AdS notations
Embedding formalism
We here consider massless Laplace equation in 
.
We realize as a hyperboloid in 
parametrized by coordinates 
.
The equation of the hyperboloid is:
.
We realize as a hyperboloid in parametrized by coordinates .
The equation of the hyperboloid is:
The Laplace operator in can be written as
can be written as
|
|
where 
is the Euler vector field (rescaling of 
and 
) and 
is the Laplace operator on .
Therefore, on harmonic functions 
:
is the Euler vector field (rescaling of and ) and is the Laplace operator on .
Therefore, on harmonic functions :
|
Eigenfunctions of the Laplace operator on the hyperboloid can be obtained as restrictions of harmonic functions on
Our space-time is not just , but 
. The formulas for Laplace operator on the sphere
are completely analogous. To distinguish between AdS and sphere, we use indices 
and 
:
, 
, 
, 
. The total Laplace operator
on 
is:
, but . The formulas for Laplace operator on the sphere
are completely analogous. To distinguish between AdS and sphere, we use indices and :
, , , . The total Laplace operator
on is:
|
Therefore, for the scalar function to be harmonic in :
:
|
This means:
|
Massless scalar in
Let us parametrize 
by a unit vector 
.
Suppose that the dependence is a harmonic polynomial 
of order . The full solution is:
by a unit vector .
Suppose that the dependence is a harmonic polynomial
of order . The full solution is:
|
where 
is a harmonic function of order 
in
parametrized by coordinates .
is a harmonic function of order in
parametrized by coordinates .Those solutions where is a polynomial form a finite-dimensional representation.
When we allow to be a rational function with denominator powers of 
,
they form an infinite-dimensional representation. That infinite-dimensional representation
is the 
of Eq. (16).