= Classicality parameter of a cosmological perturbation
{title2=$\mathcal C$}
For a Hermitian perturbation mode $X$ and its time <derivative> $\dot X$, subtract their <expectations> and use the symmetrized <covariance> to define variances $\sigma_X^2,\sigma_{\dot X}^2$. A commutator-to-fluctuation diagnostic is
$$
\mathcal C=\frac{|\langle[X,\dot X]\rangle|}{\sigma_X\sigma_{\dot X}}.
$$
For zero-mean modes this reproduces the usual expression with $\langle XX\rangle$ and $\langle\dot X\dot X\rangle$ in the denominator. A finite real <Fourier mode> or a suitably smeared field must be specified; unregulated coincident field products need not have finite <variance>.
The <Robertson uncertainty relation> gives $\mathcal C\le2$ when the denominator is nonzero. Small $\mathcal C$ means the commutator is small relative to the marginal fluctuation scales, which can justify classical approximations for some symmetrized correlators. It does not alone show that the <quantum state> is mixed or that <quantum decoherence> has occurred. For example, a correlated <Gaussian wave packet> in a <pure state> can have a large product of marginal uncertainties while retaining <quantum coherence>. Which observables and accuracy are being approximated must therefore accompany a claim of effective classicality.
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