# Short question

Is the spectrum of a field generated by a Gaussian stationary process on a manifold with constant curvature diagonal if the field is expanded in terms of eigenfunctions of the covariant Laplacian?

# Long question

In flat space, the eigenfunctions of the Laplace operator are (using Einstein summation convention) $\phi(\mathbf{x},\mathbf{k})=\exp (ix^\mu k_\mu$). A field $f(\mathbf{x})$ can be expanded as $$f(\mathbf{x})=\int d^3\mathbf{k}\,\phi(\mathbf{x},\mathbf{k})\tilde{f}(\mathbf{k}).$$ Under a translation $\mathbf{x}\rightarrow \mathbf{x}+\mathbf{a}$, the eigenfunctions transform as $\phi\rightarrow\exp(ia^\mu k_\mu)\phi$. Thus, the two-point correlation function $$\underbrace{\langle f^*(\mathbf{x})f(\mathbf{y})\rangle}_{\equiv C(\mathbf{x},\mathbf{y})}=\int d^3\mathbf{k}d^3\mathbf{q}\,\phi^*(\mathbf{x},\mathbf{k})\phi(\mathbf{y},\mathbf{q})\underbrace{\langle \tilde{f}^*(\mathbf{k})\tilde{f}(\mathbf{q})\rangle}_{\equiv \tilde{C}(\mathbf{k},\mathbf{q})}$$ transforms as $$C(\mathbf{x},\mathbf{y})\rightarrow\int d^3\mathbf{k}d^3\mathbf{q}\,\phi^*(\mathbf{x},\mathbf{k})\phi(\mathbf{y},\mathbf{q})\tilde{C}(\mathbf{k},\mathbf{q})\times\exp(ia^\mu(k_\mu-q_\mu))$$ and for the two-point function to be invariant under arbitrary translations, we require $\tilde{C}\propto\delta^3(\mathbf{k}-\mathbf{q})$. Note that $\langle\rangle$ denotes an average over realisations of the field.

However, in curved space, we cannot separate variables in Cartesian coordinates and express the eigenfunctions in terms of polar coordinates such that $\phi_{lm}(\mathbf{x},k)=Y_{lm}(\hat{\mathbf{x}})X_l(x,k)$, where $Y_lm$ are spherical harmonics and $X_l(x,k)$ are radial functions that reduce to spherical Bessel functions in the limit of flat space.

A field may be expanded as $$f(\mathbf{x})=\int dk\,\sum_{lm}\phi_{lm}(\mathbf{x},k)f_{lm}(k)$$ Is the two-point correlation function of expansion coefficients still diagonal under generalised translations (fixed-point free isometries)? I.e. is $\langle f_{lm}^*(k) f_{l'm'}(k')\rangle\propto\delta(k-k')\delta_{mm'}\delta{ll'}$? If so, why? Does the spectrum only depend on $k$ or on $k$, $l$, and $m$ if the process is stationary under rotations (i.e. isometries with a fixed point)?

I have tried my luck with google but most texts are written for mathematicians; unfortunately I am a phycisist without too much formal mathematics background.

Help is much appreciated.

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