Field equations

A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations. The term 'classical field theory' is commonly reserved for describing those physical theories that describe electromagnetism and gravitation, two of the fundamental forces of nature. Theories that incorporate quantum mechanics are called quantum field theories.

A physical field can be thought of as the assignment of a physical quantity at each point of space and time. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a vector to each point in space. Each vector represents the direction of the movement of air at that point, so the set of all wind vectors in an area at a given point in time constitutes a vector field. As the day progresses, the directions in which the vectors point change as the directions of the wind change.

The first field theories, Newtonian gravitation and Maxwell's equations of electromagnetic fields, developed in classical physics before the advent of relativity theory in 1905, and had to be revised to be consistent with that theory. Consequently, classical field theories are usually categorized as non-relativistic and relativistic. Modern field theories are usually expressed using the mathematics of tensor calculus. A more recent alternate mathematical formalism describes classical fields as sections of mathematical objects called fiber bundles .

In 1839 James MacCullagh presented field equations to describe reflection and refraction in "An essay toward a dynamical theory of crystalline reflection and refraction".

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