QuantumPhysics.dvi

10.1 The validity of perturbation theory

We shall address two questions here. First, in which physical problems can we expect to

have a small parameter naturally available in which to expand. Second, in what sense is the

perturbative expansion convergent.

10.1.1 Smallness of the coupling

There are four fundamental forces of Nature.

1. Electro-Magnetism and electrodynamics

2. Weak interactions

3. Strong interactions

4. Gravity

Compared to all the other forces, gravity is always weak at the quantum level, and we shall

ignore its effects throughout. Electrodynamics is the foremost example of a system in which

there is a small dimensionless parameter, namely the fine structureconstant,α∼ 1 / 137 .036.

As a result, perturbation theory in electrodynamics can be carriedout successfully in many

situations. One of the oldest and most famous ones is the calculationof the quantum correc-

tions to the electron and muon magnetic moments. The lowest ordervalue results directly

from the Dirac equation, and the first correction was evaluated byTomonaga, Schwinger

and Feynman, a result for which they were awarded the Nobel prize. Since then higher

order corrections have been included as well, and certain relevant effects due to the weak

and strong interactions, and the best value known to date is as follows,

μ=g

(

e

2 mc

)

gμ(exp′nt) = 2× 1 .001159652410(200)

gμ(theory) = 2× 1 .001159652359(282) (10.4)

In view of the unification of electro-magnetism and weak interactions in the Standard Model

of particle physics, the weak interactions also have roughly the coupling constantα, though at

energies well below the mass of theW±(about 80 GeV), their strength is further suppressed.

Thus, the weak interactions may also be treated perturbatively.

The situation is different for the strong interactions, as their namesuggests. At low

energies, such as in nuclear physics, the strong force dominates all other forces, including

electromagnetic. Because of this, two protons can actually bind in aHelium nucleus. Re-

markably, however, the strong interactions decrease in strength at high energies, phenomenon

referred to asasymptotic freedom. Experimentally, this property was discovered in 1968 in