Classical physics and Group theory

[ShayanJ]

You know that the current theories in particle physics are expressed in the language of group theory and the symmetries of the theory describe its properties
I don't know how is that but my question is,can we do that to classical physics too?
I mean,can we use maxwell's equations and derive a formulation of classical electromagnetism in the language of group theory for example?
If yes,how is that?
Thanks

 

[vanhees71]

Sure! Symmetries are important for all of physics. It came into the applied mathematics community through Sophus Lie, who used the groups named after him, to analyse differential equations and their solutions. A Lie group is a group whose elements build at the same time a differential manyfolds and the group operations (multiplication and inversion) are diffeomorphisms on this manifold. One very nice feature of these groups is that a whole lot can be learned about them when investigating the group in the neighborhood of the group identity by looking at the corresponding tangent space. This has in a natural way the structure of a vector space with a skew symmetric product, called a Lie algebra.

In physics this mathematical tools are very naturally applied to the most fundamental way to describe dynamical systems, the socalled Hamilton principle of least action. As has been proven by Emmy Noether in the context of General Relativity (which is a classical, i.e., a non-quantum field theory) any continuous transformation on the dynamical quantities that leave the action invariant leads to a conserved quantity. Vice versa any conserved quantity defines a generator (Lie-algebra element) of a one-parameter symmetry Lie group. Thus there is a one-to-one correspondence between symmetries and conserved quantities in classical physics, and one gets a pretty deep understanding of the physics when one analyses a mathematical model describing this physics with help of these group theoretical methods.

Classical electromagnetism even has a socalled local gauge symmetry. This is a symmetry which is given by space-time dependent transformations on the fields. Such gauge symmetries are very strong constraints for model building and determine the physics of such models completely.

As it turns out the quantum version of Maxwell electrodynamics, Quantum Electrodynamics, necessarily implies this gauge-symmetry property since masseless vector quantum fields must be described as gauge fields. But despite this convincing fact the gauge symmetry is also very useful in the domain of the classical theory.

The generalization of this electromagnetic gauge symmety to more complicated gauge groups lead to the development of the standard model of elementary particles, but this is of course already in the domain of quantum field theory rather than classical physics. Anyway, the group-theoretical view becomes the most important ingredient of these models. It not only constrains the models considerable such that one can study them systematically to find out whether they are useful for the description of nature but also has very favorable properties like the possibility to formulate renormalizable models to describe all so far known elementary particles to great accuracy.