I have some schematic notes on computing the effective action and I would like someone to help me fill the gaps.
We start with
∫Dϕe−iS[ϕ]
employing the background field method we write
ϕ=ϕ0+Δϕ
so we have
∫D(Δϕ)e−iS[ϕ0+Δϕ]
Taylor expanding around
ϕ0
S[ϕ0+Δϕ]=S[ϕ0]+∫d4x1δSδϕ(x1)Δϕ(x1)
+12∫d4x1d4x2δ2Sδϕ(x1)δϕ(x2)Δϕ(x1)Δϕ(x2)+
13!∫d4x1d4x2d4x3δ3Sδϕ(x1)δϕ(x2)δϕ(x3)Δϕ(x1)Δϕ(x2)Δϕ(x3)+…
since
ϕ0 satisfies the equations of motion the linear term in
Δϕ vanishes. Then we have
e−iS[ϕ0]∫D(Δϕ)e−i12∫d4x1d4x2δ2Sδϕ(x1)δϕ(x2)Δϕ(x1)Δϕ(x2)+…
from here on my notes neglect terms cubic,quartic... in Δϕ. Can anybody tell me why?.
Also, after this it is written
e−iS[ϕ0]det(…)
where the dots represent (I think) a functional determinant of something. Can anybody tell me what goes inside the determinant, and where this comes from?