In quantum field theory (QFT), the Feynman rules are a set of rules that allow us to calculate the probabilities of different particle interactions. They were developed by physicist Richard Feynman and are based on a perturbation series expansion known as Feynman diagrams. Here's an overview of the Feynman rules and how they can be derived:
Feynman diagrams: Feynman diagrams are graphical representations of particle interactions. They consist of lines and vertices, where lines represent particles, and vertices represent interaction points. Each line corresponds to a particle propagator, and each vertex represents an interaction term in the Lagrangian of the theory.
Propagators: Propagators describe the propagation of particles in space and time. In Feynman diagrams, lines connecting vertices represent propagators. The form of the propagator depends on the specific quantum field theory being considered. For example, in quantum electrodynamics (QED), the propagator for photons is associated with the electromagnetic field.
Vertices: Vertices in Feynman diagrams represent interactions between particles. Each vertex corresponds to an interaction term in the Lagrangian of the theory. The number of lines attached to a vertex depends on the type of interaction. For example, in QED, a vertex with three lines represents an interaction involving three photons.
External lines: External lines in Feynman diagrams represent incoming or outgoing particles. These lines connect to vertices and correspond to particles that are being observed or detected in an experiment.
Conservation laws: At each vertex, there are certain conservation laws that must be satisfied. These conservation laws include conservation of energy, momentum, charge, and other quantum numbers depending on the specific theory.
Feynman rules: The Feynman rules can be derived by systematically applying the perturbation theory to a quantum field theory. Perturbation theory involves expanding the quantum field theory's Lagrangian in terms of small interaction terms. The Feynman rules are essentially a set of prescriptions that assign specific mathematical factors to each component of a Feynman diagram based on the properties of the particles involved, such as propagators, vertices, and external lines.
By applying the Feynman rules to a given quantum field theory, one can construct and evaluate Feynman diagrams that represent different particle interactions. The amplitudes associated with these diagrams are calculated by combining the propagators, vertices, and external lines according to the Feynman rules. These amplitudes can then be used to compute the probabilities of various particle interactions or scattering processes.
In summary, the Feynman rules provide a systematic method for calculating the probabilities of particle interactions in quantum field theory. They are derived from perturbation theory and are based on the graphical representation of interactions using Feynman diagrams. By assigning mathematical factors to the different components of a Feynman diagram, the Feynman rules enable the calculation of amplitudes and probabilities for various particle interactions.