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processed by the SLAC/DESY Libraries on 29 May 1995.
HEP-TH-9505175
CERN-TH/95-138
UWThPh-19-1995
hep-th/9505175
Towards Finite Quantum Field Theory
in Non-commutative Geometry
H. Grosse
1
Institut for Theoretical Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria
C. Klimck
2
Theory Division CERN, CH-1211 Geneva 23, Switzerland
P. Presna jder
Department of Theoretical Physics, Comenius University
Mlynska dolina, SK-84215 Bratislava, Slovakia
Abstract
We describe the self-interacting scalar eld on the truncated sphere and
we p erform the quantization using the functional (path) integral approach.
The theory p osseses a full symmetry with resp ect to the isometries of the
sphere. We explicitely show that the mo del is nite and the UV-regularization
automatically takes place.
CERN-TH/95-138
May 1995
1
Part of Pro ject No. P8916-PHY of the `Fonds zur Forderung der wissenschaftlichen
Forschung in
Osterreich'.
2
Partially supp orted by the grantGA
CR 2178
1 Intro duction
The basic ideas of the non-commutative geometry were developed in [1, 2],
and in the form of the matrix geometry in [3, 4]. The applications to physical
models were presented in [2, 5], where the non-commutativitywas in some
sense minimal: the Minkowski space was not extended by some standard
Kaluza-Klein manifold describing internal degrees of freedom but just bytwo
non-commutative points. This led to a new insight on the
SU
(2)
L
N
U
(1)
R
symmetry of the standard model of electro-week interactions. The mo del was
further extended in [6] inserting the Minkowski space by pseudo-Riemannian
manifold, and thus including the gravity. Such models, of course, do not
lead to UV-regularization, since they do not introduce any space-time short-
distance behaviour.
Toachieve the UV-regularization one should introduce the non-commuta-
tivityinto the genuin space-time manifold in the relativistic case, or into
the space manifold in the Euclidean version. One of the simplest locally
Euclidean manifolds is the sphere
S
2
. Its non-commutative (fuzzy) analog
was describ ed by [7] in the framework of the matrix geometry. More general
construction of some non-commutative homogenous spaces was described in
[8] using coherent states technique.
The rst attempt to construct elds on a truncated sphere were presented
in [9] within the matrix formulation. Using more general approach the clas-
sical spinor eld on truncated
S
2
was investigated in detail in [10-11].
In this article article we shall investigate the quantum scalar eld on the
truncated
S
2
.We shall explicitely demonstrate that the UV-regularization
1
automatically app ears within the context of the non-commutative geometry.
We shall introduce only necessary notion of the non-commutative geometry
we need in our approach. In Sec. 2 we dene the non-commutative sphere
and the derivation and integration on it. In Sec. 3 weintroduce the scalar
self-interacting eld on the truncated sphere and the eld action. Further,
using Feynman (path) integrals we p erform the quantization of the mo del in
question. Last Sec. 4 contains a brief discussion and concluding remarks.
2 Non-commutative truncated sphere
A) The innite dimensional algebra
A
1
of p olynomials generated by
x
=
(
x
1
;x
2
;x
3
)
2
R
3
with the dening relations
[
x
i
;x
j
]=0
;
3
X
i
=1
x
2
i
=
2
(1)
contains all informations about the standard unit sphere
S
2
embedded in
R
3
.
In terms of spherical angles
and
'
one has
x
=
x
1
ix
2
=
e
i'
sin
; x
3
=
cos
:
(2)
As a non-commutative analogue of
A
1
we take the algebra
A
N
generated
by^
x
=(^
x
1
;
^
x
2
;
^
x
3
) with the dening relations
[^
x
i
;
^
x
j
]=
i"
ij k
^
x
k
;
3
X
i
=1
^
x
2
i
=
2
:
(3)
The real parameter
>
0characterizes the non-commutativity (later on it
will b e related to
N
). In terms of
^
X
i
=
1
^
x
i
;i
=1
;
2
;
3, eqs. (3) are changed
2
to
[
^
X
i
;
^
X
j
]=
i"
ij k
^
X
k
;
3
X
i
=1
^
X
2
i
=
2
2
;
(4)
or putting
X
=
X
1
iX
2
we obtain
[
^
X
3
;
^
X
]=
^
X
;
[
^
X
+
;
^
X
]=2
^
X
3
;
(5)
and
C
=
^
X
2
3
+
1
2
(
^
X
+
^
X
+
^
X
^
X
+
)=
2
2
:
(6)
We shall realize eqs. (4), or equivalently eqs. (5) and (6), as relations
in some suitable irreuducible unitary representations of the
SU
(2) group. It
is useful to p erform this construction using Wigner-Jordan realization of the
generators
^
X
i
;i
=1
;
2
;
3, in terms of two pairs of annihilation and creation
operators
A
;A
;
=1
;
2, satisfying
[
A
;A
]=[
A
;A
]=0
;
[
A
;A
]=
;
;
(7)
and acting in the Fock space
F
spanned by the normalized vectors
j
n
1
;n
2
i
=
1
p
n
1
!
n
2
!
(
A
1
)
n
1
(
A
2
)
n
2
j
0
i
;
(8)
where
j
0
i
is the vacuum dened by
A
1
j
0
i
=
A
2
j
0
i
= 0. The op erators
^
X
,
and
^
X
3
take the form
^
X
+
=2
A
1
A
2
;
^
X
=2
A
2
A
1
;
^
X
3
=
1
2
(
N
1
N
2
)
;
(9)
where
N
=
A
A
;
=1
;
2. Restricting to the (
N
+ 1)-dimensional subspace
F
N
=
fj
n
1
;n
2
i2Fg
;
(10)
3
we obtain for any given
N
=0
;
1
;
2
; :::
, the irreducible unitary representa-
tion in which the Casimir operator (6) has the value
C
=
N
2
N
2
+1
;
(11)
i.e. the
and
N
are related as
1
=
s
N
2
N
2
+1
:
(12)
The states
j
n
1
;n
2
i
are eigenstates of the op erator
X
3
, whereas
X
+
and
X
are rising and lowering op erators respectively
X
3
j
n
1
;n
2
i
=
n
1
n
2
2
j
n
1
;n
2
i
;
X
+
j
n
1
;n
2
i
=2
q
(
n
1
+1)
n
2
j
n
1
+1
;n
2
1
i
;
X
j
n
1
;n
2
i
=2
q
n
1
(
n
2
+1)
j
n
1
1
;n
2
+1
i
:
(13)
Since
X
i
:
F
N
!F
N
,wehave
dim
A
N
(
N
+1)
2
:
(14)
B) As a next step we extend the notions of integration and derivation to
the truncated case. The standard integral on
S
2
I
1
(
F
)=
1
4
Z
d
F
(
x
)=
1
4
Z
+
d'
Z
0
sin
dF
(
; '
) (15)
is uniquely dened if it is xed for the monomials
F
(
x
)=
x
l
+
x
m
x
n
3
. It is
obvious that
I
1
(
x
l
+
x
m
x
n
3
)=0for
l
6
=
m
, and that
x
l
+
x
l
x
n
3
=
2
l
+
n
sin
2
l
cos
n
is a p olynomial in cos
=
x
3
. An easy calculation gives
I
1
(
x
2
n
+1
3
)=0
; I
1
(
x
2
n
3
)=
2
n
2
n
+1
;
4