A Mechanical System with a Local Gauge Symmetry

Abstract

Our methodological aim is to make easy-tointerpret one of the abstract symmetries symmetry with respect to local gauge transformations. An infinite homogeneous string in three-dimensional space is considered. We first assume that free oscillations of the string are described by the function u(x, t) = cos kx exp [−ikct − iF(x, t)]. From the external observers viewpoint each point of the string rotates in the Y Z plane with an additional phase F due to direction changes of axes Y and Z in space and time. Continuously performing active Poincar´e transformations on the standing wave u(x, t) and not affecting the function F we obtain the function for the forced oscillations of a special kind U(x, t) = Ψ(x, t) cos Φ(x, t), where Ψ= exp (iS(x, t)). The phase Φ(x, t) = 0 is called “particle”. It is shown that S is the action of this particle. The particle total energy and generalized momentum that include the potential functions V, A are derived from S. Ψ reduces to an identity the Schrödinger equation with nonlocal Hamiltonian that contains the functions V, A. The identity remains valid when replacing the F with F − f(x, t). This replacement is equivalent to a local gauge transformation in the form of simultaneous replacement of Ψ with exp (if(x, t))Ψ, of V with V −∂_tf(x, t) and of A with A+∂_xf(x, t). Thus, the investigated model has local gauge symmetry.

DOI 10.14258/izvasu(2016)1-04