# 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:

• Eyes for sensing oscillations of the electromagnetic field passing through the lense,
• ears for (a) sensing oscillations of the pressure field of the air and (b) sensing the direction of the gravitational field,
• the skin for sensing stress fields and temperature fields close to the body surface,
• the tongue for sensing chemical concentration fields close to the surface of the tongue.

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:

• always a scalar mass density field telling how the mass of the object is distributed in space and changes with time,
• in case of uneven composition such as rocks, concentration fields of the various chemical substances it contains.
• in case of nonrigid objects such as fluids, a vector velocity field (or several for each chemical substance), describing the local velocity of the mass flow.
• always a scalar temperature field telling how the temperature of the object is distributed in space and changes with time,
• always a stress tensor field telling how the mechanical forces inside the object are distributed in space and changes with time.
• in case of electrically active objects such as coils or capacitors, a scalar charge density field telling how the charge of the object is distributed in space and changes with time, and a vector current field describing the local velocity of the charge flow.

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