NONCOMMUTATIVE GEOMETRY AND THE
DIFRACTION ONE DIMENSIONAL NETWORK
by
Laurian Pişcoran
Abstract. Noncommutative geometry extend the notion from classical differential geometry from differential manifold to discrete spaces and even noncommutative spaces which are given by noncommutative algebra (over R or C). Such an algebra A replace the commutative algebra of function of class
C
∞ over a smooth manifold.In this work I present some aspects about the calculus of the distance in the noncommutative geometry case. An important role in the distance calculus is playing by the Dirac operator.
MSC: 14A22
Keywords: Clifford Algebra, Dirac Operator .
1. Introduction
Definition 1.1 Let V be a finite dimensional vector space over the scalar field
K
, where K=R or C.A Quadratic form on V is a mapping
Q V
:
→
K
such that: 1) Q(λv)=λ2Q(v)2) The associated form B v w =
{
Q(v)+Q(w)−Q(v−w)}
v,w∈V2 1 ) ,
( is biliniar
In this case (V,Q)is a Quadratic space.
Definition 1.2 The pair (A,ν)is said to be a Clifford algebra for the quadratic space )
,
(V Q when :
1) A is generated as an algebra by {ν(v)|v∈V} and
{
λ λ
1| ∈K}
2) ((ν(v))2 =−Q(v)1 , v∈V
Example:1) The R-algebra of complex numbers C is generated by 1 and i, verifying the relation:
1
2=
−
2 )
(x x
Q =− , c(x)=ix
2)When we take ClR (Rp+q, Q) where Q is the quadratic form
2 2
1 2 2
1 ... ....
)
(x x xp xp xp q
Q = + + − + − +
we use the notation Cl(p,q), we put Cl(n)≡Cl(0,n) and Cl∗(n)≡Cl(n,0)
Hence, for the universal real Clifford algebra ClC (V,-Q) over the vector space V=Rn,
where Q comes from the biliniar form of the usual euclidian product in Rn, we use the notation )Cl(n , that means that if we take an orthonormal basis e1,e2,...,en in Rn, we have: 1eiej +ejei =−2δij
We have for instance:
H H ) 3 ( H, ) 2 ( , C ) 1
( = Cl = Cl = ⊕
Cl
3)Let be the Pauli matrices in C2x2:
0
1
0
0 1
σ
=
, 11
0
0
1
σ
=
−
, 20
0
i
i
σ
=
−
, 30 1
1
0
σ
=
and the associated matrices would be:
0
1
0
0 1
e
=
, 10
0
i
e
i
=
−
, 20
1
1 0
e
=
−
, 30
0
i
e
i
=
. We have:2 2 2 2
0 1 2 3
σ σ σ σ
=
=
=
=
I
,e
02=
I
,2 2 2
1 2 3
e
= = = −
e
e
I
and
σ σ
j k= −
i
σ
l,e e
j k=
e
l, where{
j k l, ,}
are cyclic permutation of the set{
1, 2, 3}
. Let A0,0 ={
λσ0,λ∈R}
, ∈
= , R
0 ,
1 x y
x y y x A , ∈ −
= , R
1 ,
0 x y
x y y x A , ∈ − = = ∈ − + − + +
= R, 0,3 C
1 2 2 1 1 0 3 2 3 2 1 0 2 ,
0 j zj
Then we have: A0,0 ≅R, A1,0 ≅R⊕R, A0,1 ≅C, A0,2 ≅H. 2. Dirac operator relative to a vector bundle
Given a riemannian manifold M, an operator ∆∈Diff(2)(E,E) of order 2 is said to be a generalized laplacian if
p
E p p
p I
2 2( )(ξ ) ξ
σ ∆ =− where
g
...
is the metric norm of M , and an operator of order 1 ,D∈Diff(1)(E,E) is said to be aDirac operator relative to the vector bundle (E,M,Kf) whenever D2is a generalized laplacian. If D is a Dirac operator we can define for any ξp∈Tp∗M an
K-endomorphism )c(ξp of Ep given by c(ξp)=σ1(D)(ξp) or, alternatively we define a map:
p p p
p T M×E →E
∗ : γ
p p p
p p
p,u ) c( )(u ) c( )u
(ξ → ξ ≡ ξ
for the tensorization of Tp∗M by C we use the above relation TC∗M ≡
( )
Tp∗M ⊗R C, soif we don’t want to precise if K =R or C we will write TK∗M ≡(Tp∗M)⊗R K, and
obviously TR∗M =Tp∗M , note that the map γpcould be defined using TKM
∗ instead of Tp∗M.
The endomorphisms c(ξp) verify the condition c p p gIEp
2 2
)
(ξ =−ξ , equivalent to
p
E p p p
p p
p c c c g I
c(ξ ) (η )+ (η ) (ξ )=−2 (ξ ,η ) , this means that we got for any p∈M a
K-bilinear map γp :(TK∗M)×Ep→Ep that is a K-linear map (TK∗M)⊗KEp→Ep,
with the preceding property.
The set of K -linear operators c(ξp):Ep →Ep generates an associative and unital
sub algebra A of the K−algebra )EndK(Ep (with 1≡IEpas element one) and there
exist a K-linear map c TKM → A ∗
: given by ξp →c(ξp) and verifying 1
) , ( 2 ) ( ) ( ) ( )
( p c p c p c p g p p
c ξ η + η ξ =− ξ η
This means that A is a Clifford algebra for the quadratic space (TK∗M,−g)and c is the Clifford mapping. The vector bundle E is a Clifford vector bundle and
E E M TK∗ × →
:
γ , given by γ(ξ,u) p =γp(ξp,up)=c(ξp)up is the coresponding
The following property exhibit the compatibility of D with the Clifford action: ))
, ( ),
( F ( )
( ) ( )
(sf c df s D s f f K M s M E
D = + ∈ ∈Γ
or alternatively, making FK(M) act multiplicatively in Γ(M,E) through f(s)=sf , we will have :
s df c f D s D f f D s D f s f D f s D sf
D( )− ( ) =( o )( )−( o )( )=(( o )−( o ))( )=[ , ]= ( ) , that is:
) ( ] ,
[D f =c df
or we can write using the previus conventions: )
( ] ,
[D f =c df .
We can say that any Dirac operator, relative to the vector bundle (E,M,Kn), with a riemannian manifold in the basis, determines in this vector bundle a structure of Clifford bundle compatible with D in the preceding sense.
Any Clifford vector bundle on a riemannian manifold endowed with a covariant derivative
) ,
( ) , (
: M E M E TKM
∗ ⊗ Γ
→ Γ
∇
compatible with a Clifford action TKM⊗E→E ∗
:
γ in a sense to be precise later, is associated to a Dirac operator acting on the sections of this vector bundle and compatible with the Clifford action, we will call such a vector bundle a Clifford-Weyl bundle.
The compatibility of ∇ with the Clifford action ξps=c(ξp)s means the
following:
s s
s X X
X ⋅ =∇ ⋅ + ⋅∇
∇ (ω ) ω ω (ω∈Ω1(M,K)) or alternatively,
s c
s c
s
c X X
X = ∇ + ∇
∇ ( (ω) ) ( ω) (ω) ))(ω∈Ω1(M,K
where )∇:ℵK(M)→ℵK(M)⊗Ω1(M,K or else ∇X :ℵK(M)→ℵK(M)
for any X ∈ℵX(M), is the Levi Cevita covariant derivative in M, determining a covariant derivative ∇:Ω1(M,K)→Ω1(M,K)⊗Ω1(M,K) or actually,
) , ( ) , (
: 1 M K 1 M K
X Ω →Ω
∇ for any X ∈ℵK(M).
The set Γ(M,E) of the sections of a Clifford-Weyl vector bundle has a structure of ))
( ), (
(ClK T∗M FK M -bimodule, with the following properties of compatibility: df
s f s
sf =∇ + ⊗ ∇( ) ( )
s s
s = ∇ ⋅ + ⋅∇ ⋅
Next, I will prove the folowing result, which is the distance between two diferent points on the straight line, in noncommutative geometry case:
d x y f x f y D f f x f y f x y
A f A
f
C = − < = − ′ < = −
∈
∈ { ( ) ( ) :[ , ] 1} sup{ ( ) ( ) : 1} sup
) , (
We know that [D,f]:A→ A
f g f g f g f g f fg Dg f fg D
g D f f D g f D g
′ = ′ − ′ + ′ = ′ − ′ = −
=
= −
= →
) ( ) ( ) (
) )( (
) ]( ,
[ o o
where I use the definition f(s)=sf .
In general for an operator T :E→E , we have : sup ( )
1 ξ
ξ T T
≤
=
In this case
f f
f f
D = ′ = ′ = ′ ≤
≤ ξ ξ ξ
ξ 1 1 sup sup
] , [ and using the inequality:
y x f y f x
f( )− ( ) ≤ ′ ⋅ − I prove the distance relation.
Now, I will use the Dirac operator to recover the distance between atoms in a periodical one dimensional diffraction network.
Let assume that we have a network in which, between two atom we have the same distance.
So, we can represent the network in this way:
1 2 n ...
Using this network we can construct the incidence matrix puting the element 1 when we have a link between two atom and puting 0 if we don’t have link.
0
...
0
0
0
0
1
...
0
0
0
0
...
...
...
...
...
...
0
...
1
0
0
0
0
...
0
1
0
0
0
...
0
0
1
0
.
We define:
=
N
D
0
...
0
0
0
0
sin
...
0
0
0
0
...
...
...
...
...
...
0
...
sin
0
0
0
0
...
0
sin
0
0
0
...
0
0
sin
0
λ
θ
λ
θ
λ
θ
λ
θ
,
where
θ
is the angle between two wave andλ
is the wave lenght.θ
... Let be N =
{
1,2,...,n}
the set of atoms.=
→
f
f
~
0 ... 0 0 0 0
... 0 0 0 0
... ... ... ... ... ...
0 ... 0 0
0
0 ... 0 0 0
0 ... 0 0 0
3 2 1
n
f f
f f
n
C ∈ ,
where
f
(
n
)
=
f
n, and with this representation we can construct the function
=
f
f
f
0
0
.
We can associate to the complex function
f
, a real functionF
with the properties: 01 =
F and Fk+1 =Fk+ fk+1− fk .
We have the operator
=
0 0 ˆ
N tr N
D D D
We know that in general, the distance in noncommutative geometry is given by:
(
,
)
=
sup
{
(
)
−
(
)
:
[
,
]
<
1
}
∈
f
x
f
y
D
f
y
x
d
A f C
In our case we can compute, and is easy to prove that
[ ] [ ]
D
ˆ
,
f
ψ
=
D
ˆ
,
F
ψ
for ψ∈Cn. So, after easy computation we will get:[ ]
−
−
=
2 1 −1sin
,....,
sin
max
,
ˆ
N N
f
f
f
f
f
D
λ
θ
λ
θ
,and using the condition [D,f] <1, we get:
θ
λ
θ
λ
θ
λ
sin
sin
...
sin
)
,
(
i
i
n
n
d
+
=
+
+
=
.References
[1].Connes A., Noncommutative Geometry, Academic Press,1990
[2].Landi G., An introduction to Noncommutative Spaces and their geometries- Springer Verlag 1997
[3].Paulo Almeida. Noncommutative Geometry, Lisboa 2001
Author:
Pişcoran Laurian
Department of Mathematics and Computer science North University of Baia Mare
