Quantization and simulation of Born-Infeld nonlinear electrodynamics on a lattice

Born-Infeld nonlinear electrodynamics arises naturally as a field theory description of the dynamics of strings and branes. Most analyses of this theory have been limited to studying it as a classical field theory. We quantize this theory on a Euclidean 4-dimensional space-time lattice and determine its properties using Monte Carlo simulations. The electromagnetic field around a static point charge is measured using Lüscher-Weisz methods to overcome the sign problem associated with the introduction of this charge. The $\mathbf{D}$ field appears identical to that of Maxwell QED. However, the $\mathbf{E}$ field is enhanced by quantum fluctuations, while still showing the short-distance screening observed in the classical theory. In addition, whereas for the classical theory, the screening increases without bound as the nonlinearity increases, the quantum theory approaches a limiting conformal field theory.

Figure 1

Euclidean action density $⟨S⟩/V$ as a function of $\beta$.

Figure 2

Wilson Lines as functions of charge $e$, for a range of $\beta =a^4{b}^2$ values: (a) $8^4$ lattice, (b) ${12}^4$ lattice.

Figure 3

Ratios of $\mathbf{D}$ to their Maxwell (free-field) values for $\beta =0.0001$ as functions of electric charge $e$, for accessible values of the on-axis separation $Z$: (a) for an $8^4$ lattice, (b) for a ${12}^4$ lattice.

Figure 4

Ratios of $\mathbf{D}$ to their Maxwell (free-field) values for $\beta =1.0$ as functions of electric charge $e$, for accessible values of the on-axis separation $Z$: (a) for an $8^4$ lattice, (b) for a ${12}^4$ lattice.

Figure 5

Ratios of $\mathbf{D}$ to their Maxwell (free-field) values for $\beta =100.0$ as functions of electric charge $e$, for accessible values of the on-axis separation $Z$: (a) for an $8^4$ lattice, (b) for a ${12}^4$ lattice.

Figure 6

Ratios of $\mathbf{E}$ to their Maxwell (free-field) values for $\beta =0.0001$ as functions of electric charge $e$, for accessible values of the on-axis separation $Z$: (a) for an $8^4$ lattice, (b) for a ${12}^4$ lattice.

Figure 7

Ratios of $\mathbf{E}$ to their Maxwell (free-field) values for $\beta =1.0$ as functions of electric charge $e$, for accessible values of the on-axis separation $Z$: (a) for an $8^4$ lattice, (b) for a ${12}^4$ lattice.

Figure 8

Ratios of $\mathbf{E}$ to their Maxwell (free-field) values for $\beta =100.0$ as functions of electric charge $e$, for accessible values of the on-axis separation $Z$: (a) for an $8^4$ lattice, (b) for a ${12}^4$ lattice.

Figure 9

$e^2$ dependence of the inverse electric field: (a) on an $8^4$ lattice, (b) on a ${12}^4$ lattice. Note that the straight lines drawn to aid the eye on (a) and (b) are identical.

Figure 10

Wilson line correlation functions for charges $e$ ranging from $0.0078125$ (top curve) to 4 (bottom curve) (a) for $\beta =100.0$ and (b) for $\beta =0.0001$.