Известия Алтайского государственного университета

Механическая система с локальной калибровочной симметрией

Гончаров Александр Иванович

Published Jan 1, 2016• 0 citations

Abstract

Our methodological aim is to make easy-to-interpret 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 YZ plane with an additional phase F due to direction changes of axes Y and Z in space and time. Continuously performing active Poincare 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)cosФ(x, t), where Ψ = exp(iS(x,t)). The phase Ф(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 Schrodinger 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 dtf (x, t) and of A with A + dxf (x, t). Thus, the investigated model has local gauge symmetry.

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