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5 Conformal Symmetry in D Dimensions
This means that angles between two vectors u and v are left invariant under the
transformation:
u · v
|u| |v|
=
u · v
|u | |v |
.
(5.3)
It is often convenient to parametrize the scale factor by an exponential
:= e
ω .
(5.4)
Considering an infinitesimal transformation
δx
μ
= ξ
μ ,
(5.5)
the condition (5.2) becomes the conformal Killing equation
δg μν = L ξ g μν = ∇ μ ξ ν + ∇ ν ξ μ =
2
d
g μν ∇ ρ ξ
ρ ,
(5.6)
such that the scale factor is
2
= 1 +
2
d
∇ ρ ξ
ρ .
(5.7)
The vector fields ξ satisfying this equation are called conformal Killing vectors
(CKV). Conformal transformations form a global subgroup of the diffeomorphism
group: the generators of the transformations do depend on the coordinates, but the
parameters do not (for an internal global symmetry, both the generators and the
parameters do not depend on the coordinates).
The conformal group contains the isometry group ISO(M) of M as a subgroup,
corresponding to the case = 1:
ISO(M) ⊂ CISO(M).
(5.8)
These transformations also preserve distances between points. The corresponding
generators of infinitesimal transformations are called Killing vectors and satisfies
the Killing equation
δg μν = L ξ g μν = ∇ μ ξ ν + ∇ ν ξ μ = 0.
(5.9)
They form a subalgebra of the CKV algebra.
An important point is to be made for the relation between infinitesimal and
finite transformations: with spacetime symmetries it often happens that the first
cannot be exponentiated into the second. The reason is that the (conformal) Killing
vectors may be defined only locally, i.e. they are well-defined in a given domain
but have singularities outside. When this happens, they do not lead to an invertible
5 Conformal Symmetry in D Dimensions
This means that angles between two vectors u and v are left invariant under the
transformation:
u · v
|u| |v|
=
u · v
|u | |v |
.
(5.3)
It is often convenient to parametrize the scale factor by an exponential
:= e
ω .
(5.4)
Considering an infinitesimal transformation
δx
μ
= ξ
μ ,
(5.5)
the condition (5.2) becomes the conformal Killing equation
δg μν = L ξ g μν = ∇ μ ξ ν + ∇ ν ξ μ =
2
d
g μν ∇ ρ ξ
ρ ,
(5.6)
such that the scale factor is
2
= 1 +
2
d
∇ ρ ξ
ρ .
(5.7)
The vector fields ξ satisfying this equation are called conformal Killing vectors
(CKV). Conformal transformations form a global subgroup of the diffeomorphism
group: the generators of the transformations do depend on the coordinates, but the
parameters do not (for an internal global symmetry, both the generators and the
parameters do not depend on the coordinates).
The conformal group contains the isometry group ISO(M) of M as a subgroup,
corresponding to the case = 1:
ISO(M) ⊂ CISO(M).
(5.8)
These transformations also preserve distances between points. The corresponding
generators of infinitesimal transformations are called Killing vectors and satisfies
the Killing equation
δg μν = L ξ g μν = ∇ μ ξ ν + ∇ ν ξ μ = 0.
(5.9)
They form a subalgebra of the CKV algebra.
An important point is to be made for the relation between infinitesimal and
finite transformations: with spacetime symmetries it often happens that the first
cannot be exponentiated into the second. The reason is that the (conformal) Killing
vectors may be defined only locally, i.e. they are well-defined in a given domain
but have singularities outside. When this happens, they do not lead to an invertible