Taming the $\alpha$-vacuum

An interacting scalar field theory in de Sitter space appears to be nonrenormalizable for a generic $\alpha$-vacuum state. This pathology arises since the usual propagator used allows for a constructive interference among propagators in loop corrections, which produces divergences that are not proportional to standard counterterms. This interference can be avoided by defining a new propagator for the $\alpha$-vacuum based on a generalized time-ordering prescription. The generating functional associated with this propagator contains a source that couples to the field both at a point and at its antipode. To one loop order, we show that a set of theories with very general antipodal interactions is causal and renormalizable.

Figure 1

The contour used to evaluate the evolution of operators over a finite time interval. Here we have shown the contour doubling for global coordinates, Eq. (2.2), but an analogous prescription is used for any coordinate system. The initial state is specified at ${\tau }_0$ and is evolved until $\tau$. We double the field content so that separate copies of the fields are used for the upper and lower parts of the contour.

Figure 2

The Penrose diagram for de Sitter space showing the region covered by inflationary coordinates which has been left unshaded. The past and future null infinity surfaces are denoted by ${\mathcal{I}}^{-}$ and ${\mathcal{I}}^{+}$. The Schwinger-Keldysh approach evolves a matrix element for an initially specified state at ${\eta }_0$ forward to an arbitrary later time $\eta$.

Figure 3

The one loop self-energy correction in a ${\Phi }^3$ theory.

Figure 4

The counterterm graphs corresponding to the possible quadratic terms. The labels $m,\alpha$ indicate whether a propagator is mixed or $\alpha$.

Figure 5

A one loop self-energy correction in a general cubic theory with an arbitrary propagator structure. Note that the external legs are either mixed or $\alpha$ propagators. The counterterm diagram removes the logarithmic divergence of the loop.