# On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

###### Abstract

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra’s formula for the radial part of the Casimir operator.

## 1 Introduction

The -dimensional sphere () and the real hyperbolic space (), which are “dual” to each other as symmetric spaces [10], are maximally symmetric. This high degree of symmetry allows one to compute explicitly the eigenfunctions of the Laplacian for various fields on these spaces. These eigenfunctions can be used in studying field theory in de Sitter and anti-de Sitter spacetimes since and are Euclidean sections of these spacetimes. Also and appear as the spatial sections of cosmological models, and various field equations and their solutions on these spaces have physical applications in this context. In addition to these applications, fields on and provide concrete examples for the theory of homogeneous vector bundles, and consequences of various theorems can explicitly be worked out.

Recently the authors presented the eigenfunctions of the Laplacian for the transverse-traceless totally-symmetric tensor fields [4] and for the totally-antisymmetric tensor fields (-forms) [5]. These fields were also analyzed in the light of group theory using the fact that they are cross sections of homogeneous vector bundles. As a continuation of these works we study in this paper the spinor fields satisfying the Dirac equation and the heat kernel for on and . The paper is organized as follows. We begin by setting up the gamma matrices used in the paper in sect. 2. Then in sect. 3 the appropriately normalized eigenfunctions of the Dirac operator on with arbitrary are presented using geodesic polar coordinates. The solutions on are expressed in terms of those on . Then we derive the degeneracies of the Dirac operator using the spinor eigenfunctions. Next, spinor eigenfunctions on are obtained by analytically continuing those on . Then they are used to derive the spinor spectral function (Plancherel measure) on . In sect. 4 the results of sect. 3 are used to write down the heat kernel for the iterated Dirac operator on these spaces. Sect. 5 is devoted to a group-theoretic analysis of spinor fields on and . We use the fact that spinor fields on these symmetric spaces are cross sections of the homogeneous vector bundles associated to the fundamental spinor representation(s) of . We first review some relevant facts about harmonic analysis for homogeneous vector bundles over compact symmetric spaces. By applying these to , in combination with the formula for the radial part of the Casimir operator given by Harish-Chandra, we derive the spinor spherical functions and the heat kernel of the iterated Dirac operator on . Then we briefly review harmonic analysis for homogeneous vector bundles over noncompact symmetric spaces, apply it to , and rederive some results of sect. 4 for this space.

## 2 -matrices in dimensions

A Clifford algebra in dimensions is generated by matrices satisfying the anticommutation relations

(2.1) |

where is the unit matrix and is the Kronecker symbol. It is well known that (2.1) can be satisfied by matrices of dimension , where for even, for odd. We give below an inductive construction of which relates easily to the spinor representations of the orthogonal groups.

For we set . For put

(2.2) |

where . For we add to and above the matrix

(2.3) |

For let

(2.4) |

where is the unit matrix, and the matrices () satisfy the Clifford algebra for and are given by the right-hand sides of (2.2)-(2.3). For we add to given in (2.4) the matrix

(2.5) |

The general pattern is now clear.

Case 1. even. Let denote the following set of matrices of dimension :

(2.6) |

where is the unit matrix of dimension , and the matrices (also of dimension ) satisfy the Clifford algebra relations in dimensions,

(2.7) |

Then (2.1) is immediately verified. This representation generalizes to even the ordinary spinor representation of gamma matrices in four dimensions, where all ’s are “off-diagonal” and is diagonal.

The matrices

(2.8) |

satisfy the commutation rules

(2.9) |

and generate a -dimensional representation of , the double cover of (for is also the universal covering group). The commutator of and is then

(2.10) |

It is easily seen that the matrix anticommutes with each of the ’s, and commutes with each of the generators . It can be shown by induction that

(2.11) |

Since is nontrivial it follows that the representation of with generators must be reducible. From (2.6) we find

(2.12) |

where and both satisfy (2.9) and are given by

(2.13) | |||||

(2.14) |

It can be shown that the matrices and generate the irreducible representations of with highest weights and respectively, where

(2.15) |

in the standard Cartan-Weyl labeling scheme. Since these weights are fundamental (see [1] p.224), and are called the two fundamental spinor representations of (of dimension each). Thus and a spinor in dimensions ( even) is reducible with respect to the orthogonal group.

In a similar way, the matrices given by (2.13) generate the unique fundamental spinor representation of (see Case 2 below), also of dimension , and one has the branching rule

(2.16) |

Case 2. odd. In this case the dimension of the -matrices is , the same as in dimensions. Let () be a set of matrices satisfying the Clifford algebra anticommutation relations (2.7) in dimensions. As observed above, the matrix

(2.17) |

anticommutes with each of the ’s. Thus if we define

(2.18) |

where the constant is such that , then the matrices

(2.19) |

satisfy the Clifford algebra (2.1) in dimensions. It can be shown by induction that the choice gives

(2.20) |

and, as a result,

(2.21) |

Again the matrices defined by (2.8) satisfy (2.9). This time they generate an irreducible representation of , namely the unique fundamental spinor representation with highest weight

(2.22) |

(see [1] p.224). If and denote the two fundamental spinor representations of (see Case 1 above), we have the branching rule

(2.23) |

Clearly the generators of are the matrices .

## 3 Spherical modes of the Dirac operator

### 3.1

The metric on may be written in geodesic polar coordinates as

(3.1) |

where is the geodesic distance from the origin (north-pole), , and are coordinates on , with metric tensor . Let be a vielbein (i.e. an orthonormal frame) on , with anolonomy and (Levi-Civita) connection coefficients

(3.2) |

(3.3) |

Let be the dual coframe to . Then . We shall work in the geodesic polar coordinates vielbein on defined by

(3.4) |

The only nonvanishing components of the Levi-Civita connection in the frame are found to be

(3.5) |

where a prime denotes differentiation with respect to the argument. Note that , as required in a vielbein for a metric connection.

Spinors are -dimensional and are associated with orthonormal frames. Under a local rotation , which transforms the covielbein as

(3.6) |

a spinor transforms by definition according to

(3.7) |

where is either one of the two elements of (more precisely of the spin representation of , see section 2) which correspond to , and is determined by

(3.8) |

A spin connection on is induced by the Levi-Civita connection . The covariant derivative of a spinor along the vielbein may be written as

(3.9) |

where summation over repeated indices is understood from now on. The Dirac operator is defined by . We shall now solve for the eigenfunctions of on by separating variables in geodesic polar coordinates.

Case 1. even. Using eqs. (2.6), (2.7), (2.12), (2.13), (2.14), and (3.5), it is straightforward to derive the following expression for the Dirac operator in the vielbein (3.4):

(3.10) |

where and

(3.11) |

is the Dirac operator on . We now project the first order Dirac equation

(3.12) |

onto the “upper” and “lower” components of . Define

(3.13) |

Then

(3.14) |

Eliminating (or ) gives the second order equation

(3.15) |

which is equivalent to the equation obtained by squaring the Dirac operator, i.e.

(3.16) |

It is well known that on a compact spin manifold is negative semidefinite (the spectrum of is purely imaginary), so that is real. Now suppose that we have solved the eigenvalue equation on , i.e.

(3.17) |

Here,
the index labels the
eigenvalues
of the Dirac operator on the -sphere,
and , ^{1}^{1}1The
spectrum of the Dirac operator on spheres is
well known (see e.g. [17]). It will be clear that our procedure
gives an independent proof of (3.26)
by induction over the dimension (see also
the remark at the end of this subsection). Eq. (3.17) (and the
analogous relation (3.51)) may then be assumed as the inductive
hypothesis in this proof.
A group-theoretic derivation of the spectrum of
on will be given in subsections 5.1 and 5.2.
while the index runs
from
to the degeneracy . Since the dimension of is the same
as the
dimension of , i.e. , one can
separate
variables in the following way:

(3.18) | |||

(3.19) |

and similarly for the “lower” spinor . Here , and labels the eigenvalues of on and as is well known. (We shall see also that this condition arises as the requirement for the absence of singularity in the mode functions.)

Substituting (3.18) in (3.15) we obtain the following equation for the scalar functions :

(3.20) |

where is the differential operator

(3.21) | |||||

A simple calculation allows us to rewrite in the following form

(3.22) |

where is the differential operator for the Jacobi polynomials [8],

(3.23) | |||||

(3.24) |

Thus, the unique regular solution to (3.20) is, up to a normalization factor,

(3.25) |

with – this condition is needed for the regularity of the eigenfunctions – and with the eigenvalues

(3.26) |

At only the modes with are nonzero. The functions are called spinor spherical functions.

By proceeding in a similar way with the functions in (3.19) we find

(3.27) | |||||

(3.28) |

They all vanish at the north pole, but for they are nonzero at the south pole ().

One can readily verify that if satisfies (3.15) and if is defined by the second equation of (3.14), then the first equation is automatically satisfied. In this way we can find the eigenfunctions of the first-order Dirac operator. We use the formulae

(3.29) | |||||

(3.30) |

to simplify the expressions. These can be proved by using the expression of the Jacobi polynomial in terms of the hypergeometric function,

(3.31) |

and the raising and lowering operators for the third entry of the hypergeometric function. Thus, we find the solutions to the first-order Dirac equation to be

(3.32) | |||||

(3.33) |

These spinors satisfy

(3.34) |

with , and are required to satisfy the normalization condition

(3.35) |

with an analogous relation for .

Suppose that the spinors are normalized by

(3.36) |

Then the normalization factor is determined by

(3.37) |

It is clear from (3.28) that the integrals of the first and the second terms are equal. Using Ref. [8] eq. 7.391 n.1 we find

(3.38) | |||||

The spinor eigenfunctions with near are then given by

(3.39) | |||||

(3.40) |

The degeneracy for the eigenvalue [or ] is given by (take , , , in (3.35) and sum over )

(3.41) | |||||

where we have used the fact that the sum over inside the integral is constant over (this is easy to prove), so that it may be calculated for , where only the term survives. The factor is the volume of ,

(3.42) |

Now the degeneracy of (or ) on (cf. (3.17)) coincides with the dimension of the representation of labelled by (see section 5). Thus in (3.41), and we have the identity

(3.43) | |||||

From this equation and eqs. (3.41) and (3.42) we obtain

(3.44) |

where we have used the doubling formula for the Gamma function

(3.45) |

The degeneracy is equal to the dimension of the spinor representation of labelled by (see section 5). Of course, the degeneracy of the eigenvalue of is .

Case 2. odd (). In this case a Dirac spinor on is irreducible under and the dimension of the -matrices is , the same as on . The Dirac operator in the geodesic polar coordinates vielbein (3.4) takes the form

(3.46) |

where and is the Dirac operator (3.11) on . Since

(3.47) |

the iterated Dirac operator is given by

(3.48) |

In order to separate variables in the eigenvalue equation (3.16) we observe that although does not commute with , the operator does, as a consequence of (3.47). Thus and can have common eigenfunctions. Furthermore, since and , the operator is hermitian (because of (3.47)), and has real eigenvalues.

Suppose that we have solved the equation

(3.49) |

for the spinor on . In order to find the possible values of we apply to both sides of (3.49) and use (3.47) to find

(3.50) |

Thus are the eigenvalues of the iterated Dirac operator on , i.e.

(3.51) |

Let satisfy

(3.52) |

Then is the eigenfunction of with eigenvalue . These are related to by

(3.53) | |||||

(3.54) |

We can now separate variables in (3.16) by letting

(3.55) | |||

(3.56) |

Substituting (3.55) in (3.16) and using (3.48) we find that the scalar functions satisfy the same equation (3.20) as in the case of even. We immediately conclude that is given by (3.25), and similarly we find that is given by (3.27).

The eigenfunctions of the first-order Dirac operator must be of the form

(3.57) |

Then, using (3.29) and (3.30) in (3.46), we find that is an eigenfunction of the first-order Dirac operator with eigenvalues if we choose . The normalization factors are given again by . The formula for the dimensionality is the same as that for even except that the representation of is -dimensional (and there is no factor of we had for even). Thus, the dimensionality is

(3.58) |

Our method shows how to separate variables in the Dirac equation written in geodesic polar coordinates. The spinor modes of on are the product of the spherical modes , times the spinor modes on [(3.17) ( even) or (3.49) ( odd)]. But again these can be obtained from the spinor modes on , and so on. Therefore we have set up an induction procedure by which the spinor modes on can be recursively calculated starting from the spinor modes on the lowest dimensional sphere .

Note also that our procedure gives an independent proof of the spectrum of the Dirac operator by induction. First consider the case . Writing , one simply has . We need to find which eigenvalues of this operator are allowed. Spinors on should transform as a double-valued spin-1/2 representation of under the rotation of zweibein. Now, the loop defined by with const. in the bundle of frames over is homotope to the -rotation at a point for our system of zweibein. Hence, the spinor field must change sign when it goes around this loop. Thus, we have