A very important concept in physics is the symmetry or invariance of the equations describing a physical system under an operation – which might be, for example, a translation or rotation in space. Intimately connected with such invariance properties are conservation laws – in the above cases, conservation of linear and angular momentum. Such conservation laws and the invariance principles and symmetries underlying them are the very backbone of particle physics. However, one must remember that their credibility rests entirely on experimental verification. A conservation law can be assumed to be absolute if there is no observational evidence to the contrary, but this assumption has to be accompanied by a limit set on possible violations by experiment.

The transformations to be considered can be either continuous or discrete. A translation or rotation in space is an example of a continuous transformation, while spatial reflection through the origin of coordinates (the parity operation) is a discrete transformation. The associated conservation laws are additive and multiplicative, respectively.

Translation and rotation operators

In an isolated physical system, free of any external forces, the total energy must be invariant under translations of the whole system in space. Since there are no external forces, the rate of change of momentum is zero and the momentum is constant. So invariance of the energy of a system under space translations corresponds to conservation of linear momentum. Similarly, invariance of the energy of a system under spatial rotations corresponds to conservation of angular momentum.