Symmetry and the principle of Nature:
In this century, I believe that Physics will enter into an another remarkable area of basic understanding which we can not discover yet. No, I am not saying about the experimental facts of CERN or Fermilab. I am saying about a completely new aspect of Physics which may be related with some symmetry of nature. Really, symmetry is a very powerful tool for physicists of understand the complicated natural phenomena more simple way.
Now, what is symmetry? we are very much concern about symmetry but it is impossible to explain symmetry in a single sentence. Why? because symmetry is not a very simple concept, some times it not very clear to us. Symmetry is itself many kinds, such as rotation, translation, mirror etc. But there are also many types of symmetries which are very much abstract and we can not realize them pictorially. Only tool to realize them is to study the Group theory which is a very powerful concept of Mathematics. Once Sir Arthur Eddington said that we require a super mathematics in which we don't know about the system of our interest but by the virtue of that super mathematics a super mathematician can extract the important properties of the system. Group Theory is such theory. Actually, I am quite wrong because Group theory is not such a theory but its representation is very likely to Eddington's imagination.
Actually Group theory is the study of symmetry in most smartest way. But how is symmetry related with nature? obviously this is the big question. We know if any symmetry is related to any matrix then the number of fundamental elements of the matrix are reduced, i.e. we can study the very same matrix in simple way if we take into account its symmetry. In very similar way we can apply the concept of symmetry in Physics. Suppose , we are studying a system which is spherically symmetric, then we can easily reduce the variation of the system into only one variable instead of three.
In Physics, symmetry also plays another important role. It can tell about some kind of conservation principle of nature. Every symmetry of nature are related to a conservation principle of nature. For example, we are now studying a system which has a translational symmetry, i.e. the system is invariant in every respect after a translation. We are considering a translation along the X- axis . The total energy or the Hamiltonian of the system is,
the only dynamical quantity of the equation is momentum or P. Now the system is invariant under the translation along X-axis, that means the Hamiltonian (H) of the system is invariant under this translation. So, the only dynamical quantity i.e. linear momentum of the equation being fixed, i.e. linear momentum of the system is conserved. If we see that system is invariant under rotation then in same logic the angular momentum will be conserved. Therefore , in every process in the nature there will be some symmetries and by the virtue of these symmetries there will be some conserved quantity( like mass, charge, spin, baryon number etc.).
Besides this, we can use the concept of symmetry to extract the complicated phenomena from a complicated system( like nucleus, quarks, neutrino etc.). That is symmetry is clue that our nature posses many kinds of mystery but those interrelated and that is why nature is so beautiful.