Scalar two-point functions at the late-time boundary of de Sitter
  We calculate two-point functions of scalar fields of mass $m$ and their
conjugate momenta at the late-time boundary of de Sitter with Bunch-Davies
boundary conditions, in general $d+1$ spacetime dimensions. We perform the
calculation using the wavefunction picture and using canonical quantization.
With the latter one clearly sees how the late-time field and conjugate momentum
operators are linear combinations of the normalized late-time operators
$\alpha_N$ and $\beta_N$ that correspond to unitary irreducible representations
of the de Sitter group with well-defined inner products. The two-point
functions resulting from these two different methods are equal and we find that
both the autocorrelations of $\alpha_N$ and $\beta_N$ and their cross
correlations contribute to the late-time field and conjugate momentum two-point
functions. This happens both for light scalars ($m<\frac{d}{2}H$),
corresponding to complementary series representations, and heavy scalars
($m>\frac{d}{2}H$), corresponding to principal series representations of the de
Sitter group, where $H$ is the Hubble scale of de Sitter. In the special case
$m=0$, only the $\beta_N$ autocorrelation contributes to the conjugate momentum
two-point function.
