Continuous symmetries and conserved currents

Abstract

The general relationship between one-parameter continuous symmetries and conserved currents in local Lagrangian field theories is reexamined. Defining a symmetry transformation of a system of fields as a one-to-one mapping of the fields carrying solutions of the equations of motion into solutions, it is noted that not all symmetry transformations imply conserved currents, and conversely, not all conserved currents can be derived immediately from symmetries. However, we develop the one-to-one correspondence between those local continuous symmetry transformations of the system which leave invariant the Euler-Lagrange equations from the original and transformed Lagrangians, and local conserved currents which can be written as linear functions of the Lagrangian derivatives. It is shown that the theorem proved by Noether, which is customarily referred to in the construction of local conserved currents for some symmetry transformations, is in general related only tangentially to the problem in question. The analysis presented for the constructions of conserved currents seems of particular interest in the case that divergence terms enter the symmetry transformation; also it illucidates the independence of conserved currents from the ambiguity always present in Lagrangian formulations due to divergence terms in the original system Lagrangian. In the elementary cases of general interest in field theory, the procedures for the construction of conserved currents do not diverge significantly from the tranditional ones, although even here the difference in attitude regarding the transformation is noted. The converse construction of symmetries from conserved currents has not been considered fully in the literature. The general analysis is applied and extended for systems of fields allowing naive canonical quantization or quantization via the Schwinger action principle. In particular, the consistency of the interpretation of integrals of the currents as quantum generators of the original symmetry transformations is explored in full generality.