx
j − k −
2 R3
Z jk
X (cid:0) (cid:1) (cid:0) (cid:1)
+ ϕ(x)ρ(x q)dx.
R3 −
Z
The main example of a variable coefficients Nelson model is obtained by replacing
intheusualNelsonmodeltheflatMinkowskimetriconR1+3 byastaticLorentzian
metric, and by allowing also the mass m to be position dependent. Recall that a
static metric on R1+3 is of the form
g (x)dxµdxν = λ(x)dtdt+λ(x) 1h (x)dxαdxβ,
µν − αβ
−
wherex=(t,x) R1+3,λ(x)>0isasmoothfunction,andh (x)isaRiemannian
α,β
metric on R3. W∈e show in Subsect. 2.3 that the natural Lagrangian for a point
particlecoupledtoascalarfieldon(R1+3,g)leads(afterachangeoffieldvariables)
to the system (1.2).
1.3. The infrared problem. Assuming reasonable hypotheses on the matrix
[ajk](x) and the functions c(x), m(x) it is easy to see that the formal expression
(1.3) can be rigorously defined as a bounded below selfadjoint operator H.
The question we address in this paper is the problem of existence of a ground
stateforH. VariablecoefficientsNelsonmodelsareexamplesofanabstractclassof
QFT Hamiltonians called abstract Pauli-Fierz Hamiltonians (see eg [G], [BD] and
Subsect. 4.1). If ω is the one-particle energy, the constant m := infσ(ω) can be
called the (rest) mass of the bosonic field, and abstract Pauli-Fierz Hamiltonians
fall naturally into two classes: massive models if m>0 and massless if m=0.
For massive models, H typically has a ground state, if we assume either that
the quantum particle is confined or a stability condition (see Subsect. 4.5). In this
paper we concentrate on the massless case and hence our typical assumption will
be that
lim m(x)=0.
x
→∞
4 C.GÉRARD,F.HIROSHIMA,A.PANATI,ANDA.SUZUKI
It follows that bosons of arbitrarily small energy may be present. The main result
ofthis paperisthatthe existenceornon-existenceofagroundstateforH depends
on the rate of decay of the function m(x). In fact we show in Thm. 4.1 that if
m(x) a x 1, for some a>0,
−
≥ h i
andifthequantumparticleisconfined,thenH hasagroundstate. Inasubsequent
paper [GHPS2], we will show that if
0 m(x) C x 1 ǫ, for some ǫ>0,
− −
≤ ≤ h i
thenH hasnogroundstate. ThereforeThm. 4.1issharpwithrespecttothedecay
rate of the mass at infinity.
(If h = ∆+λm2(x) for m(x) O( x 3/2) and the coupling constant λ is
−
− ∈ h i
sufficiently small the same result is shown in [GHPS1]).
1.4. Notation. We collect here some notation for the reader’s convenience.
If x Rd, we set x =(1+x2)12.
∈ h i
The domain of a linear operator A on some Hilbert space will be denoted by
H
DomA, and its spectrum by σ(A).
If h is a Hilbert space, the bosonic Fock space over h denoted by Γ (h) is
s
∞
Γ (h):= nh.
s ⊗s
n=0
M
We denote by a (h), a(h) for h h the creation/annihilation operators acting on
∗
∈
Γ (h). The (Segal) field operators φ(h) are defined as φ(h):= 1 (a (h)+a(h)).
s √2 ∗
If isanotherHilbertspaceandv B( , h),thenonedefinestheoperators
K ∈ K K⊗
a (v), a(v) as unbounded operators on Γ (h) by:
∗ s
K⊗
a∗(v)(cid:12)K⊗Nns h :=√n+1(cid:16)1K⊗Sn+1(cid:17)(cid:16)v⊗1Nns h(cid:17),
a(v):(cid:12)(cid:12)= a∗(v) ∗,
φ(v):=(cid:0)1 (a(v(cid:1))+a (v).
√2 ∗
They satisfy the estimates
(1.4) a♯(v)(N +1)−12 v ,
k k≤k k
where v is the norm of v in B( , h).
k k K K⊗
If b is a selfadjoint operator on h its second quantization dΓ(b) is defined as:
n
dΓ(b) := b .
(cid:12)Nns h Xj=11⊗j··1·⊗1⊗ ⊗1⊗n···j⊗1
(cid:12) − −
(cid:12)
2. The Nelson mo|del{zon s}tatic|spac{ez-tim}es
In this section we discuss the Nelson model on static space-times, which is the
main example of Hamiltonians that will be studied in the rest of the paper. It is
convenient to start with the Lagrangianframework.
2.1. Klein-Gordon equation on static space-times. Let g (x) be a Lorentz-
µν
ian metric of signature ( ,+,+,+) on R1+3. Set g = det[g ], [gµν] = [g ] 1.
µν µν −
− | |
Consider the Lagrangian
1 1
L (φ)(x)= ∂ φ(x)gµν(x)∂ φ(x)+ m2(x)φ2(x),
free µ ν
2 2
for a function m:R4 R+ and the associated action:
→
Sfield(φ)= Lfree(φ)(x)g 12(x)d4x,
R4 | |
Z
INFRARED PROBLEM FOR THE NELSON MODEL ON STATIC SPACE-TIMES 5
whereφ:R4 R. TheEuler-LagrangeequationsyieldtheKlein-Gordon equation:
→
2 φ+m2(x)φ=0,
g
for
2g = g −12∂µ g 21gµν∂ν.
−| | | |
Usually one has
1 1
m2(x)= (m2+θR(x)),
2 2
where m 0 is the mass and R(x) is the scalar curvature of the metric g ,
µν
≥
(assuming of course that the function on the right is positive). In particular if
m=0 and θ = 1 one obtains the so-called conformal wave equation.
6
We set x=(t,x) R1+3. The metric g is static if:
µν
∈
gµν(x)dxµdxν = λ(x)dtdt+λ(x)−1hαβ(x)dxαdxβ,
−
where λ(x) > 0 is a smooth function and h is a Riemannian metric on R3. We
αβ
assume also that m2(x)=m2(x) is independent on t.
Setting φ(t,x)=λh 1/4φ˜(t,x), we obtain that φ˜(t,x) satisfies the equation:
−
| |
∂t2φ˜−λ|h|−1/4∂α|h|21hαβ∂β|h|−1/4λφ˜+m2λφ˜=0.
Wenotethat h−1/4∂α h 21hαβ∂β h−1/4is(formally)self-adjointonL2(R3,dx)and
| | | | | |
istheLaplace-Beltramioperator∆ associatedtotheRiemannianmetrich (after
h αβ
the usual density change u h1/4u to work on the Hilbert space L2(R3,dx)).
7→| |
2.2. Klein-Gordonfieldcoupledto a non-relativisticparticle. Wenowcou-
ple the Klein-Gordon field to a non-relativistic particle. We fix a mass M > 0,
a charge density ρ : R3 R+ with q = ρ(y)d3y = 0 and a real potential
→ R3 6
W :R3 R. The action for the coupled system is
→ R
S =S +S +S ,
part field int
for
S = M x˙(t)2 W(x(t))dt,
part R 2 | | −
Sint = RR4φ(t,x)ρ(x−x(t))|g|21(x)d4x.
The Euler-Lagrangeequations are:
R
2 φ(t,x)+m2(t,x)φ(t,x)+ρ(x x(t))=0,
g
−
( Mx¨(t)=−∇xW(x(t))− R3φ(t,x)∇xρ(x−x(t))|g|12d3x.
Doing the same change of field variableRs as in Subsect. 2.1 and deleting the tildes,
we obtain the system:
∂2φ λ∆ λφ+m2λφ+ρ(x x(t))=0,
(2.1) t − h −
( Mx¨(t)=−∇W(x(t))− R3φ(t,x)∇ρ(x−x(t))d3x.
2.3. The Nelson model on a staticRspace-time. If the metric is static, the
equations(2.1)areclearlyHamiltonianequationsfortheclassicalHamiltonianH =
H +H +H , where:
part field int
1
H (x,ξ)= ξ2+W(x),
part
2M
H (ϕ,π)
field
= 1 π2(x) ϕ(x)λ(x)∆ λ(x)ϕ(x)+m2(x)λ(x)ϕ2(x)dx,
2 R3 − h
R H (x,ξ,ϕ,π)= ρ(y x)ϕ(y)dy.
int
R3 −
Z
6 C.GÉRARD,F.HIROSHIMA,A.PANATI,ANDA.SUZUKI
TheclassicalphasespaceisasusualR3 R3 L2(R3) L2(R3),withthesymplectic
R R
× × ×
form
(x,ξ,ϕ,π)ω(x,ξ ,ϕ,π )=x ξ x ξ+ ϕ(x)π (x) π(x)ϕ(x)dx.
′ ′ ′ ′ ′ ′ ′ ′
· − · R3 −
Z
The usual quantization scheme leads to the Hilbert space:
L2(R3,dy) Γ (L2(R3,dx)),
s
⊗
where Γ (h) is the bosonic Fock space over the one-particle space h, and to the
s
quantum Hamiltonian:
1 1 1 1
H =( ∆y+W(y)) + dΓ(ω)+ a∗(ω−2ρ( y)+a(ω−2ρ( y) ,
−2 ⊗1 1⊗ √2 ·− ·−
(cid:16) (cid:17)
where
ω =( λ∆hλ+m2λ)21,
−
dΓ(ω) is the usual second quantization of ω and a (f), a(f) are the creation/an-
∗
nihilation operators on Γ (L2(R3,dx)).
s
3. The Nelson Hamiltonian with variable coefficients
In this section we define the Nelson model with variable coefficients that will be
studied in the restofthe paper. We willdeviate slightly fromthe notationinSect.
2 by denoting by x R3 (resp. X R3) the boson (resp. electron) position. As
∈ ∈
usual we set D =i 1 , D =i 1 .
x − x X − X
∇ ∇
3.1. Electron Hamiltonian. We define the electron Hamiltonian as:
K :=K +W(X),
0
where
K = D Ajk(X)D ,
0 Xj Xk
1 j,k 3
≤X≤
acting on :=L2(R3,dX), where:
K
(E1) C [Ajk(X)] C , C >0.
0 1 0
1≤ ≤ 1
WeassumethatW(X)isarealpotentialsuchthatK +W isessentiallyselfadjoint
0
andboundedbelow. We denote byK the closureofK +W. Laterwewill assume
0
the following confinement condition :
(E2) W(X) C X 2δ C , for some δ >0.
0 1
≥ h i −
Physically this condition means that the electron is confined. As is well known
(see eg [GLL]) for the question of existence of a ground state , this condition can
be replaced by a stability condition, meaning that states near the bottom of the
spectrum of the Hamiltonian are confined in the electronic variables by energy
conservation.
We will discuss the extension of our results when one assume the stability con-
dition in Subsect. 4.5.
INFRARED PROBLEM FOR THE NELSON MODEL ON STATIC SPACE-TIMES 7
3.2. Field Hamiltonian. Let:
h := c(x) 1∂ ajk(x)∂ c(x) 1,
0 − 1 j,k d − j k −
≤ ≤
h:= hP0+m2(x),
with ajk, c, m are real functions and:
C [ajk(x)] C , C c(x) C , C >0,
0 1 0 1 0
1≤ ≤ 1 ≤ ≤
(B1) ∂αajk(x) O( x 1), α 1, ∂αc(x) O(1), α 2,
x ∈ h i− | |≤ x ∈ | |≤
∂αm(x) O(1), α 1.
x ∈ | |≤
Clearly h is selfadjoint on H2(R3) and h 0. The one-particle space and one-
≥
particle energy are:
h:=L2(R3,dx), ω :=h12.
The constant:
infσ(ω)=:m 0,
≥
can be viewed as the mass of the scalar bosons.
The following lemma is easy;
Lemma 3.1. (1) One has Kerω = 0 ,
{ }
(2) Assume in addition to (B1) that lim m(x)=0. Then infσ(ω)=0.
x
→∞
Proof. It follows from (B1) that
(uhu) C (c 1u ∆c 1u)+(c 1uc 1m2u), u H2(R3).
1 − − − −
| ≤ |− | ∈
Therefore if hu = 0 u is constant. It follows also from (B1) that c(x) 1 preserves
−
H2(R3). Therefore by the variational principle
m2 =infσ(h) C infσ( ∆+c 2(x)m2(x))=0.
1 −
≤ −
This proves (2). 2
The NelsonHamiltonian defined below will be called massive (resp. massless) if
m>0 (resp. m=0.) The field Hamiltonian is
dΓ(ω),
acting on the bosonic Fock space Γ (h).
s
3.3. Nelson Hamiltonian. Let ρ S(R3), with ρ 0, q = ρ(y)dy = 0. We
∈ ≥ R3 6
set:
R
ρ (x)=ρ(x X)
X
−
and define the UV cutoff fields as:
1
(3.1) ϕρ(X):=φ(ω−2ρX),
where for f h, φ(f) is the Segal field operator:
∈
1
φ(f):= (a (f)+a(f)).
∗
√2
Note that setting
1
ϕ(X):=φ(ω−2δX),
one has ϕ (X)= ϕ(X Y)ρ(Y)dY.
ρ
−
Remark 3.2. OnRe can think of another definition of UV cutoff fields, namely:
1
ϕ˜χ(X):=φ(ω−2χ(ω)δX),
for χ S(R), χ(0) = 1. In the constant coefficients case where h = ∆ both
∈ −
definitions are equivalent. In the variable coefficients case the natural definition
(3.1) is much more convenient.
8 C.GÉRARD,F.HIROSHIMA,A.PANATI,ANDA.SUZUKI
The Nelson Hamiltonian is:
(3.2) H :=K + dΓ(ω)+ϕ (X),
ρ
⊗1 1⊗
acting on
= Γ (h).
s
H K⊗
Set also:
H :=K + dΓ(ω),
0
⊗1 1⊗
which is selfadjoint on its natural domain. The following lemma is standard.
Lemma 3.3. Assume hypotheses (E1), (B1). Then H is selfadjoint and bounded
below on D(H ).
0
Proof. it suffices to apply results on abstract Pauli-Fierz Hamiltonians (see eg
[GGM, Sect.4]). H is an abstract Pauli-Fierz Hamiltonian with coupling operator
v B( , h) equal to:
∈ K K⊗
L2(R3,dX) u ω−12ρ(x X)u(X) L2(R3,dX) L2(R3,dx)
∋ 7→ − ∈ ⊗
1
Applying [GGM, Corr. 4.4], it suffices to check that ω−2v B( , h). Now
∈ K K⊗
ω−12v B( , h) =( sup ω−1ρX 2)21
k k KK⊗ X R3k k
∈
Usingthath CD2 andtheKato-Heinzinequality,weobtainthatω 2 C D 2,
≥ x − ≤ | x|−
hence it suffices to check that the map
L2(R3,dX) u D 1ρ(x X)u(X) L2(R3,dX) L2(R3,dx)
x −
∋ 7→| | − ∈ ⊗
is bounded, which is well known. 2
4. Existence of a ground state
Inthissectionwewillproveourmainresultabouttheexistenceofagroundstate
for variable coefficients Nelson Hamiltonians. This result will be deduced from an
abstract existence result extending the one in [BD], whose proof is outlined in
Subsects. 4.1, 4.2 and 4.3.
Theorem 4.1. Assume hypotheses (E1), (B1). Assume in addition that:
m(x) a x 1, for some a>0,
−
≥ h i
and (E2) for some δ > 3. Then infσ(H) is an eigenvalue.
2
Remark 4.2. The condition δ > 3 in Thm. 4.1 comes from the operator bound
2
ω 3 C x 3+ǫ, ǫ>0 proved in Thm. A.8.
−
≤ h i ∀
Remark 4.3. From Lemma 3.1 we know that infσ(ω) = 0 if lim m(x) = 0.
x
→∞
Therefore theNelsonHamiltonian canbemassless usingtheterminologyofSubsect.
3.2.
Remark 4.4. In a subsequent paper [GHPS2] we will show that if
0 m(x) C x 1 ǫ, for some ǫ>0,
− −
≤ ≤ h i
then H has no ground state. Therefore the result of Thm. 4.1 is sharp with respect
to the decay rate of the mass at infinity.
INFRARED PROBLEM FOR THE NELSON MODEL ON STATIC SPACE-TIMES 9
4.1. Abstract Pauli-Fierz Hamiltonians. In [BD], Bruneau and Dereziński
study the spectral theory of abstract Pauli-Fierz Hamiltonians of the form
H =K + dΓ(ω)+φ(v),
⊗1 1⊗
acting on the Hilbert space = Γ (h), where is the Hilbert space for the
s
H K⊗ K
smallsystemandhtheone-particlespaceforthebosonicfield. TheHamiltonianH
iscalledmassive(resp. massless)ifinfσ(ω)>0(resp. infσ(ω)=0). Amongother
results they prove the existence of a ground state for H if v is infrared regular.
Althoughmostoftheirhypothesesarenaturalandessentiallyoptimal,wecannot
directly applytheir abstractresults to oursituation. Infactthey assume(see [BD,
AssumptionE])thattheone-particlespacehequalsL2(Rd,dk)andtheone-particle
energy ω is the multiplication operator by a function ω(k) which is positive, with
ωbounded,andlim ω(k)=+ . Thisassumptionontheone-particleenergy
k
∇ →∞ ∞
is only needed to prove an HVZ theorem for massive (or massless with an infrared
cutoff) Pauli-Fierz Hamiltonians.
Inourcasethisassumptioncouldbededuced(modulounitaryequivalence)from
the spectral theory of h. For example it would suffices to know that h is unitarily
equivalent to ∆. This last property would follow from the absence of eigenvalues
−
for h and from the scattering theory for the pair (h, ∆) and require additional
−
decay properties of the [aij](x), m(x) and of some of their derivatives.
We willreplaceit by moregeometricassumptionsonω (see hypothesis (4.4) be-
low),similartothoseintroducein[GP],whereabstractbosonicQFTHamiltonians
wereconsidered. Sincewedonotaimforgenerality,ourhypothesesonthecoupling
operator v are stronger than necessary, but lead to simpler proofs. Also most of
the proofs will be only sketched.
Let h, two Hilbert spaces and set = Γ (h).
s
K H K⊗
We fix selfadjoint operators K 0 on and ω 0 on h. We set
≥ K ≥
infσ(ω)=:m 0.
≥
Ifm=0onehastoassumeadditionallythatKerω = 0 (seeRemark4.5forsome
{ }
explanation of this fact).
Remark 4.5. It is a real Hilbert space and ω is a selfadjoint operator on , the
X X
condition Kerω = 0 is well known to be necessary to have a stable quantization
of the abstract Klei{n-}Gordon equation ∂2φ(t)+ω2φ(t)=0 where φ(t):R .
t →X
If Kerω = 0 the phase space = for the Klein-Gordon equation
6 { } Y X ⊕ X
splits into the symplectic direct sum , for = Kerω Kerω ,
reg sing reg ⊥ ⊥
Y ⊕ Y Y ⊕
= Kerω Kerω, both symplectic spaces being invariant under the symplectic
sing
Y ⊕
evolution associated to the Klein-Gordon equation. On one can perform the
reg
Y
stable quantization. On ,if for example Kerω is d dimensional, the quantiza-
sing
tion leads to the HamiltoYnian ∆ on L2(Rd). Clearly a−ny perturbation of the form
−
φ(f) for (ω)f =0 will make the Hamiltonian unbounded from below.
0
1{ } 6
So we will always assume that
(4.1) ω 0, Kerω = 0 .
≥ { }
Let H = + dΓ(ω). We fix also a coupling operator v such that:
0
K⊗1 1⊗
(4.2) v B( , h).
∈ K K⊗
1
Thequadraticformφ(v)=a(v)+a∗(v)iswelldefinedforexampleon DomN2.
K⊗
We will also assume that:
1 1
(4.3) ω−2v(K+1)−2 is compact.
10 C.GÉRARD,F.HIROSHIMA,A.PANATI,ANDA.SUZUKI
Proposition 4.6 ([BD] Thm. 2.2). Assume (4.1), (4.3). Then H = H +φ(v)
0
is well defined as a form sum and yields a bounded below selfadjoint operator with
1 1
DomH 2 =DomH0 2.
| | | |
The operator H defined as above is called an abstract Pauli-Fierz Hamiltonian.
4.2. Existence of a ground state for cutoff Hamiltonians. We introduce as
in [BD] the infrared-cutoff objects
v =F(ω σ)v, H =K + dΓ(ω)+φ(v ), σ >0,
σ σ σ
≥ ⊗1 1⊗
whereF(λ σ)denotesasusualafunctionoftheformχ(σ 1λ),whereχ C (R),
− ∞
≥ ∈
χ(λ) 0 for λ 1, χ(λ) 1 for λ 2.
≡ ≤ ≡ ≥
An important step to prove that H has a ground state is to prove that H has
σ
a ground state. The usual trick is to consider
H˜ =K + dΓ(ω )+φ(v ),
σ σ σ
⊗1 1⊗
where:
ω :=F(ω σ)σ+(1 F(ω σ))ω =ω+(σ ω)F(ω σ).
σ
≤ − ≤ − ≤
Note that since ω σ > 0, H˜ is a massive Pauli-Fierz Hamiltonian. More-
σ σ
over it is well known≥(see eg [G], [BD]) H has a ground state iff H˜ does. The
σ σ
fact that H˜ has a ground state follows from an estimate on its essential spectrum
σ
(HVZ theorem). In[BD]thisis shownusingthe conditionthath=L2(Rd,dk)and
ω = ω(k). Here we will replace this condition by the following more abstract con-
dition, formulatedusing an additionalselfadjointoperatorr onh. Similar abstract
conditions were introduced in [GP].
We will assume that there exists an selfadjoint operator r 1 on h such that
≥
the following conditions hold for all σ >0:
(i) (z r) 1 : Domω Domω , z C R,
− σ σ
− → ∀ ∈ \
(4.4) (ii) [r,ω ] defined as a quadratic form on Domr Domω is bounded,
σ
∩
(iii) r ǫ(ω +1) ǫ is compact on h for some 0<ǫ< 1.
− σ − 2
The operator r, called a gauge, is used to localize particles in h.
We assume also as in [BD]:
1
(4.5) (K+1)−2 is compact.
This assumption means that the small system is confined.
Proposition 4.7. Assume (4.1), (4.2), (4.3), (4.4),(4.5) . Then
σ (H˜ ) [infσ(H˜ )+σ,+ [.
ess σ σ
⊂ ∞
It follows that H˜ (and hence H ) has a ground state for all σ >0.
σ σ
Proof. By (4.3), φ(v ) is form bounded with respect to H (and to K +
σ 0
dΓ(ω )) with the infinitesimal bound, hence H , H˜ are well defined as⊗b1oun1de⊗d
σ σ σ
below selfadjoint Hamiltonians.
We can follow the proof of [DG2, Thm. 4.1] or [GP, Thm. 7.1] for its abstract
version. For ease of notation we denote simply H˜ by H, ω by ω and v by v.
σ σ σ
The key estimate is the fact that for χ C (R) one has
∈ 0∞
(4.6) χ(Hext)I∗(jR) I∗(jR)χ(H) o(1), when R .
− ∈ →∞