Commun.
math. Phys. 18, 65—81 (1970)
©
by
Springer-Verlag
1970
Renormalization
of a
Quadratic Interaction
in
the
Hamiltonian Formalism
J.
GlNIBRE
Laboratoire
de
Physique Theorique, Faculte
des
Sciences dΌrsay, France*
and
G.
VELO
Istituto
di
Fisica delΓUniversita
di
Bologna
Received February
18,1970
Abstract.
The
method
of the
dressing transformation
is
used
to
perform
a
mass
re-
normalization
of a
neutral scalar free
field
in the
Hamiltonian formalism,
for
arbitrary
space dimension. The resulting situation
is
analyzed
by
means
of a
Bogoliubov transforma-
tion,
and
seen
to
yield
the
expected results.
1.
Introduction
It
is
well
known that quantum
field
theory
is
plagued with divergences
which make
the
construction
of
mathematically meaningful models
a
formidable task. These divergences
are of two
types: ultraviolet
(UV)
divergences, connected with high momentum behaviour,
and
divergences
connected
with
the
infinite volume (Haag's theorem).
A significant progress
in
circumventing
the
first
class
of
divergences
has been made recently
by
Glimm
[1,2]
using the hamiltonian formalism.
He
considered, among others,
the
case
of a
neutral scalar
field
Φ
with
a
Φ
4
interaction
in
three dimensional space time
[2],
with
a
space
cut-
off which eliminates
the
infinite volume divergences.
The
remaining
UV
divergences still make
it
impossible
to
define
the
Hamiltonian
of
the
system
in the
original Fock space. However, Glimm
was
able
to
define
a new
Hubert space
in
which
a
suitably renormalized version
of
the
Hamiltonian makes sense
as a
symmetric operator. Unfortunately,
the
construction
of the new
Hubert space
and of the
renormalized
Hamiltonian
is
fairly
complicated;
in
particular,
it is not
known whether
the
latter
is
semi-bounded
and
can
be
extended
to a
self-adjoint operator.
Moreover,
in
higher dimensional space time
or
with more singular
interactions,
higher divergences occur,
and it is not
clear
how to
extend
the
method
to
such cases.
It is
therefore
of
interest
to
test
the
method
on
a
simple model, namely
the
quadratic Hamiltonian [3,4],
for
which
nevertheless arbitrarily high divergences occur
if
one takes the dimension
*
Laboratoire associe
au
C.N.R.S.
5 Commun. math. Phys.,
Vol. 18
66
J.
Ginibre
and G. Velo:
5 of the space sufficiently large. In particular, for 5^4, UV divergences
require a change of Hubert space as in the Φ
4
theory for s = 2.
In
this paper, we perform the renormalization of the quadratic
Hamiltonian
for arbitrary space dimension in a rigorous way. Starting
from an interaction Hamiltonian with a space cut-off and a UV
cut-off,
we
first
apply Glimm's method to remove the UV cut-off and to define
a
new Hubert space and renormalized Hamiltonian as a symmetric
operator
in this space (Section 2). In order to identify the new space
and
Hamiltonian we
follow
a different path and perform a Bogoliubov
transformation, which almost diagonalizes the original Hamiltonian.
This makes it possible to define another renormalized Hamiltonian in
the
Fock space of the Bogoliubov transformed of the original creation
and
annihilation operators. The two procedures turn out to be equivalent:
in
fact, there
exists
a unitary mapping from one Hubert space to the
other,
such that the matrix elements of the operator defined by the
first
method
correspond under this mapping to the matrix elements of the
operator
defined by the second method. This identification shows that
the
renormalized hamiltonian is positive and has a unique ground state
(Section
3). We can then proceed to remove the space
cut-off.
It is re-
markable that, for this limit, no further change of Hubert space is
required. The
field
operators and the Hamiltonian converge in the strong
operator
topology on a suitable domain, and we end up with a free
field
corresponding to the renormalized mass (Section 4). Technical verifica-
tions are collected in two appendices.
2.
Renormalization
Using a Dressing
Transformation
In
this section, we perform the renormalization of the quadratic
Hamiltonian
for a neutral scalar
field
of mass m in s-dimensional space,
following
closely Glimm [2]. We use the
following
notations [5]: let §
be the Fock space; a vector ψ in § is a sequence {ψ (K); K =
k
l9
...,
k
n
\
n =
0,1,2,...}
of symmetric square integrable functions of n vectors in R
s
.
The
scalar product is defined by:
where
-\sdk
1
...dk
n
.
n
n = 0
The
vacuum state is denoted by Ω. We shall make use later on of the
set 3ι
C
9)
of vectors with a finite number of particles and with
wave
functions belonging to the Schwartz space Sf. The creation and annihila-
Renormalization
of Quadratic Interaction 67
tion
operators, the
field
operator and the free Hamiltonian are defined
respectively by:
(a(k)φ)(K)
= ψ(K,k)
where K
t
is obtained from K by omitting fc
f
.
Φ(x)
= (2π)-
s/2
J dk{2ω(k))-
1/2
e
ikx
{a(k) + a
+
(- k)),
H
0
= $dkω(k)a
+
(k)a(k) (2.0)
where ω(fc) = (fc
2
+
m
2
)
1/2
.
The interaction Hamiltonian is
given
formally by:
V=2λ$dx:Φ{x)
2
:
and
corresponds to a mass renormalization δm
2
= 4λ.
For
this operator to make sense in Jr>, it is necessary to introduce a
space cut-off and (for s ^ 2), an ultra violet (UV)
cut-off.
Let / be a
real positive even function in Sf, normalized to /(0) = 1, with positive
(even) Fourier transform / defined by:
ikx
=
(2πΓ
s
Sdxf(x)e
i
f
will
serve
as a space cut-off and
will
be kept
fixed
until Section 4,
where we shall take the limit /-> 1, i.e. f-+δ. Let χ
σ
(k) be a positive ^°°
function with compact support, equal to one for |fe| < σ. The elimination
of the UV cut-off consists in taking the limit σ->oo. Define:
Φ
σ
(x) =
(2π)"
s/2
J dk(2ω(k))-
^
2
χ
σ
(k)
e
ikx
(a(k) + a
+
(- k))
V
σ
= 2λSdxf(x):Φ
σ
(x)
2
:
V
σ
exists
as an operator in § and H
σ
= H
o
+ V
σ
is essentially
self
adjoint
on
the domain of H
o
(Appendix 1). V
σ
can be decomposed as a sum:
K=v
Oβ
+
v
lσ
+v
2σ
where
V
Oσ
= V+
σ
= λ$dk
dl\_ω{k)
ω(0] '
1/2
f(k + ΐ) χ
σ
(fe) χ
σ
(0 α(fc) a(l),
F
lσ
= 2A J dkdl lω(k) ω(t)T
1/2
/(fc- 0
X*(k)
χ.(0 α
+
(k)
a(l).
The UV divergences depend in an essential way on the dimension of
the
space. We expect the
following
situation to happen [6]: for 5 = 1,
no
divergence occurs, V^
exists
as an operator on the domain of H
θ9
68
J.
Ginibre and G.
Velo:
and
no UV
cut-off
is
needed. For
s = 2,H^ is
defined
in §, but
n@(H
0
)= {0}. For
s
= 3,
a
change
of
domain
is
again necessary,
and
an
infinite constant
has to be
added
to the
Hamiltonian.
For s^4,
it
is
no longer
possible
to
define
H^ in §,
and
a
change
of
Hubert space
is required.
In
order
to
perform the change
of
domain
or
the change
of
Hubert
space,
we
introduce
a
dressing
transformation [2,5]:
W
σ
)
(2.1)
where
W
σ
=^dkdlw
σ
(K-I)a
+
(k)a
+
(l)
Here,
vv
σ
(/c,
/)
is
a
real
L
2
-
kernel
satisfying
vv
σ
(/c,
/)
=
w
σ
(/, k)
= w
σ
( - fc, -
/),
to
be
chosen
below.
w
σ
defines
a
bounded
self
adjoint operator
in the
one
particle space
ξ)
ί
=L
2
(R
s
,dk),
which
we
shall again denote
by w
σ
.
With this definition,
the
following
identity holds:
(H
σ
+ C
σ
)
T
σ
=
T
σ
[_H
σ
-
(H
o
+ V
Oσ
+
V
lσ
)+W
σ
+1
V
Oσ
^-W^
(2.2)
where -ζ- denotes the connected product with
p
contractions [2,
5]
and:
(2.3)
Since we want
to
use (2.2)
to
define the renormalized Hamiltonian
in
the
limit σ-*oo,
we
choose
W
σ
so as to
eliminate the most singular terms
in
the
RHS, namely, in the square bracket, the terms containing two creation
operators. This
gives
the equation:
R
σ
^V
2σ
-H
0
^-W
σ
-V
lσ
^W
σ
+$V
0σ
^-W
σ
2
= 0. (2.4)
In
the next section and
in
Appendix 2,
we
shall prove that for
λ >
— m
2
/4,
Eq.
(2.4)
has a
solution
w
σ
with
the
following
properties:
for
σ^oo,
w
σ
(fc, /)
is
real symmetric bounded
^°°
function
of
(fc, /), and Suρ||wJ
< 1.
For
σ
finite, w
σ
(/c, /)
has
compact support
in
(/c, /), and therefore
w
σ
is a
Hubert
Schmidt operator: Ύrw
2
<
oo. For
σ
infinite,
Trvv
2
can
be
shown
to
be
infinite
for s ^
4. When
σ
tends
to
infinity,
w
σ
tends
to w in the
strong operator topology.
From
now on,
we
choose this specific
w
σ
.
We
shajl
now use
T
σ
to
define
a
new scalar product on
a
dense subset
of
ξ>.
For this purpose,
it
will
be
convenient to have an explicit expression
for {T
σ
ψ, T
σ
φ).
To
perform this calculation, the
following
notations
are
useful. Let
K =
(k
l9
.'.., k
n
) and
L
=
(^,...,
/J. Let α
e
L
2
(R
s
, dk)
be a
one
particle
wave
function.
We
define
a
vector £α
e§ by:
Ea(K)=f\a(k
i
).
Renormalization
of
Quadratic
Interaction
69
Let
s(/c, /)
be a
symmetric function
of
two vectors (/c, /).
We
define (cf. [7]):
E{s){K\L) =
0 for
m +
n
n
=
Σ
Π^/'W
for
m==n
where
the
sum runs over
all
permutations
π of
(1, 2,..., ή).
We finally denote
by
E(S) (K)
the sum
over
the
partitions
of K in
pairs
of
variables (this requires that
n be
even
for
E(s) (K)
to be
non zero)
of the products
of
the
s(k
h
— kj)
for all the
pairs
of
the partition.
We shall also
use the
notations E{s)(K\L)
and
E(s)(K) with
s an
operator
in
L
2
(R\dk)
represented
by a
symmetric even kernel s(fc, /)•
We can now state
the
following
result:
Lemma.
Let σ be
finite,
φe!3,ψe@.
Then:
(2.5)
dK dL
άlL
ψ(K, K')
WΛ)
(2.6)
)
Proof.
In
order
to
deal with
the
combinational problem,
it is
suffi-
cient
to
consider
the
case where
w
σ
is a
finite matrix and where
a
finite
number
of
modes occur.
For
each mode,
we
introduce coherent states
[8] \ξ
k
) defined
by
a
k
\ξ
k
)
=
\ξ
k
), (ξ
k
\ξ
k
)
= 1,
where
ξ
k
is a
complex number.
Let
|£)=(X)|£fc). These states
satisfy
the
completeness relation:
2π
where
dξ
k
dξ
k
= 2
d(Re £
fe
) d(Im 4).
We
first
consider
the
case where
φ =
£α,
ψ =
£/?, where
α and jS
are
one particle
wave
functions. Using (2.7),
we
obtain:
{T
σ
φ,T
σ
xp)
=
{T
σ
Ea,T
σ
Eβ)
=
l]\^-^{E^ξ)
k
Zπ
x
{ξ,
Eβ)
exp I
- 1 Σ
*σ*. -i(fJi + &ί i)}
I
L
k,l J