chapter  8
6 Pages

Perturbative Expansion in Field Theory

As we have seen in the previous chapter [∫ Dϕ(x) where relevant includes

ground-state factors ]

Z( J , gn) ≡< 0|U(T)|0 >= ∫ Dϕ(x) exp

( i ∫

d4x L(ϕ) )

(8.1)

will play the role of a generating functional for calculating expectation values of products of field operators, which will now be studied in more detail. In general the Lagrange density for a scalar field theory is given by

L(ϕ) = L2(ϕ) − V(ϕ) − J (x)ϕ(x), (8.2)

where L2(ϕ) is quadratic in the fields, hence for a scalar field

L2(ϕ) = 12 ( ∂µϕ(x)∂µϕ(x) − m2ϕ2(x)

) ,

(8.3) V(ϕ) = g3

3! ϕ3(x) + g4

4! ϕ4(x) + · · · .