42 From Classical Mechanics to Quantum Field Theory. A Tutorial
where the last expression gives the classical actionS. If we also define:
∫x(t)=x
x(0)=x′[Dx] = lim→ 0∫
dx 1 ...dxM− 1( m
2 πi )M 2
, (1.203)
we can eventually write Eq. (1.200) as a functional of the classical actionSas:
〈x|e−ıtH|x′〉=∫x(t)=xx(0)=x′[Dx]eıS. (1.204)This is the celebrated path integral formula for the kernel, which is telling us
that to get the quantum amplitude (1.191) we have to integrate over all possible
pathsx(t), starting atx′and ending atx, the exponential of the classical action,
evaluated on such trajectories.
Let us remark that formula (1.200) is compact and has an elegant interpreta-
tion, but it has (at least at this point of the discussion) only a formal meaning,
since we have not specified which class of trajectories, i.e. space of functions, we
work on and therefore what measure of integration we need. Looking at the way we
have got formula (1.204), it is suggestive to interpretx(t) as a trajectory obtained
in configuration space by discretizing it with the set of points{xj=x(t=j )}j.
Thus, for the very way in which it is constructed, we expect it to have no nice prop-
erties such as continuity or differentiability. We will say something more about
this in the following. For now, we use formula (1.200) in a symbolic way and refer
to Eq. (1.200) to do explicit calculations.
Example 1.3.2. The free particle.
As a first example, let us chooseV(x) = 0 and work in 1D, the calculations being
easily generalized to arbitrary dimensions. The kernel (1.200) is given by:
〈x|e−ıtH
|x′〉= lim→ 0∫
dx 1 ...dxM− 1( m
2 πı)M 2
exp[
ım
2 ∑M
n=1(xn−xn− 1 )]
.
(1.205)
Using the identity:
∫
dyexp
{
ıA
2[
(x−y)^2
α+
(y−z)^2
α′]}
=exp{
ıA(x−z)^2
2(α+α′)}(
2 πıαα′
A(α+α′))^12
,
(1.206)
it is possible to calculate recursively all integrals, to eventually find:
〈x|e−ıtH
|x′〉= lim→ 0( m
2 πı)M 2 ( 2 πı
m)M 2 −^1 (
M
t)^12
exp[
ım(x−x′)^2
2 t]
=
√
m
2 πıtexp[
ım(x−x′)^2
2 t