Quantum Field Theory-II
UZH and ETH, FS-2016
Assistants: A. Greljio, D. Marzocca, J. Shapiro
http://www.physik.uzh.ch/lectures/qft/
1
Problem Set n. 2
Prof. G. Isidori
Due: 11-03-2016
Saddle point evaluation of the Path Integral
Consider the Generating functional in Eucledean space (x2 = x20 + x21 + x22 + x23 ) of a generic scalar
field theory (~ = 1)
Z
Dφ e−SE [φ,J] ,
WE [J] = N
where
Z
4
SE [φ, J] =
dx
1
1 2 2
2
(∂µ φ) + m φ + V (φ) − J(x)φ(x) .
2
2
(1)
(2)
Define the classical field configuration φ0 ,
δSE [φ, J] =0,
δφ
φ=φ0
(3)
and expand SE [φ, J] around φ0 :
Z
(1)
δSE [φ, J] = SE [φ0 , J] + d4 x ∆SJ (x)[φ(x) − φ0 (x)]
Z
1
(2)
+
d4 xd4 y ∆SJ (x, y)[φ(x) − φ0 (x)][φ(y) − φ0 (y)] + . . .
2
(4)
I. Show that in the limit where we can neglect the dots in Eq. (4), WE [J] is given by
−1/2
WE [J] ≈ N 0 e−SE [φ0 ,J] DetK̂
(5)
where K̂ is defined by
Z
K̂f =
d4 x −∂µ2 + m2 + V 00 (φ0 ) f (x) .
(6)
Put back the ~ factors and discuss the physical meaning of this approximation.
II. Assuming
λ 4
φ ,
(7)
4!
derive the expressions of the connected Green Functions of the theory in the classical limit (leading
contribution for ~ → 0), in momentum space, up to O(λ2 ).
V (φ) =
Suggestion: expand φ0 in powers of λ.
1
2
Scalar QED
Consider the theory of a complex scalar field φ interacting with the electromagnetic field Aµ . The
Lagrangian of the system is
1 2
L = − Fµν
+ (Dµ φ)† (Dµ φ) − m2 φ† φ ,
4
(8)
where Dµ = ∂µ + ieAµ is the gauge-covariant derivative.
I. Using functional methods, show that the propagator of the free complex scalar field, in momentum space, is
i
(9)
2
p − m2 + i
as in the case of a real scalar field.
II. Using functional methods, derive the expressions of all the connected Green Functions of the
theory up to O(e2 ), in the classical limit, in momentum space.
III. Using functional methods, derive the expressions of all the connected Green Functions of the
theory in coordinate space, up to O(e2 ) beyond the classical limit, writing them as functionalproducts of the free scalar and photon propagators.
2