In Part III we deal with some essential aspects of field theory. Except in the last two sections of the present chapter, the entire discussion is in flat spacetime. After a brief introduction to field theory in general, in this chapter we deal with the classical theory of scalar fields.

Field theory

Basic concepts

A quantum field theory generally yields various species of elementary particle, each corresponding to one of the fields. In particular the photon corresponds to the electromagnetic field. Fields are classified as bosonic or fermionic according to the spin of the particle. One generally considers only fields corresponding to particles with spin-0 (scalar fields) and spin-1/2 particles (spinor fields), plus a special type of field corresponding to particles with spin-1 (vector field) known as a gauge field. The particles corresponding to gauge fields are called gauge bosons. One doesn't generally consider fields of higher spin, because these are difficult to accommodate within the theory and the corresponding elementary particles are not observed. The only exceptions are the spin-2 graviton that is supposed to correspond to a quantized weak gravitational wave, and the spin-3/2 gravitino that will be its partner in a supergravity theory.

At the quantum level, each field corresponds to an operator. Bosonic fields can be regarded as classical in a suitable regime. The fields live in curved spacetime described by the metric tensor gµν.