EXPONENTIAL REPRESENTATIONS OF THE
CANONICAL COMMUTATION RELATIONS
by
James D. Fabrey
A.B., Cornell University
(1965)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF
PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September, 1969
Signature of Author
Dep tment of Mathemats ,.JuJulyy.1969
9699
Certified by .........
Accepted by
....
. *. ....
. . .....
":Thesis
....
...
.
.................
Supervisor
e...
Chairman, Departmental Committee
on Graduate Students
Archives
EXPONENTIAL REPRESENTATIONS OF THE
CANONICAL COMMUTATION RELATIONS
James D. Fabrey
Submitted to the Department of Mathematics in July, 1969
in partial fulfillment of the requirement for the degree
of Doctor of Philosophy.
ABSTRACT
A class of representations of the canonical
commutation relations is investigated. These
representations, which are called exponential
representations, are given by explicit formulas.
Exponential representations are thus comparable to
tensor product representations in the sense that they
provide the possibility of computing useful criteria
concerning various properties. In particular, they
are all locally Fock, and non-trivial exponential
representations are globally disjoint from the Fock
representation. Also, a sufficient condition is
obtained for two exponential representations not to be
disjoint. An example is furnished by Glimm's model for
the :D : interaction for boson fields in three
space-time dimensions.
Thesis Supervisor:
Title:
James Glimm
Professor of Mathematics
ACKNOWLEDGMENTS
I am greatly indebted to Professor J. Glimm for
the suggestion of this work and for his constant interest
and advice.
Helpful discussions with Professor A. Jaffe
are gratefully acknowledged.
I wish to thank
Professor K. Hepp for his correspondence and
encouragement.
Thanks are also due to Professor R. Goodman for
useful conversations and to the National Science
Foundation for financial support.
Finally, I want to
express my appreciation to my wife, Pat, for her faith,
patience,
and understanding.
4.
TABLE OF CONTENTS
Chapter
I.
II.
III.
IV.
Page
Introduction
5
Basic Estimates
19
Renormalized Hilbert Space
31
Exponential Weyl Systems
52
Bibliography
74
Biographical Note
75
5.
CHAPTER I
Introduction
In this paper we investigate a certain class of
representations of the canonical commutation relations.
Our representations will be called exponential Weyl
systems.
A representation of the canonical commutation
relations, or a Weyl system,
H ,
from
to unitary operators
J
a complex inner product space
on a complex Hilbert space
f - W(f)
is a map
such that
W(f)W(g) = eiIm(f,g)/ 2 W(f + g)
(the Weyl relations), and
at
t = 0 .
If
{f )
t
+
(4,W(tf)4)
is continuous
is an orthonormal basis of
J
then
itP
isQ.
W(sf)
= e
3,
W(tif k )
by Stone's theorem, where
Qj,
k
k
= e
Pk
are self-adjoint.
The Weyl relations are
isQj
e
istk itP isQ
itPk
e
e
= e
e
.
which is an exponentiated version of
Q Pk
-
PkQj = 16jk
A theorem of von Neumann states that if
then
W
dim J = n <
,
is essentially (up to multiplicity) the
Schr8dinger Weyl system of quantum mechanics:
Q
= multiplication by xj, Pk
If
dim J = c
= i -
a
, H = L2(R n).
, there are uncountably many inequivalent
irreducible Weyl systems.
Fock representation.
The best known example is the
J = L 2 (Rd ) , d > 1, is called the
single particle space.
Let
H = F = EFn
be the direct
sum of Hilbert spaces
Fn = SL 2 (Rdn)
where
S
is the symmetrization projection defined on
ZL2 (Rd n ) by
... , kn) = n! - 1
Sn (kI,
Here
kIieR
,
Sn
Z
n(ka(1)3, .' , ka(n))
is
the
permutation
group on n letters,
S
cES
is the permutation group on n letters,
and
F0
= p
is the field of complex numbers.
esF
Zl
and
if
nl
nEFn
Thus
is symmetric, square integrable,
2= 11112 < m .
F
a neutral scalar boson field.
is
the Fock space for
n EFn
represents a
state of the quantum field in which there are n particles;
'l'nl1 = 1 ,
if
then
IPn(kl, ..., k )12
is the
probability density that their momenta are
Since
n
k 1 , ...
, kn.
is symmetric, the particles are indistinguishable.
= (1, 0, 0, ... )EF
is the Fock vacuum, and it is the unique (normalized)
state in which no particles are present.
In general
there are an indefinite number of particles in the state
*EF. If
Fn
+
fEL 2 (Rd), the annihilation operator
a(f)
maps
Fn-l, annihilating a particle with wave function
and the creation operator
a (f)
maps
F
f ,
+ Fn
creating a particle with wave function f:
(a(f)*n)(k l , ..., kn_)
a (f)pn = (n + 1)½S(f
= n
fI f(k )
n) .
(kl, ..., k n)dk n
The normalization constants are chosen so that
and
a (f)
are adjoints; moreover
a(T)
$(f) = 2-½[a(T) + a (f)]
is self adjoint.
W(f) = e i o (f )
is the Fock representation.
The Weyl relations follow from the commutation relations
a(f)a (g) -
a (g)a(f) = f
f(k)g(k)dk
Exponential representations are constructed as
follows.
If
operator
a
w = w(kl, ... , k )EL 2(Rdv) ,the
(w)
maps
Fn
+
F+
,
creation
creating v particles
with wave function w:
a
(w), n = [(n+l) ... (n+v)]½S(w 0
n) .
Let
D = {I
=
•nsF
: #n = 0, large n; supp "n compact}
be the set of vectors in
F
with a finite number of
particles and bounded momentum.
Let
v = v(kl, ..., k )
be a symmetric measurable function, and let
p, a
be
lower and upper cutoffs on the magnitude of the largest
momentum:
v p(k) = v(k)
max
Iki E[p,
o)
l<i<v
= 0
Note that
a > i ,
otherwise.
vpa = 0
for
p > a .
Let
va = V
*
and let
n(k) = k
,
k >
= 0
Suppose
k = 0 .
v EL2 (Rdv)
Observe that for
aV (vp)
pa
is defined.
k > k
(vn(z)o)
exp a
so that
= exp a
(v (k)a + Vn(t)(k))
= exp a*V (v
)exp a*v(v
as formal power series.
If
v
is almost in
L2
in a certain technical
sense, we construct a family of cutoff operators
T kor
,
k >- 0 ,
which are modifications of
Fix
10.
exp a* (vf(k) )
Tk
where
such that for
6eD.
We view
lim
DTYD
TkD
sED} .
,
k > k , with the identification
Then the limit
(Tk:p,
exists for
T£ i)e
0, ieD
= (Tk¢ , TZ)
"renormalized."
U
TkD .
Let
H = Fr
The subscript denotes
be the completion of
(k)fEL2 (Rd), some
p(k) = (p02 + Ik 2 )
with momentum
limit
U TkD .
k>0
Let
J = {feL 2 (Rd) :
where
r
and defines a positive definite
inner product on the set
k>0
> n(k),
Consider the sets
= Tke .
a -ý-
,
= TkoT
e
TkD = {Tk* =
T-
k
k ;
~O > 0
is
c > 0} ,
the energy of a particle
is the rest mass.
Then the
11.
lim (Tka ,W(f)T o)e
= (Tk ,Wr(f)T
C Co
defines a Weyl system
Wr(f),
exponential Weyl system.
We now define
for
Wr
is
called an
In general, the power series
Tka
OeF
exp a*Ovn(k)a)
feJ.
)r
does not converge; see [7].
Indee di
2
I la*V(vn(k)a)n.
Knn!
so that convergence cannot be expected for
Exceptions occur when
v = 2
V = i, or
has Hilbert Schmidt norm less than
12, Lemma 2].
exp a* (vn
exp a
fr1
v > 2 .
and
1/2[ll,
v
Lemma 4;
Thus we must omit portions of
Now
(v (k)a) = exp a*
(
E
n(k)<n
k)n()<
nl(k)<n7(5j)<aI
as formal power series, where
exp Vja
fl(j)min (o,n(J+l)
)
12.
Vja = a *(v
(j)min (a,n(J+l)))
n
Let
x =
exp
n(j),
>
E x /Z! .
k=0
,
Tk=
expn(j)
n(k)<n(j)<a
Tka
converges absolutely on
Thus
v,
Then for any sequence
Fr
.
a, n(j)
increases,
and
D .
depend on the parameters
Note that as
Tka
The choice of
objectives.
Wr
Vj
a
decreases and n(j)
creates more high energy particles.
n(j)
must balance two conflicting
We do not want
Tka
high energy particles that
the other hand, we want
to create so many
(*.,.)
Tka
doesn't exist.
On
to create enough high
energy particles to overcome the effect of the factor
e
a
2
require that
so that
n(j)
11-r
*
is
definite.
Thus, we
be strictly increasing but
polynomially bounded in
j
.
In Chapter II, a basic estimate controlling
products of operators of the form
aV (w)
or
aV (w)
13.
is obtained.
In the process, quantized bilinear forms
are discussed [6].
Fr
construction of
F
is
Chapter III is devoted to the
We say that an operator
.
the weak limit of an operator A in
B = lim A)
(written
t,(B) =
if
(Tk ,BTkP)r= lim
lim A
and
(Tkaop,ATRa)e
Fr ; the extension is also written
Thus if
Ira= lim I.
Ir
is the identity in
F
, then
Chapter III is also devoted to obtaining
operators in
in
F
is bounded, then it extends uniquely by
continuity to
lim A .
in
k>O
a
If
U TkD
B
Fr
as weak limits of quantized operators
F.
Chapter IV contains some results concerning
unitary operators and n-parameter unitary groups in FP
We then conclude
which are defined by weak limits.
(Theorem 2) that
Wr(f) = e
ip (f)
r
lim W(f)
, fJ .
defines a Weyl system
U
Moreover,
k>O
set of entire vectors for
r (f)
,
and
TkD
is a dense
14.
rr (f)
j
some
(f)J ,
Z lim
a
> 0,
strongly on
j >0
r(fn)
then
if
+
f-f)n -
I
Or(f)
and
lI
Wr(f
0
Wr(f)
)
U TkD .
k>0
We also show (Theorem 3) that the local systems
Wr(f),
Isupp f(k)l < p , i.e. localized in momentum space,
have a non-negative number operator in the sense of
Chaiken, and hence are unitarily equivalent to a direct
sum of Fock representations.
In Theorem 4, we prove
that if the kernel is not in
L 2 , then the global
systems
Wr(f)
,
feJ ,
are disjoint from the Fock
representation because every vector in
infinite number of particles.
Fr
has an
Finally, in Theorem 5,
it is shown that the choice of
n(j)
is somewhat a
matter of technical convenience in the sense that two
1
2
Weyl systems Wr
and Wr
with the same kernel v
and same choice of
a
are invariant subspaces
Wr l lS 1
if
Wr 2
are not disjoint.
Si
F
for
is unitarily equivalent to
n l (j) = n 2 (j)
Wri
such that
Wr 2 S2 .
for almost all j , then
are unitarily equivalent.
That is, there
Moreover,
W 1
If the kernel of
and
Wr
is
perturbed by a sufficiently small function, then the new
15.
Weyl system is not disjoint from the old one.
It is
hoped that a family of inequivalent exponential
representations may be obtained from kernels whose
pairwise differences are sufficiently large.
1) 0(f)
Remarks.
field
is related to the Newton-Wigner
and the relativistic field
0N,W
#rel by the
formulas
A
ONNW(g)
where
-
g(k)
(g),
ýtel(g) = 0([(-A+2)0
is the Fourier transform of
if we replace
9
by
#rel '
g]
g(x).
Then
the local systems (localized
in momentum space) are still unitarily equivalent to a
A
direct sum of Fock representations because Isupp gj < p
if and only if
Isupp [(-A+
02
4
^A<
.-
2) Exponential Weyl systems have some of the
advantages of tensor product representations; explicit
formulas for them are given, and one may compute criteria
for various properties of the representations.
Tensor
product representations have been useful for linear
16.
problems in quantum field theory.
M. Reed [14] has
even shown that a certain class of renormalized
Hamiltonians with nearly diagonal non-linear interactions
acts on infinite tensor product spaces.
However, tensor
product representations cannot be expected to provide
solutions to most physical problems.
This point of view
has been supported by Powers [13] in the case of the
representations of the canonical commutation relations
(CAR) provided by Fermi fields.
He proves that a
translation invariant vector state of a tensor product
representation of the CAR is a generalized free state,
i.e. the truncated n point functions vanish for
n > 2
Power's point of view may apply somewhat to
exponential representations.
Nevertheless, certain
renormalized Hamiltonians, with interactions possessing
large momentum singularities different from those of
Reed's model, are defined on exponential representation
spaces.
Indeed, exponential representations are
suitable for some superrenormalizable interactions
which may be far from diagonal.
by Glimm's model for the
An example is furnished
:4 : interaction for boson
fields in three space-time dimensions.
This model has
an infinite mass renormalization caused by the off-diagonal
1L
17.
part of the interaction in the sense that there is
fairly strong coupling between low and high energy parts
of the interaction.
Hepp has demonstrated [8] that the
renormalized Hamiltonian for a simplified version of
the interaction acts on a closed subspace, the completion
of
T 0 D, of an exponential representation space.
The
interaction is taken to be
a
Here
(-.E(k )v) + a
v = 4,
v(k l
where
,
d = 2, n(j) = J,
... , k 4 ) =
lva11 2
almost in
example,
(-Ep(k i )v
-
El(ki)
)
a = 2
.
and
(EV(ki))- h(Eki) ,
is logarithmically divergent.
L 2 , but is not close to diagonal.
v
is not in
L2
v
is
For
on the set
A
{k : 21kll <
k2 , Ik3 1, lk4 1 < 31k 1 )l. h
is the
Fourier transform of the space cutoff function, which
is any non-negative smooth function with compact support.
It would be of interest to construct a family of
inequivalent representations by choosing different space
cutoff functions.
Incidentally, Hepp has independently
18.
obtained [8,9] a Weyl system whose properties are similar
to those of
W r = W (v, 2, j).
It is conjectured that
the two representations are unitarily equivalent.
19.
CHAPTER II
Basic Estimates
In this section we obtain an estimate controlling
products of the form
involving
va
and
a
I
va
(v,) Pa
(va)
.
Products
are considered because we want
sufficient generality to compare two different exponential
Weyl systems in Chapter IV.
First we must discuss quantized
operators and bilinear forms in Fock space [6].
In order
to control the profiferation of constants in the sequel,
K>O
will denote possible different constants from one line
to the next.
Let
B
I
be a bilinear form, with distribution kernel
densely defined in
b,
FmxF n
.
quantized bilinear form
B
( ,B*)
/2t.-1
=
t=o
(tm)!.(tn)!]
defined in
b = fl®. "'fm+n, fiEL2 (Rd),
If
that
B = a (fl)...a
We associate to
FxF
then
I
the
by
m+t(x,z)b(xy)n+t(yz)dxdydz (2.1)
then one can easily check
(fm)a(fm+l)...a(fm+n).
b = w(kl'..,km)EL2 (Rdm),
B
B = a*m(w).
Also, if
Thus
B
creates M. particles and annihilates
n
particles.
is the Dirac delta function, then
8
is a distribution
6
If
20.
kernel densely defined in
FoXF 1 .
quantized operator given by
Then
6(.-k),
a(k),
is defined by
, . . .,
(a(k)4n)(k l . -.
, . kn_1 ) = n/2 n(kkl *kn
a(k)
Thus
the
_l ) .
annihilates a particle with momentum
The adjoint
a (k) of
a(k)
(2.2)
k.
is a quantized bilinear
form with distribution kernel densely defined in
F1XF O,
and it is defined by the improper operator
(a (k)4n)(kl,...kn+l )= (n+l) /2S6(k-kl)O®n(k 2 ,...,kn+l)•
(2.3)
This is,
( n+,,a (k))= (n+1)~/2(
Thus a~ (k)
a (k)
0'
"1kn+l)a(k,2
0
creates a particle with momentum k.
denote
a (k) or
B = Sb(k 1 ... , km k
La(k),a* ()]
a(k).
Let
One may verify that
I
..
kn)a (kl) ...
a (km)a(kl) ... a(kn)dkdk
= a(k)a* ()-aa*()a(k)
[a(k),a(Q)] = 0 = [a (k),a*(t)].
I.a(
= 8(k-t)
(2.4)
21.
This is,
(+,B*) = J'(Ta(ki)+, b(k,k )TTa(ki)*)dkdk',
and
J.(+,b(k)ra#(k)i*)dk = j(4,b(k)rr a (k,)*)dk
rr
= f(+,b(k)
) 8 (kj-kj+l )
a#(k i
i+j, j+1
)dk,
+ J'(,b(k)h'a#(ki)*)dk
where
ir
denotes a transposition of two factors
a(ki ), a(ki+1 ) or
a (ki), a (ki+l),
a transposition of two factors
Note that if
Fm
B
(for example, if
6(k-k ),
Nj
n
let
= nt n ,
operator.
N
and
a(kj), a*(kj+ 1
Fl ,
F.
B
then
bEL 2 (Rd(m+n)))
Indeed, if
).
to
is a densely
I
I.
is the
Then, by (2.1),
is called the number of particles
By (2.1),
(B)i
Fn
with distribution kernel
be the quantization of
N
denotes
TT
is a bounded operator from
defined unbounded operator in
identity operator on
and
= i! /2 (i-m+n)! /2(im)!
SB
i-m+n
22.
where
acts on the first
B
so that
B
variables.
ti-m+n
Hence
n
variables of
is an unsymmetrized function of
ifB
(
i-m+n
-
-
2
9??
E
il
*i-m+n'
(i-m+n)!(i-m)!-2
2pB
112
2
"2
2 E (J+l)m+n
M<B '2
< Kil B 112I(N+l)(m+n)/2
2
Thus
IIB*I! < K BI i (N+l)
Hence
Dc=(B),
so that
B
By (2.4), we may write
is densely defined.
N = JSa*(k)a(k)dk.
operators to be considered in the sequel are:
N(B ) = ikpa (k)a(k)dk,
N7 =
Ij(k )Ta*(k)a(k)dk,
Ho(B ) =
(2.5)
(m+n)/2*!.
Ik• p (k)a* (k)a(k)dk,
Similar
23.
where
and
Bp
is the ball of radius
Ho(Bp)
in
Rd.
are quantizations of operators on
by multiplication by
where
p>O
X , I(k)', and il(k)Xp
N(Bp), N ,
F1
given
respectively,
XP
is the characteristic function of Bp. By (2.1),
n
they act on Fn by multiplication by
ZE
(ki)v
n
n
E (ki ) , and Z p(k i )xp (ki ) respectively. N(B ) is
i=1
i=1
the
number. operator over
L2 (Bp)
for the Fock represen-
tation and measures the number of particles with momenta
bounded by
p. Ho(Bp)
measures the free energy of these
particles.
Suppose
B
is a quantized bilinear form
n
m
B = Jb(k,k )
To
B
a (ki) n a(ki)dkdk.
i=l
i=l
we associate a graph with
pointing to the left,
n
m
(creating) legs
(annihilating) legs pointing to
the right; all legs issue from a common vertex (4,5].
(See Figure 1.)
Fig. 1: m=2, n=3.
24.
B
is also called a Wick ordered bilinear form because
a (ki)
the
all appear to the left of the
is determined by its graph and its kernel
of two Wick ordered bilinear forms
a(kj).
b.
B l, B 2
It
The product
is not Wick
ordered, but it is a sum of Wick ordered bilinear forms,
by repeated application of the commutation relations (2.4).
The term with no
8
denoted. :B1 B2 :,
the Wick product of
term contains a
variables
8
k,t
8
functions has a kernel
function
have been contracted.
functions, denoted
from
bl®b2
8(k-4),
Bl
1 - B2 ,
B2 . If a
we say that the
The term with
j 8 functions.
is associated a graph obtained by connecting
B2 .
B1
with
j
Similarly,
(See Figure 2).
formo B,
is determined by its graph and its kernel.
B1 ...Bt
To
j
of the creating
Fig. 2: j=l.
B1 -?- B2
j
by equating contracted variables and summing
of the annihilating legs of
legs of
and
and is
has a kernel obtained
over all possible contractions with
B1 -~- B2
Bl
bl@b 2
is a sum of Wick ordered bilinear
each of which is determined by its graph and
25.
Its graph has
its kernel.
t
ordered vertices and is
obtained by specifying the number of annihilating legs
of the graph of
of
Bj, i<j.
Bi
which are connected to creating legs
Its kernel
b
is obtained from bl ...®b t
by equating contracted variables and then summing over all
possible contractions which produce the same graph.
B = fb(k)na#(ki)dk,
Then
where the product extends over the
legs of the graph which do not connect two vertices (external
legs).
If
M
is a measurable subset of the variables of
b,
then
fb(k)ra (ki)dk
M
is called a truncation.
A truncation of a product
B1 ...Bt
is given by a truncation of each of its Wick ordered terms.
A truncated power series is given by a truncation of each
of its terms.
A subgraph of a graph is a subset of the vertices of
the graph, together with all legs issuing from these vertices.
Two subgraphs are disjoint if they have no common vertices.
(They may have common legs.)
are called internal.
Legs which connect two vertices
Note that a leg may be internal in the
26.
full graph but external in a subgraph.
Pt = P(kt),
let
let
external variables,
denote the product over
rre
the product over all variables,
r
the internal variables with respect to the
I = I(G)
G,
graph
In the sequel,
and
S
the integral over internal variables
I
[5, p.8].
Let
V = a
v,v
(v), V
be symmetric, measurable functions, and
= a
(v )
the corresponding bilinear forms.
I
We shall be concerned with
v,v
which satisfy the
following property:
Let
b
be the kernel of
B = V --or
for
a>O
and
l<t<v,
there exists
E>O
V ,
l<r<v-.;
such that
Sn E-bI EL2(R2d(v-r))
I
E -a
vTPP
a.v I
t ELa(R
2tdv )
E -a
'Vp If?, !lv !I< K(log(a/p)+l)
Suppose that (2.6) is satisfied for
that
v
is almost in
is almost in
L2 ,
L2.
and
A(a) = v!I,=1
2
v = v .
In particular, an
(2.6)
Then we say
L2
function
27.
is at worst logarithmically divergent.
n(j)
Choose
a>l,
strictly increasing but polynomially bounded, and
construct
Tk'
• ) (
e exp n(j))Va, k>O,
as described in Chapter I. Tk = Tk(v,a,n(j))
an exponential dressing transformation.
is the truncation of
expVn(k)a
to at most the power
n(j), j>k,
is called
Note that
in which
Vja
Tk
appears
and does not appear for
j<k.
Now
V P(V')
q
is a sum of Wick ordered bilinear forms;
a bilinear form in this sum is called reduced if its graph
contains no
x = v -!-v
components.
Each graph is a union of its connected
components.
If the number of vertices of a connected
component is greatet than one, then the kernel is in L .
2
The improvement of the kernels, as the order of the graph
increases, demonstrates the sort of behavior to be expected
of superrenormalizable quantum field theories [5, p.10].
28.
Such estimates have been studied systematically by Eckmann
[3].
L2
The following lemma establishes geometric growth of
norms.
Lemma 2.1.
Suppose
be a reduced term from
n = p+q.
Then for
v,v
vPV
,
satisfy (2.6).
b
B
its kernel, and
there exists
a>O,
Let
K>O
such that for
sufficiently small,
E>O
eTTe
-aS
E bl
< Kn
(2.7)
I
Here
a,E
do not depend on
Proof:
p,q
or the reduced term.
We may suppose that the graph
G
of
B
is
connected because both sides of (2.7) are products of
similar expressions involving the connected components.
The cases
n = 1,2
follow from (2.6).
Let
n>3.
v
be written as a disjoint union of subgraphs of
a central vertex contracted with
1<s<v
for
use inspection.
3<n<N.
choose a subgraph
HCG
n>3.
Since
of type
G
is connected, we may
s = 1.
Then
G-H
I
disjoint union of connected components
the union of
H
single vertex.
For
Suppose we have the decomposition
n = N+l.
Let
types:
vertices [5, p.8].
This decomposition is obtained by induction on
n = 3,
G may
H . Let
is a
I
H
be
with all those Hj which consist of a
I
1 G
Clearly H
is connected. If H G,
29.
I
then apply induction to
I
If
H
= G,
H
and the components of
then one can show directly that
G
G-H
I
is
decomposable into either one or two subgraphs of the
proper type.
Thus, for
n>3,
we may write
G = UHJ,
union of subgraphs of the proper type.
where
yj
is the kernel of
inequality in the variables
a disjoint
Let
y = Tryj,
H . Then the Cauchy-Schwarz
I(G)-UI(H.)
implies that
j
-ajteTrEp yJ
0
where
K
S(G)
j I(H)
rrTT
TI.E yjr1,
compensates in the region of small
the factors
side.
<
l(k)E
Since these are a finite number of graphs of the
H
is a subgraph of type
s = 1, apply (2.6).
HoCH
for
appearing twice in the right hand
proper type, it suffices to show that
where
Jkl
of type 1. Let
vertices, and let
Multiply
y
Then for
a>O
by
For
H.
y0 , yj
s>2,
f(HrrnEIyI E L2,
s with kernel
y. For
choose a subgraph
be the graphs of the other
be the corresponding kernels.
a
n
Pa
I(H)-I(Ho)
and
yj
by
-a
r
-a.
I(H)-I(Hj)
sufficiently small, we may apply the
Cauchy-Schwarz inequality in the variables
I(H)-I(Ho)
and
30.
use the first part of (2.6) for the first factor and the
second part of (2.6) for the remaining
This completes the proof.
s-1
factors.
31.
CHAPTER III
Renormalized Hilbert Space
In this chapter we construct
process, obtain operators in
Fr
Fr, and, in the
as weak limits of
products of Wick ordered operators in
F .
We first
establish a combinatorial lemma which is sufficiently
general to be useful throughout the sequel.
Y
3
1 < i < m ,
is
Suppose
a Wick ordered operator in
F
of
the form
si
)
lyi(kl,...,ks
R a#(kj)dk
yi
where
Let
V
T=
or
fy(k)a*(k)a(k)dk,
J=l
'
is a measurable kernel and
Tk(V
= E Vj
V
,
a,
'
= Z V
n(j)),
X(a) = V* -o-V~
T
'
= T (v',
y
is measurable.
a, n'(j)),
and
' - v!(vC,
v ')
v
Note that we require
i Z j
.
Let
$eD
*,
at = a
.
,
so that
We consider
V
-o-
V
' = 0
32.
m
T
TI YiTczX
Yi
(Tka4
eX(a)
m-X(T)
=
T
1=1
(For
HYi = I
m = 0,
power series in
V
.)
Now
IT YiT
i=l
Tka HYiT
, Yi , V '
'Yl)e-
(3.1)
is a truncated
and is an infinite sum
of Wick ordered terms with distribution kernels.
A
term will be called reduced if it contains no X components
[4].
ROo
Let
be a reduced term, and let
Rja
be the
sum of all terms whose graphs differ from the graph G
of
ROa
by
graph of
with graph
E R.ja
j>o
j X components.
R.
j).
G
Let
eX(a
)RG
(G
is called the reduced
be the bilinear form
and kernel equal to the kernel of
after integration over the variables in the
X components.
Thus, (3.1) equals
E (0, RG a ), where
G
the summation extends over all reduced graphs
G.
RGa
does not necessarily have a measurable kernel because of
the 6 function in
fy(k)6(k-L)a (k)a(Z)dkd2
by (2.1) and the form of
Yi ,
(,
RGa)
.
is
However,
a sum of
integrals of the form
C
f rGa q
Cpq i~p'
q
over the variables (some coinciding) of
and
C)Opp, rGa,
Ga-Iq-
q
33.
Here
rGa
4p
is a measurable function,
p,
Let
E
q
Vq
are
4, P, and the
the p and q particleCcomponents of
choices of
and
G
depend on
be the direct sum over all reduced graphs G,
and over permissible
p, q
spaces associated with
Cp
G , of the measure
for each
fp rG
q
.
Thus (3.1) equals
I ha
E
where
ha
is a measurable function; we sometimes use
the notation
Z (¢, R0a%
G
)
ha = OrGaP .
= f/ r0a~ ,
E
function depending on
Similarly,
where
G .
Let
vv'
is non-negative.
Let
ý
rGa,
r0
rGa(S)
where
X (a)
= V
Let
m(j)
is a measurable
IROGI = IRO1
Let
bilinear form associated with
Lemma 3.1
r0a
Ir
1 = Ir 0
0
-o- V
j
be the
.
'.
Suppose
= min (n(j), n'(j)).
denote a fixed value of the variables of
.0
=
Then
Ie
m'(j) < m(j)
exp m,(j)Xj(a)roa(E)
depends on
G
and
(3.2)
( ,
m'(j)
is
34.
independent of
a ,
m'(j) = m(j)
m'(j) = 0
for
j
and
for almost all
< max (k,
j ,
) .
Let
C(j 0 ,a)
= c(v,v',j 0,a) = fl
Then for all
j
exists.
lim
uniformly in
Proof.
Yi ,
suppose that
G
(3.5)
c(v,v',j 0 ,a) = 1
S00
Suppose
G
Vj
'
< K
Xj = v!(vj,v)
a, v, v', such that
reduced if it has no
Then
(3.3)
(3.4)
< 1
c(jo,t)
->-0
and q V' vertices.
a
has p Va
It is convenient to regard
Vji,
.
Moreover
jo ÷
in
expm(j )X ()
0 ,
0 < lim
a•
-x (a)
e
Tka YiT'T
as a power series
A Wick ordered term will be called
X. = X.(w)
r 0 a(W) X 0 ,
components.
We may
else (3.2) is trivial.
is a reduced graph from
35.
n v.
NY.
P(J)/(j)
j>k
I j>9
determined by
Eq(j)
= q,
G
p(j)
Then
p(j), q(j)
Note that
q(j)
< n(j),
eX()RGo
Ga
Her e
5 .
and
of the measurable
E
G .
function associated to
/q(j) I
ja
ro0(O) is the value at
and
( j )
V.
Zp(j) =p,
< n'(j)
is the sum of all terms (after
integration over the variables in the
whose reduced graph i
x(j)
are
components)
X
Consider all such terms with
X. comnonents, j > max (k, Z) , and hence
J
p(j) + x(j) Vj and
truncations in
0 < x(j)
T
k•a
q(j)
-'
the
T
< min (n(j)-p(j),
x(j)
By
+ x(j)VQ'vertices.
= m'(j), j
n'(j)-q(j))
= 0 = m'(j),
j < max (k,
> max (k,
Z)
There are
+ x(j)
x(j)
q ( j)
+ x(j)
x(j)
ways to contract 2x(j) vertices into
x(j) Xj
components.
36.
The remaining legs are contracted according to
Summing over all such sequences
rGa(0)=e -
X(
a
HT
p(j) !q(j) ! [(p(j)+x(j))! (q(j)+x(j)) ! ]- 1r0)
j>max(k,tZ)
e-X(a)
= e
Clearly,
,
)
0<x()<m'
E
O<x(j)<m' C)
S
{x(J)}
)+x(j)\
(p(Cx(j)
q(j)+x(j)
x(j)
E
-xj () expm' (j)X (a) r 0o(,)
r ,(E)
(3.2)
has the desired properties.
It remains to prove (3.4) and (3.5).
-X(a)
aj( a)
By (2.6), for
xi.
(CF
x(j)!X.
3
<x)<
I
x ()x(J)/x(j)
x(j)
H
X
0<x(j)<m' (C) j>max(k,k)
m'(j)
Let
m()X
exD(Q)X Ca)
a > n(j)
X (a) < v!Ivn(
uniformly
G
a .
)n(j
+l)l I li v' nj)n (J+l) ii < K(log a + 1)2
Thus
Xj (0)
and
aj (o)
exist and
37.
0 < aj (a),
(W) < 1
Thus, by [1, p. 190], it suffices
to show that
z (1 - a
J
where
K
(a))
< K
is independent of
Observe that, for
a .
x>0
n < xn+l/(n+l)! .
1 - S- ee X expnx
Therefore
Z(1-aj())
j
< EXj (a)m(j )+1/(m(j)+l)!
3
< EX n(j)+1/(n(j)+l)!
j
< 2 E K1j
,
+
n'()+
·J
•3
+ ZX.T
<K
j
since
n(j), n'(j)
are strictly increasing.
The lemma
is proved.
Let
lInI
II01F be the vector with n-particle component
38.
Corollary.
(3.1) is equal to
f h
,
E
where
{h o
is a family of measurable functions which converge pointwise as
a
c .
+
Moreover,
measurable function on
Iho I<
h ,
where
h
is a
E , and
f h = Z (I I, IROG11l 1)
E
G
Proof.
where
(3.6)
c
By (3.4),
j0 = 0, rG(S)
is some constant.
pointwise as
h
a
=
-a
.
Hence
+
cr0 ()
ha = CrGao
,
converges
Let
I1Iro0 IIi
Then, by (3.2),
jrG'I
<
Ir0 1 ,
and
lhl
< h
Q.E.D.
Thus, in order to remove the momentum cutoff in
(3.1), it suffices to show that
i.e. (3.6) is bounded.
a
-+
h
is integrable,
Then (3.1) possesses a limit as
by the bounded convergence theorem.
In order to
bound (3.6), we strengthen the hypothesis. Let
Jm = {yIL2(Rdm) :
m
5 RdO})
We consider two cases:
SYEL
2
(Rdm) , some E > 01
,{yoeL
39.
S.
a)
v-v'cJv,
b)
v=v',
where
y
=
Yi
fYl
H a (k )dk,
j=l
Yii1 = Y = fy(X)a
y.iJ
Si
, l<i<m
(k)a(k)dk , l<i<m ,
is a bounded non-negative measurable function
such that
(v,
HTeYjv) = f VIT
Lemma 3.2.
Let
[ E y(k i)]v
n = G be
G , and let
V' vertices in
< Kj
(3.7)
Jj > 1
the number of
s =
Ss
or
Then the
i
i=l
following estimates are valid:
m
a) (II, IRoI 1I0 ) < Ks+n
b) (I ) IROI l0 ) < Km+nsEcn
(a,
3 = min
where
Proof.
a)
a') > 1 ,
First,
Let
v" = v - v'EJ
and
w"
be the kernels of
and
V
a-o-
V"
, and
1+6
and
v, v'
6>0
satisfy (2.6) as follows.
I < t < v -
respectively.
II-sCn
IITUy i
i=l
--oThen
V,
1.
Let
V
W,
W'
-o-
V',
I HpE'IwlcL
by
L
(2.6).
40.
Moreover, by the Cauchy-Schwarz inequality,
IIInlU Iw"I II < Il -'vl~
IIn12E v"I
I
e
for
sufficiently small.
,
t-IIw 'I <
1
i
I
HnEUwl
+ I
I
Observe that in case
IR I .
of
JR
01
a•d the kernel
We may suppose that
radius
p
in
REU Iw"IEL2
(a), Ir
01 is the
kernel of
be the bilinear form given by the graph
B
Let
Thus,
(ll,JR01 101)
,
a > 0
where
vanish off a sphere of
, JI,
>o
RdJ
fewer particles.
He -a lr I
(I, 9
and
have
n0
or
Then, by (2.5)
<
,
(nHali
a
BHnalj
l)
.,j<n 0
an
<p
< P
an 0
(101, Bil))
no
(s+vn)/2
< Ks+nI I I,- a
Since
n(j), n'(j)
11011 11 1l
l lIB'l I
Irol
are polynomially bounded,
(3.8)
41.
J-1
(n(i) + n'(i)) < jvY ,
E
i=0
for some integer
from
Tka
or
y •
.
Hence, for
Ikl
or
contains at least
T
.
for factors
K
Ir0
e'-a•
I
integer less than or equal
Tk
6 = y-
vertices
< 1,
[(n+l)/21
II <<Ks+n
Ille-allr0
I
of
jY
T;' with the magnitude of the largest
1ol
G
0
Then there are at most
momentum less than
because
j
II
R
i=l
a -E
6
[(n+l)/2] , the greatest
(n+l)/2 , vertices from one
compensates in the region of small
U(k) E
The first factor in (3.9) is bounded by
Ks+n
IIiEyli'j
, by Lemma 2.1 and the Cauchy-Schwarz
i
inequality in the contracted variables between
and
HYi .
The second factor is bounded by
[(n+l)/2]
E
Si=l
where
TkT'
-e
i
2
n/
n/2 i
.
0
c = (1+6)2 - (l+6)
case, from (3.8) and (3.9).
1 di
i
-cn
The lemma follows, in this
(3.9)
42.
b)
Here
1R0 1
does not have a measurable kernel
since it contains 6 functions.
the number of particles.
Observe that
Y
conserves
Thus, every reduced graph
from case (b) is the disjoint union of three subgraphs.
The first is the union of all components without any
V , V'
vertices; the second is the union of all
components which have exactly one V
and one V' vertex,
at least one Y vertex, and no external legs, for
example
V
-o-
(Y
v-l
-o-
V')
1
The third subgraph is a reduced graph from case (a),
s = 0; of course, the corresponding kernel does not
arise from case (a) because of the Y's.
Therefore
IR01
is a product of three factors,
corresponding to the three subgraphs.
is estimated by
YJII
<_ KII
, YJ'ipl < KJ•1
second factor is estimated by (3.7) .
has a measurable kernel
by
YJlvlI < Kj
(3.8) and (3.9)
vo l
Ir01
The first factor
.•
The
The third factor
which can be estimated
, uniformly in
a .
Thus, by
43.
(11,
IR0 l 1I1)
< Km+nh
n,-af
I
Ir'I 1
m+n-ccn1+
< Km+n-cn
The lemma is
proved.
Lemma 3.3.
or
(3.10)
There are at most
a)
Ks+n(s/2)!(vn)!2
b)
Km+nm!(vn)!
reduced graphs
Proof.
G
2
such that
IGI = n
We overestimate by bounding the number of
contraction schemes, so that schemes which produce the
same graph are counted separately.
p V vertices,
0 < p < n
G
For fixed
can have
p ,
there are at
most
)
pvn-p)
0<i<min (vp,vn-vp)
vn
i!
< (vn)!2vn
1
possible contraction schemes between the
vertices.
In case (a) there are
V
and
V'
E.
i! < (vn+l)! 2s+vn
O<i<min (s,vn)
1i
1
possible contraction schemes between
V , V'
is
vertices.
2
2 m+vn
Suppose
HYYi
(vn+l)!
HYi
and the
In case (b), the corresponding bound
has q creating legs in case (a).
Then there are at most
s-qi! < (s/2 + 1)!2 2s
min (,s-q)
O<i<min (q,s-q)
1i
i
possible contraction schemes among the
the corresponding bound is
There are
n+l
Yi .
In case (b)
(m+l)!2 2 m
choices for
p .
The lemma follows
from these estimates.
Lemma 3.4.
T'
= T (v',a,n'(j)),
Let
Tk = Tk(V,a,n(j)),
Yi
be as in Lemma 3.2.
Then for
4, 1pID
(Tkka,
H YiT ta)eX(a)=
h
E
i=l
and both limits
(311)
45.
lim f ha = f
lim
a- oo E
E -oo
exist, where
E
is
h
(3.12)
a measure space,
{h I
is
a family
of measurable functions dominated by an integrable
function
(a)
h .
f
Moreover
m
h.< KS(s/2)!
E
or
(b)
I
mH
I|I|H y ll
i=1
h < Kmm!
E
Proof.
Let
E,
{h I ,
Corollary to Lemma 3.1.
h
be given by the
The lemma then follows from
the bounded convergence theorem and a bound on (3.6).
By Lemmas 3.2 and 3.3,
(a)
(3.6)
is
m
1+6
KS(s/2)! H JjHEYijj(ZKn(K(n)!2 -Cn
i=l
or
(b)
Kn(vn)! 2 -ccnl)
Km!(EK
n
But
bounded by
n
46.
< EKne2vn log vn -Ecn+6
n
EKn( vn)!2 -cn
Kn log n -
Scn
n
K+
+
<< K
C BE
( (E c K ) / 2 ) n l +
6/ 2
n>n
<K<
Theorem 1. Let
, some n0
.
Tk = Tk(v,a,n(J)).
Then for
ý,*ED, the limit
lim (TkacJ, TUar)eA(a) = (Tk4, T 9)r
exists.
If
k > 2, a > n(k)-, then
(3.13 )
T£,i
= Tk
k
,
where
k-l
S=
exn(
a
(3.14)
(vn(j)(j+l))ED
(0,1)r provides a positive definite inner product for
U TkD , whose completion is denoted
k>0
F
.
47.
(3.13) is the special case of (3.11) in
Proof.
which
v = v',
s = 0 .
n(j) = n'(j), and
from the definition of
inner product for
Tk*
(,')r
TkD .
U
(3.14) follows
is clearly an
It remains to show that
k>0
it is positive definite, i.e. if
Ir= (Tk
STk0l
Let
Let
m
0 # *eD
,
then
Tk )r > 0
be the smallest integer such that
m
#
0 .
be a lower cutoff on the magnitude of the
T
smallest momentum:
vT(k) = v(k) ,
= 0
Then
T
,
= Tk(vT
and let
of
T
> T
is an exponential dressing
Suppose that
Rdn ,
in
E nP (Bn
n>O
N(B()) .
[ki
otherwise.
a, n(j))
transformation.
of radius
min
l<i<v
( ))
vanishes off a sphere
n > 0 .
Let
a > n(R)
> T
be the spectral decomposition
Then
IITa1 12 > l p-(B (£))TkQI
IlTn(P) *m1.
kc
2 =
2 = IIPm(B(£))T k
ITn(2) 11
2
11 2
48.
because
A
so that
Tka
does not annihilate any particles.
= v!lv
I() ( 2)
112
A = A(0) < A <
(a =
K ,
v
Let
112
independently of
k .
Now
(4m, m'
Tn(T)*Tn(2)m)
a
2ka
m)
(3.15)
Tnk Cr(R)*Tfnk () •
is a sum of terms given by graphs from
By (3.2), the sum over all graphs with the empty set
for reduced graph is given explicitly by
exPn(j)A(
IP
j>_
which is equal to
exp Az()( I Iml
by (3.5),
where
2 + 6(Z,o))
6( ,ka)
- 0
as
A
+
uniformly
We may bound the other contributions to (3.15)
a .
49.
follows.
Since
In,-a ~I
I
G 1 0
for these terms,
Irol II< Kn(P)-)
Thus, by (3.8), (3.9),
12 e- A(a)
I
I
2E
01I I
Lemmas 3.1, 3.2, and 3.3, these
contributions are bounded by
IITka*l
I11-a
Kn (M)-exp
An (9)(a)
Hence
> e-A(a)exp An( ) (a)(
)
C2-(£,a)-Kn (- E)
> exp (-v! Z
2-6(a,o)_Kn(£)- E
IIM 1mmII
I$
v2l Ihm
ll
i=1 ki <n()
> exp (-Ky(n(L)) IjIvjEv I2)(XI 1
> exp
(-Kn(Z,)E) (I 11 12-.6(,,-
l i 2 -6(,,a)-Kn(a)
)K)-E)Kn(
> 0
for
k
sufficiently large.
Therefore
IITk*l Ir>
The theorem is proved.
Lemma 3.5.
Let
lim H Y exists and
a i=1
Y,
be as in Lemma 3.4.
Then
0
-E
)
50.
a)
IIlim
a
or
b)
m
T
k
l
HiYi I -< Kss! i=l
.
i=1
m
<_Kmm!.
Illim YmTkIr
a
Proof.
lim
the limits
By Lemma
I1
(Tko,
a+oo
i
T
a +o)
exist.
IIH YiT £0
i=1
Hence (3.16)
(ITk l Ir
a
lim
+*
and a bound on (3.17)
)-A(a)
(3.16)
=1
m
lim
I I Iriiy., I I
12e
- A(a)
(3.17)
is bounded by
e-A()/2
i
i=1
is obtained in Lemma 3.4.
The
lemma follows from the Riesz representation theorem.
Corollary.
exists, and
Let
fi eJ.
Then
lim
RH
a i=l
(fi)
51.
m
m
llim H 0(f i)TlT
a i=l
Let
Y
be
Ir
N(B p)
_ Kmm!½ H Illfi
i=l
T < 0,
or H0(B ).
(3.18)
0
Then
lim Ym
exists, and
Illim ymTk I*r
m
H ¢(f
Proof.
< Kmm
)
is
(3.19)
the sum of
2m
terms of the
i=l
m
form
HIY
i=l
a( ) . Thus,
,
Yi = a'r(fi),
where
the existence of
follow from Lemma 3.5.
and
a#(f) = a (f)
lim H ¢(fi
ai=
)
and (3.18)
lim ym
The existence of
or
and
(3.19) also follow from Lemma 3.5, provided that (3.7)
is
established for
Y = N(B p ), NT, or
H 0 (Bp).
This is
verified as follows:
(v,HnCN(Bp)Jv) < KvJV(p)2a IjnllE-aVl
1
p
(v,IICN v)
Ir
SK~j Pv
2
2 KJ
<_KvJl IPIV112 <_KI
,
jV(p)2a Iv-al
(v,HyEH0(Bp )v) < KJvJV(p)
for
j > 1,
by (2.6).
12 < K j
52.
CHAPTER IV
Exponential Weyl Systems
In this chapter we prove some general results
concerning unitary operators and
groups in
Fr
which are defined by weak limits.
Particular examples are
ita
and
n-parameter unitary
lim W(tf), fEJ, lim eitN(Bp),
(f)a(f)
lim e,
fEJ.
Properties of these one-
parameter groups and exponential Weyl systems are studied
in Theorems 2, 3, and 4. A sufficient condition for two
exponential Weyl systems not to be disjoint is given in
Theorem 5.
For later purposes, we generalize the notion of weak
limit.
We say that an operator
of
into
F
Fr
Fr ,
B, which maps a subspace
is the weak limit of an operator A
(written B = Lim A)
if Z(B) = U TtD
in
and
a
(Tk4,BTj)r = lim (Tka$,AT'
If
Lim A
,)e - X().
is bounded, then it extends uniquely by continuity
the extension is also written
Fr;
r
then Lim A = lim A.
a
a
to
(4.1)
Lim A.
If
F
r
= F
,
r
53.
Lemma 4. 1.
Suppose that
is non-negative.
Suppose that
on
Lim U
Fr ,
F
such that
Fr , F r
U, V
and
I
respectively.
v ;
are unitary operators
is an operator mapping
lim V, lim V
and such that
m
on
Suppose that v-v
v-v EJ
Fr
are unitary operators
Then
I
Lim U lim V = Lir UV, lir V Lim U = Lir VI.
aO
a
a
a
Oa
a
Proof.
Choose
Tt(n) en
I
T(n)e n
I
I
V-
&
such that
I
r
#0
Then
!iLim U (Tl(n) e
and
into
- lim V T
-)r
0
(4.2)
54.
lim supl(Tk4,Lim U lim V)r-(TkUTa)e-X
a
a
a
sup l(U Tk, Tn
lim limr
< n-.
a1
e
/
<KK li
n--"
-
n-VT/)l
)rnTk am !
e-X (a)
IIT(n)aen-VTta
Tm
1/2
!r,n
( f )T'(n)e
+r!TtIt r -2Re(TL(n)en, im VT ,,)r)
K
a
2T
'2
= K(r -a VTt r) =0.
Thus Lim UV
exists and
Lim U lim V = lim UV.
a
a
Taking adjoints, we observe that
Lim VU = (Lim U lim V)
(lin V) (Lim U)
= lia V Lim U
a
Replacing
*
*
V ,U
by
a
V,U
respectively, and interchanging
t
Fr and Fr, we conclude that the second equality in
(4.2) follows from the first. The lemma is proved.
Corollary.
operators on
on
Fr .
Then
F
Suppose that
and
Ui,
Uir = lim Ui
a
l<i<m,
are unitary
are unitary operators
55.
T Ur =lim r U.
i=1
Proof.
the case
(4.4)
c i=l
The general case follows immediately from
m = 2,
which in turn follows from the lemma
t
Fr = Fr).
(with
It
Q.E.D.
is natural to ask for a sufficient condition that
Lim U be a unitary mapping from
ar
Lemma 4.2.
Suppose that
a unitary operator on
Fr
Fr .
into
F
and
Fr
v = v .
Lim U
Suppose that for each
I
en , enED and k(n),t(n)>O
F .
onto
Suppose
U
is
is an operator mapping
k,&t>O,
there exist
such that
lim lim sup!i UT•,-T (n)a
=0
e-)
and
*
'
'e-A'(o)/2
()
supliU Tk*a-Tk(n)anIe
lim lim
7*
n-ow
Then
Lim U
Proof.
mapping
Fr
isometric.
is a unitary operator mapping
By (4.1),
into
9
Fr .
(Lim U)* = Lim U
=
0.
Fr
(4.5)
onto
Fr.
i s an operator
B
It suffices to show that
Then, replacing
U
by
U
Lim U
and interchanging
is
56.
we conclude that
Fr,
and
Fr
so that
Lim U
(Lim U)
is isometric,
is unitary.
By (4.5),
nlim(Tk 4 , Lim U Ttl-Tt(n)en)r
n
a
= lim lim I (TkO,
n-ow
-T (n)an)Ie - A(( a )
UTt a
a--w
< r Tk Ir lim lim sup IIUTta
--
n-oa
*-T (n)aen!!e-A(c)/2
= 0.
Thus
L~n)
n
IfLim U TL
r
2
F-,
so that
' '2
2-1 1 1T2th
=lim IfT
='
in
T (n)e n-Lim U T
r
< lim lnm (T(
-UT
'8,
n-ana-w
+ lim limr(Tt (n)
a n
n-a a- a
< 2K li
= 0.
T (n)9
on-UTt
supfU ' -T(n)
a sp
U
*) Ie-A (a)
UTtI'a)j
T,(n)a
(
I e-A ( a )/
)le-A(a)
2
2
ene(O)/
57.
The lemma is proved.
F
unitary group on
lim U(x) = Ur(x)
and
n-parameter unitary group on
f n ei(xy)dP
n
R
R
Ur(x)
J' ei(xY)dPy,
Rn
Let
be spectral decompositions of
R ,
function on
U(x)
and
then
n = 1
Suppose
exists for all
j>0O,
g(y)dP
(4.6)
.
are the infinitesimal
A, A r
Then
respectively.
then
is a dense set of
j , j>O,
AJOlim
r a A
CO
vectors for
if
lim A
so that
Ar -
By definition,
Proof.
•)
'rdr
f
= lim
and
U, Ur
for
generators
is a bounded eontinuous
g
If
J g(y)dPry
RRn
i(x
J e - \ -' Rn
Fr .
is an
ry
respectively.
U TkD
k>O
n-parameter
is an
U(x)
Suppose that
Lemma 4.3.
i P r yT
2
r = (dk ,U
r(x)'lk t)
r
= lim(Tat,U(x)TL,,,)e-A(a)
= lim
h
'
urn I n ei(X'Y)d(pyT1
Rý
e
- C2e-
).'
58.
By the Levy continuity theorem for distributions [12,
p. 191, p.
and the Helly-Bray Theorem [12, p.
205]
, 2 =2 lim lg(y)d(, PyTla,,1 2e
jg(y)dPryT
A(c
182],
)).
cr-OCO
(4.6) is then obtained by polarization.
Suppose
n = 1
decompositions of
JAdPV,
and
J'AdPrA
R
respectively.
A, Ar
are the spectral
Then, by the
moment convergence theorem for distributions [12, p.
t
2 = lim J'xJd(IP Tta
INJdI PrhT,
so that EI(ArJ
)DU TkD
for all
184],
2
2e-A(a))
,
j>0.
Again by polarization,
k>0
(Tk ,AJT)
= J' jd(Tk'
rT
= lim fJd((Tka~ , PXTct)e
= lim
(Tk,4,
AJT.,,)
-
A(
c
))
e-A (c)
This completes the proof.
Theorem 2.
lim W(f)
GJ
= Wr(f)
Let
F r = Fr(v,a,n(j)).
= ei 4 r(f), fEJ,
Then
is a Weyl system on
Fr •
59.
U TkD is a dense set of
and
(f)J, j>O,
4r(f)JDlim
a 4
k>O
entire vectors for $r(f). If
!!kC(fn-f)ltIO,
some
E>O,
then
ýr(fn)
-r(f)
U TkD.
k>O
Definition.
and
Wr(f), fEJ,
Wr(fn)-Wr(f)
strongly on
is called an exponential
Weyl system.
Proof of theorem.
By (3.18) and the bounded
convergence theorem,
lim W(f) = lim Z (i4(f))m/m!
a
a m=O
exists.
Since
W(f)
is unitary,
Wr(f)
C lim(i4f)m/m!
E
m=O a
is bounded in
norm by one and extends uniquely to
Fr .
prove that
W(f)* = W(-f)
that
is unitary.
Now
so
Wr(f)* = Wr(-f)
Let
Here
Wr(f)
We wish to
T
p
z (i4(f ))m/m!
m=O
is an upper momentum cutoff:
$ED
and let
e =
f (k)= f(k) Jkl<T
-O
k> .
Tl(k)*ED.
(4.7)
60.
Then by Lemma 4.2, it suffices to show that, given
E>0,
c
e - A ( a)/
I IW(f)TLoa-Tk
<
e-A(cr)/2
T *!4
(i())
m=p+l
P
+
2
m
E it
A
m=l
+
for
p
e-A (a)/
E rt[ii(f )m/Im!, Tk]T *
m=l
kar(k)
p,T,,k>.e
_\ /,'n
]/m! T(r*, e-ilk)/c
2
sufficiently large.
By (3.18),
the first sum from (4.8) is bounded by
V.
(F
,) m -1/2
m=p+l1
which is less than
(f)m-4(f")m
E/3 for
p
large.
Now
= ý(f-f")
i=l
Thus, by (3.18),
the second sum from (4.8) is bounded by
(4.8)
61.
Kit••.6 (f-f'd)"I
which is less that
for
E/3
7
Ttn (k)k[(f)m, Tk
Observe that
large.
[(fT)m
k
n(k)
is the sum of all Wick ordered terms from
• )2mTo
*T~o
in which there is at least one contraction between
114
VA
Trd
\
and
+(f,)
V (k)a.
enyravii
+
C
++A
+
1a etwe
hio
Thus, by Lemma 2.1, the Cauchy-Schwarz
inequality in one of the above contracted variables, (3.8),
(3.9), Lemmas 3.1, 3.2, and 3.3 the third sum from (4.8)
is bounded by
?a+ f11,
Tt
e"
By (2.6) this is less than
E/3
for
k>t
sufficiently
large.
We have proved that
Wr
Wr(f)
satisfies the Weyl relations:
is unitary.
By (4.4),
62.
Wr(f)Wr(g) = lim W(f)W(g) = lim eim(f,g)/2W(f+g)
a
a
= eiIm(f,g)/2Wr
(fg).
By (3.18) and (4.7),
ccr
(KtCf)m:-1/2<Kt-0
.
l
E (Kt1 fPl ")
I(Tk•' (Wr(tf)-Ir)T&$)rjým=l
as
t-O.
Thus the map
at
t=O,
and
t-Wr(tf)
Wr(f), fEJ,
by Lemma 4.3 and (3.18),
is weakly continuous
is a Weyl system.
r(f) Jlim +(f)J,
j>O,
U TkD is a dense set of entire vectors for
k>O
We now prove the regularity properties.
linearity of
+
and the Weyl relations for
suffices to show that
,pC fn"-O
implies that
U TkD.
k>O
directly from (3.18) and (4.7)and
Wr(fn)-Ir
strongly on
IH r(fn)Tk*"r < K"PC fni +O
and
Moreover,
and
ýr(f).
By the real
Wr ,
it
+(fn)-O
This follows
63.
(KlMmm!-1/2
r(Wr(n)-Ir)Tk r
S41
Pe fn!*0
as
n--.
The theorem is proved.
The following result establishes a number operator
for the local system
Wr(f), fEL 2 (Bp),
which are there-
fore direct sums of Fock representations.
Theorem 3.
Let
Fr - Fr(v,in(j))
.
Then
lim eitN(Bp) defines a one parameter unitary group
a
eitNr(BP) on Fr . N(Bp) Olim N(Bp) , j'>0,
and
U TkD
k>O
is a dense set of analytic vectors for
The spectrum of
Nr(Bp)
integers, and, for
is contained in the non-negative
fEL2 (Bp)
eitNr(Bp)Wr(f)e-itNr(Bp)
Thus
Nr(Bp).
Wr(f), fEL2 (Bp),
-
Wr(eitf)
is unitarily equivalent to a
direct sum of Fock representations.
Proof.
Let
V(t) = eitN(Bp).
bounded convergence theorem,
By (3.19) and the
(4.9)
64.
lim V(t)
a
exists for
unitary,
= lim
a m=0
It 1<t 0,
Vr(t)
(itN(BP ))m/m!
P
t.>0.
some
and extends uniquely to
Fr
.
E lim(itN(B )) m/m!
m=0 O
Since
is
V(t)
We prove that
Vr(t)
unitary by essentially the same proof as that for
Let
$ED
(4.10)
bounded in norm by one
is
aT
= lim V(t)
=
is
Wr(f)
and let
p
m=0
Then by Lemma 4.2, it suffices to show that, given
E>O,
IIV(t)Tta -Tkao re
<
+
for
p,k>t
- A (a)/2
U(itN(B p))
m=p+l
E
A
/m! T atl!e - ()/2
m r [(itN(B
))m/m!,TkoT-n(k)e
1
m=l1
sufficiently large,
Itl<t- 0O *
- A(
)
(4.11)
65.
By (3.19), the first sum from (4.11) is bounded by
Z (KlIt o l)m
m=p+ 1
which is less than
Ti
)
E/2
for
p
large.
Observe that
[((k
N(Bp)m, Tka ]*[N(Bp )m,Tka ]Tn(k)
is the sum of all Wick ordered terms from
T.a.N(Bp)2mT'
in which there is at least one contraction between
and
Va(k)a
and
V,(k)o.
and at least one contraction between
N(B )
Nt(Bp)
Thus, by (3.10), Lemmas 3.1, 3.2, and 3.3,
the second sum from (4.11) is bounded by
KIIN(BP)vn(k)l I'"
By (3.7) this is less than
E/2
for
k>4
sufficiently
large.
We have proved that
Hence, by (4.4),
V(t), Itl<t 0 , is unitary.
66.
m
TTVr()
rim
I V(ti),
=
i=l
a i=l
Then we may define
Vr(t),
for all
It i
<t 0 .
t, by
Vr(t) = Vr(t/m) m = lim V(t/m)m = lim V(t),
where
m
is chosen such that
It/mi < t O.
Thus, by
(4.4),
Vr(t)Vt(t )
i.e.
Vr(t)
=
lim V(t)V(t') = lim V(t+t )
Vr (t+t,
is a one parameter unitary group on
By Lemma 4.3 and (3.19), Nr(Bp)JDlim N(B
Fr*
j,
j O,
and
U TkD is a dense set of analytic vectors for
k>O
The same lemma shows that the spectrum of
Nr(p).
N(Bp),
which is the set of nonnegative integers, coin-
cides with the spectrum of
a consequence of (4.4):
Nr(Bp).
Finally, (4.9) is
(4.12)
67.
Vr(t)Wr(f)Vr(-t) = lim V(t)W(f)V(-t) = lim W(eitf)
= Wr(eitf),
fEL2 (Bp).
Thus
Nr(Bp)
is a number operator for Wr(f),
which is therefore unitarily equivalent to a
fEL 2 (Bp),
direct sum of Fock representations [2, Theorem 1].
This
completes the proof.
Let
1Ift1 = 1,
fEJ,
a (f) = 2
2
and let
[/r(f)+i$r(if)], ar(f) = 2-/2
r(f)ir(if)]
be the creation and annihilation operators for the
representation.
Let
Nr(f) = ar(
be the number operators over
[2, Prop. 3.1, Remark 2].
lim eitN(f)
Fr
Let
for
),
N(f)
Wr, W
a (f)a(?)
respectively
They measure the number of
particles with wave function
Theorem 4.
[ff
r(*
f.
Fr = Fr(v,
,n(j)).
Then
defines a one-parameter unitary group on
and
lim eitN(f) = eitNr(f).
a
(4.13)
68.
Nr(f)JDlim N(f)j , j>O,
and
a
U TkD
is a dense set of
k>O
analytic vectors for
Nr(f).
The spectrum of
Nr(f)
is
contained in the non-negative integers.
Suppose
v*L2 (RdV).
Then every vector in
Fr
has
an infinite number of particles for the global system
Wr(f), fEJ.
Thus
Wr(f),
fEJ,
is disjoint from the Fock
representation.
Proof.
Let
Theorem 2, with
V(t) = eitN(f).
4(f,)
replaced by
is a unitary operator, for
by the proof of Theorem 3,
N(f), Vr(t) = lim V(t)
Itf<t 0,
with
By the proof of
N(f), Vr(t) = lim V(t)
some
N(Bp)
t 0 >0.
Thus,
replaced by
is a one-parameter unitary group on
Fr.
By Lemma 4.3, for
Nr(f)J
(
j>O,
r(f)+i r(if) [r(f)-i4r(if)])
3 lim (r~l(f)+in(f)][
(f-i(if)])
= lim N(f)j.
By (3.17),
for
U TkD is a dense set of analytic vectors
k>O
Nr(Bp).
Thus Vr(t) = eitN,(f).
Moreover, the
P
69.
spectrum of
N(f)
is contained in the non-negative
integers, so that by Lemma 4.3, the same statement is
true for
Nr(f).
d
vfL2 (R
).
Let
orthonormal basis of
J
,
Suppose
(fi9i=l' fiEJ,
and let
E mPrm(fi
m>O
E mPm(fi) be the spectral decompositionsof
m>O
N(fi) respectively. Let
S= (y =
i(Y): Yi
be an
)'
Nr(fi),
non-negative integers, yi = 0 almost
alwaysI,
and let
r
be assigned the measure for which each point
has measure one.
P
Then [2, p.
(Py
yET
into
= i Pr
i
i
let
yEP,
(fi),
Py =
31 and Prop. 3.1]
i
PYi
i
(f i )
.
implies that
(Pry yEF,
are both mutually orthogonal families of
projections.
F
For
R
Fix
k,p>O.
and defined by
Let
g, g
be functions mapping
70.
g(Y) = !PyTk~1 2e - A()
Eyi<p
e
YTkcr
ao
= 0
EYi>p
ZYiP
g(Y) = "PryTk1Ii!r
=.
0yi.p
U(x) = eiExjN(fJ), g -g pointwise
By Lemma 4.3, with
as
a--".
For fc J
, dimX1'<-,
let
be the spectral decompositionsof
Z mPrm(x),
E mPm(jr)
m>O
m>0
Nr(73q), N( h,) respectively.
Let
P
Q Orp)
P
E Prm(M),
Q (L)Om
m=O
Q =lim
(
rp
rp
),
m=0
(L)
lim
(
M 0.
"
Here convergence of finite dimensional subspaces is net
convergence,
and
is the projection onto
Qrp< lim q
a
Qr = strong lim Qrp
We assert that
projection
Q
subspace is zero.
= 0.
E Fm
m<p
[2].
Hence the
onto the finite particle
Thus every vector in
Fr
has an
71.
infinite number of particles, so that no subrepresentation
is unitarily equivalent to the Fock representation [2,
Theorem 2].
Let 'M'n = ( f
as
M'Y
'rpTk"
i
•
iil *
in
Noting that
lim
Qp
rp-< lim
aT
increases, we prove that
2
r
r
Qrp (M)
ý
decreases
= 0:
2
lim
Mt4-j
< lim
lim
Tr
n Tk
rp
= lim
gi(y)
i>n)
[Y:Yi.=,
=
J
r
(v)< lim inf F go(y)
7
= lim inf lim
J'
a n-0 [(y=Yi=O,
1
= lim inf
a
= limr inf
i>n
(Fatou's Lemma)
( Y)
lim fQ(mn)TkcI l2 e-A( )
n-#=
2e-A()
Tp ITka!
! Tk,,*!
< lim inf K exppA(a)e
- A(c)
(F
because A(a)-".
The theorem is proved.
= 0
72.
I
Theorem 5.
I
v-v EJ,
are not disjoint if
Moreover, if
j,
f
I
Wr = Wr(v,,n(j)), Wr = Wr(v ,a,n (j))
and
v = v
v v
and
is non-negative.
n(j) = n (j) for almost all
I
then
Wr r and
Wr
r
are unitarily equivalent.
I
Proof.
lim W(f)
By Theorem 2,
Fr,Fr
unitary operators on
and lim W(f)
By Lemma 3.4,
respectively.
I
U= Lim I isan operator mapping Fr into
Fr
and
bounded in norm by
1
exp ( 2
'2
v
)
U is non-zero because
(TO0 ,U(ToO))r = lim
by (3.2) and (3.4).
T
e-Xj(a)exp n(jXj(a)>O
Finally, by (4.2),
UWr(f) = Lim IW(f) = Lim W(f)I
=
=
Thus
U
intertwines
therefore not disjoint.
Wr(f)U.
Wr(f)U-
W
Wr(f
)
and
Wr(f),
are
which are
73.
Now suppose
always.
k>t
Let
rED
v = v
and let
and
n(j) = n (j)
almost
e
= TtT(k) *ED.
Then for
sufficiently large,
Ttcr*' TT(k)a GIIee-A (a)/2
By Lemma 4.2,
U
is unitary.
= 0.
The theorem is proved.
74.
BIBLIOGRAPHY
1.
Ahlfors L.: Complex Analysis. New York: McGraw Hill
1966.
2.
Chaiken, J.: Ann. Physics 42, 23(1967).
3.
Eckmann, J.: A theorem on kernels of superrenormalizable
theories. To appear.
4.
Friedrichs, K.: Perturbation of Spectra in Hilbert
Space. Providence: Amer. Math. Soc. 1965.
5.
Glimm, J.: Commun. Math. Phys. 10, 1(1968).
6.
Glimm, J.: Advances in Math. 3, 101(1969).
7.
Jaffe, A.: Commun. Math. Phys. 1i, 127(1965).
8.
Hepp, K.: Lecture Notes. Ecole Polytechnique (1968-1969).
9.
Hepp, K.: Proceedings of CNRS conference on systems
with an infinite number of degrees of freedom.
Paris 1969.
10. Kristensen, P., L. Mejlbo, and E. Poulsen: Math. Scand.
14, 129(1964).
11. Kristensen, P., L. Mejlbo and E. Poulsen: Commun.
Math. Phys. 6, 29(1967).
12. Loeve, M.: Probability Theory. Princeton: Van Nostrand.
1963.
13. Powers, R.: Princeton thesis (1967).
14. Reed, M.: The damped self interaction.
Commun. Math. Phys.
To appear,
75.
BIOGRAPHICAL NOTE
The author was born in New York, New York on
October 29, 1943.
He attended Cornell University as a
National Honor Society Honorary Scholar and was elected
to Phi Eta Sigman, Phi Beta Kappa, and Phi Kappa Phi.
He graduated in June,,1965, with the degree of A.B.,
with distinction in all subjects and cum laude in
mathematics.
Since September, 1965, he has been a
National Science Foundation Fellow and an Honorary
Woodrow Wilson Fellow at the Massachusetts Institute of
Technology.
He married the former Patricia Enos of
Rochester, New York, on June 3, 1967.