An inverse scattering problem for the Klein-Gordon equation with a classical source in quantum field theory

0.26 MB·English

by  Hironobu Sasaki

An inverse scattering problem for
the Klein-Gordon equation with
a classical source in quantum field theory
1
1
Hironobu Sasaki
0 ∗
2
and

n
a Akito Suzuki †
J

1
January 4, 2011

]
h
p
-
h Key words Quantum field theory, scattering theory, inverse scattering
t problem, external field problem
a
m MSC(2000) 81T10 81U40 35R30
[ Abstract

1 Aninversescatteringproblemforaquantizedscalarfieldφobeying
v
a linear Klein-Gordon equation
0
1 ((cid:3)+m2+V)φ = J in R R3
3 ×
0
is considered, where V is a repulsive external potential and J an ex-
.
1 ternal source J. We prove that the scattering operator S = S(V,J)
0
associated with φ uniquely determines V. Assuming that J is of the
1
1 form J(t,x) = j(t)ρ(x), (t,x) R R3, we represent ρ (resp. j) in
v: terms of j (resp. ρ) and S. ∈ ×
i
X
r 1 Introduction
a
We consider an inverse scattering problem for a quantized scalar field
φ interacting with an external potential V and an external source J
(see, e.g., [5, 11]) which obeys the Klein-Gordon equation
((cid:3)+m2+V(x))φ(t,x) = J(t,x) in R R3. (1.1)
×
∗Department of Mathematics and Informatics, Chiba 263-8522, Japan, E-mail:
[email protected]
†Department of Engineering, Shinshu University, Nagano, 380-8553, Japan E-mail:
[email protected]
1
Here (cid:3) = ∂2 ∆, ∆ is the Laplacian in R3, m > 0 and J, V are
∂t2 −
real functions. φ(t,x) and its conjugate field π(t,x) = ∂ φ(t,x) are
∂t
operator valued distributions (see, e.g., [18]). A typical example of
(1.1) is the nucleon-pion interaction, that is, φ describes the pion field
and J the distribution function of the nucleons (see, e.g., [5]).
Undersuitable conditions, one can show that the asymptotic fields
φ (t,x) = s- lim φ (t,x), (1.2)
out/in s
s
→±∞
π (t,x) = s- lim π (t,x) (1.3)
out/in s
s
→±∞
exist. Here π (t,x) = ∂ φ (t,x) and φ (t,x) is the solution of thefree
s ∂t s s
Klein-Gordon equation with the initial condition: φ (s,x) = φ(s,x)
s
and π (s,x) =π(s,x). Suppose that φ (x) = φ (0,x) and π (x) =
s in in in
π (0,x) give the Fock representation of the canonical commutation
in
relations (CCR):
[φ (x),π (y)] = iδ(x y), (1.4)
in in
−
[φ (x),φ (y)] = [π (x),π (y)] = 0. (1.5)
in in in in
See Section 3 for the detail. The scattering operator S = S(V,J) is
defined by the following relations (up to a constant factor):
S 1φ (x)S = φ (x), S 1π (x)S =π (x). (1.6)
− in out − in out
We prove that S uniquely determines V. Suppose that J(t,x) =
j(t)ρ(x). Then we show that ρ (resp. j) is uniquely determined by S
and j (resp. ρ).
To state our results precisely, we introduce several assumptions.
We set h= L2(R3;dx) and assume the following:
Assumption 1.1. The potential function V :R3 R is non negative
→
and satisfies V H2(R3).
∈
Then the multiplication operator V acting in h is infinitesimally
small with respect to h = ∆ since V L2(R3). Hence the operator
0
− ∈
h = h +V
0
is self-adjoint with the domain D(h) = D(h ) = H2(R3). Since V
0
is relative compact with respect to h , i.e., V(h +1) 1 is compact,
0 0 −
and V is positive, the spectrum of h is σ(h) = σ (h) = [0, ). The
ess
∞
condition V H2(R3) allows us to construct the solution of (1.1) by
∈
a Bogoliubov transformation (see Lemma 2.2 and Proposition 2.4).
We set
ω = ϕ(h), ω = ϕ(h ),
0 0
where ϕ(s) = √s+m2.
2
Assumption 1.2. We assume that h has no positive eigenvalue and
∞dR V(x)( ∆+1) 1F(x R) < , (1.7)
−
k − | | ≥ k ∞
Z0
where
1 if x R,
F(x R)= | | ≥
| | ≥ (0 if x < R.
| |
We make some comments on Assumption 1.2:
By (1.7), the following limits exist
•
w := s lim eithe ith0
−
± −t
→±∞
and the intertwining property hw = w h holds. By Enss and
0
± ±
Weder [3], we see that the scattering map for the Schro¨dinger
operator defined by
V S(V) = w w
VSR ∋ 7→ +∗ −
is injective, where
= V :R3 R V is Kato-small in h and satisfies (1.7) .
SR
V { → | }
Sincehhasnopositiveeigenvalue, (1.7)impliesthathhaspurely
•
absolutely continuous spectrum. In particular, we have w w =
∗
w w = I on h. ± ±
∗
± ±
We see that the following limits exist:
•
s- lim eitωe itω0. (1.8)
−
t
→±∞
For a proof of the existence, see Appendix A.2. By [19, Theo-
rem 1], above limits (1.8) are equal to w , respectively, i.e., the
±
invariance principal holds for ϕ.
Assumption 1.3. The function J : R R3 R satisfies
× →
(a) For each t R, the function J (x) := J(t,x) satisfies J
t t
∈ ∈
H 1/2(R3).
−
(b) The vector valued function R t e itωω 1/2J h satisfies
− − t
∋ 7→ ∈
∞ dt ω 1/2J < .
− t h
k k ∞
Z−∞
We say that V if V satisfies Assumptions 1.1 and 1.2 and that
∈ V
J if J satisfies Assumption 1.3.
∈ J
3
Theorem 1.1. Suppose that V,V and J,J . If S(V,J) =
′ ′
∈ V ∈ J
S(V ,J ), then:
′ ′
(i) S(V) = S(V ),
′
(ii) V = V ,
′
(iii) +∞dse−isωJt = +∞dse−isωJt′.
−∞ −∞
ByRthe above theoreRm, we immediately see the following:
Corollary 1.2. Let J be given. Then the map V S(V,J)
∈ J V ∋ 7→
is injective.
Proof of Theorem 1.1. (i) will be proved in Theorem 4.1. Since
V ⊂
, (ii) follows from the injectivity of the map V S(V). We will
SR
V 7→
prove (iii) in Proposition 4.5.
In order to recover the external source J, we henceforth suppose
that J is expressed by
∈J
J(t,x) = j(t) ρ(x), (1.9)
×
where j L1(R) and ρ H 1/2(R3). Let
−
∈ ∈
F(t,f) = (Ω ,φ (t,f)Ω ), f (Rd),
in out in
∈ S
where Ω is the Fock vacuum. From (1.6), we see that F(t,f) is
in
uniquely determined by S. One can show the following:
(1) For a given function j such that the Fourier transform ˆj of j
is real analytic, we represent ρ in terms of j and F(t,f). See
Theorem 4.6 for the detail.
(2) Let ρ be a given function and assume that j satisfies the follow-
ing:
For some δ > 0, eδtj(t) L1(R ). (1.10)
| | t
∈
Then we express j by means of ρ and F(t,f). See Theorem 4.8
for the detail.
From the reconstruction formulas above, we observe that the scat-
tering operator S(V,j ρ) uniquely determines j and ρ:
×
Theorem 1.3. Let V and j ρ, j ρ . Suppose that
′ ′
∈ V × × ∈ J
j,j L1(R) and ρ,ρ H 1/2(R). Then:
′ ′ −
∈ ∈
(i) Assume that that ˆj is real analytic. Then ρ= ρ if S(V,j ρ)=
′
×
S(V,j ρ).
′
×
4
(ii) Assume that (1.10)holds. Then j = j if S(V,j ρ) = S(V,j
′ ′
× ×
ρ).
Proof. See Theorem4.6. ρis uniquely determinedby j anda function
z defined in (4.6). z is uniquely determined by F(t,f). Since, as
was noted above, S determines F(t,f) uniquely, we see that ρ
7→
S(V,j ρ) is injective. Hence (i) holds. (ii) is proved similarly.
×
This paper is organized as follows. Section 2 is devoted to some
mathematical preliminaries. In Subsections 2.1 and 2.2, we review
well-known facts. The quantized Klein-Gordon field is constructed in
Subsection 2.3 and the wave operator in Subsection 2.4.. In Section
3, we discuss the scattering theory and define the scattering operator
S. Section 4 deals with theinverse scattering problem. In Subsection
4.1, we show the uniqueness of V. The reconstruction formulas of ρ
and j are given in Subsections 4.2 and 4.3, respectively. In Appendix,
we prove Lemmas 2.2 and 2.6.
2 Preliminary
In general we denote the inner product and the associated norm of a
Hilbert space by ( , ) and , respectively. The inner product
L ∗ · L k·kL
is linear in and antilinear in . If there is no danger of confusion, we
· ∗
omit the subscript in (, ) and . For a linear operator T on
L · · L k·kL
, we denote the domain of T by D(T) and, if D(T) is dense in ,
L H
the adjoint of T by T .
∗
2.1 Boson Fock space
We first recall the abstract Boson Fock space and operators therein.
The Boson Fock space over a Hilbert space h is defined by
n
∞
Γ(h) = h
n=0 s
MO
n
∞ 2
= Ψ = Ψ(n) Ψ(n) h, Ψ(n) < ,
( { }∞n=0(cid:12)(cid:12) ∈ Os Xn=0(cid:13) (cid:13)⊗nsh ∞)
(cid:12) (cid:13) (cid:13)
where nh denotes the sy(cid:12)mmetric tensor pro(cid:13)duct (cid:13)of h with the con-
⊗s (cid:12)
vention 0h = C.
⊗s
The creation operator c (f) (f h) acting in Γ(h) is defined by
∗
∈
(c (f)Ψ)(n) = √nS f Ψ(n 1)
∗ n −
⊗
(cid:16) (cid:17)
5
with the domain
∞ 2
D(c (f)) = Ψ = Ψ(n) n S f Ψ(n 1) < ,
∗ ( { }∞n=0(cid:12)(cid:12)nX=0 (cid:13) n(cid:16) ⊗ − (cid:17)(cid:13)⊗nsh ∞)
(cid:12) (cid:13) (cid:13)
whereS denotesthesymmetri(cid:12)zation(cid:13)operatoron nhs(cid:13)atisfyingS =
n (cid:12) ⊗ n
S = S2 and S ( nh) = nh.
n∗ n n ⊗ ⊗s
The annihilation operator c(f) (f h) is defined by the adjoint
∈
of c (f), i.e., c(f) := c (f) . By definition, c (f) (resp. c(f)) is
∗ ∗ ∗ ∗
linear (resp. antilinear) in f h. As is well known, the creation and
∈
annihilation operators leave the finite particle subspace
∞
D = Ψ = Ψ(n) Ψ(n) = 0, n m
f { }∞n=0 | ≥
m[=1n o
invariant. The canonical commutation relations (CCR) hold on D :
f
[c(f),c (g)] = (f,g) , [c(f),c(g)] = [c (f),c (g)] = 0. (2.1)
∗ h ∗ ∗
It follows from (2.1) that
c (f)Ψ 2 = f 2 Ψ 2+ c(f)Ψ 2, Ψ D . (2.2)
∗ f
k k k k k k k k ∈
The Segal field operator τ(f)= 1 (c(f)+c (f)) (f h) is essentially
√2 ∗ ∈
self-adjoint on D . We denote its closure by the same symbol. By
f
(2.1), the following equation holds
1
c (f)Ψ 2 = ( τ(f)Ψ 2+ τ(if)Ψ 2+ f 2 Ψ 2), Ψ D . (2.3)
∗ f
k k 2 k k k k k k k k ∈
Since D is a core for c(f), c (f) and τ(f) (f h), we observe from
f ∗
∈
(2.2) and (2.3) that
D(τ(f)) D(τ(if)) = D(c(f)) = D(c (f)). (2.4)
∗
∩
Hence the following operator equalities hold true:
1
c(f) = (τ(f)+iτ(if)),
√2
1
c (f)= (τ(f) iτ(if)).
∗
√2 −
Let
D˜ := D(c(f)).
f h
\∈
Since D˜ D , D˜ is dense in Γ(h). From (2.4), we observe that
f
⊃
D˜ = D(c (f))= D(τ(f)). (2.5)
∗
f h f h
\∈ \∈
6
The Fock vacuum Ω = Ω(n) Γ(h) is defined by Ω(0) = 1 and
{ }∞n=0 ∈
Ω(n) = 0 (n 1), which satisfies
≥
c(f)Ω =0, f h. (2.6)
∈
Ω is a unique vector satisfying (2.6) up to a constant factor.
Let A be a contraction operator on h, i.e., A 1. We define a
k k ≤
contraction operator Γ(A) on Γ(h) by
(Γ(A)Ψ)(n) = ( nA)Ψ(n), Ψ = Ψ(n)
⊗ { }∞n=0
with the convention 0A = 1. If U is unitary, i.e. U 1 = U , then
− ∗
⊗
Γ(U) is also unitary and satisfies Γ(U) = Γ(U ) and
∗ ∗
Γ(U)c(f)Γ(U) = c(Uf), Γ(U)c (f)Γ(U) = c (Uf).
∗ ∗ ∗ ∗
For a self-adjoint operator T on h, i.e., T = T∗, Γ(eitT) t R is a
{ }∈
strongly continuous one-parameter unitary group on Γ(h). Then, by
the Stone theorem, there exists a unique self-adjoint operator dΓ(T)
such that
Γ(eitT)= eitdΓ(T).
The number operator N is defined by dΓ(1).
f
2.2 Bogoliubov transformations
Let be the direct sum of two Hilbert spaces and , where
+
H H H−
= h and is a copy of it:
+
H H−
v
= = v = + v ,v h .
H H+⊕H− ( v (cid:12) + − ∈ )
(cid:20) −(cid:21)(cid:12)
(cid:12)
(cid:12)
We denote by P (resp. P ) the projection from onto (resp.
+ (cid:12) +
− H H
):
H−
v v v 0
P + = + , P + = .
+ v 0 − v v
(cid:20) −(cid:21) (cid:20) (cid:21) (cid:20) −(cid:21) (cid:20) −(cid:21)
v
A vector v RanP = is identified with one in h: + =v h.
∈ + H+ 0 + ∈
(cid:20) (cid:21)
We define an involution Q on by
H
Q = P P
+
− −
and a conjugation C on by
H
v v¯
C + = − ,
v v¯
+
(cid:20) −(cid:21) (cid:20) (cid:21)
7
wheref¯standsforthecomplexconjugationoff inh,i.e.,f¯(x) = f(x),
a.e. x R3.
∈
A bounded operator A on is written as
H
A A
A= ++ +− ,
A A
+
(cid:20) − −−(cid:21)
where Aǫǫ′ = PǫAPǫ′ (ǫ,ǫ′ = + or ). Then we observe that
−
v A v +A v
A + = ++ + +− −
v A v +A v
+ +
(cid:20) −(cid:21) (cid:20) − −− −(cid:21)
and A∗ǫǫ′ = (Aǫǫ′)∗. We introduce field operators defined on D˜ by
ψ(v) = c(P v)+c (CP v)
+ ∗
−
v
= c(v )+c (v¯ ), v = + .
+ ∗ − v ∈ H
(cid:20) −(cid:21)
One observes that ψ(v) = ψ(Cv) on D˜ and hence that ψ(v) is clos-
∗
able. We denote its closure by the same symbol. Let be the set
C
H
of vectors satisfying Cv = v:
v
= v = + v = v¯ h .
HC ( v (cid:12) + − ∈ )
(cid:20) −(cid:21)(cid:12)
(cid:12)
(cid:12)
f
Clearly, foranyv = (f h(cid:12)), theoperatorψ(v)isessentially
f¯ ∈ HC ∈
(cid:20) (cid:21)
self-adjoint on D and is equal to √2τ(f). By (2.5), we see that
f
D˜ = D(ψ(v)).
v∈\HC
v
Note that, for any v = + , the vectors
v ∈ H
(cid:20) −(cid:21)
v +v¯ iv iv¯
v+Cv = + − , i(v Cv) = +− −
v +v¯+ − iv iv¯+
(cid:20) − (cid:21) (cid:20) −− (cid:21)
belong to and the following holds on D˜:
C
H
1 i
ψ(v) = ψ(v+Cv)+ ψ(i(v Cv)).
2 2 −
It is straightforward from (2.1) that
[ψ(u),ψ(v) ] = (u,Qv)
∗
holds on D . The following is well known (see, e.g., [13]):
f
8
Lemma 2.1. (1) Let U be a bounded operator on satisfying
H
CU = UC, UQU = U QU = Q (2.7)
∗ ∗
Then there exists a unitary operator U such that
eiψ(U∗v) = U eiψ(v)U, Ueiψ(v)U = eiψ(QUQv), v
∗ ∗ C
∈ H
if and only if U is Hilbert-Schmidt. In this case, U leaves D˜
+
−
invariant.
(2) Let l be a linear functional from to C. Then there exists a
H
unitary operator U such that
l
ei(ψ(u)+l(u)) = U eiψ(u)U , v
l∗ l ∈ HC
if and only if there exists a u such that
l C
∈ H
l(u) = i(v ,Qu), u .
l C
∈ H
In this case, Ul leaves D˜ invariant and Ul = e−iψ(vl).
2.3 Quantized Klein Gordon equation
Let h = L2(R3;dx) and h = ∆ with the domain D(h )= D( ∆) =
0 0
− −
H2(R3). Then we define the Schro¨dinger operator h by
h= ∆+V(x) (2.8)
−
with the potential V : R3 R satisfying Assumption 1.1. h is self-
7→
adjoint with the domain D(h) = D(h ). Let
0
ω := (h +m2)1/2 and ω := (h+m2)1/2.
0 0
The free field Hamiltonian H is defined by
f
H = dΓ(ω ).
f 0
Since ω and ω are strictly positive, ω 1 and ω 1 is bounded. Note
0 0− −
that ωθω θ and ωθω θ are bounded operators for any 0 θ 1.
0 − 0− ≤ ≤
Indeed, since h h, we observe that ωθf ωθf . Hence ωθω θ
0 ≤ k 0 k ≤ k k 0 −
is bounded. On the other hand, since D(h) = D(h ), it follows from
0
theclosed graphtheorem that hf C (h +1)f with someC > 0.
0
k k ≤ k k
Hence we observe that ωθω θ is bounded. From this fact, we see that
0−
ω θωθ and ω θωθ can be extended to bounded operators on h. We
− 0 0−
denote the extended operators by the same symbols. The following
holds.
9
Lemma 2.2. Suppose that Assumption 1.1 holds. Then the operators
ω01/2ω−1/2 − 1, ω0−1/2ω1/2 − 1 and ω0−1/2(ω0 − ω)ω0−1/2 are Hilbert-
Schmidt.
Proof. See Appendix A.1.
For real f H 1/2(R3) and g H1/2(R3), we set
−
∈ ∈
φ (f) = ψ(u ) and π (g) = ψ(v ),
0 0 0 0
where
ω−1/2f/√2 iω+1/2g/√2
u = 0 and v = 0 .
0 "ω−1/2f/√2# 0 " iω+1/2g/√2#
0 − 0
For non real f H 1/2(R3) (resp. g H1/2(R3)), φ (f) (resp. π (g))
− 0 0
∈ ∈
is defined by of φ ((Ref)+ iφ (Imf) (resp. π ((Reg) + iπ (Img) ).
0 0 0 0
Note that for non real f H 1/2(R3) and g H1/2(R3), φ (f) and
− 0
∈ ∈
π (g) are non self-adjoint and the following equations hold on D˜:
0
1
φ0(f)= √2 c∗(ω0−1/2f)+c(ω0−1/2f¯) ,
(cid:16) (cid:17)
i
+1/2 +1/2
π (g) = c (ω g) c(ω g¯) .
0 √2 ∗ 0 − 0
(cid:16) (cid:17)
By (2.1), one can show that the CCR holds on D :
f
[φ (f),π (g)] = i(f¯,g), [φ (f),φ (f˜)] = [π (g),π (g˜)] = 0, (2.9)
0 0 0 0 0 0
for any f,f˜ H 1/2(R3) and g,g˜ H1/2(R3). It holds from (2.9)
−
∈ ∈
that
φ (f)Ψ 2 = φ (Ref)Ψ 2+ φ (Ref) 2,
0 0 0
k k k k k k
π (g)Ψ 2 = π (Reg)Ψ 2+ π (Reg) 2, Ψ D .
0 0 0 f
k k k k k k ∈
Hence we observe that, for non real f H 1/2(R3) and g H1/2(R3),
−
∈ ∈
φ (f) and π (g) are closed on the natural domain.
0 0
We introduce the bounded operator on by
H
U (t) U (t)
U(t) = ++ +− , t R,
U +(t) U (t) ∈
(cid:20) − −− (cid:21)
where
1
1/2 1/2 1/2 1/2
U++(t)= 2 ω0− cos(tω)ω0 +ω0 cos(tω)ω0−
(cid:16)i (cid:17)
ω1/2ω 1/2sin(tω)ω 1/2ω1/2
− 2 0 − − 0
(cid:16)
+ω−1/2ω1/2sin(tω)ω1/2ω−1/2
0 0
(cid:17)
10