Perturbation Theory

Perturbation theory is a mathematical framework used to approximate physical quantities when an interaction is sufficiently weak, by expanding the result in powers of the coupling constant.

In essence, if a physical problem is too complex to solve exactly but the interaction is small, that interaction can be treated as a slight “perturbation” of a simpler, non-interacting system.

1. Begin with the exact solution of an “ideal” system (with no interaction). 
2. Add progressively smaller corrections that account for the real interaction. 
3. Express the result as a power series in the coupling constant.

Note. Perturbation theory is one of the most widely used tools in theoretical physics, particularly in quantum field theory. In QED and QCD, it is employed to calculate scattering amplitudes (interaction probabilities), propagator corrections (e.g., to mass, charge, or magnetic moment), and cross sections (reaction probabilities). 

A Simple Mathematical Example

Consider a difficult function to compute, for instance:

$$ f(g) = \text{some physical quantity depending on a constant } g $$

If $g \ll 1$, meaning $ g $ is much less than 1, we can expand it as:

$$ f(g) = f_0 + g f_1 + g^2 f_2 + g^3 f_3 + \dots $$

Here:

• $f_0$ is the result with no interaction (zeroth order)
• $g f_1$ is the first correction from the interaction
• $g^2 f_2$ is the second correction, and so forth

This expansion is called a perturbative series.

The terms of a perturbative series can be represented visually using Feynman diagrams:

In quantum theories such as QED and QCD, each term in the expansion corresponds to a Feynman diagram that represents a possible particle interaction.

Each diagram contributes a term proportional to a power of the coupling constant.

Notes

A few remarks and clarifications:

• Limits of perturbation theory
  Perturbation theory is only reliable when the coupling constant is small, meaning particle interactions are weak.

  For instance, in QED (quantum electrodynamics) the coupling constant is very small ( $\alpha \approx \tfrac{1}{137}$ ), since photons are neutral. In this case, perturbation theory yields remarkably accurate results. By contrast, in QCD (quantum chromodynamics) at low energies the coupling constant is large ( $\alpha_s \sim 1$ ), so the series fails to converge. In such regimes, perturbation theory is not applicable, and one must rely on non-perturbative methods instead.

And so forth.