lecture-15-CasimirEffect - 2012 Matthew Schwartz III-1 The...

2012
Matthew Schwartz
III-1: The Casimir Effect
1 Introduction
Now we come to the real heart of quantum field theory: loops. Loops generically are infinite.
For example, the vacuum polarization diagram in scalar QED is
p
k
p
k - p
= ( − ie )
2
integraldisplay
d
4
k
(2 π ) 4
2 k
μ
− p
μ
( p + k ) 2 − m 2
2 k
ν
− p
ν
k 2 − m 2
(1)
In the region of the integral at large k μ ≫ p μ , m , this is
1
∼− 4 e 2
integraldisplay
d
4
k
(2 π ) 4
k
2
k 4
∼
integraldisplay
kdk = ∞ (2)
This kind of divergent integral appears in almost every attempt to calculate matrix elements
beyond leading order in perturbation theory: corrections to the electron mass, corrections to the
Hydrogen atom energy levels, etc. Even by the late 1930s, Dirac, Bohr, Oppenheimer and others
were ready to give up on QED because of these divergent integrals.
So what are we supposed to do about these divergences? The basic answer is very simple:
this loop is not by itself measurable. As long as we are always computing physical, measurable,
quantities, the answer will come out finite. In practice the way it works is a bit more compli-
cated – instead of computing a physical observable all along, we deform the theory in such a
way that the integrals come out finite, depending on some regulating parameter. When all the
integrals are put together, the answer for the observable turns out to be independent of the reg-
ulator and the regulator can be removed. This is the program of renormalization. Why it’s
called “renormalization” will become clear in Lecture III-4.
2 Casimir effect
Let’s start with the simplest divergence, the one in the free Hamiltonian. Recall that for a free
scalar field, which is just the sum of an infinite number of harmonic oscillators, the Hamiltonian
is
H =
integraldisplay
d
3
k
(2 π )
3
ω
k
parenleftbigg
a
k
†
a
k
+
1
2
parenrightbigg
(3)
where ω
k
= | k | . So the contribution to the vacuum energy of the photon zero modes is
E = ( 0 | H | 0 ) =
integraldisplay
d
3
k
(2 π ) 3
ω
k
2
=
1
4 π 2
integraldisplay
k
3
dk = ∞ (4)
known as the zero-point energy .
While the zero-point energy is infinite, it is also not observable. As with potential energy in
classical mechanics, only energy differences matter and the absolute energy is unphysical (with
the exception of the cosmological constant, to be discussed in Lecture III-5). To get physics out
of the zero-point energy we must consider the free theory in some context other than just sitting
there in the vacuum.
1. k
μ
≫ p
μ
can be made precise by analytically continuing to Euclidean space, where it implies | k
E
μ
| ≫ | p
E
μ
| .
For scaling arguments, we will more simply treat all the components of k
μ
as larger than all the components of
p
μ
when considering such limits.
1

Consider the zero-point energy in a box of size a . If the energy changes with a then we can
calculate F = −
dE
da
which will be a force on the walls of the box. In this case, we have a natural
infrared (low energy) cutoff: | k | >
1
a
. Of course, this does not cut off the ultraviolet (high
energy) divergence at large k