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

Abstract

We point out that if spatial information is encoded through linear operators $X_i$, or `infinite-dimensional matrices' with an involution $X_i^{\ast }=X_i$ 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.

7 pages, LaTeX

References in corpus (3)

• The quantum structure of spacetime at the Planck scale and quantum fields
• Gravitational Quantum Limit for Length Measurements
• On Nonlocality, Lattices and Internal Symmetries

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