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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
4
?
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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In order to have a very small amount of nonassociativity in the theory we
will assume that a is a very small parameter in such a way that only linear
terms in a are relevant . Putting (2.7) and (2.9) in (2.5) will then give
I =ia
2(kθh)O(p,k + h,χ,∂χ)e−iχ
2kθhei(p+k+h)x, (2.10)
where O is a function (which we will not write down explicitly )of the mo-
menta p,k,h and of χ and all of its derivatives {∂χ} . This function O is
such that it vanishes identically if ∂µχ = 0 . In other words a trivial solution
to the equation I = 0 is χ = constant which we will discard in this paper .
We would like to determine χ from the requirement that the quantum field
theory which we will eventually write down on QR4is finite . So we will
leave χ arbitrary at this stage . Clearly χ will be model depenedent and it
can generally be put in the form
χ(x) =
?
n=1
¯ hnχn(x) (2.11)
where we don’t have a tree level term because by assumption this function
will be entirely determined by the different infinities of the theory which are
generally of higher orders in ¯ h . In other words the zero order is absent in
(2.11) because QFT’s are usually finite at this order .
It is instructive to solve equation (2.7) for θ in terms of χ . We assume
that ∂µχ?=0 and rewrite the equation (2.7) in the form Cµν∂µχ = λeνwhere
λ is a small number and e is a four-vector given by (1,0,0,0) . Solving (2.7)
for θ will give the following equation
idetC
?
a3−a
detC
2
µ,ν?=0θµνθµν
= λχ3
∂0χ
−a2θ0i− iaθiµθ0µ+ θjk√detθ
= λχ3
∂iχ,
(2.12)
with
detC = χ4[detθ + a4−a2
2θµνθµν]. (2.13)
(ijk) are the even permutations of (123) and detθ is given by : detθ =
[1
8ǫµναβθµνθαβ]2. The 4 equations (2.12) provide 4 constraints on the tensor
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θ which reduce at limit λ−→0 to one constarint given by
detθ = −a4+a2
2θµνθµν
(2.14)
This is a generalization of the quantization conditions chosen in [6] . This
equation however can be thought of as giving the nonassociativity parameter
a in terms of the noncommutativity matrix θ . The solution is
a =
?1
4θµνθµν−
?
(1
4θµνθµν)2− detθ
?1
2. (2.15)
As we can see from the above analysis it is necessary and sufficient to choose θ
in such a way that (2.15) is a very small number in order for the associativity
of the star product (2.1) to be broken with the very small amount given by
(2.10) .
Using the ∗ product (2.1) we can define the Moyal bracket of any two
functions f(x) and g(x) by {f(x),g(x)} = f∗g(x)−g∗f(x) and in particular
the Moyal bracket of two coordinate functions is given by
{xµ,xν} = iχ(x)θµν. (2.16)
For self-consistency this bracket should satisfy the Jacobi identity
{xβ,{xµ,xν}} + {xν,{xβ,xµ}} + {xµ,{xν,xβ}} = 0, (2.17)
but
{xβ,{xµ,xν}} = −iaχ(∂βχ)θµν. (2.18)
Clearly at the limit of associativity (a−→0) , equation (2.18) is simply zero
and therefore (2.17) holds . We would like however to maintain Jacobi iden-
tity even for a?=0 . we then need to impose the following constraint on θ
θαβθµν+ θανθβµ+ θαµθνβ= 0.(2.19)
which will make (2.17) an identity . A class of solutions to the equation
(2.19) can be given by those antisymmetric tensors θ such that
θµν= aµ
αaν
βθαβ
0
(2.20)
where aµ
which satisfies
αare arbitrary real numbers , and θ0 is an antisymmetric tensor
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θµν
0θαβ
0
= (ηµαηνβ− ηµβηνα). (2.21)
(2.19) is the only constarint we need to impose on the tensors θ in order
to have both the associativity requirement in the sense of (2.10) and Jacobi
identiy (2.17) to be satisfied . By requiring that (2.16) should lead to a
certain kind of space-time uncertainty relations we can further restrict the
allowed antisymmetric tensors θ as we will see in the next section .
2.2 The Algebra (A,∗)
A general element f(x) of A will be defined by
f(x) =
?
d4p
(2π)4˜f(p,χ)eipx
(2.22)
where˜f is a smooth continuous function of the 4-vector p and of the fuzzyness
function χ which satisfies˜f∗(−p,χ) =˜f(p,χ) . It is of the general form
˜f(p,χ) =˜f0(p,χ)+ a˜f1(p,χ) . The ∗ product (2.1) can then be rewritten as
f ∗ g(x) =
?
d4p
(2π)4
d4k
(2π)4
?
˜f(p,χ)˜ g(k,χ)e−i
2pCk
+ a˜f(p,χ)∂˜ g(k,χ)
∂χ
O(p,k,χ,∂χ)
?
ei(p+k)x.
=
?
d4p
(2π)4˜f ∗ ˜ g(p,χ)eipx. (2.23)
O(p,k,χ,∂χ) is the function defined by the equation (2.10) . The Fourier
transform˜f ∗ ˜ g(p,χ) =˜f ∗ ˜ g(p,χ)0+a˜f ∗ ˜ g(p,χ)1is given in the other hand
by
˜f ∗ ˜ g(p,χ) =
?
d4k
(2π)4
?
˜f(p − k,χ)˜ g(k,χ)e−i
2(p−k)Ck
+ a˜f(p − k,χ)∂˜ g(k,χ)
∂χ
O(p − k,k,χ,∂χ)
?
. (2.24)
The function˜f(p,χ) can always be expanded as :˜f(p,χ) =?
which suggests that (2.22) can be rewritten in the form [6]
n=0an¯fn(χ)˜f(p)
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f(x) =
?
n=0
anfn(x)(2.25)
where
fn(x) =¯fn(χ)
?
d4p
(2π)4˜fn(p)eipx. (2.26)
fn(x) are the generators of the algebra (A,∗) written in a way which will
allow us to see the classical limit defined by χ−→0 . In this limit they must
generate the algebra (A,.) . Therefore the functions¯fn(χ) are such that they
tend to a constant when χ−→0 . This constant can always be chosen to be
1 .
2.3 Change of Generators Basis
Finally we would like to rewrite (2.16) in way which will be more suit-
able for quantization . This will involve finding a basis zµ(x) for which the
Moyal bracket {zµ,zν} is in the center of the algebra (A,∗) , in other words
{xα,{zµ,zν}} = 0 . This is not the case for the basis xµas we can see from
equation (2.18) . We then must have {zµ,zν} = iθµνC(x) where C(x) is any
function of x which does commute (in the sense of Moyal bracket) with the
elements of the algebra (A,∗) . To find such a basis we need first to find the
central elements C(x) of the algebra (A,∗) . To this end we first remark that
by using the equation (2.1) the Moyal bracket of the generator xµwith any
function f(x) is given by
{xµ,f} = iχθµν∂f
∂xν. (2.27)
It is then clear that the only obvious solutions to the equation {xµ,f} = 0
are the trivial ones , namely the constant functions . However choosing the
central element C(x) to be a constant is not good because it will lead to a
singular basis at χ(x) = 0 which can be seen from the fact that the Moyal
bracket {zµ,zν} at χ(x) = 0 will then not vanish on the contrary to what
happens to the Moyal bracket (2.16) which clearly vanishes at χ = 0 . So
we must find at least one central element which is not a constant function .
The only clear way to find such an element is to use perturbation theory .
We assume then that the quantum field theory which we will write on QR4
is relevant only up to the ¯ hNorder . The function χ(x) will then take the
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form
χ(x) =
N
?
n=1
¯ hnχn(x) (2.28)
and we would have that
χN+1(x) = 0. (2.29)
This last equation can be rewritten by using equation (2.27) as
{xµ,χN} = 0, (2.30)
in other words χNis a central element of the algebra A in the ¯ hNapprox-
imation . Actually any combination of the order of ¯ hNis central as it can
be seen from equations (2.27) and (2.29) . By choosing C(χ) = χN(x) , the
Moyal bracket of any two coordinates zµ(x) and zν(x) will then read
{zµ,zν} = iχNθµν. (2.31)
xµand zµ(x) give equivalent descriptions of the algebra (A,∗) and therefore
the quantization of (2.16) is equivalent to the quantization of (2.31) . It is
obvious however that the quantization of (2.31) is more straight forward than
the quantization of (2.16) . The new basis zµ(x) can be found in terms of
xµas follows . First we note that for the purpose of finding zµit is sufficent
to work up to the second order in C . the star product (2.1) of any two
functions f(x) and g(x) will read up to this order
f ∗ g(x) = f(x)g(x) +i
2χ(x)(θµν+ iaηµν)∂f
∂xµ
∂g
∂xν
∂2f
∂xµ∂xα
−
1
8χ2(x)(θµν+ iaηµν)(θαβ+ iaηαβ)
∂2g
∂xν∂xβ, (2.32)
and therefore the Moyal bracket of these two functions is
{f,g} = iχ(x)θµν∂f
∂xµ
∂g
∂xν−ia
2χ2(x)θαβηµν
∂2f
∂xµ∂xα
∂2g
∂xν∂xβ. (2.33)
In particular the Moyal bracket of the two coordinates zµ(x) and zν(x) is
given by
{zδ,zσ} = iχ(x)θµν∂zδ
∂xµ
∂zσ
∂xν−ia
2χ2(x)θαβηµν
∂2zδ
∂xµ∂xα
∂2zσ
∂xν∂xβ. (2.34)
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Comapring (2.31) and (2.34) will then give that
θµν∂zδ
∂xµ
=⇒∂zµ
∂zσ
∂xν
= χN−1θδσ
∂xν
= χ(x)
N−1
2ηµ
ν. (2.35)
(2.35) define scaling transformations which depend on space-time points . A
more thorough study of these transformations will be reported elsewhere .
As we can clearly see the definition (2.35) of the new basis zµin terms of
xµwill make the quadratic term in (2.34) vanishes , and for that matter all
terms which are higher orders in C will also vanish. We would like now to
rewritte (2.35) in a form which is better suited for quantization . To this
end we make use of the equation (2.27) for the case where f = zν. We then
obtain
{xµ,zν} = iχ(x)
N+1
2θµν, (2.36)
where we have used (2.35) . Equation (2.36) is actually (2.35) only written in
terms of Moyal bracket which under quantization will go to the commutator
as we will see . For the coordinates zµthe Jacobi identity {zµ,{zν,zα}} +
{zα,{zµ,zν}} + {zν,{zα,zµ}} = 0 trivially follows from (2.31) .
By using the equation (2.33) we can find that the Moyal bracket of the
generator zµwith any function f of A can be written as
{zµ,f} = iχNθµν∂f
∂zν, (2.37)
where we have made use of (2.35) . The Moyal brackets (2.31) and (2.37) do
clearly correspond to the star product
f ∗ g(z) = e
i
2Dµν(z)
∂
∂ξµ
∂
∂ηνf(z + ξ)g(z + η)|ξ=η=0
(2.38)
where now Dµν(z) = χN(θµν+ iaηµν) . This star product however is com-
pletely equivalent to (2.1) . It is simply the star product (2.1) written in the
basis zµ. A general element of the algebra (A,∗) will be written in this basis
as
f(z) =
?
d4p
(2π)4˜f(p)eipz
(2.39)
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where˜f(p) =˜f0(p) + a˜f1(p) . The star product (2.38) will then have the
form
f ∗ g(z) =
?
?
d4p
(2π)4
d4p
(2π)4˜f ∗ ˜ g(p)eipz.
d4k
(2π)4˜f(p)˜ g(k)e−i
2pDkei(p+k)z.
=
(2.40)
where˜f ∗ ˜ g(p) is given by
˜f ∗ ˜ g(p) =
?
d4k
(2π)4˜f(p − k)˜ g(k)e−i
2(p−k)Dk. (2.41)
In this case˜f ∗ ˜ g(p) is a function only of χNand not of χ . However χN
is simply a constant in the ¯ hnapproximation and therefore (2.40) is of the
same form as (2.39) .
3. Quantum Space-Time
3.1 Quantization
We will now show that the algebra (A,∗) does really describe a quantum
space-time . In other words QR4is a space-time we obtain by quantizating
R4in the following way . First of all we assume that the quantization of R4
is completely equivalent to the quantization of its underlying algebra (A,.)
[3, 4]. Then in analogy with Quantum Mechanics we will quantize (A,.)
by the usual quantization prescription of replacing the coordinate functions
xµby the coordinate operators Xµso that the algebra of functions (A,.) is
mapped to an algebra of operators (A,×) [6]. If this algebra of operators
(A,×) is to be describing the quantum space-time QR4it must be constructed
in such a way that it will be homomorphic to (A,∗) . In other words we
must construct a homomorphism X from (A,×) to (A,∗) which will map
any element F(X) of A to the element (2.22) of (A,∗) in such a way that the
operator product F(X)×G(X) is mapped to the star product (2.23) . We
would then have
F(X)−→X(F(X)) = f(x)(3.1)
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together with
F(X)×G(X)−→X(F(X)×G(X)) = f ∗ g(x)
where g(x) is the image of the operator G(X) . In particular from (3.1)
the coordinate operators Xµare mapped to the coordinate functions xµand
from (3.2) the Moyal bracket{f,g} is mapped to the commutator [F,G]×=
F×G − G×F[37]. As we will see the homomorphism X has no non trivial
kernals and therefore the arrows in (3.1) and (3.2) can go the other way .
The product × which we will call the nonassociative operator product
cannot be the ordinary dot product of operators because it is clear from the
definition (3.2) that × is nonassociative whereas the dot product of operators
is trivially an associative product . We can assume however that it will reduce
at the limit of a−→0 to the ordinary dot product of operators . The difference
∆ between the nonassociative product × and the ordinary dot product is of
the order of a and it is given by
(3.2)
∆(F,G) =F×G − F.G
a
(3.3)
where F.G is defined by
X(F(X).G(X)) = Lima−→0f ∗ g(x)
The first step in constructing this homomorphism X is to impose on the
coordinate operators Xµcommutation relations which are of the same form
as (2.16) . We then have
(3.4)
[Xµ,Xν]×= iRθµν
(3.5)
where R is an operator defined by
X(R) = χ(x). (3.6)
In terms of the ordinary commutator , equation (3.5) will simply read
[Xµ,Xν] = iRθµν. (3.7)
The contribution ∆(Xµ,Xν) − ∆(Xν,Xµ) to this commutator is identically
zero because ∆(Xµ,Xν) = −R
The operator R clearly does not commute with Xµbecause
2ηµν.
[R,Xµ]×= Rµ
(3.8)
13
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where Rµare the elements of the algebra A mapped to {χ,xµ} , i.e
X(Rµ) = {χ,xµ} = −iχθµν∂νχ
The equation (3.8) will simply mean that the Jacobi identity
(3.9)
[Xµ,[Xν,Xα]×]×+ [Xα,[Xµ,Xν]×]×+ [Xν,[Xα,Xµ]×]×= 0 (3.10)
is not satisfied unless we choose θ to satisfy (2.19) .
In general the commutator of the generator Xµwith any element F(X)
of the algebra A is found to be
[Xµ,F]×= ∆F (3.11)
where by using (3.1) and (3.2) , ∆F is the operator in A mapped to {xµ,f}
, i.e
X(∆F) = {xµ,f}.
It is clear from this equation that the central elements of the algebra A
are either those operators which are mapped to the constant functions or
the operator O which is mapped to χN. The operators mapped to the
constant functions are clearly multiples of the identity operator 1 . The
operator O in the other hand is RNwhich can be seen as follows . By using
equation (2.32) we can prove that in the ¯ hNapproximation we have that
χ∗(χ∗(χ∗(χ..∗(χ∗χ)))..) = χNwhere we have N factors in the product . This
equation will become under quantization RN+ a?N−2
O . However by using the definition (3.3) of ∆ , one can check that in the
¯ hNapproximation the second term in the expression of O is of the order of
¯ hN+1and therefore O = RN. The generators Xµwill then commute with
RN, i.e
[Xµ,RN]×= 0.
(3.12)
m=0Rm∆(R,RN−m−1) =
(3.13)
In general Xµwill commute with any element of A which is of the order of
¯ hN.
The fact that R does not commute with the algebra A makes the defini-
tion (3.5) of quantum space-time not very useful when we try to construct
explicitly the homomorphism X . To see this more clearly we first note that
general elements F(X) of the algebra A are of the form
F(X) =
?
d4p
(2π)4[˜F(p,R)eipX+ e−ipX˜F+(p,R)](3.14)
14
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The nonassociative product of any two such elements F(X) and G(X) will
involve four different terms because R dose not commute with e(ipX). So
there is no an obvious way on how to map F(X) given by (3.14) to f(x)
given by (2.22) or for that matter how to map F(X)×G(X) to the star
product f ∗ g .
For the purpose of quantization a better definition of quantum space-time
QR4is such that the commutators of the generators are in the center of the
algebra . We need then to find a basis Zµfor which we have the commutators
[Zµ,Zν]×= iθµνC where C is a central element of the algebra A . If Zµis
the operator in A mapped to the coordinate function zµintroduced in (2.31)
then C will be simply given by RN. We would then have
[Zµ,Zν]×= iθµνRN. (3.15)
The ordinary commutator will also be given by a similar equation [Zµ,Zν] =
iθµνRNbecause of the fact that ∆(Zµ,Zν) = −RN
The definition of the operators Zµin terms of Xµcan be given by the
equation
[Xµ,Zν]×= iθµνR
2ηµν.
N+1
2
×
, (3.16)
with
X(R
N+1
2
×
) = χ
N+1
2 ,(3.17)
where We clearly have used the requirement that this equation should be
mapped to (2.36) .
The coordinate operators Zµare clearly unbounded and one would like
to work with bounded operators . We will therefore consider instead the
operators eipZas the generators of the algebra A . A general element F(Z)
of A will be defined by
F(Z) =
?
d4p
(2π)4˜F(p)eipZ
(3.18)
˜F is a smooth continuous function of the 4-vector p which must satisfy
˜F+(−p) =˜F(p) in order for F(Z) to be hermitian .
The product of any two elements F(Z) and G(Z) of A can be found to
be
F(Z)×G(Z) =
?
d4p
(2π)4
d4k
(2π)4˜F(p)˜G(k)e−iRN
2
pθkei(p+k)Z
15
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