Field (physics)

In physics, a field is an assignment of a physical quantity to every point in space (or, more generally, spacetime). A field is thus viewed as extending throughout a large region of space so that its influence is all-pervading. The strength of a field usually varies over a region.

Fields are usually represented mathematically by scalar, vector and tensor fields. For example, one can model a gravitational field by a vector field where a vector indicates the force a unit mass would experience at each point in space. Other examples are temperature fields or air pressure fields, which are often illustrated on weather reports by isotherms and isobars by joining up the points of equal temperature or pressure respectively.

Field theory

Field theory usually refers to a construction of the dynamics of a field, i.e. a specification of how a field changes with time or with respect to other components of the field. Usually this is done by writing a Lagrangian or a Hamiltonian of the field, and treating it as the classical mechanics (or quantum mechanics) of a system with an infinite number of degrees of freedom. The resulting field theories are referred to as classical or quantum field theories.

In modern physics, the most often studied fields are those that model the four fundamental forces.

Classical fields

There are several examples of classical fields. The dynamics of a classical field are usually specified by the Lagrangian density in terms of the field components; the dynamics can be obtained by using the action principle.

Michael Faraday first realized the importance of a field as a physical object, during his investigations into magnetism. He realized that electric and magnetic fields are not only fields of force which dictate the motion of particles, but also have an independent physical reality because they carry energy.

These ideas eventually led to the creation, by James Clerk Maxwell, of the first unified field theory in physics with the introduction of equations for the electromagnetic field. The modern version of these equations are called Maxwell's equations. 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.

Einstein's theory of gravity, called general relativity, is another example of a field theory. Here the principal field is the metric tensor, a symmetric 2nd-rank tensor field in spacetime.

Quantum fields

It is now believed that quantum mechanics should underlie all physical phenomena, so that a classical field theory should, at least in principle, permit a recasting in quantum mechanical terms; success yields the corresponding quantum field theory. For example, quantizing classical electrodynamics gives quantum electrodynamics. Quantum electrodynamics is arguably the most successful scientific theory; experimental data confirm its predictions to a higher precision (to more significant digits) than any other theory. The two other fundamental quantum field theories are quantum chromodynamics and the electroweak theory. These three quantum field theories can all be derived as special cases of the so-called standard model of particle physics. General relativity, the classical field theory of gravity, has yet to be successfully quantized.

Classical field theories remain useful wherever quantum properties do not arise, and can be active areas of research. Elasticity of materials, fluid dynamics and Maxwell's equations are cases in point.

Continuous random fields

Classical fields as above, such as the electromagnetic field, are usually infinitely differentiable functions, but they are in any case almost always twice differentiable. In contrast, generalized functions are not continuous. When dealing carefully with classical fields at finite temperature, the mathematical methods of continuous random fields have to be used, because a thermally fluctuating classical field is nowhere differentiable. Random fields are indexed sets of random variables; a continuous random field is a random field that has a set of functions as its index set. In particular, it is often mathematically convenient to take a continuous random field to have a Schwartz space of functions as its index set, in which case the continuous random field is a tempered distribution.

As a (very) rough way to think about continuous random fields, we can think of it as an ordinary function that is $\pm\infty$ almost everywhere, but when we take a weighted average of all the infinities over any finite region, we get a finite result. The infinities are not well-defined, the last sentence is nonsense to a mathematician, but the finite values can be associated with the functions we supposedly used as the weight functions to get the finite values, and that can be well-defined. We can define a continuous random field well enough as a linear map from a space of functions into the real numbers.

Symmetries of fields

A convenient way of classifying fields (classical or quantum) is by the symmetries it possesses. Physical symmetries are usually of two types:

Spacetime symmetries

Fields are often classified by their behaviour under the symmetry transformations of spacetime. The terms used in this classification are —

Scalar fields (such as temperature) whose values are given by a single variable at each point of space. This value does not change under transformations of space.
vector fields (such as the magnitude and direction of the force at each point in a magnetic field) which are specified by attaching a vector to each point of space. The components of this vector transform between themselves as usual under rotations in space.
tensor fields, (such as the stress tensor of a crystal) specified by a tensor at each point of space. The components of the tensor transform between themselves as usual under rotations in space.
spinor fields are useful in quantum field theory.

In relativity, a similar classification holds, except that scalars, vectors and tensors are defined with respect to the Poincaré symmetry of spacetime.

Internal symmetries

Fields may have internal symmetries in addition to spacetime symmetries. For example, in many situations one needs fields which are a list of space-time scalars: (φ₁,φ₂...φ_N). For example, in weather prediction these may be temperature, pressure, humidity, etc. In particle physics, the color symmetry of the interaction of quarks is an example of an internal symmetry of the strong interaction, as is the isospin or flavour symmetry.

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.

References

Landau, Lev D. and Lifshitz, Evgeny M. (1971). Classical Theory of Fields (3rd ed.). London: Pergamon. ISBN 0-080-16019-0. Vol. 2 of the Course of Theoretical Physics.

Theoretical physics

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