Can you expand a real scalar field $\phi(t,\mathbf{x})$ in terms of spherical harmonics?

Question

A massless real scalar admits the expansion $$ \phi(t,\mathbf{x}) = \int \frac{d^3\mathbf{p}}{(2\pi)^{3/2} \sqrt{2|\mathbf{p}|}} \bigg( e^{ - i |\mathbf{p}| t + i \mathbf{p} \cdot \mathbf{x} } a_{\mathbf{p}} + e^{+ i |\mathbf{p}| t - i \mathbf{p} \cdot \mathbf{x} } a_{\mathbf{p}}^{\ast} \bigg) $$ which when quantized has ladder operators which satisfy $[a_{\mathbf{p}},a_{\mathbf{k}}^{\ast}] = \delta^{(3)}(\mathbf{p} - \mathbf{k})$ as well as $[a_{\mathbf{p}},a_{\mathbf{k}}] =[a_{\mathbf{p}}^{\ast},a_{\mathbf{k}}^{\ast}] = 0$. These are expanded in terms of a plane-wave basis.

Is it possible to expand the field in a basis of angular momentum states? The Klein-Gordon equation in spherical coordinates is a product of spherical Bessel functions and spherical harmonics which makes me think these will be involved. Is there some set of different ladder operators $b_{|\mathbf{p}|,\ell,m}$ corresponding to various angular momentum states (i.e. probably being labelled by $|\mathbf{p}|$, $\ell$ and $m$ and so on)?

Schematically, I imagine something like $$ \phi(t,\mathbf{x}) \sim \int d|\mathbf{p}| \sum_{\ell,m} \bigg( e^{ - i |\mathbf{p}| t} j_{\ell}(|\mathbf{p}r|) Y_{\ell,m}(\theta,\phi) b_{|\mathbf{p}|,\ell,m} + \text{h.c.} \bigg) $$

Answer 1

You can expand any object in any basis you want, this is precisely the point of bases. Whether a given basis is useful or not is an entirely different question. In flat space $\mathbb R^n$ the basis of plane waves is the most convenient one because it diagonalizes translations, which is a massive simplification. I cannot think of any situation where your spherical-harmonics basis is more useful than the plane-wave basis, but it is certainly a valid expansion.

Indeed, the integrand $$ f(x,\vec p):=\frac{1}{(2\pi)^{3/2}\sqrt{2|\vec p|}}e^{ipx}a_{\vec p} $$ is a certain function of $\vec p$, and you can express it in spherical coordinates and expand it in spherical harmonics: $$ f(x,r,\theta,\phi)=\sum_{\ell,m}q_{\ell,m}(x,r)Y_{\ell,m}(\theta,\phi) $$ where $$ q_{\ell,m}(x,r):=\int f(x,r,\theta,\phi)Y_{\ell,m}(\theta,\phi)^*\mathrm d\Omega $$

You can plug this into your plane-wave expansion for $\phi$, $$ \phi(x)=\int f(x,\vec p)\mathrm d\vec p+\text{c.c} $$ and simplify the $\theta,\phi$ integration somewhat but, again, this is in virtually any situation a step back, unless you have a very specific application in mind.

The plane-wave expansion is the most convenient expression for this problem. Other expansions are mathematically valid but almost invariably useless.

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When spacetime is not flat, you generically do not have translations anymore, in which case there is no point in diagonalizing them -- they are not a symmetry of your problem. In general you want to expand in the simplest, largest (abelian) isometry subgroup. If you have a curved manifold with spherical symmetry but no translation symmetry, then it definitely starts making sense expanding in spherical harmonics. And this is indeed a common technique in QFT in curved spacetime. But this goes beyond the original question in the OP so I don't have much more to say here.