Lorentz-covariant sampling theory for fields

Abstract

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.

A Discrete Analog of General Covariance -- Part 1: Could the world be fundamentally set on a lattice?

“…VII I will use these tools to provide a third and final attempt at interpreting these theories. A perspective similar to this third interpretation has been put forward in the physics literature by Achim Kempf [24][25][26][27][28][29][30][31][32][33][34] among others [35][36][37]. For an overview see [34].…”

“…These lessons are also visible in some corners of the physics literature, particularly in the work of Achim Kempf[24][25][26][27][28][29][30][31][32][33][34] among others[35][36][37]. For an overview see[34].…”

A Discrete Analog of General Covariance -- Part 2: Despite what you've heard, a perfectly Lorentzian lattice theory

“…Hence, as each state in this theory is representable on some lattice, this is a perfectly Lorentzian lattice theory. This, should be compared with compared with the Lorentz-covariant sampling theory for fields developed in [18]. There, only the second condition defining KG K Poin.inv.…”

“…Let us call this a Lorentz-covariant bandlimitation. Lorentz-covariant UV cutoffs have been considered across the physics literature recently, see [7,8,[13][14][15][16][17][18][19] among others.…”

“…This paper is largely inspired by the brilliant work of mathematical physicist Achim Kempf [5][6][7][8][9][10][11][12][13][14][15][16] among others [17][18][19]. A key feature present both here and in Kempf's work is the sampling property of bandlimited theories" can be misleading.…”