On the only three Short Distance Structures which can be described by Linear Operators

TL;DR

This paper classifies the types of space-time structures describable by linear operators, highlighting continuous, discrete, and fuzzy models, and discusses their implications at the Planck scale.

Contribution

It identifies the three fundamental short-distance space-time structures describable by linear operators and explores their properties, especially in the context of quantum gravity.

Findings

Linear operators can only describe continuous, discrete, or fuzzy space-time structures.

Fuzzy space-time may be relevant at the Planck scale.

Infinite-dimensional matrices exhibit weaker properties like symmetry and isometry, affecting space-time modeling.

Abstract

We point out that if spatial information is encoded through linear operators $X_i$, or `infinite-dimensional matrices' with an involution $X_i^*=X_i$ then these $X_i$ can only describe either continuous, discrete or certain "fuzzy" space-time structures. We argue that the fuzzy space structure may be relevant at the Planck scale. The possibility of this fuzzy space-time structure is related to subtle features of infinite dimensional matrices which do not have an analogue in finite dimensions. For example, there is a slightly weaker version of self-adjointness: symmetry, and there is a slightly weaker version of unitarity: isometry. Related to this, we also speculate that the presence of horizons may lead to merely isometric rather than unitary time evolution.