A velocity dependent gauge invariance
The paper will discuss mathematical consequences of the application of derived variables in gauge fields. Applying the property of the second Noether theorem, that allows generalised variables - and discusses invariances of physical quantities that depend directly on arbitrary parameters and through them indirectly on the space-time coordinates - this paper extends an article by Al-Kuwari and Taha (1991) with a new conclusion. They concluded, that there are no extra conserved currents associated with local gauge invariance. We will show, that in a more general case, there are further conserved Noether currents. For simplicity we discuss that case, when the mentioned parameters are velocities. Physics is aware of several phenomena, which depend first of all on velocities (like e.g., the force caused by charges moving in a magnetic field, or the Lorentz transformation). In its method the paper will reconstruct the clue, introduced by Utiyama (1956, 1959) and followed by Al-Kuwari and Taha (1991), in the presence of a gauge field that depends on the co-ordinates of the velocity space. In this course we apply certain analogies with Mills (1989) paper. We show, that handling the space-time co-ordinates as indirect variables in the gauge field, reproduces the same results that have been derived in the configuration space, while the proposed new treatment gives additional information extending those. The result is an extra conserved Noether current.
H. A. Al-Kuwari and M. O. Taha, Am. J. Phys. 59, 363 (1991).
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R. Utiyama, Phys. Rev. 101, 1597 (1956).
R. Utiyama, Progr. of Theor. Phys. Suppl. 9, 19 (1959).
R. Mills, Am. J. Phys. 57, 493 (1989).