Gauge-theoretical origin of mechanics

Abstract

The momentum of a particle is interpreted as a gauge potential that defines the phase factors and gauge field in terms of the properties of the space–time manifold. Equating the field and the phase factors with the corresponding electromagnetic quantities results in the Lorentz and the Klein–Gordon equation respectively.

Current State of Quantum Theory

“…The Weyl-London gauge transformation is widely used in quantum mechanics in its original as well as generalized to multidimensional cases. Further significant progress was made by formulating and extending Hamilton's action principle in the framework of gauge transformations with a ieS µ µ φ′ =− , where S is the classical action, for a charged particle in an electromagnetic field [8]. This formulation was later extended to include the multi-dimensional gauge fields [9].…”

“…Thus, London's assumption of a being purely imaginary [7], is deduced here as a result. Since a is the same constant for all cases, (8) Left side of (8) is the length acquired by a unit vector transported along xy ρ . Thus, for a trajectory to be physical, i.e., an allowed particle path, (8) requires a vector attached to the particle to regain its length at some point along the path.…”

“…It is clear from (8) and (9) that the assigned gauges participate in describing the observable effects.…”

“…First such attempt was made by London [7] but little progress was made. However, some recent developments based on this concept, have been successful in making major advances [8,9]. The formulation including such developments is referred to as the geometrical formulation.…”

Essentially Pure Particle Formulation of Quantum Mechanics

“…The left side under the same conditions reduces to (1−(p µṗν −ṗ µ p ν )dσ µν ) where p µ are the components of the canonical momentum. This equality is equivalent to the Lorentz equation, equivalently, Newton's second law [6]. The gauge mechanical principle may also be interpreted in terms of Weyl's original notion of gauging a rigid measuring rod as follows.…”