On the Equivalence between the Superposition Principle and a Global Gauge Invariance

Michel Bawin, André Burnel

Jan Zaanen, Aron Beekman

V. J. Stenger

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.

Gabriel Catren

N Erez

We consider a particle in a coherent superposition of states with different electric charges moving in the vicinity of a magnetic flux. Formally, it should acquire a (gauge-dependent) Aharonov–Bohm (AB) relative phase between the charge states, even for an incomplete loop. If measurable, such a geometric, rather than topological, AB phase would seem to break gauge invariance. Wick, Wightman and Wigner argued that since (global) charge-dependent phase transformations are physically unobservable, charge state superpositions are unphysical (‘charge superselection rule’). This would resolve the apparent paradox in a trivial way. However, Aharonov and Susskind disputed this superselection rule: they distinguished between such global charge-dependent transformations and transformations of the relative inter-charge phases of two particles, and showed that the latter could in principle be observable! Finally, the paradox again disappears once we consider the ‘calibration’ of the phase measured by the Aharonov–Susskind phase detectors, as well as the phase of the particle at its initial point. It turns out that such a detector can only distinguish between the relative phases of two paths if their (oriented) difference forms a loop around the flux.

Fernando Sols

A general study of electron linear transport in systems with open boundary conditions is presented. It is shown that a consistent treatment of bulk and boundary contributions is only achieved in a strictly gauge-invariant formulation of the electomagnetic linear response. Invariance under large gauge transformations requires the use of a generalized version of the continuity equation in which current sources at infinity are explicitly included. These sources are described by delta functions located at the points of infinity and they guarantee global charge conservation. The formulations of steady-state transport in terms of global conductance coefficients and nonlocal conductivity tensors are shown to be strictly equivalent for interacting electron systems in the presence of a magnetic field. As a corollary, the reciprocity theorem for a four-lead resistance is proved in its full generality for systems with arbitrary many-body interactions.