Field (physics)

In physics, a field is an assignment of a quantity to every point in space. For example, one can speak of a gravitational field, which assigns a gravitational potential to each point in space. The isotherms shown in weather bulletins every day on TV are a picture of a temperature field on the surface of the earth. Fields are classified by space-time symmetries or by internal symmetries.

Field theory usually refers to a construction of the dynamics of a field, ie, a specification of how a field changes with time. 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.

Contents

1 Space-time symmetries 2 Internal symmetries 3 Classical and quantum fields 4 See also

Space-time symmetries

Fields are often classified by their behaviour under the symmetry transformations of space, ie, under rotations and translations. 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 Poincare 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.

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 fields as scalars, vectors or tensors under internal symmetries.

Classical and quantum fields

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 as fields of force which dictate the motion of particles, but also have 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 (in the 19th century) 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 (rank 2) tensor field in spacetime.

Einstein's theory of gravity, called general relativity is another example of a field theory. It involves a (rank 4) tensor field in spacetime.

One recognizes now that the universe is essentially quantum mechanical. So any field theory of real phenomena must be at base a quantum field theory. The quantum field theory arising from Maxwell's equations is called quantum electrodynamics. Nowadays one recognizes two more fundamental field theories of particle physics — quantum chromodynamics and 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.

See also

• Quantum field theory and gauge theory
• Electromagnetic field and Maxwell's equations
• General relativity and the standard model of particle physics
• Elasticity and fluid dynamics

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