Homework # 2, Quantum Field Theory I: 7640
Due Monday, October 6
Problem 1. Gaussian integration. [10 points]
Prove that
Z ∞
dN x ei
xT Ax
=
−∞
(iπ)N/2
(det A)1/2
for a symmetric real N × N matrix A. Here x is a real column vector of length N , so
that its transpose reads xT = (x1 , x2 , ..., xN ).
Problem 2. Prove the identity
sin x
x
= Π∞
n=1 1 −
For this, form a function f (x) = ln( sinx x ) =
derivative g(x) = f 0 (x) =
x2
π 2 n2
P∞
n=1
. [10 points]
ln 1 −
x2
π 2 n2
and then introduce its
P∞
2x
n=1 x2 −π 2 n2 .
Next we consider a contour integral I =
R
C
1
dz z22x
over the grand circle C of radius
+x2 e2z −1
R → ∞ in the complex z-plane. Show that the integral converges for all |z| = R so that
contribution from that path is zero. Also show that singularities of the integrand are inside
the grand circle. As a result, I = 0. On the other hand, show, with a help of Cauchi’s
residue theorem, that I = g(x) +
1
x
−
cos x
,
sin x
which establishes simple relation for the desired
Ry
function, f 0 (x) =
integrate both sides of this relation over
that f (y) =
the identity listed in the first line of the problem. (You
cos x
− x1 . Finally,
sin x
ln( siny y ) and thus prove
0
dx... to find
also need to argue why f (0) does not contribute to the final result.)
Problem 3. Driven harmonic oscillator. [10 points]
Consider a driven harmonic oscillator (mass m and frequency Ω) Lagrangian of which
is given by L = 12 mẋ2 − 12 mΩ2 x2 + J(t)x. Note that J(t) is an arbitrary function of time
(physically, it represents an external electric field acting on a charged harmonic oscillator).
Your task is to find propagator D(x0 , T ; x, 0) for this system, by splitting the path x(t) =
xcl (t) + y(t), where xcl (t) is the classical trajectory in the presence of external source J(t).
Show that it obeys mẍcl + mΩ2 xcl = J(t). As a result, xcl = xhom + xinhom is the sum
of the homogeneous and inhomogeneous solutions. The inhomogeneous solution is written
in terms of the time-ordered Green’s function G(t) as xinhom (t) = −
RT
0
dt0 G(t − t0 )J(t0 )/m.
2
The Green’s function solves ( dtd 2 + Ω2 )G(t − t0 ) = −δ(t − t0 ). Use this, as well as explicit
form of G(t) (to be worked out in the class) to find xcl which satisfies the required boundary
conditions xcl (0) = x and xcl (T ) = x0 . Use this to calculate Scl and write down explicit
result for D(x0 , T ; x, 0). (For this you also need to carry out the integral over ‘quantum
trajectories’ y(t) which in fact we have worked out previously - you should however explain
what this result is and why it can be used in this problem.)