2024/11/23 by Savaliya, Abhi, Bidlan, Ayush
#Classical Physics (physics.class-ph) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Group Theory (math.GR) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2411.16731
We present a theoretical framework on non-local classical field theory using fractional integrodifferential operators. Due to the lack of easily manageable symmetries in traditional fractional calculus and the difficulties that arise in the formalism of multi-fractional calculus over ℝD space, we introduce a set of new fractional operators over the \mathbbS3 × ℝ1 space. The redefined fractional integral operator results in the non-trivial measure canonically, and they can account for the spacetime symmetries for the underlying space \mathbbS3 × ℝ1 with the Lorentzian signature (+, -, -, -, -). We conclude that the field equation for the non-local classical field can be obtained as the consequence of the optimisation of the action by employing the non-local variations in the field after defining the non-local Lagrangian density, namely, L(ϕa(x), \mathbbðαϕa(x)), as the function of the symmetric fractional derivative of the field, e.g. in the context of the kinetic term, and the field itself.