At the end of the 19th century, the electromagnetic field was understood as a collection of two vector fields in space. Nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime.
A convenient way of classifying a field (classical or quantum) is by the symmetries it possesses. Physical symmetries are usually of two types:
Fields are often classified by their behaviour under transformations of spacetime. The terms used in this classification are:
If there is a symmetry of the problem, not involving spacetime, under which these components transform into each other, then this set of symmetries is called an internal symmetry. One may also make a classification of the charges of the fields under internal symmetries.
Much like statistical mechanics has some overlap between quantum and classical mechanics, statistical field theory has links to both quantum and classical field theories, especially the former with which it shares many methods. One important example is mean field theory.