A Non-Riemannian Metric on Space-Time Emergent From Scalar Quantum Field Theory

TL;DR

This paper demonstrates that the two-point function of a scalar quantum field defines a non-Riemannian metric on space-time, which differs from classical metrics at large scales but aligns at short distances, offering new insights into quantum field theory geometry.

Contribution

It introduces a novel non-Riemannian metric derived from the two-point function of scalar quantum fields, expanding the geometric understanding of space-time in quantum theories.

Findings

The two-point function forms a metric satisfying the triangle inequality.

At large distances, the metric differs significantly from Euclidean geometry.

The metric's Lipschitz class is cutoff-independent.

Abstract

We show that the two-point function \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'| at large distances, yet agrees with it at short distances. For example, space-time has finite diameter which is not universal. The Lipschitz equivalence class of the metric is independent of the cutoff. \sigma(x,x') is not the length of the geodesic in any Riemannian metric. Nevertheless, it is possible to embed space-time in a higher dimensional space 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.