Mass Dependence of the Araki-Uhlmann Relative Entropy Across Dimensions

TL;DR

This work addresses how the Araki-Uhlmann relative entropy $S(\psi_f|\Omega)$ between a localized coherent state and the vacuum in a free massive scalar QFT on $(1+d)$-dimensional Minkowski spacetime depends on the mass $m$ and spatial dimension $d$ (with $d=1,2,3$). The authors compute the relative entropy via the smeared Pauli-Jordan distribution, $S(\psi_f|\Omega) = -\frac{1}{2}\Delta_{\rm PJ}(f,f_s|_{s=0})$, using explicit Pauli-Jordan functions in each dimension and a numerically evaluated test-function setup implemented with Quasi-Monte Carlo integration. They find a dimension-dependent pattern: monotonic decay with mass in $(1+1)$ and $(1+3)$ dimensions accompanied by oscillations whose amplitudes scale as $m^{-1/2}$ and $m^{1/2}$ respectively, while in $(1+2)$ dimensions the relative entropy is non-monotonic with a local maximum near $m\alpha\approx 2.2$ and rapidly damped oscillations; these results closely tie the mass dependence to the high-mass behavior of the Pauli-Jordan function in each dimension. The study advances modular theory in QFT by quantifying how parameter choices and geometry affect state distinguishability and suggests future work on universal scaling, interactions, holographic connections, and other causal domains.

Abstract

We investigate the mass dependence of the Araki-Uhlmann relative entropy between a localized coherent excitation and the vacuum state of a free scalar quantum field on the $(1+d)$-dimensional Minkowski spacetime for $d = 1, 2, 3$. In this context, the relative entropy admits a closed expression in terms of the smeared Pauli-Jordan distribution, whose analytic structure is sensitive to both the mass and the spacetime dimensionality. Prior studies in $(1+1)$ dimensions have shown a monotonic decay of the relative entropy with increasing mass. We extend that analysis to higher dimensions using numerical techniques and elucidate how the interplay between dimensionality and mass controls the behavior of the relative entropy. Our results provide new insights for the study of the Araki-Uhlmann relative entropy in QFT and its dependence on physical parameters.

Figures (4)

• Figure 1: Density plot of $f$ and $f'_s|_{s = 0}$ from Eqs.\ref{['eq:test_function']} and \ref{['eq:test_function_boosted']}, respectively, for the $(1+1)$-dimensional case. In this plot, $\alpha = 1 = \beta$ and $l = 3$. As can be seen, both are compactly supported in the right Rindler wedge $W_{\rm R}$ defined in Eq.\ref{['eq:Right_Rindler_Wedge']} and represented here as the gray shaded region.
• Figure 2: The relative entropy $S(\psi_f|\Omega)$ between $\ket{\psi_f}$ and $\ket{\Omega}$ as a function of the mass $m\alpha$ of the field on a $(1+d)$-dimensional Minkowski spacetime and the corresponding relative errors are presented on the left and right plots, respectively. Here $\alpha = 1 = \beta$ and $l = 3\alpha$. The red, green and blue curves represent the $d = 3$, $d = 2$ and $d = 1$ cases in both plots. In each case, $\eta$ is chosen such that the entropies peak at $1$. For $d = 1$ and $d = 3$ the entropies peak as $m \to 0$ (for $d = 1$, the massless case $m = 0$ is unphysical due to the infrared divergence of the theory in this case) and decays with increasing mass with dwindling oscillations in the process. For $d = 2$, the relative entropy is minimum as $m\to 0$ (up to the discontinuity at $m = 0$), increases towards a maximum near $m\alpha \approx 2.2$ then decays slower than in the $d=1$ and $d = 3$ cases. The $d=2$ case also presents an oscillation while decaying, which is dampened faster than those for the $d=1$ and $d=3$ cases. For this reason, we can only see a trace of an oscillation around $m=7$ in the green-curve on the left plot. The oscillations in the relative error follow the oscillations in the relative entropy. When the relative entropy becomes closer to zero, the absolute error becomes more pronounced, which produces the oscillations we see on the right plot. With the evaluation settings we used, the absolute error stabilizes around $10^{-5}$, $10^{-3}$ and $10^{-2}$ for $d = 1, 2$ and 3, respectively. Concomitantly, the relative entropy decays with increasing mass. For this reason, the relative error grows with increasing mass.
• Figure 3: The log-log versions of the relative entropy plot from \ref{['fig:Results_1']}. The left panel shows the relative entropy for $m\alpha \in [10^{-10},20]$ . The right panel shows it for $m\alpha \in [1,20]$. As can be seen in the left panel, the relative entropy for $d = 1,3$ varies very slowly until $m\alpha \approx 1$. On the other hand, the relative entropy for $d = 2$ presents a power-law increase. For larger values of $m\alpha$ the relative entropy presents a power-law tail with visible oscillations for $m\alpha \gtrsim 5$ for $d = 1,3$. This decay is noticeably slower for $d = 2$ and has only one oscillation at $m\alpha \approx 7$ that is more visible in \ref{['fig:Results_1']}. A discontinuity appears in the plot for $d = 3$ around $m\alpha \approx 10$ because the numerical value of the relative entropy becomes negative. This is not at odds with the positive-definiteness of the relative entropy because our result is still compatible with positive values, as can be seen in the error bands presented in \ref{['fig:Results_tail']}.
• Figure 4: Linear-scale view of the $m\alpha \gtrsim 5$ region from \ref{['fig:Results_1']}. Here the oscillations in the power-law decay of the relative entropy for $d = 3$ (left plot) and $d = 1$ (right plot) are more clearly visible. The red-shaded region for $d = 3$ represents the error bands in the relative entropy which were derived from the numerical integration error estimates. The error bands for $d = 1$ are not visible in this scale as the absolute error for this case is of order $\sim 10^{-4}$. For $m\alpha \gtrsim 8$, the relative entropy is comparable to the absolute error, and while the data remains consistent with a decaying, oscillatory tail, the detailed structure in this regime cannot be definitively resolved with our present numerical precision.