Perturbation theory is a powerful mathematical tool used to calculate the probability of particle interactions in quantum field theory. It allows physicists to make approximate calculations by expanding the interaction terms in a series of increasing complexity.
In quantum field theory, particle interactions are described by Feynman diagrams. Each diagram represents a possible interaction process involving incoming and outgoing particles, with lines representing particles and vertices representing interactions. Perturbation theory provides a systematic way to calculate the probability amplitudes associated with these diagrams.
The basic idea of perturbation theory is to start with a simple, solvable theory called the "free theory" and then treat the interactions as small perturbations. The probability amplitude for a given process is then expanded as a series in powers of the coupling constant, which measures the strength of the interactions.
The series expansion is usually written in terms of Feynman diagrams, with each term corresponding to a specific diagram. The lowest-order term corresponds to the free theory and represents the simplest, non-interacting process. Higher-order terms involve additional vertices and loops, which account for the interactions between particles.
The probability amplitude for a specific process is obtained by summing over all the diagrams contributing to that process, with each diagram weighted by its corresponding coefficient in the series expansion. This summation includes an infinite number of diagrams, but in practice, physicists often calculate only a finite number of terms, neglecting higher-order contributions.
The calculations within perturbation theory involve evaluating integrals and performing algebraic manipulations associated with the Feynman diagrams. Each vertex and propagator in the diagram contributes a specific mathematical expression, and the overall amplitude is obtained by multiplying these contributions together.
Perturbation theory provides increasingly accurate results as more terms in the series are included. However, it is important to note that perturbation theory may break down in certain situations, such as at very high energies or for strongly interacting systems. In these cases, alternative techniques, such as non-perturbative methods or numerical simulations, are needed to study particle interactions.
Despite its limitations, perturbation theory remains a crucial tool for calculating particle interactions in quantum field theory, providing insights into the behavior of elementary particles and enabling predictions that can be tested against experimental data.