Lorentz-covariant sampling theory for fields

Sampling theory is a discipline in communications engineering involved with the exact reconstruction of continuous signals from discrete sets of sample points. From a physics perspective, this is interesting in relation to the question of whether spacetime is continuous or discrete at the Planck scale, since in sampling theory we have functions which can be viewed as equivalently residing on a continuous or discrete space. Further, it is possible to formulate analogues of sampling which yield discreteness without disturbing underlying spacetime symmetries. In particular, there is a proposal for how this can be adapted for Minkowski spacetime. Here we will provide a detailed examination of the extension of sampling theory to this context. We will also discuss generally how spacetime symmetries manifest themselves in sampling theory, which at the surface seems in conflict with the fact that the discreteness of the sampling is not manifestly covariant. Specifically, we will show how the symmetry of a function space with a sampling property is equivalent to the existence of a family of possible sampling lattices related by the symmetry transformations.

Automated summary

Sampling theory is a discipline that deals with reconstructing continuous signals from discrete sets of sample points in communications engineering. It is also relevant in the question of spacetime discreteness at the Planck scale, as sampling theory can be applied to continuous or discrete spaces. There is a proposal to apply this to Minkowski spacetime and this article explores the extension of sampling theory in this context. Additionally, the article discusses how spacetime symmetries are manifested in sampling theory, despite the discreteness of the sampling not being manifestly covariant.