About

Usage.

For questions where the dynamical variables are fields, that is, functions of several variables (typically, one time coordinate and several space coordinates). Comprises both classical field theory and quantum field theory. Use this tag when the question applies to both classical and quantum phenomena. Otherwise, use the specific tag instead.

Background.

In mathematical terms, a field is a (possibly local) section on a fibre bundle (whose base is at least two dimensional). In simpler terms, a field is a function of at least two variables, such as $f(x,y)$ or $f(t,\boldsymbol x)$.

In physics, fields are used to describe phenomena whose dynamical degrees of freedom depend on several variables. For example, the temperature of an extended body $T(t,\boldsymbol x)$ or the electromagnetic field of a certain system $F_{\mu\nu}(t,\boldsymbol x)$. In both cases, the variables depend on both time and space. When the variables are time-independent the system is said to be static.

In principle, the coordinates need not be space and time. In general terms, the coordinates may be taken in an arbitrary (abstract) manifold. In principle, the manifold is taken to be at least two-dimensional; for otherwise one usually refers to the situation as point-mechanics (where the variables evolve in time only).

In any case, the fields are usually taken to satisfy certain equations of motion, which are partial differential equations in an initial-value problem. These equations are typically derived through a stationary principle, either in the Lagrangian form (lagrangian-formalism) or in the Hamiltonian form (hamiltonian-formalism). In both cases, the equations of motion may be under-determined, in which case one says that the theory is a gauge-theory.

Finally, classical-field-theory refers to fields that are regular functions, while quantum-field-theory refers to operator-valued functions.