Differential Form - Applications in Physics

Applications in Physics

Differential forms arise in some important physical contexts. For example, in Maxwell's theory of electromagnetism, the Faraday 2-form, or electromagnetic field strength, is

where the are formed from the electromagnetic fields and, e.g., or equivalent definitions.

This form is a special case of the curvature form on the U(1) principal fiber bundle on which both electromagnetism and general gauge theories may be described. The connection form for the principal bundle is the vector potential, typically denoted by A, when represented in some gauge. One then has

The current 3-form is

where are the four components of the current-density. (Here it is a matter of convention, to write instead of i.e. to use capital letters, and to write instead of . However, the vector rsp. tensor components and the above-mentioned forms have different physical dimensions. Moreover, one should remember that by decision of an international commission of the IUPAP, the magnetic polarization vector is called since several decades, and by some publishers i.e. the same name is used for totally different quantities.)

Using the above-mentioned definitions, Maxwell's equations can be written very compactly in geometrized units as

where denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general.

The 2-form which is dual to the Faraday form, is also called Maxwell 2-form.

Electromagnetism is an example of a U(1) gauge theory. Here the Lie group is U(1), the one-dimensional unitary group, which is in particular abelian. There are gauge theories, such as Yang-Mills theory, in which the Lie group is not abelian. In that case, one gets relations which are similar to those described here. The analog of the field F in such theories is the curvature form of the connection, which is represented in a gauge by a Lie algebra-valued one-form A. The Yang-Mills field F is then defined by

In the abelian case, such as electromagnetism, but this does not hold in general. Likewise the field equations are modified by additional terms involving wedge products of A and F, owing to the structure equations of the gauge group.