Non-Markovian Abraham-Lorentz-Dirac equation: Radiation Reaction without Pathology

Casting EFT in the QOS framework is a subject of importance in theoretical physics worthy of serious exploration. Closer to what has been done, namely, using QOS concepts and nonequilibrium techniques (see, e.g., UW12 ; CH08 ) to explore radiation phenomena, we mention two veins, one we are pursuing now, and the other, one of us has worked on with collaborators.

Motion of a charged particle in a field is a standard example of reduced dynamics in the open system framework, where the charged particle is the system of our interest and the electromagnetic field serves as its environment. Backreactions of the field enter into the equation of motion of the charged particle in the forms of radiation reaction (damping) and the Lorentz force. In a quantum field the Lorentz force is not only time-dependent and also fluctuating.

The electromagnetic field, due to the way it couples with the charged particle, acts like a supra-Ohmic environment, although the vector potential itself has an Ohmic spectral density, shown in Sec. II. The frequency spectrum of the field typically ranges from 0 to infinity, but from physical considerations, a cutoff frequency exists which reflects the range of validity of the electromagnetic theory or the structure of the system it interacts with. When the charge is assumed to be pointlike, the cutoff frequency is usually taken to infinity to simplify calculations, if quantum phenomena like pair creation of charged particles are excluded from one’s considerations. The equation of motion for the point charge thus obtained is the ALD equation, which is known to contain third-order time derivative of the position variable. It accounts for damping due to the back-reaction of emitted radiation in the field from the charge. The presence of this term leads to the following pathologies:

1. 1.

   Third initial condition: since the equation of motion is a third-order ordinary differential equation, it needs three conditions to uniquely determine the solution, in contrast to two conditions in typical Newtonian dynamics.

2. 2.

   Runway behavior: the solution contains a component that grows unbounded exponentially.

3. 3.

   Pre-acceleration: to avoid the runaways, the ALD equation is often rephrased as an integro-differential equation. Although the corresponding solution does not exhibit the runaway behavior, the charge described by this solution will accelerate before the force is applied. It violates causality.

We show in this paper that the popular narrative about a point charge moving in an electromagnetic field is a special limiting case of the fully non-Markovian dynamics of the reduced system in the open systems conceptualization and formulation, namely, the formal Markovian limit. From the broader perspectives of non-Markovianity in the open systems dynamics, most of these commonly encountered pathologies can be eliminated. Namely,

1. 1.

   The presence of the third-order time derivative term is innocuous. There is no need for an additional third initial condition beyond the usual two.

2. 2.

   Presence of third-order time derivative term does not necessarily lead to runaways. The system’s dynamical stability hinges more on the choice of the cutoff scale, whose inverse can also be understood as the memory time of the environment. The instability (runaway) emerges only when the memory time is shorter than a critical value.

3. 3.

   Pre-acceleration is not an issue in this context.

To identify how the conventional formulation leads to these pathologies, we will in Sec. II briefly go over the derivation of the equation of motion of a Brownian oscillator (Unruh-DeWitt detector) in electrodynamics in the open systems framework. In this way we can compare with the more proper and correct treatment from the fully non-Markovian dynamics, which is our central theme.

In Sec. III, we discuss the general characteristics and dynamical stability of the supra-Ohmic Brownian oscillator, coupled to a non-Markovian bath. We show the shared features in common with a charged oscillator in scalar electrodynamics. Then in Sec. IV we use a specific examples of bath spectral density to illustrate the ideas discussed formally in Sec. III. In Appendix A we briefly summarize the Ohmic dynamics for comparison, and in Appendix B, we address the ambiguities that arise from treating the delta function in taking the Markovian limit. Then we provide more examples of bath spectral densities in Appendix C for discussions in Sec. III. In particular, we highlight some unusual properties of the hard-cutoff bath spectrum. In Appendix D we comment on the higher-order supra-Ohmic systems.

To avoid the complexity from the vectorial nature of the electromagnetic field, it is sufficient for our purpose to consider scalar electrodynamics. In the case we consider here, the results of both descriptions differ by a numerical factor, caused by the number of polarizations, angular weight, and the choice of the unit system in electromagnetism. The scalar field discussed here is analogous to the vector potential in electromagnetism, and that the equation of motion of the scalar charge oscillator is parallel to the Lorentz equation, including the consideration of the reactive force from (scalar field) radiation.

Suppose that the internal degree of freedom $\chi(t)$ of an Unruh-DeWitt detector is modeled by a one-dimensional, nonrelativistic oscillator, which carries the scalar charge $e$. The external or mechanical (edf) degree of freedom $\bm{z}(t)$ of the detector may follow a given trajectory in $3+1$ Minkowski space, but for our purpose with focus on the internal degrees of freedom we can assume it will stay at rest for simplicity. The internal degree of freedom (idf) of the detector is coupled to an ambient massless scalar field $\phi(\bm{x},t)$. The oscillator’s velocity $\dot{\chi}(t)$ is coupled to the field $\phi(\bm{x},t)$ in the same way as in electrodynamics. For the moment assume that the detector is pointlike, and the spectrum of the field extends to infinity. We shall return to this assumption later.

The action of the oscillator-field system as described is given by

$$S=\int\!ds\;\biggl{[}\frac{m_{\textsc{b}}^{\vphantom{2}}}{2}\dot{\chi}^{2}(s)-\frac{m_{\textsc{b}}^{\vphantom{2}}\omega_{\textsc{b}}^{2}}{2}\,\chi^{2}(s)\biggr{]}+e\int\!d^{4}y\;\phi(\bm{y},s)\,\dot{\chi}(s)\,\delta(\bm{y}-\bm{z})+\int\!d^{4}y\;\Bigl{\{}\frac{1}{2}\bigl{[}\partial_{s}\phi(\bm{y},s)\bigr{]}^{2}-\frac{1}{2}\bigl{[}\partial_{\bm{y}}\phi(\bm{y},s)\bigr{]}^{2}\Bigr{\}}$$

(II.1)

where $y^{\mu}=(s,\bm{y})$ is the external (spacetime) coordinate. The overhead dot represents taking the derivatives with respect to time. The mass and the oscillator’s natural frequency take on bare values, denoted by the subscript b, for the moment. Divergent contributions from the oscillator-field interaction will be dealt with accordingly. The action (II.1) is seen to describe essentially the scalar version of electromagnetic interaction, with $\phi$ playing the role of the electromagnetic vector potential, and $e\dot{\chi}$ the role of (internal) electric current.

This action produces a simultaneous set of equations of motion

	$\displaystyle m_{\textsc{b}}^{\vphantom{2}}\,\ddot{\chi}(t)+m_{\textsc{b}}^{\vphantom{2}}\omega_{\textsc{b}}^{2}\,\chi(t)$	$\displaystyle=-e\,\dot{\phi}(\bm{z},t)\,,$		(II.2)
	$\displaystyle\ddot{\phi}(\bm{x},t)-\partial^{2}_{\bm{x}}\phi(\bm{x},t)$	$\displaystyle=e\,\dot{\chi}(t)\,\delta(\bm{x}-\bm{z})\,,$		(II.3)

in analogy with the Lorentz equation of the charge and the wave equation of the vector potential, respectively. If the oscillator-field interaction is switched on at $t=0$, then the formal solution to (II.3) is given by

$\displaystyle\phi(\bm{x},t)=\phi_{h}(\bm{x},t)+e\int_{0}^{t}\!d^{4}y\;G_{R,0}^{(\phi)}(\bm{x},t;\bm{y},s)\,\dot{\chi}(s)\,\delta(\bm{y}-\bm{z})\,,$

(II.4)

where $\phi_{h}(\bm{x},t)$ denotes the homogeneous solution to the wave equation, (II.3). The subscript $0$ in $G_{R,0}^{(\phi)}(\bm{x},t;\bm{y},s)$ tells us that it is the retarded Green’s function of the free scalar field $\phi_{h}$, not the full field $\phi$. The second term on the righthand side of (II.4) gives the radiation field emitted by the charge oscillator. We then substituted (II.4) into (II.2) to find an equation of motion for the charged oscillator under the influence of the environment,

$\displaystyle m_{\textsc{b}}^{\vphantom{2}}\,\ddot{\chi}(t)+m_{\textsc{b}}^{\vphantom{2}}\omega_{\textsc{b}}^{2}\,\chi(t)+e^{2}\frac{\partial}{\partial t}\int_{0}^{t}\!ds\;G_{R,0}^{(\phi)}(\bm{z},t;\bm{z},s)\,\dot{\chi}(s)=-e\,\frac{\partial}{\partial t}\hat{\phi}_{h}(\bm{z},t)\,.$

(II.5)

The term on the righthand side plays the role of the Lorentz force, while the nonlocal expression will account for radiation damping by the scalar field radiation. In the coincident spatial limit, we usually write $G_{R,0}^{(\phi)}(\bm{z},t;\bm{z},s)$ as $G_{R,0}^{(\phi)}(t-s)$ to avoid cluttering of notations.

Although the scalar field itself has an Ohmic spectrum, the velocity coupling can induce supra-Ohmic effects by the presence of additional time derivatives in the nonlocal term in (II.5). This can be made more explicit if we write the nonlocal term as

	$\displaystyle\frac{\partial}{\partial t}\int_{0}^{t}\!ds\;G_{R,0}^{(\phi)}(t-s)\,\dot{\chi}(s)$	$\displaystyle=G_{R,0}^{(\phi)}(0)\,\dot{\chi}(t)+\int_{0}^{t}\!ds\;\partial_{t}G_{R,0}^{(\phi)}(t-s)\,\dot{\chi}(s)$
		$\displaystyle=-\ddot{\Gamma}^{(\textsc{e})}(0)\,\chi(t)+\ddot{\Gamma}^{(\textsc{e})}(t)\,\chi(0)+\int_{0}^{t}\!ds\;\partial^{3}_{s}\Gamma^{(\textsc{e})}(t-s)\,\chi(s)\,.$		(II.6)

To place it in the context of supra-Ohmic dynamics we have used the substitution $G_{R,0}^{(\phi)}(\tau)=-\partial_{\tau}\Gamma^{(\textsc{e})}(\tau)$ where $\Gamma^{(\textsc{e})}(\tau)$ is an Ohmic kernel function, discussed in fuller details in Appendix A. The third-order time derivative in the last term will give a factor of $\kappa^{3}$ in the Fourier representation of $\Gamma^{(\textsc{e})}(\tau)$,

$$\partial^{3}_{\tau}\Gamma^{(\textsc{e})}(\tau)=\partial^{3}_{\tau}\int_{-\infty}^{\infty}\!\frac{d\kappa}{2\pi}\;\frac{I(\kappa)}{2\pi\kappa}\,e^{-i\kappa\tau}=i\int_{-\infty}^{\infty}\!\frac{d\kappa}{2\pi}\;\frac{\kappa^{3}I(\kappa)}{2\pi\kappa}\,e^{-i\kappa\tau}\,,$$

(II.7)

and thus $\partial^{3}_{\tau}\Gamma^{(\textsc{e})}(\tau)$ accounts for the supra-Ohmic effect. The function $I(\kappa)$ is the spectral density of the scalar field $\phi$ and has the Ohmic form.

In the Markovian limit where the cutoff frequency $\Lambda$ is taken to infinity, the kernel function $\Gamma^{(\textsc{e})}(\tau)$ is given by

$$\Gamma^{(\textsc{e})}(\tau)=\frac{1}{2\pi}\,\delta(\tau)\,,$$

(II.8)

and thus the retarded Green’s function of the free field $G_{R,0}^{(\phi)}(\tau)$ in the spatial coincident limit has a very simple form

$$G_{R,0}^{(\phi)}(\tau)=-\theta(\tau)\,\frac{1}{2\pi}\,\dot{\delta}(\tau)\,,$$

(II.9)

where $\theta(\tau)$ is the Heaviside function. This is highly singular. Hereafter we will discuss the confusions and difficulty arisen in interpreting the supra-Ohmic dynamics, if the Markovian limit of this kernel function is treated prima facie as a Dirac-delta function.

Before formally finding the solution, let us first address the renormalization of the parameters. Inserting (II.9) into the non-local term yields

	$\displaystyle e^{2}\frac{\partial}{\partial t}\int_{0}^{t}\!ds\;G_{R}^{(\phi)}(t-s)\,\dot{\chi}(s)$	$\displaystyle=\frac{e^{2}}{2\pi}\frac{\partial}{\partial t}\int_{0}^{t}\!ds\;\Bigl{[}\frac{\partial}{\partial s}\delta(t-s)\Bigr{]}\,\dot{\chi}(s)$
		$\displaystyle=\frac{e^{2}}{2\pi}\frac{\partial}{\partial t}\biggl{[}\delta(0)\,\dot{\chi}(t)-\delta(t)\,\dot{\chi}(0)-\int_{0}^{t}\!ds\;\delta(t-s)\,\ddot{\chi}(s)\biggr{]}$
		$\displaystyle=\frac{e^{2}}{2\pi}\biggl{[}\delta(0)\,\ddot{\chi}(t)-\dot{\delta}(t)\,\dot{\chi}(0)-\partial_{t}\int_{0}^{t}\!ds\;\delta(t-s)\,\ddot{\chi}(s)\biggr{]}$
		$\displaystyle=\frac{e^{2}}{2\pi}\biggl{[}\delta(0)\,\ddot{\chi}(t)-\dot{\delta}(t)\,\dot{\chi}(0)-\delta(t)\,\ddot{\chi}(0)-\frac{1}{2}\dddot{\chi}(t)\biggr{]}\,.$		(II.10)

Here we have given more details in derivations to show the subtleties: (A) Probably better accepted, the integral $\displaystyle\int_{0}^{t}\!ds\;\delta(t-s)\,\ddot{\chi}(s)$ only gives half of the contribution because heuristically half of the “peak” of the delta function is used; however, (B) it is less known that when a delta function is involved in an expression like $\displaystyle\partial_{t}\int_{0}^{t}\!ds\;\delta(t-s)\,\ddot{\chi}(s)$ should give a result like

$$\partial_{t}\int_{0}^{t}\!ds\;\delta(t-s)\,\ddot{\chi}(s)=\delta(t)\,\ddot{\chi}(0)+\frac{1}{2}\dddot{\chi}(t)\,.$$

(II.11)

The first term on the righthand side may be argued to be irrelevant when $t>0$, but its very existence is needed for a consistent (formal) Markovian limit and in the consideration of the number of the initial conditions. It is often omitted inadvertently. A more detailed discussion on consistent implementation and interpretation of the delta function in the context of taking the Markovian limit is discussed in Appendix B. A short conclusion is that we should interpret the delta function by its asymptotic form, commensurate with the specified physical setting.

The first term in (II.10) corresponds to mass renormalization, so by the following identifications

$\displaystyle m_{\textsc{b}}^{\vphantom{2}}$

$\displaystyle\mapsto m_{\textsc{p}}^{\vphantom{2}}=m_{\textsc{b}}^{\vphantom{2}}+\frac{e^{2}}{2\pi}\,\delta(0)\,,$

$\displaystyle m_{\textsc{b}}^{\vphantom{2}}\omega_{\textsc{b}}^{2}$

$\displaystyle\mapsto m_{\textsc{p}}^{\vphantom{2}}\omega_{\textsc{p}}^{2}\,,$

(II.12)

the equation of motion (II.5) then reduces to²²2One may construct an equation of motion which has the form $m_{\textsc{p}}^{\vphantom{2}}\,\ddot{\chi}(t)+m_{\textsc{p}}^{\vphantom{2}}\omega_{\textsc{p}}^{2}\,\chi(t)-2m_{\textsc{p}}^{\vphantom{2}}\gamma\,\dddot{\chi}(t)=-e\,\frac{\partial}{\partial t}\hat{\phi}_{h}(\bm{z},t)$ even when $t=0$, but it describes slightly different dynamics from (II.13) and will not correspond to the formal Markovian limit of (III.11) or (III.1). Instead, we will have the Laplace transform of the solution given by $\displaystyle\tilde{\chi}(\sigma)$ $\displaystyle=\frac{(1-2\gamma\sigma)\sigma}{\sigma^{2}+\omega_{\textsc{p}}^{2}-2\gamma\,\sigma^{3}}\,\chi(0)+\frac{1-2\gamma\sigma}{\sigma^{2}+\omega_{\textsc{p}}^{2}-2\gamma\,\sigma^{3}}\,\dot{\chi}(0)-\frac{2\gamma}{\sigma^{2}+\omega_{\textsc{p}}^{2}-2\gamma\,\sigma^{3}}\,\ddot{\chi}(0)+\text{the particular solution}\,,$ slightly different from (II.2).

$\displaystyle m_{\textsc{p}}^{\vphantom{2}}\,\ddot{\chi}(t)+m_{\textsc{p}}^{\vphantom{2}}\omega_{\textsc{p}}^{2}\,\chi(t)-2m_{\textsc{p}}^{\vphantom{2}}\gamma\,\dddot{\chi}(t)-4m_{\textsc{p}}^{\vphantom{2}}\gamma\Bigl{[}\dot{\delta}(t)\,\dot{\chi}(0)+\delta(t)\,\ddot{\chi}(0)\Bigr{]}=-e\,\frac{\partial}{\partial t}\hat{\phi}_{h}(\bm{z},t)\,,$

(II.13)

with $\gamma=e^{2}/(8\pi m)$. Here we see the appearance of $-\dddot{\chi}(t)$. It accounts for radiation damping taking the form of a self-force. It implies that the solution may depend on $\ddot{\chi}(0)$, in addition to $\chi(0)$ and $\dot{\chi}(0)$. Furthermore, this equation of motion itself contains a few terms that explicitly depends on the initial conditions. Their presence naturally arises in the framework of open systems when we re-write (II.4) as (II.13) in terms of physical parameters by means of the repeated integrations by parts in (II.10). Emphatically they are needed in correctly treating the initial-condition issues. When $t>0$, these terms disappear, and we arrive at the nonrelativistic ALD equation. The appearance of the $\dddot{\chi}(t)$ term in the ALD equation which is not seen in the typical Newtonian dynamics has drawn much attention to the invention of schemes to revise the equation of motion by, order reduction, interpretation of the solutions, and discussions of their consequences.

The formal solution to (II.13) can be given by means of the Laplace transformation. However, there is a catch. The Laplace transformation of the delta function is typically assigned to be 1 because the lower bound in (II.15) is replaced by $0^{-}$ in the standard reference. If we use this convention, we find

	$\displaystyle\tilde{\chi}(\sigma)$	$\displaystyle=\frac{(1-2\gamma\sigma)\sigma}{\sigma^{2}+\omega_{\textsc{p}}^{2}-2\gamma\,\sigma^{3}}\,\chi(0)+\frac{1+2\gamma\sigma}{\sigma^{2}+\omega_{\textsc{p}}^{2}-2\gamma\,\sigma^{3}}\,\dot{\chi}(0)+\frac{2\gamma}{\sigma^{2}+\omega_{\textsc{p}}^{2}-2\gamma\,\sigma^{3}}\,\ddot{\chi}(0)$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\text{the particular solution}\,,$		(II.14)

where the particular solution denotes contribution from the force term, and the Laplace transform $\tilde{f}(\sigma)$ of a function $f(t)$ is defined by

$$\tilde{f}(\sigma)=\int_{0}^{\infty}\!dt\;e^{-t\sigma}\,f(t)\,,$$

(II.15)

with $\operatorname{Re}\sigma>0$, and the inverse transform is given by

$$f(t)=\frac{1}{2\pi i}\int_{C}\!d\sigma\;\tilde{f}(\sigma)\,e^{t\sigma}\,,$$

(II.16)

where the closed contour $C$ encompasses all poles of $\tilde{f}(\sigma)$. Sometimes, we use the notation like $\pounds f(t)$ to denote applying the Laplace transformation to the function $f(t)$. We see that the solution in (II.2) needs the input from three initial conditions $\chi(0)$, $\dot{\chi}(0)$ and $\ddot{\chi}(0)$.

However, from the discussion in Appendix B, the following seems to us more consistent with the Markovian limit, if we still set the lower bound in (II.15) to 0, and heuristically imagine that only half of the delta function “peak” is used to evaluate the Laplace transformation. Then the third subtlety in this supra-Ohmic Markovian dynamics is that (C) we will find the Laplace transform of the delta function is $1/2$, and likewise the Laplace transform of $\dot{\delta}(t)$ is $-\delta(0)+\dfrac{\sigma}{2}$, instead of $\sigma$. Using this algorithm, we obtain instead that

$\displaystyle\tilde{\chi}(\sigma)=\frac{(1-2\gamma\sigma)\sigma}{\sigma^{2}+\omega_{\textsc{p}}^{2}-2\gamma\,\sigma^{3}}\,\chi(0)+\frac{1-4\gamma\sigma\,\delta(0)}{\sigma^{2}+\omega_{\textsc{p}}^{2}-2\gamma\,\sigma^{3}}\,\dot{\chi}(0)+\text{the particular solution}\,.$

(II.17)

This result depends on only two initial condition and conforms to Newtonian dynamics. The appearance of $\delta(0)$ may look alarming, but its presence is necessary when we take the Markovian limit of non-Markovian dynamics, as will be shown later. Moreover, this solution is purely formal and is the runaway kind. From the fuller perspective of the non-Markovian dynamics we shall present below, the Markovian limit of this supra-Ohmic non-Markovian equation of motion is inherently unstable and it is not surprising that it possesses runaway solutions. Finally, as we stress earlier that it will be more physically consistent to interpret the delta function by its asymptotic form. Here we remind again that the runaway behavior of the solution has nothing to do with the presence of $\delta(0)$ or its asymptotic form; the runaway results from the existence of the positive real root in $\sigma^{2}+\omega_{\textsc{p}}^{2}-2\gamma\,\sigma^{3}=0$.

Earlier in (II.10), we mentioned that a term $-\dfrac{e^{2}}{2\pi}\delta(t)\,\ddot{\chi}(0)$ is often omitted. The restoration of this term is important because without its presence, the solution will still depend on the initial condition $\ddot{\chi}(0)$ no matter what convention of the Laplace transformation of the delta function we take.

The previous discussions on treating the delta function expressions, in particular in terms of its asymptotic form, will be better motivated and justified once we learn more about supra-Ohmic, non-Markovian dynamics in Sec. (III).

The salient features revealed in our analysis of the dynamics of the charge particle in scalar electrodynamics are: 1) the appearance of the $\dddot{\chi}(t)$ in the equation motion does not always imply dynamical instability and the necessity of the additional initial condition $\ddot{\chi}(0)$ to uniquely determine the solution, 2) instability arises corresponding to the length of the memory time of the bath, and 3) the description of the Markovian limit in the supra-Ohmic dynamics is more complicated than in the Ohmic case. The meaning of the Markovian limit can only be reached and better understood by adopting a broader scope, starting from the full non-Markovian dynamics of the reduced system, as we shall show in the next section.

The non-Markovianity, or the finite cutoff scale, in the spectrum of the field can naturally emerge from various considerations. For example, it can be induced by the finite-size effect of the charge particle, or by the scale inherited in the modeling of the field configuration, such as in the presence of a dielectric or under spatial confinement. It may even reflect the energy scale associated with the validity of the theory. We will see that in the current discussion $2\gamma=e^{2}/(4\pi m_{\textsc{p}})$ plays such a role. It gives the “radius” of the charge and defines a critical scale so that when the length scale corresponding to the memory time is shorter than this radius, the dynamics of the charge begins to show runaway behavior. This is yet another reason why one should refrain from treating the charge as a point particle.

This observation, that the charge should be treated as an extended object, is not new Yaghjian ; Erber ; MonShar . We reached this conclusion from the memory time or non-Markovianity considerations. When the size of the charge is larger than the classical radius there are no runaway solutions or pre-acceleration issues. In fact, these difficulties can be effectively mitigated by introducing a suitable scale into the theory or the calculations such as, for example, the switching-on time scale of the charge-field interaction, the length scale in the electron form factor due to the shape or the charge distribution, or the finite-difference numerical algorithm³³3This is perhaps a good point to compare the work of Ref. FOC91 with the present treatment: Both start the non-Markovianity consideration from an equation of motion like (II.5), but then the two approaches differ. The authors of FOC91 convert the integro-differential equation (II.5) into an ordinary, but higher-order differential equation to make connection with the ALD equation. This is possible for the specific form of spectral density chosen there. However, for more general bath spectral functions, it is not a simple task to make such a conversion. These authors identify the role of the cutoff, the origin of non-Markovianity, in dynamical stability based on causality requirement. In contrast, we stay in the non-Markovian formalism, and this allows us to address instability in a more general context in terms of the factor $1-8\pi\gamma\,\Gamma^{(\textsc{e})}(0)$, discussed in Sec. III, and to better identify subtleties surrounding the Markovian limit in supra-Ohmic dynamics. JH1 ; FOC91 ; Pena ; GLR ; Lan .

In the next section, we shall focus on a class of non-Markovian dynamics often encountered in Brownian motion, and provide a systematic elaboration on its dynamical characteristics. In addition, we will establish its connection with the (nonrelativistic) non-Markovian ALD equation and then discuss its Markovian limit. We also provide a few examples to give concrete pictures of these dynamical features.

An intuitive and likely the simplest way to implement non-Markovian dynamics in a supra-Ohmic bath is to choose a standard class of equation of motion for Brownian motion

$$m_{\textsc{b}}^{\vphantom{2}}\,\ddot{\chi}(t)+m_{\textsc{b}}^{\vphantom{2}}\omega_{\textsc{b}}^{2}\,\chi(t)-e^{2}\int_{0}^{t}\!ds\;G_{R,0}^{(\textsc{e})}(t-s)\,\chi(s)=e\,\xi(t)\,,$$

(III.1)

but to implement the spectral density of the bath by a supra-Ohmic form $J(\kappa)=\kappa^{\lambda}\,\mathcal{P}_{\Lambda}(\kappa)$ with $\lambda>1$. Eq (III.1) describes the motion of a Brownian oscillator, which is the reduced system of interest to us, coupled to a supra-Ohmic thermal Gaussian bath. The influences of the bath are summarized into a noise force $\xi(t)$ and a nonlocal expression in (III.1) that depends on the retarded Green’s function of the free bath. The latter contains the reaction to the disturbance to the bath generated by the oscillator when it is driven by the noise force, and also includes corrections to or renormalization of the oscillator’s physical parameters. The retarded Green’s function of the free bath is given by

$$G_{R,0}^{(\textsc{e})}(\tau)=i\,\theta(\tau)\int_{0}^{\infty}\!\frac{d\kappa}{2\pi}\;\frac{J(\kappa)}{2\pi}\,\Bigl{[}e^{-i\kappa\tau}-e^{+i\kappa\tau}\Bigr{]}\,,$$

(III.2)

where $\theta(\tau)$ is the Heaviside theta function. To comply with the fluctuation-dissipation relation of the free bath, the noise $\xi(t)$ of the bath is required to satisfy the Gaussian statistics:

$\displaystyle\langle\xi(t)\rangle$

$\displaystyle=0\,,$

and

$\displaystyle\frac{1}{2}\langle\bigl{\{}\xi(t),\xi(t^{\prime})\bigr{\}}\rangle$

$\displaystyle=G_{H,0}^{(\textsc{e})}(\tau)=\int_{-\infty}^{\infty}\!\frac{d\kappa}{2\pi}\;\frac{J(\kappa)}{4\pi}\coth\frac{\beta\kappa}{2}\,e^{-i\kappa\tau}\,,$

(III.3)

where $\tau=t-t^{\prime}$, and $\beta$ is the inverse initial bath temperature. Here we will only deal with the $\lambda=3$ case. We will give a brief description of the higher supra-Ohmic system in Appendix D.

Since from time to time, we would like to compare the supra-Ohmic dynamics with the Ohmic one, we will express the supra-Ohmic kernel functions in terms of $\Gamma^{(\textsc{e})}(\tau)$, which is often seen in the context of Ohmic Brownian motion. The kernel function $\Gamma^{(\textsc{e})}(\tau)$ takes the form

$$\Gamma^{(\textsc{e})}(\tau)=\int_{-\infty}^{\infty}\!\frac{d\kappa}{2\pi}\;\frac{I(\kappa)}{2\pi\kappa}\,e^{-i\kappa\tau}\,,$$

(III.4)

such that supra-Ohmic spectral density $J(\kappa)$ is related to the Ohmic spectral density $I(\kappa)$ by $J(\kappa)=\kappa^{2}\,I(\kappa)$, and the supra-Ohmic retarded Green’s function $G_{R,0}^{(\textsc{e})}(\tau)$ is expressed as

$$G_{R,0}^{(\textsc{e})}(\tau)=\theta(\tau)\,\frac{\partial^{3}}{\partial\tau^{3}}\Gamma^{(\textsc{e})}(\tau)\,.$$

(III.5)

For a bath consisting of a continuum of modes, the kernel functions can be potentially ill-defined. To circumvent this ambiguity, the function $\mathcal{P}_{\Lambda}(\kappa)$ is introduced to do the job. In this paper, we assume that it contains only one scale, the cutoff frequency $\Lambda$. The contributions from the modes of the bath whose frequencies are higher than $\Lambda$ will be suppressed. We also assume that $\mathcal{P}_{\Lambda}(\kappa)$ is an even function of $\kappa$, and approaches unity when $\Lambda$ goes to infinity. This is necessary to ensure a unique Markovian limit, independent of the functional forms of the spectral densities. The following three common forms of $\mathcal{P}_{\Lambda}(\kappa)$ will be discussed in this paper,

$\displaystyle\mathcal{P}_{\Lambda}(\kappa)$

$\displaystyle=e^{-\frac{\lvert\kappa\rvert}{\Lambda}}\,,$

$\displaystyle\mathcal{P}_{\Lambda}(\kappa)$

$\displaystyle=\frac{\Lambda^{2}}{\Lambda^{2}+\kappa^{2}}\,,$

$\displaystyle\mathcal{P}_{\Lambda}(\kappa)$

$\displaystyle=\Theta(\Lambda-\kappa)\,\Theta(\kappa+\Lambda)\,.$

(III.6)

In the first case, the high frequency modes are exponentially suppressed, while the second form, called Lorentzian, only algebraically. Thus in certain cases, the latter may not be enough to ease off the divergence due to the high-frequency bath modes. The last case, called hard cutoff, is probably the most naïve way to apply the cutoff. Any mode with a frequency higher than $\Lambda$ is outright discarded. We mention this case because we want to warn the danger of it having some peculiar, undesirable properties when $\Lambda$ is not extremely large. The Lorentz spectrum will be treated in Sec. IV, and the other two examples are relegated to Appendix C.

Due to the presence of finite $\Lambda$ in the spectral density, the nonlocal expression in (III.1) cannot be reduced to a local form⁴⁴4For example, if the kernel function $\Gamma^{(\textsc{e})}(\tau)$ is proportional to $\delta(\tau)$, then the integral expression like $\displaystyle\int_{0}^{t}\!ds\;\delta(t-s)\,\chi(s)$ reduces to $\dfrac{1}{2}\chi(t)$, local in term. We do not mean to re-write the whole nonlocal equation of motion, into the so-called time-local form. The latter can in principle be achieved for the linear non-Markovian equation of motion., and thus it will allow the past history to affect the current state of motion. Then for any finite $\Lambda$, the dynamics described by (III.1) will be history-dependent, and hence is non-Markovian⁵⁵5In this paper, non-Markovianity has a broader connotation, referring to memories in the open system’s dynamics, not an approximation used in expressing the two-point function. For example, the Hadamard function $G_{H,0}^{(\textsc{e})}(\tau)$ in (III.3) in the high-temperature limit is approximately given by (the derivative of) the delta function, and is said to have a Markovian form. We refrain from this narrower sense of usage.. From the duality properties of the Fourier transformation, we will expect that $\Lambda^{-1}$ plays the role of memory time. It determines how far back past history of the motion can affect the current motion. Taking the Markovian limit by letting $\Lambda\to\infty$ can then be understood as letting the memory time approach zero, and the system dynamics becomes memoryless; it forgets everything that happens before the current moment.

We first take care of the renormalization issue in the supra-Ohmic equation of motion (III.1). The supra-Ohmic spectral density of the bath implies that the non-local term in (III.1) will have a more serious divergence issue than the Ohmic bath. Inserting (III.5) into (III.1), we can write the nonlocal term as

	$\displaystyle\quad-e^{2}\int_{0}^{t}\!ds\;G_{R}^{(\textsc{e})}(t-s)\,\chi(s)$
	$\displaystyle=e^{2}\int_{0}^{t}\!ds\;\frac{\partial^{3}}{\partial s^{3}}\Gamma^{(\textsc{e})}(t-s)\,\chi(s)$		(III.7)
	$\displaystyle=e^{2}\biggl{\{}\ddot{\Gamma}^{(\textsc{e})}(0)\,\chi(t)-\ddot{\Gamma}^{(\textsc{e})}(t)\,\chi(0)+\dot{\Gamma}^{(\textsc{e})}(0)\,\dot{\chi}(t)-\dot{\Gamma}^{(\textsc{e})}(t)\,\dot{\chi}(0)+\Gamma^{(\textsc{e})}(0)\,\ddot{\chi}(t)-\Gamma^{(\textsc{e})}(t)\,\ddot{\chi}(0)\biggr{\}}$
	$\displaystyle\qquad\qquad\qquad\qquad\qquad-e^{2}\int_{0}^{t}\!ds\;\Gamma^{(\textsc{e})}(t-s)\,\dddot{\chi}(s)\,.$		(III.8)

The term proportional to $\ddot{\chi}(t)$ in (III.8) will be absorbed into mass renormalization. This is not seen in the treatment of typical Ohmic Brownian motion, where only the natural frequency needs renormalizing. The term proportional to $\chi(t)$ is then associated with the frequency renormalization in the supra-Ohmic case, such that the physical mass and the physical frequency are respectively given by

$\displaystyle m_{\textsc{p}}$

$\displaystyle=m_{\textsc{b}}+e^{2}\Gamma^{(\textsc{e})}(0)\,,$

$\displaystyle m^{\vphantom{2}}_{\textsc{p}}\omega^{2}_{\textsc{p}}$

$\displaystyle=m^{\vphantom{2}}_{\textsc{b}}\omega^{2}_{\textsc{b}}+e^{2}\ddot{\Gamma}^{(\textsc{e})}(0)\,.$

(III.9)

For a finite $\Lambda$, the values of $\ddot{\Gamma}^{(\textsc{e})}(0)$, $\dot{\Gamma}^{(\textsc{e})}(0)$, ${\Gamma}^{(\textsc{e})}(0)$ may not be humongous at all, so we may justifiably view the parameters in (III.9) as the effective ones, which have absorbed the corrections induced by the interaction between the system and the bath. Then the equation of motion (III.1) becomes

$\displaystyle m_{\textsc{p}}\,\ddot{\chi}(t)+m^{\vphantom{2}}_{\textsc{p}}\omega^{2}_{\textsc{p}}-e^{2}\int_{0}^{t}\!ds\;\Gamma^{(\textsc{e})}(t-s)\,\dddot{\chi}(s)-e^{2}\Bigl{[}\ddot{\Gamma}^{(\textsc{e})}(t)\,\chi(0)+\dot{\Gamma}^{(\textsc{e})}(t)\,\dot{\chi}(0)+\Gamma^{(\textsc{e})}(t)\,\ddot{\chi}(0)\Bigr{]}=e\,\xi(t)\,,$

(III.10)

in terms of physical or effective parameters. Note that $\dot{\Gamma}^{(\textsc{e})}(0)=0$ when the power of the monomial before $\mathcal{P}_{\Lambda}(\kappa)$ in the bath spectral density is an odd integer. Here, the appearance of $\chi$ and its time derivatives at $t=0$ in the equation of motion is a consequence of sudden switch-on and casting the original equation of motion (III.1) in terms of physical parameters like mass and natural frequency CFdiss ; Fleming . On the other hand, if instead we introduce a gradual switching by explicitly implementing a time-dependent coupling constant $e(t)$ which increases sufficiently smoothly from the zero value at the initial time $t=0$ to full strength over a finite time interval, then the terms like $\chi$ and its time derivatives at $t=0$ in (III.10) may disappear, depending on the time derivatives of the coupling constant at $t=0$, as can be inspected by replacing $e\chi(s)$ with $e(s)\chi(s)$ in (III.8).

Before delving deeply into the issues associated with supra-Ohmic non-Markovian dynamics, let us take a quick look at how the non-Markovian version of ALD equation is related to (III.10) and what common features they may share.

We have stated in the previous section that from the viewpoint of non-Markovian dynamics, the dynamics described by the equation of motion (II.5) is destined to be unstable. In addition, we have come across quite a few subtleties and obscurities in interpreting this Markovian equation. The origin of these difficulties arises from the Markovian spectrum of the scalar field. This can be most clearly understood if we replace in (II.5) the delta function kernel by a more general Ohmic non-Markovian kernel function $\Gamma^{(\textsc{e})}$, (III.4). It also allows us to see the connection with the supra-Ohmic dynamics discussed earlier. Without repeating the derivation, we find that, from (II.5), the non-Markovian ALD equation of motion corresponding to (II.13) takes the form

$\displaystyle m_{\textsc{p}}^{\vphantom{2}}\,\ddot{\chi}(t)+m_{\textsc{p}}^{\vphantom{2}}\omega_{\textsc{p}}^{2}\,\chi(t)$

$\displaystyle-e^{2}\int_{0}^{t}\!ds\;\Gamma^{(\textsc{e})}(t-s)\,\dddot{\chi}(s)-e^{2}\biggl{[}\Gamma^{(\textsc{e})}(t)\,\ddot{\chi}(0)+\dot{\Gamma}^{(\phi)}(t)\,\dot{\chi}(0)\biggr{]}=-e\,\dot{\phi}_{h}(\bm{x},t)\,,$

(III.11)

with

$\displaystyle m_{\textsc{p}}^{\vphantom{2}}$

$\displaystyle=m_{\textsc{b}}^{\vphantom{2}}+e^{2}\Gamma^{(\textsc{e})}(0)\,,$

$\displaystyle m_{\textsc{p}}^{\vphantom{2}}\omega_{\textsc{p}}^{2}$

$\displaystyle=m_{\textsc{b}}^{\vphantom{2}}\omega_{\textsc{b}}^{2}\,.$

(III.12)

Comparing with (III.10), we readily see that (III.11) does not have the $\ddot{\Gamma}^{(\textsc{e})}(t)\,\chi(0)$ term, and the force terms on their righthand sides are not alike due to the different forms of the oscillator-field coupling.

To see the relation of the non-Markovian ALD equation (II.5) with the supra-Ohmic equation of motion (III.1), we write the non-local term in (II.5) as

$\displaystyle e^{2}\frac{\partial}{\partial t}\int_{0}^{t}\!ds\;G_{R,0}^{(\phi)}(t-s)\,\dot{\chi}(s)$

$\displaystyle=-e^{2}\ddot{\Gamma}^{(\textsc{e})}(0)\,\chi(t)+e^{2}\ddot{\Gamma}^{(\textsc{e})}(t)\,\chi(0)+e^{2}\int_{0}^{t}\!ds\;\partial^{3}_{s}\Gamma^{(\textsc{e})}(t-s)\,\chi(s)\,,$

(III.13)

with the use of integration by parts and the substitution $G_{R,0}^{(\phi)}(\tau)=-\partial_{\tau}\Gamma^{(\textsc{e})}(\tau)$. The third term on the right hand side gives (III.7), the whole nonlocal term in (III.1), but there are two additional terms in the non-Markovian ALD equation. The first term will account for additional frequency renormalization, not seen in (III.12), and the second term gives the contribution missing in (III.11) but present in (III.10). Thus we see both systems have slightly different dependence on the initial conditions.

The Laplace transform of the formal solution to (III.11) is given by

$\displaystyle\tilde{\chi}(\sigma)$

$\displaystyle=\tilde{d}_{1}(\sigma)\,\chi(0)+\tilde{d}_{2}(\sigma)\,\dot{\chi}(0)-\frac{e}{m_{\textsc{p}}}\frac{\tilde{d}_{2}(\sigma)}{1-8\pi\gamma\,\sigma\Gamma^{(\textsc{e})}(0)}\,\bigl{[}\sigma\,\tilde{\phi}_{h}(\sigma)-\phi_{h}(0)\bigr{]}\,,$

(III.14)

with

$\displaystyle\tilde{d}_{1}(\sigma)$

$\displaystyle=\frac{[1-8\pi\gamma\,\sigma\tilde{\Gamma}^{(\textsc{e})}(\sigma)]\sigma}{\sigma^{2}+\omega^{2}_{\textsc{p}}-8\pi\gamma\,\sigma^{3}\tilde{\Gamma}^{(\phi)}(\textsc{e})}\,,$

$\displaystyle\tilde{d}_{2}(\sigma)$

$\displaystyle=\frac{1-8\pi\gamma\,\sigma\Gamma^{(\textsc{e})}(0)}{\sigma^{2}+\omega^{2}_{\textsc{p}}-8\pi\gamma\,\sigma^{3}\tilde{\Gamma}^{(\textsc{e})}(\sigma)}\,.$

(III.15)

We see that in a more general non-Markovian setting, even though the equation of motion has terms that depend on $\dddot{\chi}(t)$ and the initial condition $\ddot{\chi}(0)$, its solution actually depends only on two initial conditions $\chi(0)$ and $\dot{\chi}(0)$, thus supporting our earlier interpretation of the delta function in deriving (II.17). Taking a closer look at (III.11), we notice that when we perform the Laplace transformation, the nonlocal term in (III.11) will yield a contribution to cancel the $\ddot{\chi}(0)$ term. Since (III.15) has the same denominator in $\tilde{d}_{i}(\sigma)$ as (III.17) below, both sets of $\tilde{d}_{i}(\sigma)$ will be subjected to the same criterion of stability, and have the same relaxation dynamics.

In summary, compared with the generic supra-Ohmic non-Markovian dynamics described by (III.1), the non-Markovian ALD equation is shown to have different dependence on the initial conditions, but they have the same relaxation dynamics and stability criterion. These features of the generic supra-Ohmic non-Markovian dynamics we shall discuss below will equally apply to the non-Markovian ALD equation. At the end of this section, we will revisit the Markovian limit of (III.11).

To better understand this supra-Ohmic Langevin equation we comment on some of its uncommon characteristics. Superficially, Eq. (III.9) contains a third-order time derivative $\dddot{\chi}(t)$ of $\chi(t)$ and depends explicitly on the initial condition $\ddot{\chi}(0)$. These two features are not seen in the typical equations of motion, and are often considered as signaling something unphysical. They also create difficulties in implementing numerical solutions of this equation because it seems that an additional initial condition $\ddot{\chi}(0)$ is needed as well. However, this does not really pose any problem. When we carry out the Laplace transformation of (III.10) to find the Laplace transform $\tilde{\chi}(\sigma)$, we obtain

$\displaystyle\tilde{\chi}(\sigma)=\tilde{d}_{1}(\sigma)\,\chi(0)+\tilde{d}_{2}(\sigma)\,\dot{\chi}(0)+\frac{e}{m_{\textsc{p}}}\frac{\tilde{d}_{2}(\sigma)}{1-8\pi\gamma\Gamma^{(\textsc{e})}(0)}\,\tilde{\xi}(\sigma)\,,$

(III.16)

with

$\displaystyle\tilde{d}_{1}(\sigma)$

$\displaystyle=\frac{[1-8\pi\gamma\Gamma^{(\textsc{e})}(0)]\,\sigma}{\sigma^{2}+\omega_{\textsc{p}}^{2}-8\pi\gamma\tilde{\Gamma}^{(\textsc{e})}(\sigma)\,\sigma^{3}}\,,$

$\displaystyle\tilde{d}_{2}(\sigma)$

$\displaystyle=\frac{1-8\pi\gamma\Gamma^{(\textsc{e})}(0)}{\sigma^{2}+\omega_{\textsc{p}}^{2}-8\pi\gamma\tilde{\Gamma}^{(\textsc{e})}(\sigma)\,\sigma^{3}}\,.$

(III.17)

The inverse Laplace transformations of $\tilde{d}_{1}(\sigma)$ and $\tilde{d}_{2}(\sigma)$ give a special set of fundamental solutions that in the time domain satisfy

$\displaystyle d_{1}(0)$

$\displaystyle=1\,,$

$\displaystyle\dot{d}_{1}(0)$

$\displaystyle=0\,,$

$\displaystyle d_{2}(0)$

$\displaystyle=0\,,$

$\displaystyle\dot{d}_{2}(0)$

$\displaystyle=1\,,$

(III.18)

by construction. In particular $\dfrac{1}{m_{\textsc{p}}}\dfrac{d_{2}(\tau)}{1-8\pi\gamma\Gamma^{(\textsc{e})}(0)}$ can be identified as the retarded Green’s function $G_{R}^{(\chi)}(\tau)$ of $\chi$ associated with the equation of motion, (III.10).

Eq. (III.16) does not depend on $\ddot{\chi}(0)$, so the appearance of $\ddot{\chi}(0)$ in (III.10) is nothing but superficial. The Laplace transform of the nonlocal term will yield an additional $\ddot{\chi}(0)$ to cancel with the corresponding term in the second line of (III.10). Thus, we only actually need two initial conditions $\chi(0)$ and $\dot{\chi}(0)$, instead of three. More importantly, we emphasize that even when we deal with the nonlocal equation of motion, we still need only two initial conditions, instead of the initial history, commonly met for the delayed differential equation. The latter is often seen in the context of spatial non-Markovianity due to the case, for example, when the oscillator is in the proximity of a boundary HL08 or when the constituents of the system is spatially separated HH16 .

The presence of $\dddot{\chi}(t)$ in the equation of motion (III.10) does not always imply a runaway result. Here in particular from (III.17), we note that its solution depends on the factor $1-8\pi\gamma\Gamma^{(\textsc{e})}(0)$. The extra term $-8\pi\gamma\Gamma^{(\textsc{e})}(0)$ arrises from $-e^{2}\,\dot{\Gamma}^{(\textsc{e})}(t)\,\dot{\chi}(0)$ in the second line of (III.10), and this type of term is not seen in the Ohmic case because we only apply integration by parts once on the corresponding nonlocal term in the Ohmic case. The factor $1-8\pi\gamma\Gamma^{(\textsc{e})}(0)$, though odd looking, is necessary to ensure the proper normalization of $d_{1}(t)$ and $d_{2}(t)$ such that the conditions $d_{1}(0)=1$ and $\dot{d}_{2}(0)=1$ are satisfied. In turn, these ensure the correct behavior of the nonequilibrium Robertson-Schrödinger function at early times.

Furthermore, this factor has a special significance with regard to the dynamical stability of this class of supra-Ohmic, non-Markovian Langevin equation (III.1) under consideration. When this factor is positive, the fundamental solution $d_{1}(t)$ and $d_{2}(t)$ in general exhibit damping behavior, at least for the classes of bath spectral densities we are considering. On the other hand, if this factor is negative, then the solution increases without bound. To understand why the dynamics is unstable when $1-8\pi\gamma\,\Gamma^{(\textsc{e})}(0)<0$, let us examine the denominator of $\tilde{d}_{2}(\sigma)$, $\tilde{\mathfrak{J}}(\sigma)=\sigma^{2}+\omega_{\textsc{p}}^{2}-8\pi\gamma\,\tilde{\Gamma}^{(\textsc{e})}(\sigma)\,\sigma^{3}$, along the real $\sigma$ axis. Suppose $\tilde{\Gamma}^{(\textsc{e})}(\sigma)$ is well defined in the limit $\sigma\to 0$. Thus we find $\displaystyle\lim_{\sigma\to 0}\tilde{\mathfrak{J}}(\sigma)=\omega_{\textsc{p}}^{2}>0$. On the other hand $\displaystyle\lim_{\sigma\to\infty}\tilde{\Gamma}^{(\textsc{e})}(\sigma)$ in the $\sigma$ domain corresponds to $\displaystyle\lim_{t\to 0}\Gamma^{(\textsc{e})}(0)\,\theta(t)$ in the $t$ domain. Thus we have

$$\lim_{\sigma\to\infty}\tilde{\Gamma}^{(\textsc{e})}(\sigma)\sim\pounds\Gamma^{(\textsc{e})}(0)\,\theta(t)=\frac{\Gamma^{(\textsc{e})}(0)}{\sigma}\,,$$

(III.19)

so that in the limit $\sigma\to\infty$, we have

$$\lim_{\sigma\to\infty}\tilde{\mathfrak{J}}(\sigma)\simeq\bigl{[}1-8\pi\gamma\,\Gamma^{(\textsc{e})}(0)\bigr{]}\sigma^{2}+\mathcal{O}(\sigma)\,,$$

(III.20)

going unbounded quadratically with $\sigma$. When $1<8\pi\gamma\,\Gamma^{(\textsc{e})}(0)$, we find $\tilde{\mathfrak{J}}(\sigma)<0$ in the $\sigma\to\infty$ limit. Now we obtain $\displaystyle\lim_{\sigma\to 0}\tilde{\mathfrak{J}}(\sigma)>0$ but $\displaystyle\lim_{\sigma\to\infty}\tilde{\mathfrak{J}}(\sigma)<0$, we conclude that $\tilde{\mathfrak{J}}(\sigma)$ must have at least one real root between $\sigma=0$ and $\sigma\to\infty$. That is, $\tilde{d}_{2}(\sigma)$ will have a real positive pole. This will cause the runaway behavior in the solution to the equation of motion (III.10).

Following the previous analysis, we observe that the presence of the factor $1-8\pi\gamma\,\Gamma^{(\textsc{e})}(0)$ will make $\tilde{d}_{2}(\sigma)$ assume the form $+1/\sigma^{2}$ as $\sigma\to\infty$. This implies that as $t>0$, $d_{2}(t)$ will grow like $+t$. The plus sign here is physically necessary. Suppose we have the initial condition $\chi(0)=0$ but $\dot{\chi}(0)\neq 0$, then around the initial time, in the absence of any external force, the solution $\chi(t)$ is roughly given by $d_{2}(t)\,\dot{\chi}(0)$. Physically, we expect that in the absence of external forces, if the system is initially and momentarily at rest, then it should be displaced in the same direction as the initial velocity. It implies $d_{2}(t)$ must take on positive values around the initial time. It would be odd if it moves in a direction opposite to the initial speed. Thus the factor $1-8\pi\gamma\,\Gamma^{(\textsc{e})}(0)$ ensures physical consistency. Finally we mentioned that the nonequilibrium fluctuation-dissipation relation of the system, valid when $1>8\pi\gamma\,\Gamma^{(\textsc{e})}(0)$, still takes on the standard expression $\bar{G}_{H}^{(\chi)}(\kappa)=\coth\dfrac{\beta\kappa}{2}\,\operatorname{Im}\bar{G}_{R}^{(\chi)}(\kappa)$ after the system reaches its final equilibrium state. Its form is not affected by the presence of the factor.

Since the cutoff scale $\Lambda$ is contained in $\Gamma^{(\textsc{e})}(0)$, it implies that there exists a critical value $\Lambda_{c}$ such that when $\Lambda>\Lambda_{c}$, the supra-Ohmic non-Markovian system we consider here becomes unstable. When phrased in terms of the memory time, if the memory time $\Lambda^{-1}$ is shorter than the critical time scale $\Lambda_{c}^{-1}$, the system exhibits runaway behavior. Therefore, the Markovian limit of this supra-Ohmic system is purely formal, and in principle does not exist physically. Later when we take a closer look at the effects of various bath spectral density, we will give a more vivid demonstration of this important point.

The previous discussion reveals that the existence of the $\dddot{\chi}(t)$ in the equation of motion of a supra-Ohmic system does not automatically imply the necessity of additional initial condition, nor dynamical instability of the system. The latter primarily hinges on the duration of the memory time of the bath. This modus operandi should work equally well in the Markovian limit. However this is not the case in the conventional treatment of the non-relativistic ALD equation (see discussion in Sec. II). The aforementioned consistency consideration thus motivates our interpretation of the delta function arising from taking the Markovian limit in terms of its asymptotic form which supports our analysis expounded in Appendix B.

Finally, we write down the expression for the dispersion $\langle p^{2}(t)\rangle$ of the canonical momentum $p=m\dot{\chi}$ conjugated to $\chi$,

$$\langle p^{2}(t)\rangle=m^{2}\dot{d}_{1}^{2}(t)\,\langle\chi^{2}(0)\rangle+\dot{d}_{2}^{2}(t)\,\langle p^{2}(0)\rangle+\frac{e^{2}}{[1-8\pi\gamma\,\sigma\Gamma(0)]^{2}}\int_{0}^{t}\!ds\,ds^{\prime}\;\dot{d}_{2}(t-s)\,\dot{d}_{2}(t-s^{\prime})\,G_{H,0}^{(\textsc{e})}(s,s^{\prime})\,,$$

(III.21)

because we need its late times $t\to\infty$ limit,

$\displaystyle\langle p^{2}(\infty)\rangle=\frac{m_{\textsc{p}}}{1-8\pi\gamma\,\Gamma^{(\textsc{e})}(0)}\,\operatorname{Im}\int_{-\infty}^{\infty}\!\frac{d\kappa}{2\pi}\;\kappa^{2}\coth\frac{\beta\kappa}{2}\,\bar{d}_{2}(\kappa)=m_{\textsc{p}}\operatorname{Im}\int_{-\infty}^{\infty}\!\frac{d\kappa}{2\pi}\;\kappa^{2}\coth\frac{\beta\kappa}{2}\,\bar{G}_{R}^{(\chi)}(\kappa)\,,$

(III.22)

with the Fourier transform $\bar{d}_{2}(\kappa)$ of $d_{2}(t)$ given by $\bar{d}_{2}(\kappa)=\tilde{d}_{2}(-i\kappa)$, to explore the effect of the bath spectrum. Here we have used the identity

$\displaystyle\operatorname{Im}\bar{d}_{2}(\kappa)=\frac{\bar{d}_{2}^{\vphantom{*}}(\kappa)-\bar{d}_{2}^{*}(\kappa)}{2i}=8\pi\gamma\,\bigl{[}1-8\pi\gamma\,\Gamma^{(\textsc{e})}(0)\bigr{]}^{-1}\lvert\bar{d}_{2}(\kappa)\rvert^{2}\operatorname{Im}\bar{G}_{R,0}^{(\textsc{e})}(\kappa)\,,$

(III.23)

to simplify the reuslt. Thus $\langle p^{2}(\infty)\rangle$ takes the same form as in the Ohmic case because the factor $1-8\pi\gamma\,\Gamma^{(\textsc{e})}(0)$ is canceled.