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arXiv:hep-th/0003232v4 14 Dec 2000

SU/4240-718

Noncommutative Geometry As A Regulator

Badis Ydri1.

Physics Department , Syracuse University.

Syracuse , N.Y , 13244-1130 , U.S.A.

Abstract

We give a perturbative quantization of space-time R4in the case

where the commutators Cµν= [Xµ,Xν] of the underlying algebra

generators are not central . We argue that this kind of quantum

space-times can be used as regulators for quantum field theories .

In particular we show in the case of the φ4theory that by choosing

appropriately the commutators Cµνwe can remove all the infinities

by reproducing all the counter terms . In other words the renormal-

ized action on R4plus the counter terms can be rewritten as only a

renormalized action on the quantum space-time QR4. We conjecture

therefore that renormalization of quantum field theory is equivalent

to the quantization of the underlying space-time R4.

1E-mail: ydri@suhep.phy.syr.edu

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1. Introduction

Noncommutative geometry [1] allows one to define the geometry of a

given space in terms of its underlying algebra . It is therefore more general

than the ordinary differential geometry in the sense that it enables us to

describe algebraically the geometry of any space whether or not it is smooth

and/or differentiable . It is generally believed that NCG can be used to

reformulate if not to solve many problems in particle physics and general

relativity such as the problem of infinities in quantum field theories and its

possible connection to quantum gravity [2, 3, 4, 5, 6, 7] . The potential of

constructing new nonperturbative methods for quantum field theories using

NCG is also well appreciated [2, 3, 4, 8, 9, 10, 11, 12, 13, 14, 15, 16] . The

recent major interest in NCG however was mainly initiated by the work of

[17] on Yang-Mills theory on noncommutative torus and its appearance as

a limit of the matrix model of M-theory . The relevance of NCG in string

theory was further discussed in [18] .

Quantum field theories on noncommutative space-time was extensively

analysed recently in the literature [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]

and it was shown that divegences although not completely removed they

are considerably softened . The reason is that the quantization of R4or

R2by replacing the coordinate functions xµby the coordinate operators Xµ

in the sense of [6] will only modify vertices in the quantum theory and not

propagators . On compact spaces in the other hand such as the 4−sphere

S4[8] , the 2− sphere S2[9] and CP2[15] divegences are automatically

cancelled out when we quantize the space and that is because on comapct

spaces (which was not the case on noncompact spaces) quantization leads to

a finite number of degrees of freedom (points).

It is hoped that noncommutative geometry will shed new lights on the

meaning of renormalization because it provides a very powerfull tools to for-

mulate possible physical mechanisms underlying the renormalization process

of quantum field theories . One such mecahnism which was developed by

Deser [34] , Isham et al.[35] and pursued in [30, 31] is Pauli’s old idea that

the quantization of gravity should give rise to a discrete structure of space-

time which will regulate quantum field theories . As one can immediately

see the typical length scale of Pauli’s lattice is of the order of Planck’s scale

λpwhich is very small compared to the weak scale and therefore corrections

to the classical action will be very small compared to the actual quantum

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corrections. This idea however is still very plausible especially after the di-

covery made in [19] of an UV-IR mixing which could be used in a large extra

diemension-like activity to solve the above hierarchy problem .

The philosphy of this paper will be quite different . We will assume that

space-time is really discrete and that the continuum picture is only an ap-

proximation [30]. The discreteness however is not given a priori but it is a

consequence of the requirement that the quantum field theory under consid-

eration is finite . The noncommutativity parameter θ is therefore expected

to be a function of both the space-time and the Quantum field theory and it

is completely determined by the finiteness requirement . This simply mean

that the quantization of space-time is achieved by replacing the coordinate

functions xµby the coordinate operators Xµas in [33] but , and to the con-

trary to what was done in [6] , these operators will not satisfy the centrality

conditions [Xµ,[Xν,Xα]] = 0 .

The paper is organized as follows : In section 1 we introduce the star

product [36] for the case where the noncommutativity parameter θ is not

a constant . The necessary and sufficient condition under which this star

product is associative turns out to be simply [Xµ,[Xν,Xα]] = 0 .

associativity requirement however is relaxed and allowed to be broken to the

first order in this double commutator . This relaxation is necessary because

one can check that we can not generalize [6] , by making the commutators

[Xµ,Xν] not central , while simultaneously preserving associativity . The

algebra (A,∗) where A is the algebra of functions on R4is then defined .

In section 2 we quantize perturbatively the algebra (A,∗) . In other words

we find the homomorphism (A,∗)−→(A,×) order by order in perturbation

theory where A is the algebra of operators generated by the coordinate oper-

ators Xµ. The star product becomes under quantization the nonassociative

operator product × and the corresponding Moyal bracket becomes the com-

mutator [.,.]×[37] . The difference between × and the ordinary dot product

of operators is of the order of the double commutator[Xµ,[Xν,Xα]] . This

is basically an example of deformation quantization [36, 37, 38, 39] and in

particular it shows explicitly the result of [38] that Doplicher et al.[6] quan-

tization prescription of space-time is a deformation quantization of R4. We

rederive also the space-time uncertainty relations given in [6] . In section 3

we construct a Dirac opertor on the quantum space-time QR4, write down

the action integrals of a scalar field in terms of the algebra (A,×) as well

as in terms of the algebra (A,∗) . Finiteness requirement is then used to

fix θ in the two loops approximation of the φ4theory . Section 4 contains

The

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conclusions and remarks .

2. The Star Product

2.1 Associativity

Let R4be the space-time with the metric ηµν= (1,1,1,1) . The algebra

underlying the whole differential geometry of R4is simply the associative

algebra A of functions f on R4. It is generated by the coordinate functions

xµ, µ = 0,1,2,3 . This algebra is trivially a commutative algebra under the

pointwise multiplication . A review on how the algebra (A,.) captures all

the differential geometry of R4can be found in [2, 3, 4, 5] .

It is known that we can make the algebra A non-commutative if we replace

the dot product by the star prduct [36] . The pair (A,∗) is then describing

a deformation QR4of space-time which will be taken by definition to be the

quantum space-time . The ∗ product is defined for any two functions f(x)

and g(x) of A by [18]

f ∗ g(x) = e

i

2Cµν(x)

∂

∂ξµ

∂

∂ηνf(x + ξ)g(x + η)|ξ=η=0

(2.1)

where Cµνform a rank two tensor C which in general contains a symmetric

as well as an antisymmetric part[11] . It is assumed to be a function of x of

the form

Cµν(x) = χ(x)(θµν+ iaηµν)(2.2)

where χ(x) is some function of x . θ is the antisymmetric part and it is an

x-independent tensor . a is as we will see the non-associativity parameter

and it is determined in terms of the tensor θ as follows . The requirement

that the star product (2.1) is associative can be expressed as the condition

that I = 0 where I is given by :

I = (eipx∗ eikx) ∗ eihx− eipx∗ (eikx∗ eihx).

eipxare the generators of the algebra A written in their bounded forms .

Using the definition (2.1) we can check that

(2.3)

eipx∗ eikx= e−i

2pCkei(p+k)x

(2.4)

and therefore (2.3) takes the form

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Ie−i(p+k+h)x

= e−1

− e−1

2Cµν(x)hν

∂

∂ξµ[e−i

2pC(x+ξ)k+i(p+k)ξ]|ξ=0

∂ξµ[e−i

2Cνµ(x)pν

∂

2kC(x+ξ)h+i(k+h)ξ]|ξ=0.(2.5)

To see clearly what are the kind of conditions we need to ensure that the

equation I = 0 is an identity , we first expand both sides of (2.5) in powers

of C and keep terms only up to the second order . It will then read

I =i

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?

Cµνhνp∂C

∂xµk − Cνµpνk∂C

∂xµh

?

.(2.6)

As we can clearly see the associativity of the star product at this order is

maintaned if and only Cµν∂µC = 0 and Cνµ∂µC = 0 . The two consequences

of these two conditions are given by the equations aηµν∂µχ = 0 and θµν∂µχ =

0 . The first equation is simply a = 0 because the solution χ = constant will

be discarded in this paper . The second equation in the other hand means

as we can simply check that the noncommutativity matrix θ is singular , i.e

detθ = 0 . We can aslo check that the two above conditions are necessary

and sufficient to make the star product (2.1) associative at all orders because

of the identities θµ1ν1θµ2ν2..θµnνn∂n

If we would like to avoid the singularity of the noncommutativity matrix

θ we have then to relax the requirement of associativity . We can start by

reducing the associativity of the star product (2.1) by imposing only one of

the above two conditions , say

µ1,µ2,..,µnχ = 0.

Cµν∂C

∂xµ

=0

=⇒

=

Cµν∂χ

∂xµ

0.(2.7)

Before we analyze further this equation , we remark that this condition on

the tensor C will lead to the identities

Cµ1ν1Cµ2ν2..Cµnνn∂n

µ1,µ2,..,µnCαβ= 0.(2.8)

(2.7) will also lead to the equation

Cνµ∂C

∂xµ= i∂νχ2

?

aθ + ia2η

?

.(2.9)

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