Non-Riemannian metric emergent from scalar quantum field theory

We show that the two-point function $\sigma (x,x^{\prime })=\sqrt{⟨[\varphi (x)-\varphi (x^{\prime }){]}^2⟩}$ 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 $|x-x^{\prime }|$ at large distances, yet agrees with it at short distances. For example, space-time has a finite diameter that is not universal. The Lipschitz equivalence class of the metric is independent of the cutoff. $\sigma (x,x^{\prime })$ 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 $\sigma (x,x^{\prime })$ is the length of the geodesic in the ambient space. $\sigma (x,x^{\prime })$ should be useful in constructing the continuum limit of quantum field theory with fundamental scalar particles.

Figure 1

$\sqrt{a}\sigma (x,x^{\prime })$ is plotted as a function of Euclidean distance scaled by lattice length.

Figure 2

This plot shows the dependence of $\sigma$ with Euclidean distance in dimensions three and four.

Figure 3

This plot shows the ratio of distance using two cutoffs on the Euclidean distance.

Figure 4

The trajectory in the $xy$ plane is shown by dotted line.

Figure 5

The diagram shows how tangents are approximated by chord lengths.