The harmonic oscillator problem in SED has been studied by many leaders in the field; see, for example, [

5,

6,

7,

8,

9,

10,

11], it is also discussed, at length, in the book [

3]. This is typically done by taking, in frequency integrals, the contributions from resonances and not bothering much about high- or low-frequency peculiarities. It is our purpose to clarify where regularizations are needed and which form they should have, in order to derive these physically relevant results in a proper fashion.

Let the stochastic field and the particle position have the frequency representation

The equation of motion in time and frequency read, respectively,

The damping term, if taken from Equation (

4), would be

${\mathbf{D}}_{\omega}\approx i{\beta}^{2}{\omega}^{3}{\mathbf{r}}_{\omega}$. Fortunately, it has been derived, from first principles, in Equation (3.110) of [

3]. In our notation, this exact result reads

where we assumed a similar exponential cut-off

$exp(-|{\omega}^{\prime}|{\tau}_{c})$ as in the stochastic spectrum. This structure is a truncated convolution, so that it still leads to a product in Fourier space. With

${(\dot{\mathbf{r}})}_{\omega}=-i\omega {\mathbf{r}}_{\omega}$, one gets

with the function

${D}_{\omega}$, not to be confused with

${\mathbf{D}}_{\omega}$ or the integration measure

$\mathrm{D}\omega $ of Equation (

6), given by

where PV denotes the principal value. Transforming to the time domain, this yields

where

$\theta $ is the Heaviside step function. Indeed, for

$u<0$, the contour can be closed in the upper half of the complex

$\omega $ plane, where

$i\omega {D}_{\omega}$ is analytic, as is evident from the middle expression in the first line of (

14). Hence,

$D(u)=0$ for

$u<0$. Likewise, for

$u>0$, the contour can be closed in the lower half-plane, yielding

$D(u)=\overline{D}(u)$. This explains the causality relation

$D(t-s)=0$ for

$s>t$. After replacing

$\overline{D}\to D$ in Equation (

12) for

$\mathbf{D}(t)$, we may extend the

s and

u integrals from

$-\infty $ to

∞.

For small

$\omega {\tau}_{c}$, the last expression in (

14) yields

The

${\delta}_{m}$ term corresponds to a mass renormalization, due to the presence of the electromagnetic field modes. In the units of the hydrogen problem,

$\delta {m}_{e}/{m}_{e}={\delta}_{m}=4\alpha /3\pi =0.0031$, independent of

Z (as it should). The term

$i{\beta}^{2}\omega $ corresponds to the Lorentz damping term

$\mathbf{D}\approx {\beta}^{2}\stackrel{\u20db}{\mathbf{r}}$ in (

11). This approximation is known to have run-away solutions

$\mathbf{r}\sim expt/{\beta}^{2}$, artifacts that are absent in our exact treatment. For large

$\left|\omega \right|{\tau}_{c}$, one finds that the mass renormalization drops out and, instead, we have that

${D}_{\omega}\to -(4{\beta}^{2}/\pi {\tau}_{c})/{(\omega {\tau}_{c})}^{2}$ becomes negligible.

The poles of

${G}_{\omega}$ determine the complex eigenfrequencies. They follow from

Contrary to the approximation (

16), which formally allows

${D}_{\omega}\sim -1$ at a large imaginary

$\omega =i\left|\omega \right|$ with

$\left|\omega \right|\sim (1+{\delta}_{m})/{\beta}^{2}+{\beta}^{2}{\omega}_{0}^{2}$, the exact function

${D}_{\omega}$ is small for all real and complex

$\omega $, so there appear no such spurious eigenvalues and run-away solutions

$\sim exp(t/{\beta}^{2})$ in the exact treatment. As expected on physical grounds, there are only solutions near

$\pm {\omega}_{0}$. For small

$\beta $ and

${\delta}_{m}$, they read

with the approximations giving the leading terms in

${\beta}^{2}$.

#### The Steady State

We now get, from (

6) and (

17), the steady value

where the first expression can be verified from the frequency-discretization of [

13]. The integral is finite for

${\tau}_{c}\to 0$ and dominated by the narrow resonance region around

${\omega}_{0}$, with the result

Likewise,

There is a similar resonance around

$\omega \approx {\omega}_{0}$, yielding

$\langle {\dot{\mathbf{r}}}^{2}(t)\rangle =\frac{3}{2}{\omega}_{0}$ and

${\mathcal{E}}_{0}=\frac{1}{2}\langle {\dot{\mathbf{r}}}^{2}(t)\rangle +\frac{1}{2}{\omega}_{0}^{2}\langle {\mathbf{r}}^{2}(t)\rangle =\frac{3}{2}{\omega}_{0}$, which is the ground state energy of the

$3d$ quantum oscillator. However, the large

$\omega $ limit is only suppressed by the exponential. To evaluate its contribution to the leading order,

${\omega}_{0}^{2}$ can be set to zero, while

${D}_{\omega}\sim -({\beta}^{2}/{\tau}_{c})/{(\omega {\tau}_{c})}^{2}$ can be neglected. The remaining integral in (

26) is trivial, and brings

For

${\omega}_{0}=0$, this result follows immediately from the free-particle solution

${(\dot{\mathbf{r}})}_{\omega}=\beta {\mathbf{A}}_{\omega}/(1+{D}_{\omega})$, leading to

$\dot{\mathbf{r}}(t)\approx \beta \mathbf{A}(t)$ and

$\langle {\dot{\mathbf{r}}}^{2}(t)\rangle \approx 3{\beta}^{2}{C}_{AA}(0)$. The large term

${\beta}^{2}{C}_{AA}(0)\sim 1/\alpha {Z}^{2}$ comes from large frequencies

$\sim 1/{\tau}_{c}$, which are cut off—but not enough—by the factor

$exp(-|\omega |{\tau}_{c})$.

To further suppress the large

$\omega $ contributions, we propose to subtract the free propagator

${G}_{\omega}^{0}$,

The expression

${\mathbf{r}}_{\omega}={\overline{G}}_{\omega}{\mathbf{E}}_{\omega}$ can be written as

${\mathbf{r}}_{\omega}={G}_{\omega}{\overline{\mathbf{E}}}_{\omega}$, with the renormalized stochastic field

Now, the extra factor

${({\omega}_{0}/\omega )}^{4}{|1+{D}_{\omega}|}^{2}$ assures enough suppression of the large

$\omega $ contributions, so that the resonance at

${\omega}_{0}$ is dominant. This yields to the leading order

In the new expression for the position fluctuations,

the correction

$\varphi $ inherits the logarithmic divergency at

$\omega =0$ from the free particle. This, likely, is addressed by accounting for soft photon emission, leading to a small

${\beta}^{2}log1/\beta \sim {\alpha}^{3}log1/\alpha $ “Lamb” correction.

The average energy now agrees, to leading order, with that of the ground state of the QM oscillator,