A brief introduction to field theory

Processes in which particles can be created or annihilated are best treated in the language of quantum field theory, using field operators j x) that are linear superpositions of operators a p) and a p) which respectively create and annihilate particles of momentum p when they act upon any state vector. If the state happens not to contain the relevant particle of momentum p then a p) acting on it just gives zero (the detailed mathematical relationship can be found in Appendix 1). 

The field operators (f) x) obey equations of motion that are derived from a Lagrangian L via a variational principle, in direct analogy to classical mechanics, and interactions between different fields are produced by adding to the free Lagrangian Lq an interaction term V containing products of the various field operators that are to influence each other. The equations of motion thereby become coupled equations relating the different fields to each other. Usually L is written as an integral over all 

The techniques of field theory are highly sophisticated involving subtle questions of renormalization etc. (see e.g. Bjorken and Drell, 1964, 1965 Itzykson and Zuber, 1980). At this point we require only a simple heuristic appreciation of certain features. 

If the interaction term is small, in the sense that it is proportional to a small coupling constant, a perturbative approach may work. The successive terms are usually shown graphically as Feynman diagrams—a line representing the firee propagation of a particle, a vertex the interaction between particles, the structure of which is controlled by the form of the interaction term in the Lagrangian. 

By far the best known field theory is quantum electrodynamics (QED) in which the coupling of the electron field to the electromagnetic 

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