The conformally invariant vector field on $R \times S^3$

0.32 MB·English

by  Zurab Ratiani

The Conformally Invariant Vector Field
3
on R S
×
Zurab Ratiani
5
0
0
2
Diploma Thesis at the
n
a
J Institut fu¨r Theoretische Physik

8
der Universit¨at Hannover
2

1
v
May 2002
0
3
2
1
0
5
0
/
h
t
-
p
e
h
:
v
i
X
r
a

Contents
1. Introduction 5
2. Mathematical background 7
2.1. The Conformal transformations . . . . . . . . . . . . . . . . . . . . . . . 7
2.2. Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3. Covariant Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4. The manifold R S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
×
2.5. Curvature of R S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
×
2.6. Generators of the conformal transformations . . . . . . . . . . . . . . . . 14
2.7. Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.8. The flat coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.9. Infinitesimal transformations . . . . . . . . . . . . . . . . . . . . . . . . . 20
3. Classical field theory 21
3.1. The vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2. The gauge-fixing term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4. Quantum field theory 27
4.1. Propagator quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2. Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . 28
4.3. The Interactions with a scalar density field . . . . . . . . . . . . . . . . . 31
4.4. The energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5. The Physical States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5. Summary 43
A. The Lorentz generator 45
B. Checking the commutation relation 47
3
Contents
4
1. Introduction
Conformal symmetry is the highest space-time symmetry. In addition to the trans-
formations of the Poincare group it includes the transformations which preserve angles
between worldlines:dilatationsandthespecialconformaltransformations.Thelasttrans-
formation is singular in the Minkowski space, but on R S3 it is well defined. Global
×
space-time may differ from Minkowski space, while being locally very similar to it, and
the new ”energy”, the translation operator P from the conformal algebra, will corre-
0
spondingly differ from the special relativistic energy, although again being very close to
it in the case of sufficiently localized phenomena. The difference in energy-eigenvalues
should be very small for localized states, but there is a considerable lost of symmetry in
the replacement of R S3 by Minkowski space.
×
The second chapter deals with geometrical objects, with curvature related quantities:
Riemann tensor, Ricci tensor, Ricci scalar. The conformal Killing-equations are pre-
sented. Next, the electromagnetic field will be constructed on the manifold R S3. This
×
manifold is a special case of the Robertson-Walker spaces. R S3 has 15 generators of
×
the conformal group and therefore is a maximal conformally symmetric manifold. The
space of the rays on the lightcone in R2,4 is isomorphic to the (S S3)/Z . This fact is
2
×
helpful to find out the conformal Killing-vectors, that describe the conformal symmetry
of the curved manifold. R S3 is a universal covering space of (S S3)/Z .
2
× ×
In the third chapter the electromagnetic field is considered. The Lagrangian that is
invariant undertheconformaltransformationsofthedynamicalfieldsA onlyiswritten.
m
The Maxwell field in four dimensions is a vector field and in contrast to the scalar field
there is no need to introduce densities in order to obtain a conformally invariant theory.
For the eigenmode expansion of the field A the eigenfunctions of the ”four-momentum”
m
operator, the operator that plays the role of the momentum in the conformal algebra,
will be used. At the end of this chapter, a gauge-fixing term is found, which makes
strongly apparent the similarity to the flat case.
In the fourth chapter the Green’s function is found. It allows us to carry out the
quantization. Here the creation and annihilation operators of photons on the curved
space will be discussed. The interaction of the Maxwell field with the charged scalar
density fields φ(x) is considered. The question whether such terms break the conformal
invariance, or give raise to some new vertex-factors for the Feynman diagrams is inves-
tigated. Finally the energy-spectrum of the dynamical field A (x) will be considered.
m
Due to the gauge invariance of the field there exist the unphysical degrees of freedom
which lead to the negative-norm states. The BRS method will be used to solve this
problem.
5
1. Introduction
6
2. Mathematical background
2.1. The Conformal transformations
Suppose we have a region M of the space Rn with coordinates x ,...,x , and some
1 n
mapping T of points x to points x : x = T(x). On M tangential vectors to curves are
′ ′
defined :
dxm ∂xm dxn
′ ′
= (2.1)
ds ∂xn ds
|x′(x) |x
An invertible transformation is called conformal if the length-square of all vectors will
change at most by some coordinate dependent factor eλ(x):
∂xm dxk ∂xn dxl dxk dxl
′ ′ g (x(x)) = eλ(x) g (x) (2.2)
∂xk ds ∂xl ds mn ′ ds ds kl
x x x x
| | | |
This must be valid for every vector, so:
∂xm ∂xn
′ ′ g (x(x)) = eλ(x)g (x) (2.3)
∂xk ∂xl mn ′ kl
The last formula shows us that after a conformal transformation the new metric ten-
sor g (x) is proportional to the original one g (x) if both are expressed in the same
m′ n ′ mn ′
coordinates. The transformation is called isometry if the conformal factor eλ(x) = 1 and
hence length-square of the vector does not change. Equation (2.3) describes the finite
conformal transformations.
The one parameter-group of transformations T maps points x to points x of the
α ′
manifold. If one varies the group parameter α, one gets a curve with the tangential
vectors:
(T x)m
ξm(T (x)) = α . (2.4)
α
dα
Inserting α = 0 one obtains vector at the original point x.
Considering infinitesimally small transformations: xm = xm+ξm, if one differentiates
′
(2.3) with respect to the transformations parameter at α = 0 and uses the equations:
∂(T x)m ∂(T x)m
α = δm, ∂ α = ∂ ξm (2.5)
∂xk k α ∂xk k
α=0 α=0
| |
7
2. Mathematical background
one gets:
∂ ξmδng +δm∂ ξng +δmδn∂ g (x)+ǫg = 0. (2.6)
k l mn k l mn k l α mn kl
Here we have used the notation: ǫ = ∂ λ .
− α |α=0
Finally one receives the conformal Killing equation :
ξk∂ g +∂ ξkg +∂ ξkg +ǫg = 0. (2.7)
k mn m kn n mk mn
From the last equations one can find the conformal Killing vectors, if the components
of the metric tensor are known. This vectors describe the symmetry of the manifold.
Becauseofthesymmetryofthemetrictensor(2.7)thereareatmostd(d+1)/2equations,
where d is the dimension of the manifold.
2.2. Isometries
Isometries aretransformationsthat do not changelengths ofvectors. The Killing vectors
generatesuch transformations. ThisvectorscanbefoundfromtheKillingequation. The
last one is received from the equation (2.7) if one sets the parameter ǫ = 0:
ξk∂ g +∂ ξkg +∂ ξkg = 0. (2.8)
k mn m kn n mk
If the components of the metric tensor are known one can find the Killing vectors from
equations (2.8). We will compute the components of one Killing vector. In our case, for
manifold R S3 we get the following equations:
×
(11) ∂ ξ1 = 0 (2.9)
1
(22) ξ1cosα∂ ξ2sinα = 0 (2.10)
2
(33) ξ1cosαsinθ+ξ2sinαcosθ∂ +ξ3sinαsinθ = 0 (2.11)
3
(12) ∂ ξ2sin2α+∂ ξ1 = 0 (2.12)
1 2
(13) ∂ ξ3sin2αsin2θ+∂ ξ1 = 0 (2.13)
1 3
(23) ∂ ξ3sin2αsin2θ+∂ ξ2sin2α = 0. (2.14)
2 3
8
2.3. Covariant Differentiation
The numbers in brackets indicate the indices of the metric tensor. Here we assume
that the numbers 0,1,2,3 correspond to the coordinates t,α,θ,φ accordingly. To solve
this system of equations we will use the Ansatz:
ξ1 = Asin(θ+a)
ξ2 = Acos(θ +a)cotα+b (2.15)
Here A,a,b are some unknown constants. From the equation (2.11) one gets:
ξ3 = φ Asin(θ +a)cotα+(Acos(θ+a)cotα+b)cotθ . (2.16)
−
(cid:16) (cid:17)
If we insert this into (2.13) we will receive the following condition onto the a’s: a = πn,
2
where n = 0, 1, 2, 3,.... From equation (2.14) we receive for a = π/2 that b = 0.
± ± ±
Now we can insert this values for the constants a and b into the last expression for ξ3
and we get ξ3 = 0. So one can now write down the following Killing vector:
ξm = (cosθ, sinθcotα,0). (2.17)
−
2.3. Covariant Differentiation
The gradient of a tensor does not behave like a tensor under general coordinate trans-
formations xi = xi(y1,...yn),i = 1...n. The components of a tensor T˜ in the yi
k
coordinates satisfy:
∂T˜ ∂ ∂xj ∂Ti∂yj ∂ ∂yj
T˜j = k = Ti = +Ti
,l ∂yl ∂yl ∂yi! ∂yl ∂xi ∂yl ∂xi
∂Ti∂xp ∂yj ∂2yj ∂xk
= +Ti . (2.18)
∂xp ∂yl ∂xi ∂xi∂xk ∂yl
The second term vanishes under linear coordinate transformations yi = aixj, but not
j
under general coordinate transformations. So, it is necessary to generalize the concept
of the gradient. The covariant derivative of (1,1) tensor has the form:
∂Ti
D Ti = j +Γi Ti Γl Ti, (2.19)
k j ∂xk lk j − jk l
and there exist similar expressions for tensors of different rank. Covariant differentiation
is a linear operation and commutes with the operation of contraction. For a zero-rank
tensor it is the usual gradient of the function. The covariant derivative of a product of
tensors is calculated using the Leibniz product rule.
The Γ-s are called Christoffel symbols or affine connections. The coordinates are
said to be (Psuedo)Euclidean (with respect to a given connection), if in terms of those
coordinates the Γ’s are identically zero:
Γi 0 i,j,k = 1,...,n. (2.20)
kl ≡
9
2. Mathematical background
On the other hand, the coordinates x1,...,xn are said to be Euclidean if in terms of
these coordinates the metric g is given by:
ij
g = δ , (2.21)
ij ij
where δ is the Kronecker symbol.
A connection Γi is said to be compatible with a metric g if the covariant derivative
jk ij
of the metric tensor g is identically zero:
ij
D g 0 i,j,k = 1,...,n. (2.22)
k ij
≡
For a vector field :
∂Ti ∂ ∂Ti ∂Tp
D D Ti = D +Γi Tn = +Γi Tn +Γi +Γp Tn
k l k ∂xl nl ! ∂xk ∂xl nl ! pk ∂xl nl !−
∂Ti
Γp +Γi Tn
− lk ∂xp np !
∂2Ti ∂Tn ∂Tp ∂Ti ∂Γi
= + Γi +Γi Γp +Tn nl +
∂xk∂xl ∂xk nl pk ∂xl − lk∂xp ∂xk
+Γi Γp Tn Γp Γi Tn. (2.23)
pk nl − lk np
If one exchanges the order of differentiation and after that subtracts this two expres-
sions from each other, one gets:
∂Γi ∂Γi
(D D D D )Ti = nl nk Tn + Γi Γp Γi Γp Tn
k l − l k ∂xk − ∂xl ! pk nl − pl nk
(cid:16) (cid:17)
∂Ti
(Γp Γp ) +Γi Tn . (2.24)
− lk − kl ∂xp np !
Introducing the notation:
∂Γi ∂Γi
Ri = nl nk +Γi Γp Γi Γp , (2.25)
− nkl ∂xk − ∂xl pk nl − pl nk
Tp = Γp Γp , (2.26)
kl lk − kl
finally one obtains :
(D D D D )Ti = Ri Tn +TpD Ti (2.27)
k l − l k − nkl kl n
One can write analogous equations for the tensors with lower indices:
(D D D D )T = Rn T +TpD T . (2.28)
k l − l k i ikl n kl n i
The Ri are the components of so-called Riemann curvature tensor and Tp is the
nkl kl
torsion tensor. If the connection is symmetric Tp 0.
kl ≡
10