The symmetries of a physical system manifest themselves through its conservation laws; they are encoded in the Hamiltonian, the generator of time translations. In Quantum Mechanics, the hermitian operators which commute with the Hamiltonian generate the symmetries of the system.

Similar considerations apply to local Quantum Field Theory. There, the main ingredient is the Dirac-Feynman path integral taken over the exponential of the action functional. In local relativistic field theory, the action is the space-time integral of the Lagrange density, itself a function of fields, local in space-time, which represent the basic excitations of the system. In theories of fundamental interactions, they correspond to the elementary particles. Through E. Noether's theorem, the conservation laws are encoded in the symmetries of the action (or Lagrangian, assuming proper boundary conditions at infinity).

Physicists have identified four different forces in Nature. The force of gravity, the electromagnetic, weak, and strong forces. All four are described by actions which display stunningly similar mathematical structures, so much so that the weak and electromagnetic forces have been experimentally shown to stem from the same theory. Speculations of further syntheses abound, unifying all three forces except gravity into a Grand Unified Theory, or even including gravity in Superstring or M Theories!

In his remarkable 1939 James Scott lecture, Dirac speaks of the mathematical quality of Nature and even advocates a principle of mathematical beauty in seeking physical theories!