Applications of Group Theory ~~ A Brief Intro!

Group theory you see has many interesting applications in varied fields of science and technology because group theory is inherently a theory of symmetry and structure. I will not go through the details of any of these applications but rather discuss how I, as a physicist, personally use and appreciate group theory on daily basis.

It is very fascinating that everything in this universe loves symmetry, including me. In fact the very fundamental laws of nature inhibit a certain symmetry structure (some apparently don't, giving rise to more interesting phenomenon such as the electrical dipole moment of a single electron). And this is very beautiful and one derives an infinite amount of joy simply by thinking about it. More fascinating (and very joyful indeed) is the fact that we can study symmetry in a very systematic and beautiful manner using group theory.

Suppose we have the set 'sym(T)' which is the set of all bijections (or automorphisms if you may call it) of a set T onto itself. The elements of this set can be combined to form compositions of mappings which actually define binary operations on the set. Moreover, since these are bijections, one can think of these elements as being transformations (eg. rotation, translation etc.) of coordinate systems if T is a set of infinite order (transformation of real vector spaces if T is the set of real numbers). One can easily see that the binary operation on this set is the composition of transformations. The transformations of a coordinate system (basis set) are represented by linear matrices which are invertible. Therefore, we see that there is a one-to-one correspondence between this set and the set of all invertible general linear matrices. They are actually the same thing in essence (isomorphic is the right word). I think this set sym(T) is the most general example (and perhaps a loose definition) of a group: “it is the set of all transformations of coordinate systems”. See this has got a very beautiful physical picture now. I like this particular group very much that I have given it a name, 'the Father Group'. All other groups in this world can be thought of as sub-groups of this Father Group. Because all groups arise as a result of constraining a subset of the Father Group to preserve certain structure or symmetries. (Please note that different groups may have different elements and different binary operations as well, but in essence they may represent the same structure. This is the basic notion of isomorphism. So in essence, all groups come from the Father Group.)

The above was just an illustration of how general and therefore beautiful group theory is. 

Coming to more interesting applications of group theory that I use on daily basis are the following: (1) Fourier transforms and (2) numerically solving Schrodinger's equation. The later is particularly heavily dependent on group theory to simplify computational procedures. For example, think of some symmetry operation (such as parity i.e. space inversion, or time reversal symmetry etc.) that leaves the Hamiltonian invariant. This is same as saying that the symmetry operator commutes with the Hamiltonian. If that is the case, the matrix elements of the Hamiltonian between eigenvectors of the symmetry operator will vanish. One can thus eliminate calculations of these matrix elements from computational steps which would greatly improve upon speed and efficiency of these calculations. Moreover, many times study of symmetry structure of these operators reveals very interesting rather unusual or unknown facts about the system under investigation. So it is a good thing indeed to study symmetry.

I love it. So will you, I am sure. Grab a book (get a nice one, warn you!) or browse through the internet for learning more about group theory. It will be an awesome experience I promise you (that is assuming you're passionate about science like me).

Good luck! :)