Geometry and quantum field theory - 07 Quantum mechanics.pdf

MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 37
7. Quantum mechanics
So far we have considered quantum field theory with 0-dimensional spacetime (to make a joke, one
may say that the dimension of the space is −1). In this section, we will move closer to actual physics:
we will consider 1-dimensional spacetime, . the dimension of the space is 0. This does not mean that
we will study motion in a 0-dimensional space (which would be really a pity) but just means that we
will consider only point-like quantum objects (particles) and not extended quantum objects (fields). In
other words, we will be in the realm of quantum mechanics.
. The path integral in quantum U(q ) be a smooth function on the real line
(the potential). We will assume that U(0) = 0, U
(0) = 0, and U

(0) = m2,wherem> 0.
Remark. In quantum field theory the parameter m in the potential is called the mass parameter.
To be more precise, in classical mechanics it has the meaning of frequency ω of oscillations. However, in
quantum theory thanks to Einstein frequency is identified with energy (E =
ω/2π), while in relativistic
theory energy is identified with mass (again thanks to Einstein, E = mc2).
We want to construct the theory of a quantum particle moving in the potential field U(q). According
to what we discussed before, this means that we want to give sense to and to evaluate the normalized
correlation functions
iS(q)/

q(t1) ...q(tn)e Dq
<q(t1) ...q(t ) >:= ,
n eiS(q)/
Dq

where S(q)= L(q)dt,andL (q)=q ˙2/2 − U(q).
As we discussed, such integrals cannot be handled rigorously by means of measure theory if
is a
positive number; so we will only define these path integrals “in perturbation theory”, . as formal
series in
.
Before giving this (fully rigorous) definition, we will explain the motivation behind it. We warn the
reader that this explanation is heuristic and involves steps which are mathemat