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

A. Higuchi; R. Camporesi

REVIEW

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

R. Camporesi, A. Higuchi

General Relativity and Quantum Cosmology · Submitted 9 May 1995 · 19 references

Pith's verdict: A careful, honest reference paper: the arbitrary-N spinor eigenfunctions and the group-theoretic rederivation are the real contributions, the heat kernel is a restatement of the authors' own earlier work, and the only defects are two typos and two asserted-but-standard steps.

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arXiv IDgr-qc/9505009 (version 1)

Pith IDYYCGLR74, Pith's permanent record of this paper and its review

CitationA. Higuchi, R. Camporesi. "On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces." arXiv:gr-qc/9505009 (1995). https://pith.science/paper/YYCGLR74

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Bottom line: this is a careful, useful reference paper, and its two halves genuinely check each other. Nothing here needs a fight, and nothing here is a breakthrough either — it's a solid brick in an established wall, and the authors are honest about which brick is new and which is recycled from their own earlier work [3]. That honesty is worth something; most papers in this genre overclaim the kernel and bury the overlap in a footnote. Here the overlap is stated twice, in plain text, with the equation number of the earlier paper. Good. What's actually new: the explicit spinor eigenfunctions on S^N and H^N in geodesic polar coordinates for arbitrary N (Eqs. 3.32-3.33, 3.63), the recursion that builds them from the S^{N-1} modes, and the group-theoretic rederivation in section 5. The heat kernel itself is not new — the authors say so, pointing at ref. [3] eq. 5.14 — so the honest novelty claim is the eigenfunctions plus the representation-theoretic treatment, not the kernel. What's good: the cross-checks are real. Eq. (5.48) reproduces (3.26) and (5.51) reproduces (3.44) from completely different machinery — Casimir eigenvalues and Weyl's dimension formula versus separation of variables. That's the kind of agreement that makes me trust the normalization constants, which are the part of this business people usually get wrong. The odd-N case in section 5.2, where the spherical functions don't vanish at the south pole and the constants c, d come out ±i from the Schur orthogonality relations, is handled with more care than I'd expect. Soft spots, in proportion: they're small. Eq. (4.6) is printed as a product D_{N-1}(l)D_{N-1}(0)Ω_{N-1} where (4.7) and the trace argument require the quotient D_{N-1}(l)/(D_{N-1}(0)Ω_{N-1}); that's a typo, not an error, since the final kernel is right. The duplication formula (3.45) has the exponent as 2x-1 where it should be 1-2x — again a typo, (3.44) is correct and independently confirmed. The induction proof of (4.6) is asserted rather than given, and the analytic continuation (3.59) is asserted rather than justified. Both are standard, both are the sort of thing a referee would ask for in a sentence, neither is load-bearing. Who it's for: anyone computing spinor determinants, Casimir energies, or one-loop effects on S^N, H^N, dS, AdS. It's a working reference, not a conceptual advance. Recommendation: send it out. It's a clean, honest paper in an established program, and the referee's job is mostly to confirm the two typos and ask for the (4.6) induction.

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Minor objections (5)

[§3.1, Eq. (3.45)] The duplication formula is printed as Γ(x)Γ(x+1/2) = √π 2^{2x-1} Γ(2x); the standard form is Γ(x)Γ(x+1/2) = √π 2^{1-2x} Γ(2x). Since Eq. (3.44) is correct and is independently reproduced by the Weyl dimension formula in Eq. (5.51), the exponent in Eq. (3.45) appears to be a typographical error and should be corrected.
[§4.1, Eq. (4.6)] The right-hand side is printed as D_{N-1}(l)D_{N-1}(0)Ω_{N-1} 1, but Eq. (4.7) uses the coefficient D_{N-1}(l)/(D_{N-1}(0)Ω_{N-1}). The intended relation is Σ_m χ̂χ̂† = [D_{N-1}(l)/(D_{N-1}(0)Ω_{N-1})]·1, which at l=0 is consistent with Eq. (3.43). This should be corrected.
[§4.1, Eq. (4.6)] The text states that Eq. (4.6) 'can be proved by induction in N' without supplying the induction. While the result is standard (it follows from Schur orthogonality and the homogeneity of the sum), a short proof or a reference would help the reader.
[§5.2, Eq. (5.81)] The identity is proved explicitly for N=3 and the proof for N>3 'will be omitted.' Since this identity is used to derive the coupled equations (5.89) and hence the spherical functions (5.106)-(5.107), a brief indication of the general proof would strengthen the group-theoretic derivation.
[§4.1, Eqs. (4.8)-(4.9)] The heat kernel is written with f_N(θ,θ',t) in Eq. (4.8) and with f_N(θ,t) = f_N(0,θ,t) in Eq. (4.9). It would help to state explicitly that the dependence on θ' drops out after the sum over m, so that the notation is consistent.

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The paper

The paper claims

The Dirac equation is solved in closed form on spheres and hyperbolic spaces of every dimension, and the resulting modes are summed to give an explicit heat kernel.

The reading

The paper sets out to write down, explicitly and in any dimension $N$, the spinor wavefunctions that solve the Dirac equation on the round sphere $S^N$ and on real hyperbolic space $H^N$ — the two maximally symmetric spaces of constant positive and negative curvature. The authors separate variables in geodesic polar coordinates, so that a spinor on $S^N$ is built recursively from a spinor on $S^{N-1}$ times two radial functions obeying a Jacobi-type equation. The payoff is a spectrum $i(n+N/2)$ on the sphere with an explicit degeneracy, and a heat kernel for the squared Dirac operator that factorizes into a parallel-transport matrix times a single scalar function of geodesic distance. A second, independent derivation via homogeneous vector bundles and the radial part of the Casimir operator reproduces the same answers, which is what makes the result more than a calculation. Anyone needing propagators or mode sums for spinors in de Sitter, anti-de Sitter, or spatially curved cosmological models has these functions available off the shelf.

What carries the argument

The load-bearing device is the recursive separation of variables in geodesic polar coordinates. Writing the polar vielbein as $e_N=partial_theta$, $e_j = tilde e_j/sin theta$, the Dirac operator on $S^N$ takes the form $(partial_theta + (N-1)/2 cot theta)Gamma^N + (1/sin theta)tilde{/!!nabla}$, where $tilde{/!!nabla}$ is the Dirac operator on the equatorial $S^{N-1}$. This splits the $N$-dimensional problem into a one-dimensional radial equation — reducible to the Jacobi differential operator $D^{(a,b)}_x$ — plus a lower-dimensional Dirac problem already solved. The second device is the group-theoretic re-derivation: spinor fields are sections of the homogeneous bundle $Spin(N+1)/Spin(N)$ associated to a fundamental spinor representation, the Laplacian's eigenvalue is $C_2(tau)-C_2(lambda)$ in Casimir numbers, and Harish-Chandra's formula for the radial part of the Casimir operator becomes an ordinary differential equation on the one-dimensional maximal torus. That the two routes agree is the paper's real verification.

Extended reading notes

Core claim

The central claim is that the eigenfunctions of the Dirac operator on $S^N$ and $H^N$ are known in closed form for arbitrary $N$. On the sphere the regular solutions of $/!!nabla psi = i lambda psi$ exist only for $lambda = pm(n+N/2)$ with $n=0,1,2,dots$, and the corresponding spinors are written as products of a radial factor built from a Jacobi polynomial, $(cos(theta/2))^{l+1}(sin(theta/2))^{l}P^{(N/2+l-1,,N/2+l)}_{n-l}(cos theta)$, times a spinor harmonic on $S^{N-1}$. The degeneracy is $D_N(n) = 2^{N/2}(N+n-1)!/(n!(N-1)!)$ for even $N$ and $2^{(N-1)/2}(N+n-1)!/(n!(N-1)!)$ for odd $N$. Hyperbolic space is reached by the substitution $theta to iy$, $n to -i lambda - N/2$, turning Jacobi polynomials into hypergeometric functions and the discrete spectrum into a continuous one labelled by real $lambda$. Summing the modes gives the heat kernel of $/!!nabla^2$ in the factorized form $K(x,x',t)=U(x,x'),f_N(d(x,x'),t)$, where $U$ is the parallel spinor propagator and $f_N$ depends only on the geodesic distance.

Load-bearing premise

The hyperbolic-space modes and the factorized form of the heat kernel are obtained by analytic continuation and by a mode sum that the paper says can be proved by induction in $N$ but does not actually carry out, so both rest on steps asserted rather than shown.

Editorial extensions

If this is right

Spinor mode sums and propagators on $S^N$ and $H^N$ are available in closed form for any dimension, removing the need for case-by-case construction in de Sitter and anti-de Sitter Euclidean sections.
The heat kernel $K(x,x',t)=U(x,x')f_N(d,t)$ gives a direct route to spinor heat-kernel coefficients and to zeta-function determinants of the iterated Dirac operator on these spaces.
The spinor spectral function (Plancherel measure) on $H^N$, $mu(lambda) = frac{pi}{2^{2N-4}}|Gamma(N/2+i lambda)Gamma(1/2+i lambda)/Gamma(N/2)|^2$, is fixed explicitly, giving the continuous spectral weight needed for any spinor loop computation.
The degeneracy formula $D_N(n)$ coincides with the dimension of the $Spin(N+1)$ spinor representation of highest weight $(n+1/2,1/2,dots,1/2)$, so the mode count is fixed by representation theory rather than by hand.
The recursive construction gives an independent inductive proof of the Dirac spectrum on spheres, starting from the explicit $S^2$ case where the eigenvalues are half-integers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the same radial equation governs both curvature signs, the sphere and hyperbolic results form a single analytic family; one could treat the curvature sign as a parameter and read off how spinor propagators deform as the background passes through zero curvature.
The factorization of the heat kernel into parallel transport times a scalar is exactly the statement that the spinor heat kernel is a zonal function, which suggests the same factorized form should hold for any bundle whose typical fibre is irreducible under the isotropy group — a template worth testing on $p$-form and transverse-traceless tensor cases.
The odd-dimensional case is structurally different: the spinor spherical functions do not vanish at the antipode, and the fibre representation there is equivalent to the original one. That asymmetry between even and odd $N$ is a concrete place where dimension enters qualitatively rather than quantitatively.
Since the authors note the factorization is stated for non-conjugate points only, a reader could ask whether the parallel propagator's direction dependence at conjugate points is a genuine obstruction or an artifact of the polar frame.

Original 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.

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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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates.	method 0.46	0.90	0.39	asserted 1.00
s1	These eigenfunctions are used to derive the heat kernel of the iterated Dirac operator on these spaces.	result 0.50	0.91	0.21	asserted 1.00
s2	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.	contribution 0.86	0.92	0.27	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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References

Works cited (19)

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R. Camporesi and A. Higuchi, The Plancherel measure for p-forms in real hyperbolic spaces, J. Geom. Phys. 15 (1994) 57-94

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R. Camporesi, The spherical transform for homogeneous vector bundles ove r hyper- bolic spaces, preprint 1993

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A. Knapp, Representation Theory of Semisimple Groups, an Overview ba sed on Examples. (Princeton University Press, 1986)

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T.H. Koornwinder, A note on the multiplicity free reduction of certain orthogo nal and unitary groups , Indag. Math. 44 (1982) 215-218 44

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K. Nomizu, Invariant aﬃne connections on homogeneous spaces , Amer. J. Math. 76 (1954) 33-65

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Followed this paper's inductive construction, expressing gamma matrices in (N-1)+1 dimensions in terms of spacelike gamma matrices in (N-1) dimensions.

The eigenmodes for spinor quantum field theory in global de Sitter space-time (2020) · 0.98 · paraphrased: the licence does not allow quoting

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Papers citing it · 122 sentences without a confident label

Uses its method or tool 18 papers

We get the following system of partial differential equations [equation] where $κ$ are the eigenvalues of the Dirac operator on the $D$-dimensional sphere, that is, $κ= ±i (l + (D-2)/2)$ and $l=0,1,2,…$, [ref] .

The Dirac equation in D-dimensional spherically symmetric spacetimes · Quasinormal modes of the Dirac field in the $D$-dimensional Nariai spacetime · 0.86
It was explicitly demonstrated in [ref] that for a wide class of fields, the coincident zeta function in AdS$_5$ may be computed from the corresponding S$^5$ answer by means of the following analytic continutation [equation] As a result the coincident zeta function on AdS$_5$ is given by [equation] We now provide an explicit example of computing the coincident zeta function on AdS$_5$ for a scalar field without using the analytic continuation proposed above.

Exploring Free Matrix CFT Holographies at One-Loop · Harmonic Analysis on Spheres, Hyperboloids, and their Quotients · 0.85
Although the analysis is straightforward using the results of [ref] , we shall go through it carefully, since, as we shall see in the next two sections, these results need to be further corrected due to mixing between scalar, vector and tensor fields in the black hole near horizon geometry.

Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function · Introduction · 0.80

Uses its data or values 13 papers

The 4-d Dirac operator on $S^4$, $Γ̂^μ∇_μ$, has eigenvalues $±i(l+2)/$ $(l=0,1,2,…)$ with degeneracy $D_l(4,1/2)= 2/3 (l+1)(l+2)(l+3)$ [ref] .

One-loop Effective Action of the Holographic Antisymmetric Wilson Loop · Fermionic heat kernels · 1.00
The eigenvalues of the Dirac operator on $S^2$ take integer values [ref] , so that _ℓ (S^2)= ℓ R^-1 , ℓ=1,2,…For the 1+1 dimensional case, fermion energies on the $S^1⊂S^3$ are half-integer valued _ℓ (S^1)= (ℓ-12) R^-1 , ℓ=1,2,…Interestingly, the probe D-branes exhibit the same kind of instabilities we saw above for the D7-branes, but now at values of the chemical potential set by the fermion energy levels on $S^2$ and $S^1$.

Spinning flavour branes and fermion pairing instabilities · D5 and D3$^'$ flavour branes in global $AdS_5$ · 0.81
Equation ([ref]) contains the degeneracy for the eigenvalues $±ił$ computed in [ref] .

Phases of theories with fermions in AdS · Zero temperature: Euclidean AdS$_d+1$ · 0.89

Compares with it 4 papers

For zero coupling ($g=0$) the energy spectrum is $E=±(N+32)$ and the degeneracy was found to be [equation] in agreement with [ref] .

Fermions, Skyrmions and the 3-Sphere · Conclusion · 0.95
[ref] (up to shifts in the exponent which come from mass terms), as well as with the general expression for the zeta function in [ref] .

The Heat Kernel on AdS_3 and its Applications · The Coincident Heat Kernel · 0.99
In the conventions that we use here the eigenvalues of the Dirac operator on the hyperbolic space are purely imaginary as in Ref. [ref] (and therefore $K ∈ℝ$), whereas in Refs. [ref] , [ref] the eigenvalues of the Dirac operator on the hyperbolic space are real numbers.

Quasinormal frequencies of the Dirac field in the massless topological black hole · QNF of the Dirac field · 0.98

Cites it as support 1 paper

Since this type of compactification does not allow for a chiral zero-mode fermion in the four-dimensional (4D) effective spectrum [ref] , we restrict ourselves to a non-universal extra dimensional scenario.

Multi-State Dark Matter from Spherical Extra Dimensions · Introduction · 0.97

Cites it as background 11 papers

The explicit expressions for $ψ^⊥_λ,ω$ and $ψ_λ,ω$ on $ℍ^N$ are well known and can be found, e.g., in [ref] .

Propagators in AdS for higher-derivative and nonlocal gravity: Heat kernel approach · Heat kernel representation of D(box) · 0.83
A free fermion on a curved background which is conformally equivalent to $ℝ^d$ is studied in e.g. [ref] .

Free energy and defect $C$-theorem in free fermion · Boundary condition of fermion on HxS · 0.99
Its spectrum has been discussed in various coordinate systems [ref] .

Bulk-boundary entanglement correspondence and the Ryu-Takayanagi conjecture in an $AdS_2/CFT_1$ setup · Heat kernel and spectral density · 0.85

Discussion

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