1207.0748 (Arnab Kar et al.)

A Non-Riemannian Metric on Space-Time Emergent From Scalar Quantum Field
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Arnab Kar, S. G. Rajeev

We show that the standard deviation \sigma(x,x') = \sqrt{<[\phi(x) - \phi(x')]^2>} of a scalar quantum field theory is a metric (i.e., a symmetric positive function satisfying the triangle inequality) on space-time (with imaginary time). It is very different from the Euclidean metric |x-x'|: for four dimensional free scalar field theory, \sigma(x,x') \to \frac{\sigma_{4}}{a^{2}} -\frac{\sigma_{4}'}{|x-x'|^{2}} + \mathrm{O}(|x-x'|^{-3}), as |x-x'|\to\infty. According to \sigma, space-time has a finite diameter \frac{\sigma_{4}}{a^{2}} which is not universal (i.e., depends on the UV cut-off a and the regularization method used). The Lipschitz equivalence class of the metric is independent of the cut-off. \sigma(x,x') is not the length of the geodesic in any Riemannian metric, as it does not have the intermediate point property: for a pair (x,x') there is in general no point x" such that \sigma(x,x')=\sigma(x,x")+\sigma(x",x'). Nevertheless, it is possible to embed space-time in a higher dimensional space of negative curvature so that \sigma(x,x') is the length of the geodesic in the ambient space. \sigma(x,x') should be useful in constructing the continuum limit of quantum field theory with fundamental scalar particles.

View original: http://arxiv.org/abs/1207.0748