Curving Space, Effective Gravity and Simultaneous Measurements

Abstract

We present a way of understanding the curvature of space-time, the basic philosophy being that the (linear) geometry of any space is determined by the (linear) functionals on the algebra(s) of any fields defined on the space. It is known that quantum states or hypothetical measurements on a quantum system may be regarded as unit or normalized positive (or non-degenerate to be more generic) linear functionals on a $C^\ast$-algebra which contains the variable events (or variables to be brief) that describe the dynamics of the quantum system. We consider linear projection geometry and differential calculus on a $C^\ast$-algebra $\mathcal{A}$ by means of the positive linear functionals $\mathcal{A}^\ast$ on $\mathcal{A}$. The analysis is done with the help of linear projections based on a version of the Cauchy-Schwarz inequality. We mention the relation between uncertainties or errors involved in the simultaneous measurement of more than one statistical or quantum variable. We also attempt to obtain Einstein's theory of the gravitational field by analytic continuation.