Non-Local Classical Field Theory with Fractional Operators on ${\mathbb{S}}^3\times {\mathbb{R}}^1$ Space

Abstract

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 ${\mathbb{R}}^{\text{D}}$ space, we introduce a set of new fractional operators over the ${\mathbb{S}}^3\times {\mathbb{R}}^1$ 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 ${\mathbb{S}}^3\times {\mathbb{R}}^1$ 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, $\mathcal{L}({\varphi }_a\left(x\right),{\mathbb{ð}}^{\alpha }{\varphi }_a\left(x\right))$, as the function of the symmetric fractional derivative of the field, e.g. in the context of the kinetic term, and the field itself.

The work lacks the necessary physical depth!