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BENTHAM SCIENCE PUBLISHERS eBooks

Gauge Symmetry

P. Enders

Published Feb 12, 2019• 0 citations
Publisher

Abstract

Gauge symmetry respectively gauge invariance plays a crucial role in modern quantum field theory. Historically, the first locally gauge-invariant equations of motion are Maxwell’s original equations for the electrical charges and currents and the electromagnetic fields created by them. In order to keep the relation to classical mechanics as close as possible, this chapter starts with a critical examination of global gauge invariance, basing on Helmholtz’s interpretation of the potential energy as “disposable work storage”. Then, Helmholtz’s explorations of the relationships between forces and energies are sketched and extended. Their generalization to quantum mechanics leads to the gauge invariance of the stationary and timedependent Schrödinger equations. An important application is the Ehrenberg-Siday- Aharonov-Bohm effect and its gravito-electromagnetic analog. For the sake of the unity of physics, it is shown, how the (gravito-)electromagnetic equations can be deduced. Here, Newton’s imaginations about the field of gravity are also exploited.

DOI
10.2174/9781681084497119010019

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Gauge symmetry respectively gauge invariance plays a crucial role in modern quantum field theory. Historically, the first locally gauge-invariant equations of motion are Maxwell’s original equations for the electrical charges and currents and the electromagnetic fields created by them. In order to keep the relation to classical mechanics as close as possible, this chapter starts with a critical examination of global gauge invariance, basing on Helmholtz’s interpretation of the potential energy as “disposable work storage”. Then, Helmholtz’s explorations of the relationships between forces and energies are sketched and extended. Their generalization to quantum mechanics leads to the gauge invariance of the stationary and timedependent Schrödinger equations. An important application is the Ehrenberg-Siday- Aharonov-Bohm effect and its gravito-electromagnetic analog. For the sake of the unity of physics, it is shown, how the (gravito-)electromagnetic equations can be deduced. Here, Newton’s imaginations about the field of gravity are also exploited.
Maxwell’s equations, formulated in 1865, preceded quantum mechanics by six decades. When the logical direction is reversed, however, the electromagnetic field structure can be interpreted as the form required for a quantum theory of charged matter once three standard ingredients are imposed: local gauge invariance, conserved electric current, and a dynamical electromagnetic field. Quantum mechanics supplies the charged-particle state, probability amplitudes, quantum currents, and—through the Dirac equation—a relativistically consistent description of electron charge and spin. Gauge symmetry then organizes the electromagnetic interaction, while current conservation constrains the admissible source structure. The homogeneous Maxwell identities follow naturally from expressing the field through a four-potential, whereas the sourced Maxwell equations require an additional electromagnetic field action or equivalent dynamical assumption. The resulting framework clarifies both the strength and the limit of a “Maxwell from quantum mechanics” program: quantum theory and gauge structure strongly constrain electromagnetism, but Maxwell field dynamics are not obtained from the electron wave equation alone.
The gauge invariance of the Aharonov-Bohm (AB) effect with a quantum treatment for the electromagnetic field is demonstrated. We provide an exact solution for the electromagnetic ground energy due to the interaction of the quantum electromagnetic field with the classical charges and currents that act as sources of the potentials in a classical description, in the Lorenz gauge. Then, we use first-order perturbation theory to compute an extra change on the electromagnetic ground energy due to the presence of a quantum charged particle with known wave function in the system. This energy in general depends on the quantum particle path in an interferometer, which results in an AB phase difference between the paths. The gauge invariance of this AB phase difference is then shown for the magnetic, electric, and the recently proposed electrodynamic versions of the AB effect. However, the AB phase difference could depend on the gauge for nonclosed paths, which reinforces the view that it only can be measured in closed paths.
The laws of physics were not handed down from above. Neither are they rules somehow built into the structure of the universe. They are ingredients of the models that physicists invent to describe observations. Rather than being restrictions on the behavior of matter, the laws of physics are restrictions on the behavior of physicists. If the models of physics are to describe observations based on an objective reality, then those models cannot depend on the point of view of the observer. This suggests a principle of point-of-view invariance that is equivalent to the principle of covariance when applied to space-time. As Noether showed, space-time symmetries lead to the principles of energy, linear momentum, and angular momentum conservation--essentially all of classical mechanics. It also leads to Lorentz invariance and special relativity. When generalized to the abstract space of functions such as the quantum state vector, point-of-view invariance is identified with gauge invariance. Quantum mechanics is then just the mathematics of gauge transformations with no additional assumptions needed to obtain its rules, including the superposition and uncertainty principles. The conservation and quantization of electric charge follow from global gauge invariance. The electromagnetic force is introduced to preserve local gauge invariance. Although not discussed here, the other forces in the standard model of elementary particles are also fields introduced to preserve local gauge invariance. Gravity can also be viewed as such a field. Thus practically all of fundamental physics as we know it follows directly from the single principle of point-of-view invariance.
This paper examines how Maxwell electrodynamics can be recovered as the field-theoretic structure required to consistently couple quantum matter to electromagnetism, while distinguishing what follows from quantum mechanics from what must be supplied independently. Local phase invariance introduces electromagnetic potentials through minimal coupling and yields the homogeneous Maxwell equations as geometric identities, while quantum probability conservation provides a conserved electric current and Ehrenfest dynamics recover the Lorentz-force form. The inhomogeneous Maxwell equations require an additional Lorentz- and gauge-invariant electromagnetic field action, whose vacuum constants satisfy μ0ε0=1/c2\mu_0\varepsilon_0=1/c^2, linking the construction to the SQC/MSC electric–magnetic prefactor relation. Hydrogen is used as an explicit microscopic source: the 1s1s state generates a quantum charge distribution, 2p2p states provide orbital currents and magnetic moments, and 1s1s–2p2p superpositions produce radiative dipole transitions. The treatment is then extended from individual atoms to a hydrogen gas, where polarization and magnetization generate the macroscopic fields DD and HH. The resulting framework shows how quantum sources and material response fit consistently within Maxwell electrodynamics without claiming that quantum mechanics alone uniquely derives the complete electromagnetic field dynamics.
A nominally legitimate set of electromagnetic potentials to represent electric and magnetic fields can cause violation of basic symmetries in both classical and quantum applications, creating errors in dynamical calculations. Preservation of symmetries places constraints on the choice of potentials and thus constitutes a robust reason to regard electromagnetic potentials as more fundamental than electromagnetic fields. This represents a major extension of the quantum-only example of the Aharonov-Bohm effect for establishing the primacy of potentials over fields. Symmetry violation caused by inappropriate gauge choice means that gauge transformations are not necessarily unitary, since they do not ensure preservation of the values of physical observables. Unitarity is examined in the Schr\odinger equation and in transition amplitudes.
The Aharonov-Bohm (AB) effect, wherein an electron acquires a phase shift in a field-free region due to electromagnetic potentials, poses a profound challenge to the ontology of quantum mechanics and gauge theories. This paper demonstrates that gauge-invariant explanations, which attribute the phase to measurable quantities, rely on nonlocal and discontinuous mechanism and fail to account for its continuous accumulation along the electron's path—a process evident in the generalized AB effect, where a time-varying magnetic flux induces a phase that builds gradually over time, as predicted by quantum mechanics. Through a new analysis spanning quantum mechanics and quantum electrodynamics, I argue that the electromagnetic potentials AμA_\muAμ​, fixed in the Lorenz gauge, emerge as the fundamental physical reality, offering a local, relativistically consistent account of the phase’s generation. This exclusion of gauge-invariant paradigms reverberates across gauge theories: it redefines the Higgs mechanism, favoring dynamic potentials over static invariants, and extends to general relativity, where gravitational potentials gμνg_{\mu\nu}gμν​ may anchor spacetime’s substantival reality via a gravitational AB effect. By unraveling the AB phase’s continuous generation—locally mediated by potentials—this study not only addresses a long-standing conundrum but also bridges electromagnetic, gravitational, and Yang-Mills frameworks, offering a unified potential-centric perspective with novel implications for physics and philosophy.