I.1.1 Conservation Laws and Noether Theorem

Conservation laws are other ways to describe the symmetries that govern the system, especially during its evolution in time. For example, if some quantity  remains constant throughout motion, it is described as conserved or invariant, and its rate of change, which is its first derivative with respect to time, is zero: . The earliest conserved quantities discovered in physics were momentum and energy.

In 1788, with the development of Lagrangian mechanics, which is related to the principle of least action, where the state of the system can be described by any type of generalized coordinates , unlike the customary approach in Newtonian mechanics, where the laws are expressed in Cartesian coordinate system. The action is defined as the time integral  of a function known as the Lagrangian : , where the dot over  signifies the rate of change of the coordinates : .

Hamilton s principle states that the physical path , as the one actually taken by the system, is a path for which infinitesimal variations in that path cause no change in , at least up to first order. This principle results in the Euler-Lagrange equations: .

Thus, if one of the coordinates, say , does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side requires that: , where the momentum  is conserved throughout the motion on the physical path.

Thus, the absence of the ignorable coordinate  from it implies that the Lagrangian is unaffected by changes or transformations of ; the Lagrangian is invariant, and is said to exhibit a symmetry under such transformations. This is the seed idea generalized in Noether s theorem, which can be stated in simple words as:if a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time, or:to every differentiable symmetry generated by local actions, there corresponds a conserved current.

William Hamilton developed a theory of canonical transformations which allowed changing coordinates so that some coordinates disappeared from the Lagrangian, as above, resulting in conserved canonical momenta. Another approach, and perhaps the most efficient for finding conserved quantities, is the Hamilton-Jacobi equation.

In 1918, Noether s theorem gave a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. This means that in a transformation of a physical system that acts the same way everywhere and at all times, there exists an associated time independent quantity called a conserved charge. Conversely, each conserved quantity has a corresponding symmetry. For example, the isometry of space gives rise to conservation of linear momentum, whereas the isometry of time gives rise to conservation of energy.

It must be noted, however, that Noether s theorem applies only on differentiable space-time. Therefore, it is essential that the symmetry be continuous, so that it is specified by a set of parameters that can be varied continuously, and that the symmetry transformation can be arbitrarily close to the group identity. This means that all discrete symmetries of nature, such as time reversal invariance or mirror reflection, do not lead to new conserved quantities. However, there are other quantum counterparts of this theorem, expressed in the Ward-Takahashi identities. Generalizations of Noether s theorem to super-spaces also exist.