User:Mpatel/sandbox/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 it's influence is all-pervading. The strength of a field usually varies over a region.

Fields are usually represented mathematically by vector fields 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.

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's now understood that quantum mechanics underlies all physical laws, so we attempt to treat a classical field as a quantum mechanical system, thus giving a quantum field theory. Quantizing classical electrodynamics gives quantum electrodynamics. It's noteworthy that quantum electrodynamics is the most successful scientific theory ever — it's predictions match experimental data to an unprecedented level of accuracy never seen before.

There are two more fundamental field theories of particle physics — quantum chromodynamics and the electroweak theory. These three interactions are unified into the standard model of particle physics. General relativity has not yet been successfully quantized.

There remain many areas in which classical field theory is useful, since the quantum nature of the universe may not manifest itself in every situation. Elasticity of materials, fluid dynamics and Maxwell's equations are some of many useful classical field theories. Some of them remain active areas of research.

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:

Continuous symmetries

Fields are often classified by their behaviour under the symmetry transformations of space, i.e., under rotations and translations. As a result, most continuous symmetries tend to be spacetime symmetries. 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.

Discrete symmetries

Fields may have discrete 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 a discrete 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 also called an internal symmetry. One may also make a classification of the charges of the fields under internal symmetries.