The Gauge Principle

A pioneering advance was made by Hermann Weyl with his proposal that the parallel displacement of a vector does not necessarily leave its length invariant, but changes it by an amount 

As the gauge factors depend on the path of displacement, equation (4.13) can be integrated only if the circulation vectors of the type 

The formal resemblance of (4.14) and (4.15) with Maxwell s equations (4.4) and (4.5) prompted the identification of with the electromagnetic 4-potential and the tensor (4.14) as the electromagnetic field. The absence of an electromagnetic field [F = 0) is the necessary condition for the validity of general relativity which only accounts for gravitation. 

An obvious objection to Weyl s theory is that an atom carried around a closed path in an electromagnetic field would radiate at a different wavelength when reaching the end of the loop. This is refuted by experiment. It was shown by London how to address this problem quantum-mechanically. 

Accepting the Bohr postulate of an electronic orbit on a hydrogen atom, stabilized by a balance between mechanical and electrostatic forces. 

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The principle of taking energy from the vacuum is the gauge principle, and this is illustrated as follows on the U(l) level. The U(l) gauge equations in the vacuum are [6]... 

The principle behind this derivation is the gauge principle, and so is the same for all gauge groups. The equivalence (456) was first demonstrated on the 0(3) level [15], but evidently exists for all gauge group symmetries. The gauge principle in electrodynamics therefore leads to the energy and momentum of the photon and classical field. The 4-current J appears in both Eqs. (443) and (444) and is self-dual, a result that is echoed in the self-duality of the vacuum field equations ... 

This modified equation is just the Schrodinger equation that describes the interaction of a charged particle with the elctromagnetic field. This appearance of interaction with a field is known as the gauge principle. A vector field such as A, introduced to guarantee local phase invariance, is called a gauge field. The local invariance of Schrodinger s equation ensures that quantum mechanics does not conflict with Maxwell s field. 

A deep reason for this fanatic belief is the gauge principle. I would say that all empirical conservation laws not protected by the gauge principle are doomed to be violated at some level of strength. The only relevant question is at which energy scales these conservation laws are violated. Both theory and experiment should give definitive signatures for this energy scale. We already seem to have some hint on this. 

The synthesis of general relativity and quantum theory is embodied in the gauge principle that emerges as a natural feature of projective relativity and explains the unihcation of the electromagnetic and gravitational helds. A brief introduction to the concept of gauge invariance is provided in a second Appendix. 

Despite its general utility the gauge principle remains an empirical assumption. It clearly identifies all fields as manifestations of space-time configurations, but not characterized more closely in any way. Each field is... 

Because of the close connection of this concept with the concept of Weyl s geometry with the same name I may call it the gauge principle. The parameter which 1 simply called, following J.H.C. Whitehead, a factor, 1 may now call a gauge variable. A transformation such as (5) we call a gauge transformation. 

The mathematics of infinity is crucial and in order to avoid this unphysical situation projective geometry and a topologically closed cosmos are adopted at the outset. Projective topology is shown to satisfy the demands of both special and general relaticity. The periodic properties of both quantum and chemical systems arise naturally from closed topology and the gauge principle within projective relativity. 

The subsequent discovery (Boeyens, 2003) of the grand periodicity of atomic matter put these speculations into sufficient perspective to allow definite conclusions about the projective topology of space-time and the universe. In the final analysis, all conclusions reached in this work can be reduced to the gauge principle, as summarized in Appendix B. Some readers may like to set the scene by reading this appendix before the main text. 

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