# Field theory (physics)

The term field theory is used to summarize the teaching of physical fields , i.e. for classical field theory ( potential and vector fields ) and quantum field theory .

The field theories have developed from the potential theory of the earth's gravity field , which emerged around 1800, and are the mathematical basis for the description of all those physical effects that are caused by forces or interactions. As such, they are a central component of theoretical physics , geophysics and also other geosciences .

A distinction is made between scalar , vector and tensor fields : A scalar field assigns a scalar to each point in space , i.e. a real number such as temperature or electrical potential. In contrast, a vector field assigns to each point in space, a vector to be, such as in the electric field or the velocity field of a flow. Tensor fields are the subject of continuum mechanics and general relativity . In quantum field theories , the fields are quantized.

Various cross-relationships exist between the individual field types. For example, there are force, i. H. Vector fields , the individual vectors of which result from an underlying scalar field (the scalar potential ) by derivation according to the location, e.g. B. the gravitational field as a derivative ( gradient ) of the gravitational potential , the gravity field as a derivative of the gravity potential , the electric field as a derivative of the electrical potential , etc. Conversely, scalar fields can be derived from certain vector fields by means of the divergence operator or other vector fields can be derived from certain vector potentials with the rotation operator , such as the magnetic flux density .

## Classical field theories

The classical field theories emerged in the 19th century and therefore do not yet take into account the effects that are only known from quantum physics . The best-known classical theories are the potential theory - developed around 1800 from research into the shape of the earth and the earth's gravitational field  - and electrodynamics , which was developed by Maxwell around 1850. The gravity in the general theory of relativity is a classical field theory. Forces act continuously.

Historically, two hypotheses were first put forward about fields: the short- range hypothesis and the long-range hypothesis . In the short-range action hypothesis, it is assumed that both the bodies involved in the interaction and the field involved have an energy, whereas in the long-range action theory only the bodies involved. In addition, according to the remote control hypothesis, disturbances would occur instantaneously, i. H. spread infinitely fast. This discussion started with Isaac Newton , Pierre-Simon Laplace and Michael Faraday . The two possibilities cannot be experimentally differentiated for static or slowly changing fields. Therefore, the question could only be resolved by Heinrich Hertz 's discovery of electromagnetic waves in favor of the near effect: Electromagnetic waves can only propagate when the field itself has an energy.

A distinction is also made between relativistic and non-relativistic field theories.

## formalism

All field theories can be described with mathematical formulas of the Lagrangian . These extend the Lagrange formalism of mechanics. Is a field theory a Lagrangian density known, then a leading variation of the effect${\ displaystyle {\ mathcal {L}} = {\ mathcal {L}} (\ phi _ {i}, \ partial \ phi _ {i})}$

${\ displaystyle S = \ int \ mathrm {d} ^ {n} x {\ mathcal {L}} (\ phi _ {i}, \ partial \ phi _ {i})}$

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

${\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi _ {i}}} - {\ partial _ {\ mu}} {\ frac {\ partial {\ mathcal {L} }} {\ partial (\ partial _ {\ mu} \ phi _ {i})}} = 0 \ quad i = 0.1, \ dotsc}$

These equations form a system of differential equations that uniquely determine the behavior of the fields. Therefore they are also called the equations of motion of a field theory. In order to describe a certain physical system , it is necessary to define the boundary conditions appropriately. However, many physical problems are so complex that a general solution to the problem is impossible or only accessible through numerical methods. Nevertheless, the Lagrangian in field theory enables a systematic investigation of symmetries and conservation quantities.

## The equation of motion for fields

Just as the Lagrangian equations of the 2nd kind are obtained from Hamilton's principle, the Lagrangian equations for fields can be obtained from Hamilton's principle for fields .

To do this, you vary the field

${\ displaystyle \ phi (x, t) \ rightarrow \ phi (x, t) + \ delta \ phi (x, t),}$

which also varies the spatial and temporal derivatives to:

${\ displaystyle {\ frac {\ partial \ phi} {\ partial x}} \ rightarrow {\ frac {\ partial \ phi} {\ partial x}} + \ delta {\ frac {\ partial \ phi} {\ partial x}} = {\ frac {\ partial \ phi} {\ partial x}} + {\ frac {\ partial} {\ partial x}} \ delta \ phi}$
${\ displaystyle {\ frac {\ partial \ phi} {\ partial t}} \ rightarrow {\ frac {\ partial \ phi} {\ partial t}} + {\ frac {\ partial} {\ partial t}} \ delta \ phi}$

As with the derivation of the Lagrangian equations of the 2nd kind, the integral is written in first order as:

${\ displaystyle \ delta \ int \ mathrm {d} t \ int \ mathrm {d} x \, {\ mathcal {L}}}$
${\ displaystyle = \ int \ mathrm {d} t \ int \ mathrm {d} x \ left [{\ mathcal {L}} \ left (\ phi + \ delta \ phi, {\ frac {\ partial \ phi} {\ partial t}} + {\ frac {\ partial} {\ partial t}} \ delta \ phi, {\ frac {\ partial \ phi} {\ partial x}} + {\ frac {\ partial} {\ partial x}} \ delta \ phi, t \ right) - {\ mathcal {L}} \ left (\ phi, {\ frac {\ partial \ phi} {\ partial t}}, {\ frac {\ partial \ phi} {\ partial x}}, t \ right) \ right]}$
${\ displaystyle = \ int \ mathrm {d} t \ int \ mathrm {d} x \ left [{\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi}} \ delta \ phi + { \ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial t}}}} {\ frac {\ partial} {\ partial t}} \ delta \ phi + {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial x}}}} {\ frac {\ partial} {\ partial x}} \ delta \ phi \ right]}$

Now one carries out a partial integration in the spatial and temporal integrals , so that the derivatives of the variation terms are passed on. The following applies to the temporal integration

${\ displaystyle \ int _ {t_ {1}} ^ {t_ {2}} \ mathrm {d} t \ int \ mathrm {d} x {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial t}}}} {\ frac {\ partial} {\ partial t}} \ delta \ phi = \ left [\ int \ mathrm {d} x {\ frac { \ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial t}}}} \ delta \ phi \ right] _ {t_ {1}} ^ {t_ {2} } - \ int \ mathrm {d} t \ int \ mathrm {d} x {\ frac {\ mathrm {d}} {\ mathrm {d} t}} {\ frac {\ partial {\ mathcal {L}} } {\ partial {\ frac {\ partial \ phi} {\ partial t}}}} \ delta \ phi.}$

It is used here that

${\ displaystyle \ delta \ phi (x, t_ {1}) = \ delta \ phi (x, t_ {2}) = 0}$

applies because the start and end points are recorded. Therefore the following applies to the boundary terms:

${\ displaystyle \ left [\ int \ mathrm {d} x {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial t}}}} \ delta \ phi \ right] _ {t_ {1}} ^ {t_ {2}} = 0}$

The spatial derivation is analogous, with the boundary terms disappearing because the fields approach zero at a great distance (e.g. when the Lagrange density describes a particle) or, in the case of a vibrating string, they are fixed at the ends; d. This means that in these points the deflection described by disappears. ${\ displaystyle \ phi (x, t)}$

This ultimately results

${\ displaystyle \ delta \ int \ mathrm {d} t \ int \ mathrm {d} x \, {\ mathcal {L}} = \ int \ mathrm {d} t \ int \ mathrm {d} x \ left [ {\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi}} - {\ frac {\ mathrm {d}} {\ mathrm {d} t}} {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial t}}}} - {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial x}}}} \ right] \ delta \ phi.}$

Since now appears as a factor of the entire integral and is arbitrary, the integral can only vanish according to the principle of variation if the integrand itself vanishes. The following applies: ${\ displaystyle \ delta \ phi}$

${\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi}} - {\ frac {\ mathrm {d}} {\ mathrm {d} t}} {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial t}}}} - {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial x}}}} = 0}$

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

${\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi}} - {\ frac {\ mathrm {d}} {\ mathrm {d} t}} {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial t}}}} - \ sum _ {i = 1} ^ {3} {\ frac {\ mathrm {d} } {\ mathrm {d} x_ {i}}} {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial x_ {i}}}}} = 0,}$

or in the above representation and in the generalization for scalar fields${\ displaystyle N}$${\ displaystyle \ Phi _ {i}}$

${\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi _ {i}}} - {\ partial _ {\ mu}} {\ frac {\ partial {\ mathcal {L} }} {\ partial \ partial _ {\ mu} \ phi _ {i}}} = 0, \ quad i = 1, \ dots, N.}$

## Field types

In field theory, a distinction is made between source fields and vortex fields. Source fields have sources and sinks as their cause , on which their field lines originate and end. Vortex fields are caused by so-called vortices, around which their field lines contract in a closed form, although such a clear form of the vortex field is by no means absolutely necessary: ​​It is sufficient if the orbital integral along any self-contained path within the field is at least one zero provides different values ​​(see below), for example in so-called laminar flows .

Source field
${\ displaystyle \ exists A: \ oint _ {A} {\ vec {X}} \ cdot \ mathrm {d} {\ vec {A}} \ neq 0}$
There is at least one envelope surface A for which the
revolution integral does not vanish. ${\ displaystyle {\ vec {X}} \ cdot \ mathrm {d} {\ vec {A}}}$
Open source field
${\ displaystyle \ forall A: \ oint _ {A} {\ vec {X}} \ cdot \ mathrm {d} {\ vec {A}} = 0}$
There is no envelope surface A for which the
orbital integral does not vanish. ${\ displaystyle {\ vec {X}} \ cdot \ mathrm {d} {\ vec {A}}}$

### Source field

For a general field size X , a source field is given if the divergence is not equal to zero and the rotation is equal to zero:

${\ displaystyle \ mathbf {\ operatorname {div}} \ mathbf {X} = \ nabla \ cdot \ mathbf {X} \ neq 0, \ qquad \ mathbf {\ operatorname {red}} \ mathbf {X} = \ nabla \ times \ mathbf {X} = \ mathbf {0}}$

Source fields can be divided into Newton fields and Laplace fields depending on their boundary value setting. Newton fields such as the gravitational field exist in a spatially unlimited environment of a source or sink, whereas Laplace fields only exist in the finite environment of a combination of sources and sinks, which leads to corresponding boundary value problems. An example of such a Laplace field is the electrostatic field between two electrodes that are electrically charged in opposite directions . Newton and Laplace fields can also occur in a mixed configuration.

Vortex field
${\ displaystyle \ exists S: \ oint _ {S} {\ vec {X}} \ cdot \ mathrm {d} {\ vec {s}} \ neq 0}$
There is at least one boundary curve S for which the
orbital integral does not vanish. ${\ displaystyle {\ vec {X}} \ cdot \ mathrm {d} {\ vec {s}}}$
Invertebrate field
${\ displaystyle \ forall S: \ oint _ {S} {\ vec {X}} \ cdot \ mathrm {d} {\ vec {s}} = 0}$
There is no boundary curve S for which the
orbital integral does not vanish. ${\ displaystyle {\ vec {X}} \ cdot \ mathrm {d} {\ vec {s}}}$

### Vortex field

The field lines of vortex fields are self-contained or infinitely long and not tied to the existence of sources and sinks. The areas around which the field lines contract are called vortices (English curl ) and the following applies:

${\ displaystyle \ mathbf {\ operatorname {red}} \ mathbf {X} = \ nabla \ times \ mathbf {X} \ neq \ mathbf {0}, \ qquad \ mathbf {\ operatorname {div}} \ mathbf {X } = \ nabla \ cdot \ mathbf {X} = 0}$

Similar to source fields, vortex fields can also be subdivided into the class of Newton fields and Laplace fields . An example of a Newton field is the density of the displacement current of an electromagnetic wave , an example of a Laplace field, on the other hand, is the electrical vortex field, which develops around a time-varying magnetic flux .

### General

In the general case there is any space X from a superposition of a source field X Q and a vortex field X W . Because of its central position, this relationship is called the fundamental theorem of vector analysis or the Helmholtz theorem:

${\ displaystyle \ mathbf {X} = \ mathbf {X} _ {Q} + \ mathbf {X} _ {W}}$

Each summand can be made up of a superposition of a Newton and a Laplace field, so that the equation can have four components.

In field theory, a field is clearly specified if both its source and vortex densities and statements about possibly existing edges and the edge values ​​prevailing there are available. The practical significance for the split is based on the easier accessibility of the individual problem. In addition, in many practically significant applications, the problems can be reduced to just one component.

## literature

• Ashok Das: Field theory - a path integral approach. World Scientific, Singapore 2006, ISBN 981-256-848-4 .
• Lev D. Landau: The Classical Theory of Fields. Elsevier / Butterworth-Heinemann, Amsterdam 2005, ISBN 0-7506-2768-9 .
• Günther Lehner: Electromagnetic field theory: for engineers and physicists. Springer, Berlin 2008, ISBN 978-3-540-77681-9 .
• Parry Moon et al: Field theory handbook. Springer, Berlin 1971, ISBN 0-387-02732-7 .
• Arnim Nethe: Introduction to Field Theory. Köster, Berlin 2003, ISBN 3-89574-520-0 .
• Adolf J. Schwab: Conceptual world of field theory . 6th edition. Springer, Berlin / Heidelberg 2002, ISBN 3-540-42018-5 .
• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .