Faster-than-light signaling in the rotating-wave approximation

We present new results on the causality violations introduced by the rotating wave approximation commonly used in quantum optics and high-energy physics. We find that the causality violations and faster-than-light signaling induced by the approximation have “fat tails,” i.e., they are polynomially decaying with the distance from the light-cone of the emitter. Furthermore, we also show that the fundamental problems with the incompatibility between the approximation and relativity are not cured even in the long interaction time regime (where the approximation is often taken). This renders the approximation unsuitable for any regime where we are concerned about relativistic causality and information transmission via the electromagnetic field.

Figure 1

Energy density distribution immediately following $\delta$-coupling interaction with no approximations. Here $G(\mathit{\zeta })=\mathrm{\Theta }(1-\mathit{\zeta })$, i.e., a hard sphere. Note that the interaction has no nonlocal field consequences.

Figure 2

Energy density distribution immediately following $\delta$-coupling interaction under the RWA. Here $G(\mathit{\zeta })=\mathrm{\Theta }(1-\mathit{\zeta })$, i.e., spherically symmetric with a sudden cutoff. The spike at $|\mathit{x}|=R$ is a consequence of the $F(\mathit{x})$ discontinuity. Note the polynomially decaying tail for $|\mathit{x}|>R$.

Figure 3

$:{\widehat{\varphi }}^2:$ distribution immediately following $\delta$-coupling interaction under the RWA. Here $G(\mathit{\zeta })=\mathrm{\Theta }(1-\mathit{\zeta })$, i.e., spherically symmetric with a sudden cutoff. Note the polynomially decaying tail for $|\mathit{x}|>R$.

Figure 4

Energy density distribution from a second order perturbative interaction where $\chi (t)=\mathrm{\Theta }(t)\mathrm{\Theta }(T-t)$ with no approximations and $T=150R$. Here $G(\mathit{\zeta })=\mathrm{\Theta }(1-\mathit{\zeta })$, i.e., spherically symmetric with a sudden cutoff. Note that the interaction has no nonlocal field consequences, i.e., no effect beyond $|\mathit{x}|>151R$. The vertical line at $|\mathit{x}|=151R$ indicates the locality limit.

Figure 5

$:{\widehat{\varphi }}^2:$ density distribution from a second order perturbative interaction where $\chi (t)=\mathrm{\Theta }(t)\mathrm{\Theta }(T-t)$ with no approximations and $T=150R$. Here $G(\mathit{\zeta })=\mathrm{\Theta }(1-\mathit{\zeta })$, i.e., spherically symmetric with a sudden cutoff. Note that the interaction has no nonlocal field consequences, i.e., no effect beyond $|\mathit{x}|>151R$ (to numerical precision). The vertical line at $|\mathit{x}|=151R$ indicates the locality limit.

Figure 6

Energy density distribution from a second order perturbative interaction where $\chi (t)=\mathrm{\Theta }(t)\mathrm{\Theta }(T-t)$ under the RWA and $T=150R$. Here $G(\mathit{\zeta })=\mathrm{\Theta }(1-\mathit{\zeta })$, i.e., spherically symmetric with a sudden cutoff. Note the polynomial decaying tail for $|\mathit{x}|>151R$. The vertical line at $|\mathit{x}|=151R$ indicates the locality limit.

Figure 7

$:{\widehat{\varphi }}^2:$ density distribution from a second order perturbative interaction where $\chi (t)=\mathrm{\Theta }(t)\mathrm{\Theta }(T-t)$ under the RWA and $T=150R$. Here $G(\mathit{\zeta })=\mathrm{\Theta }(1-\mathit{\zeta })$, i.e., spherically symmetric with a sudden cutoff. Note the polynomial decaying tail for $|\mathit{x}|>151$. The vertical line at $|\mathit{x}|=151R$ indicates the locality limit.

Figure 8

Integrated RWA detector response function. Here we used $F_{\mathrm{A}}(\mathit{x})=\mathrm{\Theta }(R_{\mathrm{A}}-|\mathit{x}|)$ and $F_{\mathrm{B}}(\mathit{x})=\mathrm{\Theta }(R_{\mathrm{B}}-|\mathit{x}-\mathit{d}|)$. In addition the detector interaction times where ${\chi }_{\mathrm{B}}(t_2)=1$ for $t_2{\mathrm{\Omega }}_{\mathrm{B}}\in (0,10)$ and zero otherwise; and ${\chi }_{\mathrm{A}}(t_1)=1$ for $t_1{\mathrm{\Omega }}_{\mathrm{A}}\in (13,23)$ and zero otherwise. Given that both detectors have a radius of $R$ the lightcone should only reach $|\mathit{d}|=25R$. The polynomial decay beyond this is a consequence of the RWA. The vertical line at $|\mathit{d}|=R$ indicates the superior limit of the strong Huygen’s principle and the vertical line at $|\mathit{d}|=25R$ indicates the causal limit. Here $R_{\mathrm{A}}=R_{\mathrm{B}}=R$ and ${\mathrm{\Omega }}_{\mathrm{A}}={\mathrm{\Omega }}_{\mathrm{B}}=R^{-1}$.

Figure 9

Integrated no approx detector response function. Here we used $F_{\mathrm{A}}(\mathit{x})=\mathrm{\Theta }(R_{\mathrm{A}}-|\mathit{x}|)$ and $F_{\mathrm{B}}(\mathit{x})=\mathrm{\Theta }(R_{\mathrm{B}}-|\mathit{x}-\mathit{d}|)$. In addition the detector interaction times where ${\chi }_{\mathrm{B}}(t_2)=1$ for $t_2{\mathrm{\Omega }}_{\mathrm{B}}\in (0,10)$ and zero otherwise; and ${\chi }_{\mathrm{A}}(t_1)=1$ for $t_1{\mathrm{\Omega }}_{\mathrm{A}}\in (13,23)$ and zero otherwise. Given that both detectors have a radius of $R$ the lightcone should only reach $|\mathit{d}|=25R$. Note how when considering the full model then causality is maintained. The vertical line at $|\mathit{d}|=R$ indicates the superior limit of the strong Huygen’s principle and the vertical line at $|\mathit{d}|=25R$ indicates the causal limit. Here $R_{\mathrm{A}}=R_{\mathrm{B}}=R$ and ${\mathrm{\Omega }}_{\mathrm{A}}={\mathrm{\Omega }}_{\mathrm{B}}=R^{-1}$.