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Non-Abelian gauge field theory in scale relativity

### Abstract

Gauge field theory is developed in the framework of scale relativity. In this theory,space-time is described as a nondifferentiable continuum, which implies it is fractal, i.e., explicitly dependent on internal scale variables. Owing to the principle of relativity that has been extended to scales, these scale variables can themselves become functions of the space-time coordinates. Therefore, a coupling is expected between displacements in the fractalspace-time and the transformations of these scale variables. In previous works, an Abelian gauge theory (electromagnetism) has been derived as a consequence of this coupling for global dilations and/or contractions. We consider here more general transformations of the scale variables by taking into account separate dilations for each of them, which yield non-Abelian gauge theories. We identify these transformations with the usual gauge transformations. The gauge fields naturally appear as a new geometric contribution to the total variation of the action involving these scale variables, while the gauge charges emerge as the generators of the scale transformation group. A generalized action is identified with the scale-relativistic invariant. The gauge charges are the conservative quantities, conjugates of the scale variables through the action, which find their origin in the symmetries of the “scale-space.” We thus found in a geometric way and recover the expression for the covariant derivative of gauge theory. Adding the requirement that under the scale transformations the fermion multiplets and the boson fields transform such that the derived Lagrangian remains invariant, we obtain gauge theories as a consequence of scale symmetries issued from a geometric space-time description.

© 2006 American Institute of Physics

Received 21 November 2005
Accepted 18 January 2006
Published online 13 March 2006

Acknowledgments:
The authors are grateful to Dr. J. E. Campagne for helpful remarks on an earlier version of the paper and to a referee for his useful comments.

Article outline:

I. INTRODUCTION
II. SCALE RELATIVITY AND QUANTUM MECHANICS: SUMMARY
A. Foundations of scale relativity
B. Metric of a fractalspace-time
C. Geodesics of a fractalspace-time
1. Infinity of geodesics
2. Discrete symmetry breaking
3. Quantum-covariant derivative
D. The Dirac equation as a geodesics equation in a fractalspace-time
III. SCALE-RELATIVISTIC THEORY OF ELECTROMAGNETISM: SUMMARY
A. Electromagnetic field and electric charges
B. Quantum electrodynamics
C. Gauge invariance
IV. NON-ABELIAN GAUGE FIELDS
A. Scale-relativistic description
1. Introduction
2. General scale transformations
3. Multiplets
4. Rotations in “scale-space”
B. Yang-Millstheory with the scale-relativity tools
1. Simplified notation
2. Scale relativistic tools for Yang-Millstheory
3. Yang-Millstheory
V. CONCLUSION

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2006-03-13

2016-08-31

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