A Linear Response Analysis of the Semiclassical Approximation to Spin $\nicefrac{{1}}{{2}}$ Quantum Electrodynamics in 1+1 Dimensions

In the massless limit, substitution of (13) into (17) gives

and therefore $\eqref{sb}$ becomes

which is the equation for a harmonic oscillator with frequency $|q|/\sqrt{\pi}$ and source $J_{C}$.

For the classical source $\eqref{Jasymp}$ with initial conditions $A(0)=0$ and $E(0)=0$, the solution to $\eqref{Amassless}$, is given by PlaNewsomeAnderson

The electric field is

where $\mathrm{ci}(x)=-\int_{x}^{\infty}\frac{\cos{(t)}}{t}dt$ and $\mathrm{si}(x)=-\int_{0}^{x}\frac{\sin{(t)}}{t}dt$ are the cosine and sine integral functions respectively. There is no convenient analytic form for solutions to $\eqref{Amassless}$ for the Sauter pulse classical source (16a). However, it is expected that solutions will be characterized by similar harmonic behavior as is evidenced numerically in Sec. V.1.

It is interesting to note that the retarded current density two-point correlation function is nonzero even when the classical current is zero. Substituting (12) into (26) one finds

where

In $\eqref{adiabaticJJ}$ and $\eqref{adiabaticchi}$ the notation has been slightly modified to distinguish $\omega^{2}_{k}\equiv k^{2}+m^{2}$ and $\omega^{2}_{k^{\prime}}\equiv k^{\prime\,2}+m^{2}$. It follows that $\eqref{twopoint1}$ reduces to

Since no particles are present prior to activation of the classical current $J_{C}$, the fact that the two-point function is nonzero indicates the existence of vacuum polarization effects in the absence of a background electric field.

When the time-dependent mode functions $h^{(I,II)}_{k}$ take the form $\eqref{h1h2massless}$, the current density two-point function $\eqref{twopoint2}$ vanishes. It then follows from $\eqref{masslessJq}$ that $\eqref{deltaJqrenorm}$ simplifies to

Thus the linear response equation (22) with $\eqref{masslesscurrent}$ becomes

which is of similar harmonic character as $\eqref{Amassless}$ with equivalent frequency. The form of $\eqref{LREmassless}$ guarantees for the massless limit that perturbations in the background field remain bounded with fixed amplitude for both forms of $\delta J_{C}$ in (23) and (24). For the case of $\eqref{deltaJclass}$, the solution to $\eqref{LREmassless}$ with initial conditions $\delta A(0)=0$ and $\delta E(0)=0$ yields the same result as in $\eqref{Anomass}$ and $\eqref{Enomass}$, but with the replacement $E_{0}\to\delta E_{0}$.

It is interesting to note that given the limit $\eqref{masslesscurrent}$, the non-local term in $\eqref{deltaJqrenorm}$ carries the additional interpretation of being a measure of the extent to which $\delta\langle J_{Q}\rangle_{\mathrm{ren}}$ differs from that of its massless counterpart, since

For the classical source terms $\eqref{Jasymp}$ and $\eqref{JSauter}$ associated with the asymptotically constant and Sauter pulse profiles, both the backreaction process, represented through the electric field $E$, the renormalized current density $\langle J_{Q}\rangle_{\mathrm{ren}}$, and their associated linear response to perturbations, $\delta E$ and $\delta\langle J_{Q}\rangle_{\mathrm{ren}}$, are shown in Fig. 1 and Fig. 2 respectively. The results for $E$ and $\langle J_{Q}\rangle_{\mathrm{ren}}$ were previously shown in PlaNewsomeAnderson . Note that in this subsection the focus is on the behaviors of solutions to the linear response equation. In Sec. VB, a discussion is given between these solutions and the validity of the semiclassical approximation.

For $qE_{0}/m^{2}=1$, the deviation of the electric field from the classical solution for the asymptotically constant profile increases monotonically over the times considered. In contrast for the Sauter pulse profile, after the maximum of the pulse occurs at time $qt=0$, the electric field undergoes long period oscillations which are modulated by shorter-period, smaller-amplitude oscillations. For $qE_{0}/m^{2}=10^{3}$, backreaction effects are significantly stronger. In all cases plasma oscillations occur at sufficiently late times. The strongest backreaction effects occur in the massless limit, $qE_{0}/m^{2}\to\infty$, resulting in pure harmonic behavior for the electric field.

The current density $\langle J_{Q}\rangle_{\mathrm{ren}}$ in the case $qE_{0}/m^{2}=1$ for the asymptotically constant profile exhibits small oscillations at early times which decay away, likely having their origin in the sudden activation of the background current $J_{C}$. At late times, large timescale oscillations are present. For the Sauter pulse profile, once the pulse occurs at $qt=0$ there is an initial increase in the current density which begins to dampen soon after. The higher frequency modulations present are similar in nature to those seen for very early times in the case of the asymptotically constant profile. However, for the Sauter pulse these modulations do not damp away. This appears to be related to the fact that the classical source term $\eqref{JSauter}$ is small after the initial pulse. For $qE_{0}/m^{2}=10^{3}$, the current density for both profiles undergoes a more rapid increase with larger amplitude, indicative of an increase in pair production events, and begins to oscillate with an approximately constant frequency.

The linear response solutions $\delta E$ to $\eqref{fullLRE}$ for the asymptotically constant profile with $qE_{0}/m^{2}=1$ undergo oscillatory behavior with an amplitude that grows in time, indicative of an instability. For the Sauter pulse when $qE_{0}/m^{2}=1$, after the maximum of the pulse occurs at time $qt=0$, the solution $\delta E$ undergoes long period oscillations which are modulated by shorter-period, smaller-amplitude oscillations which grow in time. For the case of $qE_{0}/m^{2}=10^{3}$, both solutions are characterized by approximate simple harmonic behavior with a frequency similar to that of the electric field.

The associated current density perturbation $\delta\langle J_{Q}\rangle_{\mathrm{ren}}$ in the case of $qE_{0}/m^{2}=1$ for the asymptotically constant profile exhibits relatively large amplitude, long time-scale oscillations as well as smaller amplitude, higher frequency modulations. The Sauter pulse current density perturbation for $qE_{0}/m^{2}=1$ after $qt=0$ initially exhibits sporadic oscillatory behavior before settling down to constant frequency oscillations which grow in amplitude, generating the modulations seen at late times for $\delta E$. Both profiles for the case of $qE_{0}/m^{2}=10^{3}$ exhibit approximate simple harmonic behavior for $\delta\langle J_{Q}\rangle_{\mathrm{ren}}$, growing to a relatively constant amplitude with oscillations of constant frequency similar to that of $\langle J_{Q}\rangle_{\mathrm{ren}}$.

There are various way in which one can perturb the semiclassical backreaction equation, each yielding a modified solution $E_{2}$ that differs from the original solution $E_{1}$. The expansion of the electric field $E_{2}$ in terms of the unperturbed field $E_{1}$ is of the form $E_{2}=E_{1}+\delta E+\mathcal{O}(\delta E^{2})$, where $\delta E$ is a solution to the linear response equation (22). For two solutions $E_{1}$ and $E_{2}$ to the backreaction equation $\eqref{sb}$, whose initial conditions differ by a sufficiently small amount, one can construct $\Delta E\equiv E_{2}-E_{1}$ such that to linear order the perturbations can be approximated as $\Delta E\approx\delta E$. One would expect such an approximation to hold at early times prior to significant particle production. Here, two solutions are considered with slightly different values of $E_{0}$ for either the asymptotically constant classical profile (15b) or the Sauter pulse (16b). For both profiles, $E_{0}$ is the maximum value the classical electric field will have.

The difference between two solutions to the backreaction equation $\eqref{sb}$ satisfies

with

In order for $\Delta E\approx\delta E$, it is clear that $\Delta\langle J_{Q}\rangle_{\textnormal{ren}}\approx\delta\langle J_{Q}\rangle_{\textnormal{ren}}$ must hold, since one can set $\Delta J_{C}=\delta J_{C}$ for all times.

To analyze the behaviors of solutions to $\eqref{LRE}$ and compare them with $\eqref{approxLRE}$, it is useful to isolate the quantum contribution to the electric field $E_{Q}$ by subtracting from the exact electric field $E$ the corresponding solution to the classical equation $E_{C}$ as

From the structure of $\eqref{LRE}$ and $\eqref{approxLRE}$, the quantum contributions to the exact and approximate solutions to the linear response equation $\delta E$ and $\Delta E$ can be similarly isolated, i.e. $\delta E_{Q}\equiv\delta E-\delta E_{C}$ and $\Delta E_{Q}\equiv E-E_{C}$. The criterion for the validity of the semiclassical approximation can therefore be modified to state that if quantities constructed from either $\delta E_{Q}$ or $\Delta E_{Q}$ grow significantly during some period of time then the semiclassical approximation is considered to be invalid PlaNewsomeAnderson .

In order to provide a meaningful description of the growth in time for $\delta E_{Q}$, a useful quantity to consider is the relative difference between $\delta E_{Q}$ or $\Delta E_{Q}$ and the quantum contribution to the associated background field $E_{Q}$, formulated as¹¹1For ease of comparison with the solutions to the linear response equation, we use a slightly different definition of $R_{Q}$ than was used in PlaNewsomeAnderson .

The degree to which $\Delta E_{Q}\approx\delta E_{Q}$ will be characterized in a scale invariant way by the degree to which $R_{Q}\approx\bar{R}_{Q}$. In PlaNewsomeAnderson , $R_{Q}$ was used to characterize the growth of the approximate solutions to the linear response equation. For the classical solutions one can define in the same way as above

The early time behaviors of both $R_{Q}$ and $\bar{R}_{Q}$ are shown in Fig. 3 for the asymptotically constant background profile and in Fig. 4 for the Sauter pulse background profile. For both profiles, the perturbed electric field present in $\eqref{deltaJclass}$ and $\eqref{deltaJclass2}$ was set to $\delta E_{0}=10^{-3}$. The classical quantity $R_{C}$ is also included for comparison.

For the case $qE_{0}/m^{2}=1$, the approximate solution $\Delta E_{Q}$ consistently undervalues $\delta E_{Q}$ for both profiles. Given the form of the perturbative expansion $\Delta E_{Q}=\delta E_{Q}+\mathcal{O}(\delta E_{Q}^{2})$, the extent to which $\Delta E_{Q}\neq\delta E_{Q}$ directly reflects the impact of higher-order perturbative terms. The early time regime for which growth in $R_{Q}$ was investigated in PlaNewsomeAnderson can be adequately attenuated at $qt=5$, at which point the relative difference between $R_{Q}(qt=5)$ and $\bar{R}_{Q}(qt=5)$ is of order $10^{-1}$ for both profiles. For $qE_{0}/m^{2}=10^{3}$, the same relative difference is of order $10^{-2}$ for the asymptotically constant profile and of order $10^{-3}$ for the Sauter pulse profile.

Numerical results were also obtained for the cases $qE_{0}/m^{2}=0.5,2,10,100$. The results provide evidence that if $qE_{0}/m^{2}\gg 1$, then $\Delta E_{Q}\approx\delta E_{Q}$ for an extended period of time, whereas $\Delta E_{Q}\approx\delta E_{Q}$ only for relatively early times if $qE_{0}/m^{2}\sim 1$. Since the validity analysis in PlaNewsomeAnderson was for early times, the conclusion in that paper that the semiclassical approximation breaks down for both the asymptotically constant and Sauter pulse profiles if $qE_{0}/m^{2}\sim 1$ is verified here. The conclusion that the criterion used for the validity of the semiclassical approximation is not violated at eary times for large values of $qE_{0}/m^{2}$ is also verified.

What is also indicated by the above results is that over an extended period of time, it is likely that all solutions to the linear response equation $\eqref{fullLRE}$ are unstable for $qE_{0}/m^{2}<\infty$, and only in the true massless limit $\eqref{LREmassless}$ is pure harmonic motion achieved and no instability present in $R_{Q}$. However, consideration of sufficiently late times is not physically realistic given the two major simplifications made of one spatial degree of freedom and ignoring spin $\nicefrac{{1}}{{2}}$ particle interactions. Major modifications to the backreaction process and its linear response to perturbations are expected to occur if one, or both, of these limitations are relaxed.

The question of whether or not quantum fluctuations associated with the current density two-point function $\langle[J_{Q}(t,x),J_{Q}(t^{\prime},x^{\prime})]\rangle$ are responsible for the instability observed in the solutions of the linear response equation (22), and hence the validity of the semiclassical approximation, will now be addressed. For simplicity, the asymptotically constant classical source (15a) will be considered here. However, similar qualitative results hold for the Sauter pulse classical source (16a).

From (35), one can see that $\delta\langle J_{Q}\rangle_{\mathrm{ren}}$ can be written in terms of its massless limit plus a non-local term. In the massless limit, one can see from (34) that $\langle[J_{Q}(t,x),J_{Q}(t^{\prime},x^{\prime})]\rangle=0$. This results in bounded harmonic behavior, leading to a constant value for $\bar{R}_{Q}$ in (39b). Thus for $qE_{0}/m^{2}<\infty$, if there is growth in $\bar{R}_{Q}$ it must be driven by the non-local term, which depends in part on $\langle[J_{Q}(t,x),J_{Q}(t^{\prime},x^{\prime})]\rangle$.

The non-local term introduces a memory effect in the evolution of perturbations in the background field $\delta A$, the behavior of which depends not only on its current value at time $t$, but also on its entire history for past times $t^{\prime}$. Initially, $\delta A(t^{\prime}=0)=0$, but as $\delta A(t)$ evolves according to (22), the time-dependent integration $\int_{0}^{t}dt^{\prime}$ over the product of both $\int dx^{\prime}\langle[J_{Q}(t,x),J_{Q}(t^{\prime},x^{\prime})]\rangle$ and $\delta A(t^{\prime})$ accumulates contributions from the current density two-point function. Additionally, the quantity $\delta A(t^{\prime})$ itself depends on the two-point function for all past times $t^{\prime}$, creating a feedback loop where earlier perturbations in $\delta A(t^{\prime})$ exert a delayed influence on $\delta A(t)$. As the upper limit of integration grows with $t$, the system continuously incorporates the effects of quantum fluctuations associated with the spin $\nicefrac{{1}}{{2}}$ field, causing the current density two-point function to act as a type of amplifying source term. This drives the growth in $\delta A(t)$ leading to instability due to the influence of quantum fluctuations.

The rate of this growth depends on the relative strength of $qE_{0}/m^{2}$. For $qE_{0}/m^{2}\gg 1$, the time-dependent modes $h^{(I,II)}_{k}$ approach $\eqref{h1h2massless}$, thereby reducing the contribution of $\langle[J_{Q}(t,x),J_{Q}(t^{\prime},x^{\prime})]\rangle$. This implies that quantum fluctuations as measured by this particular two-point correlation function diminish as the field strength grows, which is consistent with the behavior discusses previously for the massless case. However, one expects particle production to increase in this limit, emphasizing the point that the criterion for the validity of the semiclassical approximation used here is a necessary, but not sufficient condition. As the critical scale $qE_{0}/m^{2}\sim 1$ is approached, the contribution of $\langle[J_{Q}(t,x),J_{Q}(t^{\prime},x^{\prime})]\rangle$ is expected to be significant, indicating that the associated quantum fluctuations are large.

These features can be seen numerically in Fig. 5, where the contents of the temporal integrand present in $\eqref{deltaJqrenorm}$ are shown as a function of the past time $t^{\prime}$ up to the current time $t$, which is chosen to be $qt=25$. For both characteristic cases $qE_{0}/m^{2}$ shown, we include for the spatial integral of the current density two-point function both situations in which the background field $E_{0}\neq 0$ as well as $E_{0}=0$ prior to activation.

For $qE_{0}/m^{2}=1$, the spatial integral of the current density two-point function has an amplitude of oscillation which grows rapidly to a maximum as $t^{\prime}\to t$ before terminating to zero when $t^{\prime}=t$. This property is shared by both cases where $E_{0}=1$ and $E_{0}=0$, with there being good agreement between the two. However, for earlier values of $t^{\prime}$ the amplitude of oscillations for $E_{0}=1$ are damped significantly with an increase in the frequency relative to $E_{0}=0$ occurring, resulting in a smaller contribution from earlier times for the integral over time $t^{\prime}$. The amplitude of the quantity $\delta A(t^{\prime})$ also grows significantly over the same time scale, which is a consequence of past contributions from the current density two-point function driving this growth. The magnitude of their product therefore results in rapid growth as $t^{\prime}\to t$ which, when integrated in time $t^{\prime}$ up to the current time $t$, drives the instability seen for $\bar{R}_{Q}$ in Fig. 3.

For $qE_{0}/m^{2}=10^{3}$, the spatial integral of the current density two-point function is characterized by sporadic oscillations, with an relative amplitude approximately two orders of magnitude smaller than the critical case. Also, the maximum amplitude occurs for much earlier times $t^{\prime}$, with damping occurring as $t^{\prime}\to t$. Furthermore, it is only when $|t^{\prime}-t|\ll 1$ that there is any agreement between the $E_{0}=1$ and $E_{0}=0$ cases, with the latter case having oscillations which take place over a much longer timescale. Consequently, the amplitude of $\delta A(t^{\prime})$ does not grow significantly over the timescale considered, maintaining an approximately harmonic behavior. The integration of their product therefore yields a minimal contribution to the linear response equation, which in turn leads to much slower growth in $\bar{R}_{Q}$.