Vacuum induced dispersions on the motion of test particles in $D+1$ dimensions

When the vacuum state of a scalar or electromagnetic field is modified by the presence of a reflecting boundary, an interacting test particle undergoes velocity fluctuations. Such effect is regarded as a sort of quantum analog of the classical Brownian motion. Several aspects about this system have been recently investigated in the literature, for instance, finite temperature effects, curved spacetime framework, near-boundary regime, late time behavior, and subvacuum phenomena. Here, further steps are given in this analysis by considering the effect of vacuum fluctuations of a scalar field in the presence of a perfectly reflecting flat boundary over the motion of a scalar test particle when the background field does not satisfy the Huygens’ principle. Specifically, the background field is allowed to have mass and the system is studied in $D+1$ dimensions. A method of implementing a smooth transition between distinct states of the field is also developed, rendering regularized analytic expressions describing the velocity fluctuations of the test particle. This method is applied to study some special behaviors of the system. Possible applications include fields known to occur in nature as, for instance, the massive Higgs’ field, for which the velocity fluctuations are here predicted to acquire a characteristic oscillation, thus behaving differently from their electromagnetic counterparts.

Figure 1

Some representative plots for the propagator tail as function of $\sigma$. Here, $D=3$.

Figure 2

Some representative plots for the propagator tail as function of $\sigma$ when $D=2$. In this case the plot corresponding to the massless field works as an envelope providing a maximum for the oscillation amplitude.

Figure 3

Smooth switching behavior of $F_{{\tau }_s,\tau }^{(3)}(t)$. The sudden limit $F_{\tau }^{(0)}(t)$ is recovered when ${\tau }_s\to 0$, as suggested by the plots, and verified from the definition of $F_{{\tau }_s,\tau }^{(3)}(t)$.

Figure 4

Velocity dispersions $⟨{(\mathrm{\Delta }{v}_{\perp }{)}^2⟩}_1^{(1)}$ as function of $\tau /x$ for some representative values of $mx$, where we set $n=20$. The solid curve recovers the regularization obtained in [6]. As $mx$ grows, the dashed and dotted curves show that the dispersions start to oscillate, a fingerprint of the non-Huygensian character of the background field coming from its mass.

Figure 5

Velocity dispersions modeled with the switching mechanism $F_{n,\tau }^{(1)}$ for $D=2$. Here, $n=20$. As $D=2$, the velocity has dispersions parallel to the wall (a), and perpendicular to the wall (b). Again, if $mx>0$, the oscillatory behavior is recovered.

Figure 6

Late time behavior of the velocity dispersion in $1+1$ modeled with the switching mechanism $F_{{\tau }_s,\tau }^{(3)}$ as a function of $mx$. As $mx\to 0$, we recover the expected late time divergence of this model. For finite $mx$, the theory is regularized, showing that the divergence comes from the infrared sector of the theory.

Figure 7

Comparison of both switching process $F_{{\tau }_s,\tau }^{(3)}$ and $F_{n,\tau }^{(1)}$ for $D=1$ and $mx=1$. Here, $n=10$ and ${\tau }_s/x=0.1$. For small $\tau /x$, both process are indistinguishable, as shown by the dashed and dotted curves. As the late time regime is approached, the dispersion modeled by $F_{{\tau }_s,\tau }^{(3)}$ oscillates about its late time regime value, given by the dot-dashed curve.