Non-commutative Geometry & the Holographic Principle

Abstract

Non-commutative geometries represent position coordinates on a bounding surface of space in terms of non-commuting variables, thereby unifying relativity theory with quantum theory in a natural way. This procedure mathematically formalizes the smallest possible distance scale, called the Planck length, and gives a fundamental explanation for how any possible space-time geometry is quantized. The bounding surface of space is an event horizon that naturally arises as an observer enters into an accelerated frame of reference. The holographic principle is a natural consequence of non-commutative geometries since quantized bits of information are encoded on a bounding surface of space in a pixelated way. The natural pixel size is about a Planck area. This holographic formulation describes whatever appears to happen in the three dimensional space bounded by a two dimensional bounding surface of space. Holography is deeply ingrained in the geometrical nature of relativity theory and the wave-interference nature of quantum theory. No overarching theory is necessary to understand this connection between non-commutative geometries and the holographic principle. An argument is made that an overarching theory is not even possible since such a theory constrains the observer's frame of reference. The principle of equivalence gives the observer the freedom to enter into any possible frame of reference. In a freely falling frame of reference the bounding surface of space disappears. This mechanism, called horizon complementarity, fundamentally connects holography to non-dual metaphysics.