Field theory (physics)

The term field theory is used collectively for the terms classical field theory ( potential and vector fields) and quantum field theory.

The field theories have evolved from the resulting 1800 potential theory of the Earth's gravity field and the mathematical basis for the description of all those physical effects, which are caused by forces or interactions. As such, they are a central part of theoretical physics, geophysics and other geoscience.

A distinction in the fields between so-called scalar and vector fields: a scalar field assigns to each point in space a scalar, ie a real number to be as in the case of temperature or electrical potential. Contrast, fields that map each point in space a vector is called vector fields such as the electric field or the velocity field of a flow. In the general theory of relativity beyond relative fields play a role, and in quantum field theory, finally, there are quantum fields.

Between individual fields also exist various cross- relationships. For example, there is strength, that is, vector fields, whose individual vectors from an underlying scalar field ( the scalar potential ) arise by differentiation with respect to the location, such as the gravitational field as the derivative (gradient ) of the gravitational potential, the gravitational field as a derivation the severity of potential, the electric field as the derivative of the electric potential, etc. Conversely, again scalar be derived, or, finally, by means of a rotation vector potential from certain fields other vector fields, such as the magnetic flux density vector from specific areas by means of the so-called divergence.

Classical Field Theories

The classical field theories developed in the 19th century and therefore still neglect the effects of quantum mechanics. The best-known classical theories are the potential theory - emerged in 1800 from the exploration of Earth's figure and Earth's gravity - and the electrodynamics developed by Maxwell in 1850. The gravity in the framework of general relativity is a classical field theory. Forces act this continuously.

Historically, two hypotheses about fields were first established: the Nahwirkungshypothese and the distance effect hypothesis. In the Nahwirkungshypothese is assumed that both parties to the interaction of body and the involved field have an energy, while in the action at a distance theory only the body involved. In addition, interference would be according to the distance effect hypothesis instantaneously spread. This discussion was based on Isaac Newton, Pierre- Simon Laplace and Michael Faraday. As in the case of static (ie: time constant ) fields can not be distinguished experimentally between the two hypotheses, the question remained undecided until the discovery of electromagnetic waves by Heinrich Hertz. Electromagnetic waves can in fact only be spread if the field itself has an energy.

A further distinction between relativistic and non-relativistic field theories.

Formalism

All field theories can be described with mathematical formulas of Lagrangians. These expand the Lagrangian formalism of mechanics. If a Lagrangian density for a field theory known then performs a variation of the action

Similarly to the procedure in classical mechanics ( including partial integration) to the Euler -Lagrange equation of field theory:

These equations form a system of differential equations that define the behavior of the fields clearly. Therefore, they are known as equations of motion of a field theory. In order to describe a particular physical system, it is necessary to define the boundary conditions suitable. However, many physical problems are so complex that a general solution of the problem is impossible or accessible only through numerical methods. Nevertheless, the Lagrangians in field theory allow a systematic study of symmetries and conserved quantities.

The equation of motion for fields

So, how to get the second kind Lagrange equations from Hamilton's principle, one can obtain the Lagrange equations for the fields from the Hamilton 's principle for fields.

These will vary to the field

Thus, the spatial and temporal derivatives are varied to

As in the derivation of Lagrange equations 2nd type to write the integral in the first order with

Now one leads to the spatial and temporal integrals of a partial integration, so that the leads are passed by the variation terms. Is then valid for the time integration

Here is used that

, since the start and end point are recorded. Therefore subject to the boundary terms

With the spatial derivative of the procedure is analogous, where the boundary terms vanish, because the fields go a great distance towards zero ( eg, if the Lagrangian density describes a particle ) or they are in the case of a vibrating string fixed at the ends; i.e., that in these points the displacement of which is described by, disappears.

This ultimately results

Now that occurs as a factor of the total integral and is arbitrary, the integral can only disappear with the variational principle, if the integrand itself vanishes. It is therefore

In the three-dimensional case, just added the terms for y and z. The full equation of motion is therefore

Or in the table above and in the generalization of scalar fields

Field types

In the field theory, a distinction is made between source fields and vortex fields. Source fields have sources and sinks as the cause on which originate and terminate their field lines. Vortex fields have to be the cause so-called vertebrae to which contract their field lines in a closed form, although such a vivid form of the eddy field is mandatory not: It is sufficient if the contour integral along any self-contained path within the field at least once a non-zero value gives different (see below), for example in so-called. laminar flows.

Source field

For a general field size X is a source field is given when the divergence is equal to 0 and the rotation is equal to 0:

Source fields can be classified according to their boundary-value position in Newton- Laplace fields and fields. Newton fields such as the gravitational field exist in a spatially unlimited environment of a source or sink, Laplace fields, however, only in the finite environment of a combination of sources and sinks, resulting in corresponding boundary value problems. Example of such a Laplacian is the electrostatic field between two oppositely electrically charged electrodes. Newton and Laplace fields this can result in mixed configuration.

Vortex field

The field lines of eddy fields are self-contained and not tied to the existence of sources and sinks. The areas to which contract field lines are called vertebrae (English curl ) and we have:

Similar to source fields can also be rotational fields divide in the class of Newton - Laplace fields and fields. Examples of Newton's field is the density of the displacement current of an electromagnetic wave example of a Laplace field, however, the electric eddy field that develops around a time-varying magnetic flux.

Generally

In the general case there is any field X from a superposition of a source field XQ and a vortex field XW. This relationship is called the fundamental theorem of vector analysis or as Helmholtz 's theorem because of its central position:

Each summand may be, again composed of a superposition of a Newton and a Laplacian field, so that the equation may have four components.

In the field theory, a field is then uniquely specified if both its source and vortex densities as well as statements about any existing margins and prevailing there exist boundary values ​​. The practical significance for the splitting is due to the ease of accessibility to the individual problem. Moreover, in many practically important applications, the problems reduce to only one component.

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