Symmetry Constraints in Inflation, $\alpha$-vacua, and the Three Point
  Function
  The Ward identities for conformal symmetries in single field models of
inflation are studied in more detail in momentum space. For a class of
generalized single field models, where the inflaton action contains arbitrary
powers of the scalar and its first derivative, we find that the Ward identities
are valid. We also study a one-parameter family of vacua, called
$\alpha$-vacua, which preserve conformal invariance in de Sitter space. We find
that the Ward identities, upto contact terms, are met for the three point
function of a scalar field in the probe approximation in these vacua.
Interestingly, the corresponding non-Gaussian term in the wave function does
not satisfy the operator product expansion. For scalar perturbations in
inflation, in the $\alpha$-vacua, we find that the Ward identities are not
satisfied. We argue that this is because the back-reaction on the metric of the
full quantum stress tensor has not been self-consistently incorporated. We also
present a calculation, drawing on techniques from the AdS/CFT correspondence,
for the three point function of scalar perturbations in inflation in the
Bunch-Davies vacuum.
