What is a field?

From the way fields are actually used in physics and engineering, and consistent with the mathematical definition, fields are properties of any extended part of the universe with well-defined spatial boundaries. (The latter may be missing in case of infinitely extended objects, e.g., the universe as a whole - if it is infinitely extended.)

Causality is reflected in the fact (that makes physical predictions - and indeed life, which is based on the predictability of Nature - possible) that to a meaningful (and sometimes extremely high) accuracy, changes with time in the complete set of fields relevant for a particular application are determined by the current values of these fields.

Being properties of objects, fields cannot be touched but they can be sensed by appropriate sensors. In particular, several human senses probe properties of fields close the surface of the corresponding sensors:

More specifically, a field is a numerical property of an extended part of the universe, which depends on _points_ characterized by position and time (though the time dependence may be trivial). It is called a scalar, vector, tensor, operator field etc., depending on whether the numerical values at each point are scalars, vectors, tensors, operators, etc., and a real or complex field depending on whether these objects have real or complex coefficients.

Fields are the natural means to characterize numerically the detailed properties of extended macroscopic objects. This can be seen on a very elementary level. (It also applies to microscopic objects, but there the characterization is much more technical.)

All macroscopic objects possess a number of fields, most of them natural in the sense that all humans in our current technological culture experience in their daily life aspects of these fields either with their own sensors, or with technical gadgets known to be sensitive to these. There are:

Not tangible objects such as the space between material objects also have space-time dependent properties, and hence associated fields, namely the (in nonrelativistic case scalar) gravitational field, the (vector) electric field and the (vector) magnetic field.

Hardly visible in everyday life, but very important in physics is an additional field, the (scalar) energy density field telling how the internal energy of the object is distributed in space and changes with time.

Additional fields are employed by physicists whenever the above fields are either not sufficient to give a complete description of the phenomenology they are interested in, or not sufficient to give a tractable theoretical description of the processes.

Causality is implemented by means of parabolic or hyperbolic
differential equations relating the derivatives of the fields.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ