Embedding formalism Massless scalar in

Embedding formalism

We here consider massless Laplace equation in . We realize as a hyperboloid in parametrized by coordinates . The equation of the hyperboloid is:
 (17)
The Laplace operator in can be written as
where 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:
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:
where 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).