coarse graining consists of functionally integrating out the higher modes. As we would like to generate vacuum expectation values from the effective action rather than the matrix elements one should use the in-in or closed-time-path (CTP) version of the effective action (Calzetta and Hu 1987) rather than the in-out version. The CTP coarse-grained effective action in our case is given by (for details, see Sinha and Hu 1991) e\^[iS\_[eff]{}( a\^+ ,\_[0]{}\^[+]{},a\^-,\_[0]{}\^[-]{} )]{} = f\_[k]{}\^[+]{} [D]{}f\_[k]{}\^[-]{} e\^[i( S(a\^+ ,[\_0]{}\^+, f\_[k]{}\^[+]{}) - S\^[\*]{}( a\^- ,[\_0]{}\^[-]{} ,f\_[k]{}\^[-]{}) )]{} $S^{*}$ indicates that in this functional integral, $m^{2}$ carries an $i\epsilon$ term. $a^{\pm}, \chi_{0}^{\pm}, f_{k}^{\pm}$ are the fields in the positive (negative) time branch running from $\eta =
\stackrel{-}{+}\infty$ to $\pm\infty$. The path integral is over field configurations that coincide at $t = \infty$. ${\cal D} {f_k}$ symbolizes the functional integration measure over the amplitudes of the higher modes of the scalar field.

We have derived the one loop renormalized coarse-grained effective action given as follows (omitting terms that involve $-$ fields only) S\_[eff]{} & = & S\_[g+]{} + [12]{}d{ [[\^+]{}’]{}\^2 - \^2 [\^+]{}\^2}\
& &-[ł4!]{}d\^4 + [13ł48]{}d+ [116]{}d\^4\
& &+ [132]{} [d\_1 d\_2]{}(\_1) K(\_1 - \_2) (\_2)\
& & +[132]{}[d\_1 d\_2]{}(\_1) |[K]{}(\_1 - \_2)(\_2) 