Historical roots of gauge invariance

Abstract

Gauge invariance is the basis of the modern theory of electroweak and strong interactions (the so called Standard Model). The roots of gauge invariance go back to the year 1820 when electromagnetism was discovered and the first electrodynamic theory was proposed. Subsequent developments led to the discovery that different forms of the vector potential result in the same observable forces. The partial arbitrariness of the vector potential A brought forth various restrictions on it. div A = 0 was proposed by J. C. Maxwell; 4-div A = 0 was proposed L. V. Lorenz in the middle of 1860's . In most of the modern texts the latter condition is attributed to H. A. Lorentz, who half a century later was one of the key figures in the final formulation of classical electrodynamics. In 1926 a relativistic quantum-mechanical equation for charged spinless particles was formulated by E. Schrodinger, O. Klein, and V. Fock. The latter discovered that this equation is invariant with respect to multiplication of the wave function by a phase factor exp(ieX/hc) with the accompanying additions to the scalar potential of -dX/cdt and to the vector potential of grad X. In 1929 H. Weyl proclaimed this invariance as a general principle and called it Eichinvarianz in German and gauge invariance in English. The present era of non-abelian gauge theories started in 1954 with the paper by C. N. Yang and R. L. Mills.Comment: final-final, 34 pages, 1 figure, 106 references (one added with footnote since v.2); to appear in July 2001 Rev. Mod. Phy

Spontaneous breaking of Lorentz invariance

“…For enlightening treatments of the principle of gauge invariance and its historical role in the development of modern physical theories, see [17,18]. 3 In the bargain, this scheme would appear to offer an unorthodox avenue to a renormalizable quantum theory of linearized gravity, because the fermion self-interactions could be interpreted as coming from the integrating out, at low energies, of gauge bosons that have acquired large masses via the Higgs mechanism, so that linearized gravity would be the low energy behavior of a renormalizable theory.…”

Decontracted double BRST symmetry on the lattice

“…Beyond perturbation theory, however, there is a problem with such a construction that has not been fully and comprehensively addressed as yet. It relates to the famous Gribov ambiguity [3], the existence of so-called Gribov copies that satisfy the Lorenz condition [4] (or any other local gauge-fixing condition) but are related by gauge transformations, and are thus physically equivalent. As a result of this ambiguity, the usual definitions of a BRST charge fail to be globally valid.…”

The Connection between Inertial Forces and the Vector Potential

“…More recently, Mead (1997) derived standard results of electromagnetic theory of the direct interaction of macroscopic quantum systems assuming solely the Einstein-de Broglie relations, the discrete nature of charge, the Green's function for the vector potential, and the continuity of the wave function -without any reference to Maxwell's equations. Holding an opposite view are Heaviside and Hertz who envisaged in the vector potential merely as an auxiliary artifact to computation (Semon and Taylor, 1996;Coïsson, 1973;Konopinski, 1978;Calkin, 1979;Gingras, 1980;Jackson and Okun, 2001;Iencinella and Matteucci, 2004;Fowles, 1980). In this paper inertia is discussed in terms of the "potential momentum", or vector potential created by the particle, as the primary source for the inertia force.…”

“…As the current must be evaluated at the retarded time we follow a formalism developed by Lorentz to understand the action of each part of a particle on the others since we assume it is not punctual (see Jackson and Okun (2001) for more details on the self-reaction force). The retarded quantity has an expansion in Taylor's series: This is still a pre-relativistic formulation restraining the validity of its results to low particle velocities.…”