An Introduction to Renormalization in Atomic Physics
Lecture notes for the $\bm{2}^{\bf nd}$ International Summer School on
Advanced Quantum Mechanics
(Prague, September 02-11, 2021)
Milan Šindelka
Laser Plasma Department, Institute of Plasma Physics of the Czech Academy of
Sciences,
U Slovanky 2525/1a, 18200 Prague 8, Czech Republic
<EMAIL_ADDRESS>
()

The purpose of these notes is to provide a pedagogical introduction to the
concept of renormalization in atomic physics. We study quantum dynamics of a
model of a nonrelativistic single electron atom coupled to the quantum
radiation field in the dipole approximation. An interaction between the
electron and the radiation field is regularized, using a suitable cutoff
prescription which eliminates the coupling of the electron to the highly
ultraviolet (UV) field modes. Subsequently, we analyze the corresponding
Heisenberg picture equations of motion, and focus on behavior of physical
observables in the limit of a pointlike electron (i.e., in the limit when the
above mentioned UV cutoff is gradually lifted to infinity). We identify the
radiation reaction force acting on the electron, and show that this force
actually diverges towards infinity when gradually removing the UV cutoff. Such
an analysis leads ultimately to an unique renormalization prescription for the
electron mass, as well as to determination of the Abraham-Lorentz force acting
on the electron. Dependence of the just sketched renormalization procedure
upon the spatial dimension is highlighted. Finally, we present an example
calculation of the atomic level shifts by means of the renormalized
perturbation theory, and formulate also the renormalized mean field theory.

A. Basic theoretical setup

  * •

A.1 Introducing our model problem (preliminaries)
Our theoretical model lives in the $d$-dimensional Euclidean space
${\mathbb{R}}^{d}$.
We shall consider the cases of $d=2,3$.
Subsystem #1:
Model of a nonrelativistic atom containing a single electron. Defined by the
usual Hamiltonian

$\hat{H}_{{\rm A}}\;=\;\frac{\hat{\bm{p}}^{2}}{2\,m}\;+\;V(\hat{\bm{x}})\hskip
14.22636pt.$ (1)

Explanations:
$m$ = physical mass of the electron.
$\hat{\bm{x}}$ and $\hat{\bm{p}}$ = standard canonical observables
$\hat{\bm{x}}$ and $\hat{\bm{p}}$ = corresponding to the position and momentum
of the electron.
$V({\bm{x}})$ = an external scalar potential acting on the electron.
$V({\bm{x}})$ = We assume that $V({\bm{x}})$ is strictly bound (diverging to
$+\infty$ at spatial infinities).
Note that the electron spin is ignored in our subsequent considerations.
Subsystem #2:
Quantized radiation field (continuous ensemble of harmonic oscillators).
Defined by the Hamiltonian

$\hat{H}_{{\rm R}}\;=\;\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\hbar\omega_{k}\;\hat{a}_{{\bm{k}}\wp}^{\dagger}\,\hat{a}_{{\bm{k}}\wp}\hskip
14.22636pt.$ (2)

Explanations:
$({\bm{k}}\wp)$ = modes of the radiation field, carrying modal frequencies
$\omega_{k}=c|{\bm{k}}|$ and polarization $\wp$.
$({\bm{k}}\wp)$ = $[\,$The electromagnetic field modes are transverse. No
longitudinal photons.
$({\bm{k}}\wp)$ = $[\,$Thus, for each ${\bm{k}}\in{\mathbb{R}}^{d}$, there
exists just a single polarization in the case of $d=2$,
$({\bm{k}}\wp)$ = $[\,$and two linearly independent mutually orthogonal
polarizations in the case of $d=3$.$\,]$
$\hat{a}_{{\bm{k}}\wp}$ = modal annihilators, characterized by their standard
commutation properties

$\hskip
14.22636pt\Bigl{[}\,\hat{a}_{{\bm{k}}\wp}\,,\,\hat{a}_{{\bm{k}}^{\prime}\wp^{\prime}}\,\Bigr{]}\;=\;\hat{0}\hskip
7.11317pt,\hskip
7.11317pt\Bigl{[}\,\hat{a}_{{\bm{k}}\wp}\,,\,\hat{a}_{{\bm{k}}^{\prime}\wp^{\prime}}^{\dagger}\,\Bigr{]}\;=\;\delta^{d}({\bm{k}}-{\bm{k}}^{\prime})\;\delta_{\wp\wp^{\prime}}\;\hat{1}\hskip
7.11317pt,\hskip
7.11317pt\Bigl{[}\,\hat{a}_{{\bm{k}}\wp}\,,\,\hat{\bm{x}}\,\Bigr{]}\;=\;\Bigl{[}\,\hat{a}_{{\bm{k}}\wp}\,,\,\hat{\bm{p}}\,\Bigr{]}\;=\;\hat{0}\hskip
7.11317pt.$ (3)

$\hat{a}_{{\bm{k}}\wp}$ = The field mode $({\bm{k}}\wp)$ contains a zero point
energy (ZPE) equal to $\hbar\omega_{k}/2$,
$\hat{a}_{{\bm{k}}\wp}$ = this contribution has been discarded in (2),
$\hat{a}_{{\bm{k}}\wp}$ = in spite of the fact that the total ZPE
$\int_{{\mathbb{R}}^{d}}{\rm d}^{d}k\,\sum_{\wp}\,\hbar\omega_{k}/2$ is
infinite.
$\hat{\bm{A}}({\bm{y}})$ = the Coulomb gauge vector potential, given by its
standard modal expansion

$\hat{\bm{A}}({\bm{y}})\;=\;\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\frac{\hbar\,c^{2}}{\omega_{k}}}\;\,e^{+i{\bm{k}}\bm{\cdot}{\bm{y}}}\,\hat{a}_{{\bm{k}}\wp}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\;+\;{\rm
c.c.}\hskip 14.22636pt.$ (4)

${\bm{\varepsilon}}_{{\bm{k}}\wp}$ = unit polarization vector of the field
mode $({\bm{k}}\wp)$.
${\bm{\varepsilon}}_{{\bm{k}}\wp}$ = Vectors ${\bm{\kappa}}={\bm{k}}/k$ and
$\\{{\bm{\varepsilon}}_{{\bm{k}}\wp}\\}_{\wp}$ form an orthonormal basis set
of ${\mathbb{R}}^{d}$. Closure property

${\bm{\kappa}}\,{\bm{\kappa}}^{T}\;+\;\sum_{\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}^{T}\;=\;{\mathbb{I}}\hskip
14.22636pt.$ (5)

${\bm{\varepsilon}}_{{\bm{k}}\wp}$ = Here ${\mathbb{I}}$ stands of course for
the $d$-by-$d$ unit matrix.
Subsystems #1 and #2 put together:
When the subsystems #1 and #2 are mutually uncoupled (non-interacting),
the total Hamiltonian of our problem is of course just a sum

$\hat{H}_{\rm free}\;=\;\hat{H}_{{\rm A}}\;+\;\hat{H}_{{\rm R}}\hskip
14.22636pt.$ (6)

  * •

A.2 Introducing our model problem (coupling and related matters)
Interacting subsystems #1 and #2:
Nonrelativistic single electron atom coupled
to the quantum radiation field in the dipole approximation.
Schrödinger picture. The minimum coupling Hamiltonian

${\color[rgb]{0,0,1}\hat{H}_{t}\;=\;\frac{1}{2\,m_{o}^{t}(\alpha)}\left(\hat{\bm{p}}\,-\,\frac{q_{t}}{c}\int_{{\mathbb{R}}^{d}}\varrho({\bm{y}})\;\hat{\bm{A}}({\bm{y}})\;{\rm
d}^{d}y\right)^{2}+\;V(\hat{\bm{x}})\;+\;\hat{H}_{{\rm R}}\hskip 14.22636pt.}$
(7)

Explanations:
$q_{t}$ = electric charge of the electron, varying infinitely slowly
(adiabatically) in time.
$q_{t}$ = Vanishing in the infinite past, and reaching its physical value $q$
at finite times.
$q_{t}$ = Stated mathematically,

$\bm{\dot{}}{q}_{t}\;\equiv\;\frac{{\rm d}q_{t}}{{\rm d}t}\;\to\;0\hskip
14.22636pt;$ (8) $q_{t\to-\infty}\;=\;0\hskip 14.22636pt.$ (9)

$\varrho({\bm{y}})$ = charge density profile of the electron, we shall assume
an unit normalized Gaussian

$\varrho({\bm{y}})\;=\;\left(\frac{\alpha}{\pi}\right)^{\frac{d}{2}}e^{-\alpha{\bm{y}}^{2}}\hskip
14.22636pt.\hskip 14.22636pt[\,\alpha>0\,]$ (10)

$\alpha$ = positive real parameter serving as an UV regulator
$\alpha$ = (this terminology will be clarified shortly, see equation (13)
below).
$\alpha$ = $[\,$Later on we shall be concerned with the limit of
$\alpha\to+\infty$ for which $\varrho({\bm{y}})\to\delta^{d}({\bm{y}})$.
$\alpha$ = $[\,$The limit of $\alpha\to+\infty$ will be taken only after the
adiabatic limit concerning $q_{t}$.$\,]$
$m_{o}^{t}(\alpha)$ = bare mass of the electron,
$m_{o}^{t}(\alpha)$ = depending in an as yet unspecified manner upon the UV
regulator $\alpha$.
$m_{o}^{t}(\alpha)$ = Varying infinitely slowly (adiabatically) with time,
$m_{o}^{t}(\alpha)$ = again in an as yet unspecified manner,
$m_{o}^{t}(\alpha)$ = and reducing to the physical electron mass $m$ in the
infinite past.
$m_{o}^{t}(\alpha)$ = Stated mathematically,

$\bm{\dot{}}{m}_{o}^{t}(\alpha)\;\equiv\;\frac{{\rm
d}\,m_{o}^{t}(\alpha)}{{\rm d}t}\;\to\;0\hskip 14.22636pt;$ (11)
$m_{o}^{t\to-\infty}(\alpha)\;=\;m\hskip 14.22636pt.$ (12)

$m_{o}^{t}(\alpha)$ = $[\,$Later on, the adiabatic temporal variations of
$m_{o}^{t}(\alpha)$ will be linked to those of $q_{t}$,
$m_{o}^{t}(\alpha)$ = $[\,$and the $\alpha$-dependence of $m_{o}^{t}(\alpha)$
will be fixed by a renormalization prescription.
$m_{o}^{t}(\alpha)$ = $[\,$You may have a look at equations (87) and (91)
below.
$m_{o}^{t}(\alpha)$ = $[\,$Let us conveniently point out in advance
$m_{o}^{t}(\alpha)$ = $[\,$that the just mentioned renormalization
prescription is constructed in such a way
$m_{o}^{t}(\alpha)$ = $[\,$as to restore physical meaningfulness of our model
$m_{o}^{t}(\alpha)$ = $[\,$in the highly nontrivial limit of
$\alpha\to+\infty$.$\,]$
Direct calculation utilizing (10) and (4) yields explicitly

$\displaystyle\int_{{\mathbb{R}}^{d}}\varrho({\bm{y}})\;\hat{\bm{A}}({\bm{y}})\;{\rm
d}^{d}y$ $\displaystyle=$ $\displaystyle\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\frac{\hbar\,c^{2}}{\omega_{k}}}\;\,e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\,\hat{a}_{{\bm{k}}\wp}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\;+\;{\rm
c.c.}\hskip 14.22636pt.$ (13)

Formula (13) shows that the highly UV field modes
(i.e., those modes for which the factor $e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}$
is essentially zero)
are actually decoupled (cut off) from the electronic degrees of freedom.
This explains why is the parameter $\alpha$ named as an UV regulator.
The limit of $\alpha\to+\infty$ implies removing gradually the UV cutoff.
Note also that properties (9) and (12) imply having

$\hat{H}_{t\to-\infty}\;=\;\hat{H}_{\rm free}\;=\;(\ref{H-free-def})\hskip
14.22636pt.$ (14)

  * •

A.3 Why such a model system ?
The ultimate goal of these lecture notes is to provide an accessible
introduction to renormalization.
Our chosen model defined by the Hamiltonian (7) lends itself very well for
this purpose:

    * $\star$

The model is relatively simple, this enables us
to  pursue an analytic and non-perturbative treatment of mass renormalization,
without being distracted by tedious technical elaborations.
$[\,$Perturbative or numerical treatments become inevitable in fully
relativistic QED,
$[\,$as well as in other QFT theories, or in various theories of condensed
matter physics.
$[\,$Moreover, extremely tedious technicalities do emerge along the way
$[\,$both in QFT and in condensed matter physics.
$[\,$For a newcomer, it is often not easy to understand the essence of
renormalization
$[\,$when presentation of the subject is obscured by computational
technicalities.$\,]$

    * $\star$

The model is closely related to atomic physics, it can even be used
as the first approximation to explore a selected class of QED phenomena in
atomic systems
(such as QED theory of harmonic generation, not shown in these lecture notes).

B. Heisenberg picture quantum dynamics

  * •

B.1 Basic elaborations
We wish to derive the Heisenberg equations of motion for:

    * $\circ$

the electronic degrees of freedom $\hat{\bm{x}}(t)$, $\hat{\bm{p}}(t)$ ;

    * $\circ$

the field degrees of freedom $\hat{\bm{A}}(t,{\bm{y}})$, these are
equivalently represented by $\hat{a}_{{\bm{k}}\wp}(t)$.

Proceeding in the standard way, one gets

$\displaystyle\frac{{\rm d}}{{\rm d}t}\,\hat{\bm{x}}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{i\,\hbar}\,\left[\,\hat{\bm{x}}(t)\,,\hat{H}_{t}\,\right]\;=\;\frac{1}{m_{o}^{t}(\alpha)}\left(\hat{\bm{p}}(t)\,-\,\frac{q_{t}}{c}\int_{{\mathbb{R}}^{d}}\varrho({\bm{y}})\;\hat{\bm{A}}(t,{\bm{y}})\;{\rm
d}^{d}y\right)\hskip 14.22636pt;$ (15) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\,\hat{\bm{p}}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{i\,\hbar}\,\left[\,\hat{\bm{p}}(t)\,,\hat{H}_{t}\,\right]\;=\;-\,\bm{\nabla}\,V(\hat{\bm{x}}(t))\hskip
14.22636pt;$ (16) $\displaystyle{\color[rgb]{0,0,1}\frac{{\rm d}}{{\rm
d}t}\,\hat{a}_{{\bm{k}}\wp}(t)}$ $\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}\frac{1}{i\,\hbar}\,\left[\,\hat{a}_{{\bm{k}}\wp}(t)\,,\hat{H}_{t}\,\right]\;=\;-\,i\,\omega_{k}\;\hat{a}_{{\bm{k}}\wp}(t)\;+\;i\,\frac{q_{t}}{2\,\pi}\,\frac{1}{\sqrt{\hbar\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\,\Bigl{(}{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}\bm{\dot{}}{\hat{\bm{x}}}(t)\Bigr{)}\hskip
14.22636pt.}$ (17)

Here $\bm{\dot{}}{\hat{\bm{x}}}(t)=\frac{{\rm d}}{{\rm
d}t}\,\hat{\bm{x}}(t)=(\ref{Heisenberg-x})$.
In (15) one encounters the Heisenberg picture vector potential

$\hat{\bm{A}}(t,{\bm{y}})\;=\;\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\frac{\hbar\,c^{2}}{\omega_{k}}}\;\,e^{+i{\bm{k}}\bm{\cdot}{\bm{y}}}\,\hat{a}_{{\bm{k}}\wp}(t)\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\;+\;{\rm
c.c.}\hskip 14.22636pt.$ (18)

Different velocity components $\bm{\dot{}}{\hat{x}}_{j}(t)$ and
$\bm{\dot{}}{\hat{x}}_{j^{\prime}}(t)$ do not commute with each other for
$j\neq j^{\prime}$.
In (17) we have taken advantage of a general operator identity

$\Bigl{[}\,\hat{f},\hat{g}^{2}\,\Bigr{]}\;=\;\Bigl{\\{}\,\Bigl{[}\,\hat{f},\hat{g}\,\Bigr{]},\hat{g}\,\Bigr{\\}}\hskip
14.22636pt;$ (19)

and of the commutation property

$\left[\,\hat{a}_{{\bm{k}}\wp}(t)\,,\left(\hat{\bm{p}}(t)\,-\,\frac{q_{t}}{c}\int_{{\mathbb{R}}^{d}}\varrho({\bm{y}})\;\hat{\bm{A}}(t,{\bm{y}})\;{\rm
d}^{d}y\right)\,\right]\;=\;-\,\frac{q_{t}}{2\,\pi}\;\sqrt{\frac{\hbar}{\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\hskip
14.22636pt;$ (20)

which follows immediately from (3) and (13), (18).

  * •

B.2 The electric field
For our subsequent purposes,
it is convenient to introduce at this point the Heisenberg picture electric
field operator

$\hat{\bm{E}}(t,{\bm{y}})\;=\;-\,\frac{1}{c}\;\partial_{t}\,\hat{\bm{A}}(t,{\bm{y}})\hskip
14.22636pt.$ (21)

Let us determine explicitly the modal expansion of $\hat{\bm{E}}(t,{\bm{y}})$
by plugging into (21) the formulas (18) and (17). One obtains a preliminary
outcome

$\displaystyle\hat{\bm{E}}(t,{\bm{y}})$ $\displaystyle=$ $\displaystyle
i\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\hbar\,\omega_{k}}\;\,e^{+i{\bm{k}}\bm{\cdot}{\bm{y}}}\,\hat{a}_{{\bm{k}}\wp}(t)\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\;+\;{\rm
c.c.}$ $\displaystyle+$
$\displaystyle\frac{q_{t}}{2\,\pi^{2}}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\;\,\omega_{k}^{-1}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\,\sin\Bigl{(}{\bm{k}}\bm{\cdot}{\bm{y}}\Bigr{)}\,\sum_{\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\left({\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}\bm{\dot{}}{\hat{\bm{x}}}(t)\right)\hskip
14.22636pt.$

The closure property (5) enables us to rewrite the second line of (•) as

$\frac{q_{t}}{2\,\pi^{2}}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\;\,\omega_{k}^{-1}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\,\sin\Bigl{(}{\bm{k}}\bm{\cdot}{\bm{y}}\Bigr{)}\,\Bigl{(}{\mathbb{I}}\,-\,{\bm{\kappa}}\,{\bm{\kappa}}^{T}\Bigr{)}\,\bm{\dot{}}{\hat{\bm{x}}}(t)\hskip
14.22636pt.$ (23)

However, an integrand of (23) is an odd function of ${\bm{k}}$, hence the
whole term (23) vanishes for all ${\bm{y}}$.
Correspondingly, one has simply

$\displaystyle\hat{\bm{E}}(t,{\bm{y}})$ $\displaystyle=$ $\displaystyle
i\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\hbar\,\omega_{k}}\;\,e^{+i{\bm{k}}\bm{\cdot}{\bm{y}}}\,\hat{a}_{{\bm{k}}\wp}(t)\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\;+\;{\rm
c.c.}\hskip 14.22636pt.$ (24)

Coupling of the radiation field to our model atom enters inside (24) solely
through $\hat{a}_{{\bm{k}}\wp}(t)$.

  * •

B.3 Newton-Heisenberg equation of motion for $\hat{\bm{x}}(t)$
Now we are ready to write down the Newton-Heisenberg equation of motion for
$\hat{\bm{x}}(t)$.
Combination of (11), (15), (16), (21) yields immediately

${\color[rgb]{0,0,1}m_{o}^{t}(\alpha)\;\frac{{\rm d}^{2}}{{\rm
d}t^{2}}\,\hat{\bm{x}}(t)\;=\;-\,\bm{\nabla}V(\hat{\bm{x}}(t))\;+\;q_{t}\int_{{\mathbb{R}}^{d}}\varrho({\bm{y}})\;\hat{\bm{E}}(t,{\bm{y}})\;{\rm
d}^{d}y\hskip 14.22636pt.}$ (25)

The just derived Newton-Heisenberg equation of motion (25)
possesses an appearance which is much expected (based on physics intuition).
Indeed, on the r.h.s. of (25) one recognizes the scalar potential term and the
electric force term.
In passing we note that an additional elaboration (not shown here) would
provide
the Heisenberg-Maxwell wave equation for $\hat{\bm{A}}(t,{\bm{y}})$ or
$\hat{\bm{E}}(t,{\bm{y}})$. Such a wave equation
turns out to be inhomogeneous, containing a source term associated with
the electric current generated by motion of the electron.

  * •

B.4 An intermediate summary and additional reflections
Quantum dynamics of our studied system
(= model of a nonrelativistic atom coupled to the quantized radiation field)
is described by the following Heisenberg equations of motion:

    * Eq. (25) governing quantum dynamics of the electron;

    * Eq. (17) governing quantum dynamics of the radiation field.

Equations (25) & (17) are mutually coupled,
and need to be solved starting from certain properly specified initial
conditions:

$\hat{\bm{x}}(t_{0})\;=\;\hat{\bm{x}}\hskip 14.22636pt,\hskip
14.22636pt\bm{\dot{}}{\hat{\bm{x}}}(t_{0})\;=\;(\ref{Heisenberg-x})|_{t=t_{0}}\;=\;\frac{1}{m_{o}^{t_{0}}(\alpha)}\left(\hat{\bm{p}}(t_{0})\,-\,\frac{q_{t_{0}}}{c}\int_{{\mathbb{R}}^{d}}\varrho({\bm{y}})\;\hat{\bm{A}}(t_{0},{\bm{y}})\;{\rm
d}^{d}y\right)\hskip 14.22636pt;$ (26)

and

$\hat{a}_{{\bm{k}}\wp}(t_{0})\hskip 14.22636pt.$ (27)

Here $t_{0}$ stands for an arbitrary initial time instant.
Let us hereafter push $t_{0}\to-\infty$, taking advantage of the fact that
our model atom and the radiation field become mutually uncoupled in the
infinite past,
see above equations (9), (12), (14).
Consistently with property (14) we identify $\hat{\bm{x}}(t_{0}\to-\infty)$,
$\hat{\bm{p}}(t_{0}\to-\infty)$, and $\hat{a}_{{\bm{k}}\wp}(t_{0}\to-\infty)$
with the corresponding interaction picture operators. This amounts to set

$\displaystyle{\color[rgb]{0,0,1}\hat{\bm{x}}(t\to-\infty)}$
$\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}e^{+\frac{i}{\hbar}\hat{H}_{\rm
free}t}\;\hat{\bm{x}}\;e^{-\frac{i}{\hbar}\hat{H}_{\rm
free}t}\;=\;e^{+\frac{i}{\hbar}\hat{H}_{{\rm
A}}t}\;\hat{\bm{x}}\;e^{-\frac{i}{\hbar}\hat{H}_{{\rm A}}t}\hskip
14.22636pt;}$ (28)
$\displaystyle{\color[rgb]{0,0,1}\hat{\bm{p}}(t\to-\infty)}$
$\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}e^{+\frac{i}{\hbar}\hat{H}_{\rm
free}t}\;\hat{\bm{p}}\;e^{-\frac{i}{\hbar}\hat{H}_{\rm
free}t}\;=\;e^{+\frac{i}{\hbar}\hat{H}_{{\rm
A}}t}\;\hat{\bm{p}}\;e^{-\frac{i}{\hbar}\hat{H}_{{\rm A}}t}\hskip
14.22636pt\,;}$ (29)
$\displaystyle{\color[rgb]{0,0,1}\hat{a}_{{\bm{k}}\wp}(t\to-\infty)}$
$\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}e^{+\frac{i}{\hbar}\hat{H}_{\rm
free}t}\;\hat{a}_{{\bm{k}}\wp}\;e^{-\frac{i}{\hbar}\hat{H}_{\rm
free}t}\;=\;e^{+\frac{i}{\hbar}\hat{H}_{{\rm
R}}t}\;\hat{a}_{{\bm{k}}\wp}\;e^{-\frac{i}{\hbar}\hat{H}_{{\rm
R}}t}\;=\;\hat{a}_{{\bm{k}}\wp}\;e^{-i\omega t}\hskip 14.22636pt.}$ (30)

The r.h.s. of (28), (29), (30) contains of course
the Schrödinger picture operators $\hat{\bm{x}}$, $\hat{\bm{p}}$, and
$\hat{a}_{{\bm{k}}\wp}$ of subsection A.1.
Note also that

${\color[rgb]{0,0,1}m\,\bm{\dot{}}{\hat{\bm{x}}}(t\to-\infty)\;=\;\hat{\bm{p}}(t\to-\infty)\hskip
14.22636pt.}$ (31)

So far, so good.
All the elaborations presented above in Section B are absolutely standard,
conventionally looking, and yield intuitively plausible resulting formulas.
Recall, however, that our starting Hamiltonian (7) is parameterized by the UV
regulator $\alpha$
which controls the width of the charge density profile
$\varrho({\bm{y}})=(\ref{varrho-def})$ of the electron,
and which thus decouples the highly UV field modes from the electronic
motions.
Our ultimate interest consists in exploring physical properties of the studied
system
(= model of a nonrelativistic atom coupled to the quantized radiation field)
in the limit of $\alpha\to+\infty$, for which
$\varrho({\bm{y}})\to\delta^{d}({\bm{y}})$,
and for which thus all the UV field modes are coupled to the electronic
degrees of freedom in (7).
One would perhaps be tempted to anticipate at this point that the sought limit
should be trivial.
One would perhaps be tempted to set $m_{o}^{t}(\alpha)=m$
$[\,$i.e., to identify the bare mass $m_{o}^{t}(\alpha)$ with the physical
mass $m$ of the electron$\,]$,
to reduce (25) simply into

$m\;\frac{{\rm d}^{2}}{{\rm
d}t^{2}}\,\hat{\bm{x}}(t)\;=\;-\,\bm{\nabla}V(\hat{\bm{x}}(t))\;+\;q_{t}\;\hat{\bm{E}}(t,{\bm{0}})\hskip
14.22636pt;$ (32)

and to reduce (17) simply into

$\displaystyle\frac{{\rm d}}{{\rm d}t}\,\hat{a}_{{\bm{k}}\wp}(t)$
$\displaystyle=$
$\displaystyle-\,i\,\omega_{k}\;\hat{a}_{{\bm{k}}\wp}(t)\;+\;i\,\frac{q_{t}}{2\,\pi}\,\frac{1}{\sqrt{\hbar\omega_{k}}}\;\Bigl{(}{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}\bm{\dot{}}{\hat{\bm{x}}}(t)\Bigr{)}\hskip
14.22636pt.$ (33)

Most importantly, the just mentioned speculative anticipations turn out to be
totally wrong.
The limit of $\alpha\to+\infty$ is actually highly nontrivial.
An appropriate theoretical analysis of the limit of $\alpha\to+\infty$ gives
rise
to the story of mass renormalization and to the concept of Abraham-Lorentz
force.
The adventure is just beginning!

C. Electric field of the radiation reaction

  * •

C.1 Formal solution of equation (17)
Importantly, the field part of the studied problem lends itself to an analytic
treatment.
Indeed, the Heisenberg equation of motion (17) possesses the following formal
solution:

$\displaystyle\hat{a}_{{\bm{k}}\wp}(t)$ $\displaystyle=$
$\displaystyle\hat{a}_{{\bm{k}}\wp}\,e^{-i\omega_{k}t}$ $\displaystyle+$
$\displaystyle\frac{i}{2\,\pi}\,\frac{1}{\sqrt{\hbar\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\int_{-\infty}^{t}{\rm
d}t^{\prime}\;\,q_{t^{\prime}}\;e^{-i\omega_{k}(t-t^{\prime})}\,\Bigl{(}{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})\Bigr{)}\hskip
14.22636pt.$

The initial condition
$\hat{a}_{{\bm{k}}\wp}(t\to-\infty)=\hat{a}_{{\bm{k}}\wp}\;e^{-i\omega t}$ of
equation (30) has been incorporated, cf. also (9).

  * •

C.2 The electric field
Having in hand formula (•), we may plug it into (24)
and evaluate explicitly the corresponding Heisenberg picture electric field
$\hat{\bm{E}}(t,{\bm{y}})$.
One has

$\hat{\bm{E}}(t,{\bm{y}})\;=\;\hat{\bm{E}}_{0}(t,{\bm{y}})\;+\;\hat{\bm{E}}_{\rm
RR}(t,{\bm{y}})\hskip 14.22636pt;$ (35)

where by definition

$\displaystyle\hat{\bm{E}}_{0}(t,{\bm{y}})$ $\displaystyle=$ $\displaystyle
i\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\hbar\,\omega_{k}}\;\,e^{-i(\omega_{k}t-{\bm{k}}\bm{\cdot}{\bm{y}})}\,\hat{a}_{{\bm{k}}\wp}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\;+\;{\rm
c.c.}\hskip 7.11317pt;$ (36)

and

$\displaystyle\hat{\bm{E}}_{\rm RR}(t,{\bm{y}})$ $\displaystyle=$
$\displaystyle-\,\frac{1}{2\,\pi^{2}}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\;q_{t^{\prime}}\,\cos\Bigl{(}\omega_{k}(t-t^{\prime})-{\bm{k}}\bm{\cdot}{\bm{y}}\Bigr{)}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\,\sum_{\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\left({\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})\right)\hskip
7.11317pt.$ (37)

The closure property (5) implies

$\displaystyle\hat{\bm{E}}_{\rm RR}(t,{\bm{y}})$ $\displaystyle=$
$\displaystyle-\,\frac{1}{2\,\pi^{2}}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\;q_{t^{\prime}}\,\cos\Bigl{(}\omega_{k}(t-t^{\prime})-{\bm{k}}\bm{\cdot}{\bm{y}}\Bigr{)}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\,\Bigl{(}{\mathbb{I}}\,-\,{\bm{\kappa}}\,{\bm{\kappa}}^{T}\Bigr{)}\,\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})\hskip
7.11317pt.$ (38)

Physical interpretation of the sum (35):
$\hat{\bm{E}}_{0}(t,{\bm{y}})=(\ref{hat-E-0-t-y-def})$ is just the interaction
picture counterpart of the Heisenberg electric field (24).
Meaning that $\hat{\bm{E}}_{0}(t,{\bm{y}})$ evolves in time freely,
as if there was no charged particle (electron) generating a current.
$\hat{\bm{E}}_{\rm RR}(t,{\bm{y}})=(\ref{hat-E-RR-t-y-def-T})$ corresponds to
the  electric field of the radiation reaction (RR).
Meaning that $\hat{\bm{E}}_{\rm RR}(t,{\bm{y}})$ describes the radiation
which has been emitted/absorbed by our charged particle (electron) during its
motion
at the time instants prior to $t$.

  * •

C.3 Equation of motion for the electron revisited
Let us return to the Newton-Heisenberg equation of motion (25) for the
electron.
After incorporating the splitting (35) one gets

$m_{o}^{t}(\alpha)\;\frac{{\rm d}^{2}}{{\rm
d}t^{2}}\,\hat{\bm{x}}(t)\;=\;-\,\bm{\nabla}V(\hat{\bm{x}}(t))\;+\;\hat{\bm{F}}_{0}^{\rm
el}(t)\;+\;\hat{\bm{F}}_{\rm RR}^{\rm el}(t)\hskip 14.22636pt;$ (39)

where by definition

$\hat{\bm{F}}_{0}^{\rm
el}(t)\;=\;q_{t}\int_{{\mathbb{R}}^{d}}\varrho({\bm{y}})\;\hat{\bm{E}}_{0}(t,{\bm{y}})\;{\rm
d}^{d}y\hskip 14.22636pt;$ (40)

and

$\hat{\bm{F}}_{\rm RR}^{\rm
el}(t)\;=\;q_{t}\int_{{\mathbb{R}}^{d}}\varrho({\bm{y}})\;\hat{\bm{E}}_{\rm
RR}(t,{\bm{y}})\;{\rm d}^{d}y\hskip 14.22636pt.$ (41)

Entity $\hat{\bm{F}}_{0}^{\rm el}(t)=(\ref{electric-force-E0})$ represents the
interaction picture contribution
to the electric force acting on the electron. Formulas (36) and (10) imply
having

$\hat{\bm{F}}_{0}^{\rm el}(t)\;=\;q_{t}\;\,i\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\hbar\,\omega_{k}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\,e^{-i\omega_{k}t}\,\hat{a}_{{\bm{k}}\wp}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\;+\;{\rm
c.c.}\hskip 14.22636pt;$ (42)

cf. similar integration performed in equation (13). If so, then

$\hat{\bm{F}}_{0}^{\rm
el}(t)\;=\;q_{t}\;\hat{\bm{E}}_{0}(t,{\bm{0}})\;+\;{\cal O}(\alpha^{-1})\hskip
14.22636pt;$ (43)

with the ${\cal O}(\alpha^{-1})$ correction becoming insignificant for large
enough $\alpha$.
Note that $\hat{\bm{E}}_{0}(t,{\bm{0}})=(\ref{hat-E-0-t-y-
def})|_{{\bm{y}}={\bm{0}}}$ is a well defined interaction picture operator
whose time evolution is explicitly known and independent upon
$\hat{\bm{x}}(t)$.
Entity $\hat{\bm{F}}_{\rm RR}^{\rm el}(t)=(\ref{electric-force-ERR})$
represents the RR contribution
to the Heisenberg picture electric force acting on the electron.
After plugging (38) into (41) one finds that

$\hat{\bm{F}}_{\rm RR}^{\rm
el}(t)\;=\;-\,\frac{q_{t}}{2\,\pi^{2}}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\;q_{t^{\prime}}\,\cos\Bigl{(}\omega_{k}(t-t^{\prime})\Bigr{)}\;e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\Bigl{(}{\mathbb{I}}\,-\,{\bm{\kappa}}\,{\bm{\kappa}}^{T}\Bigr{)}\,\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})\hskip
14.22636pt.$ (44)

The r.h.s. of our Newton-Heisenberg equation of motion (39) for the electron
is now fully specified.
Our next step will consist in evaluating explicitly the electric RR force
$\hat{\bm{F}}_{\rm RR}^{\rm el}(t)=(\ref{electric-force-ERR-T})$
while assuming that the UV regulator $\alpha$ approaches its limiting value
$\alpha\to+\infty$.
This amounts to integrate analytically both over ${\bm{k}}\in{\mathbb{R}}^{d}$
and over $t^{\prime}\in(-\infty,t)$
in the regime of $\alpha\to+\infty$.
Since both technical details and the ultimate outcome
depend crucially upon the spatial dimension $d$ of the problem,
it is inevitable to discuss the cases of $d=2$ and $d=3$ separately,
this is done in the two subsequent paragraphs.
We shall begin with the $d=3$ case which is simpler to deal with than the
$d=2$ case.

  * •

C.4 An explicit evaluation of the electric RR force for $\bm{d}\bm{=}\bm{3}$
Integration over ${\bm{k}}\in{\mathbb{R}}^{3}$ is conveniently performed in
the spherical polar coordinates
$(k,\varphi_{\bm{\kappa}},\vartheta_{\bm{\kappa}})$.
Geometric identity

$\int_{0}^{2\pi}{\rm
d}\varphi_{\bm{\kappa}}\,\int_{0}^{\pi}\sin\vartheta_{\bm{\kappa}}\;{\rm
d}\vartheta_{\bm{\kappa}}\;\,{\bm{\kappa}}\,{\bm{\kappa}}^{T}\;=\;\frac{4\,\pi}{3}\;{\mathbb{I}}$
(45)

converts equation (44) into

$\displaystyle\hat{\bm{F}}_{\rm RR}^{\rm
el}(t)\;=\;-\,\frac{4}{3}\,\frac{q_{t}}{\pi}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\int_{0}^{\infty}k^{2}\,{\rm
d}k\;\cos\Bigl{(}kc(t-t^{\prime})\Bigr{)}\;e^{-\frac{k^{2}}{2\,\alpha}}\;q_{t^{\prime}}\,\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})\hskip
14.22636pt.$ (46)

Define now conveniently an auxiliary function

$\rho_{\alpha}\Bigl{(}c(t-t^{\prime})\Bigr{)}\;=\;\frac{1}{\pi}\,\int_{0}^{\infty}{\rm
d}k\;\cos\Bigl{(}kc(t-t^{\prime})\Bigr{)}\;e^{-\frac{k^{2}}{2\,\alpha}}\;=\;\sqrt{\frac{\alpha}{2\,\pi}}\;e^{-\frac{\alpha}{2}\bigl{(}c(t-t^{\prime})\bigr{)}^{2}}\hskip
14.22636pt.$ (47)

$[\,$Note that $\rho_{\alpha}(c(t-t^{\prime}))$ approaches
$\delta(c(t-t^{\prime}))$ in the limit of $\alpha\to+\infty$.$\,]$
Formula (46) simplifies accordingly, we have

$\displaystyle\hat{\bm{F}}_{{\rm RR}}^{\rm el}(t)$ $\displaystyle=$
$\displaystyle\frac{4}{3}\,\frac{q_{t}}{c^{2}}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\;\Bigl{(}\partial_{t^{\prime}t^{\prime}}\,\rho_{\alpha}(c(t-t^{\prime}))\Bigr{)}\;q_{t^{\prime}}\,\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})\hskip
14.22636pt.$ (48)

In the regime of $\alpha\to+\infty$, the term
$\Bigl{(}\partial_{t^{\prime}t^{\prime}}\,\rho_{\alpha}(c(t-t^{\prime}))\Bigr{)}$
falls off rapidly to zero
unless $t^{\prime}$ is very close to $t$. This observation enables us to
proceed further:

    * $*$

The adiabatic switching property (8) enables us to replace $q_{t^{\prime}}$ by
$q_{t}$ in (48).

    * $*$

The electron velocity $\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})$ can be expanded
into power series around $t^{\prime}=t$.
Such that

$\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})\;=\;\bm{\dot{}}{\hat{\bm{x}}}(t)\;+\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\,(t^{\prime}-t)\;+\;\frac{1}{2}\,\dddot{\hat{\bm{x}}}(t)\,(t^{\prime}-t)^{2}\;+\;{\cal
O}\Bigl{(}(t^{\prime}-t)^{3}\Bigr{)}\hskip 14.22636pt.$ (49)

Equation (48) becomes thus

$\displaystyle\hat{\bm{F}}_{{\rm RR}}^{\rm el}(t)$ $\displaystyle=$
$\displaystyle\frac{4}{3}\,\frac{q_{t}^{2}}{c^{2}}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\;\Bigl{(}\partial_{t^{\prime}t^{\prime}}\,\rho_{\alpha}(c(t-t^{\prime}))\Bigr{)}$
(50)
$\displaystyle\left\\{\bm{\dot{}}{\hat{\bm{x}}}(t)\;+\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\,(t^{\prime}-t)\;+\;\frac{1}{2}\,\dddot{\hat{\bm{x}}}(t)\,(t^{\prime}-t)^{2}\;+\;{\cal
O}\Bigl{(}(t^{\prime}-t)^{3}\Bigr{)}\right\\}\;=$ $\displaystyle=$
$\displaystyle\frac{4}{3}\,\frac{q_{t}^{2}}{c^{2}}\,\int_{0}^{+\infty}{\rm
d}\tau\;\Bigl{(}\partial_{\tau\tau}\,\rho_{\alpha}(c\tau)\Bigr{)}\;\left\\{\bm{\dot{}}{\hat{\bm{x}}}(t)\;-\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\;\tau\;+\;\frac{1}{2}\,\dddot{\hat{\bm{x}}}(t)\;\tau^{2}\;+\;{\cal
O}(\tau^{3})\right\\}\hskip 14.22636pt.$

Recall that $\rho_{\alpha}(c\tau)$ is given by above prescription (47).
The integrals
$\int_{0}^{+\infty}\tau^{n}\,\partial_{\tau\tau}\,\rho_{\alpha}(c\tau)\;{\rm
d}\tau$ are doable analytically for all $n\in\\{0,1,2,3,\bm{\cdots}\\}$.
One gets a sequence of outcomes

$\displaystyle\int_{0}^{+\infty}{\rm
d}\tau\;\Bigl{(}\partial_{\tau\tau}\,\rho_{\alpha}(c\tau)\Bigr{)}\;\tau^{0}$
$\displaystyle=$ $\displaystyle 0$ (51) $\displaystyle\int_{0}^{+\infty}{\rm
d}\tau\;\Bigl{(}\partial_{\tau\tau}\,\rho_{\alpha}(c\tau)\Bigr{)}\;\tau^{1}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{\alpha}{2\,\pi}}$ (52)
$\displaystyle\int_{0}^{+\infty}{\rm
d}\tau\;\Bigl{(}\partial_{\tau\tau}\,\rho_{\alpha}(c\tau)\Bigr{)}\;\tau^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{c}$ (53)
$\displaystyle\int_{0}^{+\infty}{\rm
d}\tau\;\Bigl{(}\partial_{\tau\tau}\,\rho_{\alpha}(c\tau)\Bigr{)}\;\tau^{3}$
$\displaystyle=$ $\displaystyle\frac{3}{c^{2}}\,\sqrt{\frac{2}{\pi\,\alpha}}$
(54) $\displaystyle\bm{\vdots}\hskip 56.9055pt$ $\displaystyle\hskip
28.45274pt\bm{\vdots}$

Correspondingly, our electric RR force (50) takes the following final
appearance,

$\displaystyle{\color[rgb]{0,0,1}\hat{\bm{F}}_{{\rm RR}}^{\rm el}(t)}$
$\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}-\,\frac{4}{3}\,\frac{q_{t}^{2}}{c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\;+\;\frac{2}{3}\,\frac{q_{t}^{2}}{c^{3}}\,\dddot{\hat{\bm{x}}}(t)\;+\;{\cal
O}(\alpha^{-1/2})\hskip 14.22636pt.\hskip 14.22636pt[\,d=3\,]}$ (55)

Most importantly, the first term on the r.h.s. of (55) is proportional to
$\alpha^{+1/2}$.
Hence our just evaluated electric RR force $\hat{\bm{F}}_{{\rm RR}}^{\rm
el}(t)$
possesses an infinite magnitude in the limit of $\alpha\to+\infty$.
This highly unsettling conclusion is bringing us already to the essence of
renormalization.
We will carefully analyze the implications of formula (55) later on in Section
D.

  * •

C.5 An explicit evaluation of the electric RR force for $\bm{d}\bm{=}\bm{2}$
Integration over ${\bm{k}}\in{\mathbb{R}}^{2}$ is conveniently performed in
the polar coordinates $(k,\varphi_{\bm{\kappa}})$.
Geometric identity

$\int_{0}^{2\pi}{\rm
d}\varphi_{\bm{\kappa}}\;\,{\bm{\kappa}}\,{\bm{\kappa}}^{T}\;=\;\pi\;{\mathbb{I}}$
(56)

converts equation (44) into

$\hat{\bm{F}}_{\rm RR}^{\rm
el}(t)\;=\;-\,\frac{q_{t}}{2\,\pi}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\int_{0}^{\infty}k\;{\rm
d}k\;\cos\Bigl{(}kc(t-t^{\prime})\Bigr{)}\;e^{-\frac{k^{2}}{2\,\alpha}}\;q_{t^{\prime}}\,\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})\hskip
14.22636pt.$ (57)

Define now conveniently an auxiliary function

$\Xi_{\alpha}\Bigl{(}c(t-t^{\prime})\Bigr{)}\;=\;\int_{0}^{\infty}{\rm
d}k\;\sin\Bigl{(}kc(t-t^{\prime})\Bigr{)}\;e^{-\frac{k^{2}}{2\,\alpha}}\hskip
14.22636pt.$ (58)

$[\,$Note that $\Xi_{\alpha}(c(t-t^{\prime}))=(\ref{Xi-alpha-def})$ can be
defined even for any negative $c(t-t^{\prime})$.$\,]$
Formula (57) simplifies accordingly, we have

$\hat{\bm{F}}_{\rm RR}^{\rm
el}(t)\;=\;\frac{q_{t}}{2\,\pi\,c}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\,\Bigl{(}\partial_{t^{\prime}}\,\Xi_{\alpha}(c(t-t^{\prime}))\Bigr{)}\,q_{t^{\prime}}\,\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})\hskip
14.22636pt.$ (59)

Proceeding further, we integrate in (59) by parts,
while taking advantage of properties (8)-(9), and while noting also that
$\Xi_{\alpha}(0)=0$.
One gets an intermediate outcome

$\hat{\bm{F}}_{\rm RR}^{\rm
el}(t)\;=\;-\,\frac{q_{t}}{2\,\pi\,c}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\;\,\Xi_{\alpha}(c(t-t^{\prime}))\;q_{t^{\prime}}\,\bm{\ddot{}}{\hat{\bm{x}}}(t^{\prime})\hskip
14.22636pt.$ (60)

Next, let us make a small digression, and gather two auxiliary insights:

    * 1)

Direct calculation yields explicitly

$\Xi_{\alpha}(c(t-t^{\prime}))\;=\;-i\;\sqrt{\frac{\pi\alpha}{2}}\;\,{\rm
erf}\left(i\,\sqrt{\frac{\alpha}{2}}\;c(t-t^{\prime})\right)\,e^{-\frac{\alpha}{2}(c(t-t^{\prime}))^{2}}\hskip
14.22636pt.$ (61)

We shall actually not use the formula (61) directly in our mainstream
considerations.
However, we shall notice that

$\Xi_{\alpha}(c(t-t^{\prime}))\;=\;\frac{1}{c(t-t^{\prime})}\;+\;{\cal
O}(\alpha^{-1})\hskip 14.22636pt;$ (62)

valid as long as $c(t-t^{\prime})\neq 0$.
Convergence of $\Xi_{\alpha}(c(t-t^{\prime}))=(\ref{f-alpha-take-2})$ to its
leading order approximation $\frac{1}{c(t-t^{\prime})}$ is not uniform.
$[\,$This can be seen most easily by plotting
$\Xi_{\alpha}(c(t-t^{\prime}))=(\ref{f-alpha-take-2})$ for some fixed finite
$\alpha$.$\,]$

    * 2)

Our massive charged particle (electron) possesses the Compton wavelength

$\lambda\;=\;\frac{\hbar}{mc}\hskip 14.22636pt.$ (63)

Here $m$ stands for the physical mass,
and not for the bare mass $m_{o}^{t}(\alpha)$ appearing in the Hamiltonian
$\hat{H}=(\ref{H})$.
The associated Compton time equals then to

$\tilde{t}\;=\;\frac{\lambda}{c}\;=\;\frac{\hbar}{mc^{2}}\hskip 14.22636pt.$
(64)

In addition, let $\eta>0$ be any dimensionless constant. We define for
convenience a cutoff time

$t_{c}\;=\;\eta\,\tilde{t}\hskip 14.22636pt.$ (65)

Now we are ready to return to (60) and set

${\cal
F}_{\alpha}(t-t^{\prime})\;=\;\int_{t_{c}}^{t-t^{\prime}}\Xi_{\alpha}(c\tau)\;{\rm
d}\tau\hskip 14.22636pt;$ (66)

such that

$-\,\partial_{t^{\prime}}\,{\cal
F}_{\alpha}(t-t^{\prime})\;=\;\Xi_{\alpha}(c(t-t^{\prime}))\hskip 14.22636pt.$
(67)

The electric RR force $\hat{\bm{F}}_{\rm RR}^{\rm el}(t)=(\ref{electric-force-
ERR-T-take-4-d=2})$ is subsequently transformed into

$\displaystyle\hat{\bm{F}}_{\rm RR}^{\rm el}(t)$ $\displaystyle=$
$\displaystyle\frac{q_{t}}{2\,\pi\,c}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\,q_{t^{\prime}}\,\bm{\ddot{}}{\hat{\bm{x}}}(t^{\prime})\;\partial_{t^{\prime}}\,{\cal
F}_{\alpha}(t-t^{\prime})\;=$ (68) $\displaystyle=$
$\displaystyle\frac{q_{t}^{2}}{2\,\pi\,c}\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\,{\cal
F}_{\alpha}(0)\;-\;\frac{q_{t}}{2\,\pi\,c}\,\int_{-\infty}^{t}{\rm
d}t^{\prime}\,q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;{\cal
F}_{\alpha}(t-t^{\prime})\hskip 14.22636pt.$

In the second line we have integrated by parts while exploiting (8)-(9).
For large values of $\alpha$, one can take advantage of an expansion (62), and
write correspondingly

${\cal
F}_{\alpha}(t-t^{\prime})\;=\;\frac{1}{c}\,\ln\left(\frac{t-t^{\prime}}{t_{c}}\right)\;+\;{\cal
O}(\alpha^{-1})\hskip 14.22636pt.\hskip 14.22636pt[\,(t-t^{\prime})>0\,]$ (69)

Formula (69) enables us to write down another important intermediate outcome

$\displaystyle\hat{\bm{F}}_{\rm RR}^{\rm el}(t)$ $\displaystyle=$
$\displaystyle\frac{q_{t}^{2}}{2\,\pi\,c}\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\,{\cal
F}_{\alpha}(0)\;-\;\frac{q_{t}}{2\,\pi\,c^{2}}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\,\ln\left(\frac{t-t^{\prime}}{t_{c}}\right){\rm
d}t^{\prime}$ $\displaystyle+$ $\displaystyle{\rm an}\;\,\alpha{\rm-
dependent\;\,correction\;\,vanishing\;\,as}\;\,\alpha\to+\infty\hskip
14.22636pt\hskip 14.22636pt.$

Observe that the second (logarithmic) term of (•) is independent upon the UV
regulator $\alpha$.
The reader may be puzzled here by the singularities of $\ln(t-t^{\prime})$
which occur at $t^{\prime}\to t$ and $t^{\prime}\to-\infty$.
Fortunately, the just mentioned singularities are anticipated to be benign:
mzStated briefly, the $t^{\prime}\to t$ singularity is weak enough to be
integrable,
mzas long as the fourth time derivative $\ddddot{\hat{\bm{x}}}(t^{\prime})$
remains bounded in magnitude.
mzFurthermore, the slow divergence of $\ln(t-t^{\prime})$ at
$t^{\prime}\to-\infty$ is overcompensated
mzby an oscillatory behavior of $\dddot{\hat{\bm{x}}}(t^{\prime})$. Time
oscillations of $\bm{\dot{}}{\hat{\bm{x}}}(t^{\prime})$,
$\bm{\ddot{}}{\hat{\bm{x}}}(t^{\prime})$, $\dddot{\hat{\bm{x}}}(t^{\prime})$
etc. do take place,
mzsince in our model the electron moves in a restricted spatial region
mzdetermined by a strictly bound potential $V({\bm{x}})$, see A.1.
mzA more careful mathematical treatment of the just discussed singularities of
$\ln(t-t^{\prime})$
mzis relegated to Appendix A.
In the second line of (•) we wrote just ”an $\alpha-$dependent correction
vanishing as $\alpha\to+\infty$”,
this is sufficient for our purposes.
$[\,$Writing hastily ${\cal O}(\alpha^{-1})$ seems to be unsafe, since in
equation (69)
$[\,$the convergence of ${\cal F}_{\alpha}(t-t^{\prime})$ to its leading order
approximation $\frac{1}{c}\,\ln\left(\frac{t-t^{\prime}}{t_{c}}\right)$ is not
uniform.$\,]$
Proceeding further in our elaborations, we shall focus on analyzing the first
term of equation (•),
or, equivalently, the quantity

${\cal F}_{\alpha}(0)\;=\;\int_{t_{c}}^{0}\Xi_{\alpha}(c\tau)\;{\rm
d}\tau\;=\;\int_{t_{c}}^{0}{\rm d}\tau\int_{0}^{+\infty}{\rm
d}k\;\sin\Bigl{(}kc\tau\Bigr{)}\;e^{-\frac{k^{2}}{2\,\alpha}}\hskip
14.22636pt.$ (71)

An integration over $\tau$ can be performed first, this yields

${\cal F}_{\alpha}(0)\;=\;\int_{0}^{+\infty}{\rm
d}k\;\,e^{-\frac{k^{2}}{2\,\alpha}}\;\frac{\Bigl{(}\cos(k\,c\,t_{c})\,-\,1\Bigr{)}}{kc}\hskip
14.22636pt.$ (72)

The integral (72) obviously diverges to $-\infty$ for $\alpha\to+\infty$.
A more explicit insight is gained by calculating the $\alpha$-derivative

$\displaystyle\frac{{\rm d}}{{\rm d}\alpha}\,{\cal F}_{\alpha}(0)$
$\displaystyle=$ $\displaystyle\int_{0}^{+\infty}{\rm
d}k\;\,e^{-\frac{k^{2}}{2\,\alpha}}\;\frac{k^{2}}{2\,\alpha^{2}}\;\frac{\Bigl{(}\cos(k\,c\,t_{c})\,-\,1\Bigr{)}}{kc}\;=$
(73) $\displaystyle=$
$\displaystyle\frac{i\,t_{c}}{2}\,\sqrt{\frac{\pi}{2\,\alpha}}\;\,{\rm
erf}\left(i\,\sqrt{\frac{\alpha}{2}}\,(c\,t_{c})\right)e^{-\frac{\alpha}{2}(ct_{c})^{2}}\;=$
$\displaystyle=$
$\displaystyle-\,\frac{1}{2\,c\,\alpha}\;-\;\frac{1}{2\,c^{3}\,t_{c}^{2}\,\alpha^{2}}\;+\;{\cal
O}(\alpha^{-3})\hskip 14.22636pt.$

Formula (73) implies that, inevitably,

${\cal
F}_{\alpha}(0)\;=\;-\,\frac{1}{2\,c}\,\ln\alpha\;+\;\zeta(\eta)\;+\;\frac{1}{2\,c^{3}\,t_{c}^{2}\,\alpha}\;+\;{\cal
O}(\alpha^{-2})\hskip 14.22636pt.$ (74)

Here $\zeta(\eta)$ stands for an as yet undetermined finite constant, which is
equal formally to

$\displaystyle\zeta(\eta)$ $\displaystyle=$
$\displaystyle\lim_{\alpha\to+\infty}\left\\{{\cal
F}_{\alpha}(0)\;+\;\frac{1}{2\,c}\,\ln\alpha\right\\}\;=$ (75)
$\displaystyle=$
$\displaystyle\lim_{\alpha\to+\infty}\,\left\\{\int_{0}^{+\infty}{\rm
d}k\;\,e^{-\frac{k^{2}}{2\,\alpha}}\;\frac{\Bigl{(}\cos(k\,c\,\eta\,\tilde{t}\,)\,-\,1\Bigr{)}}{kc}\;+\;\frac{1}{2\,c}\,\ln\alpha\right\\}\hskip
14.22636pt.$

The dependence of $\zeta(\eta)$ upon $\eta$ can be found via differentiating
(75) with respect to $\eta$.
One gets

$\frac{{\rm d}\zeta(\eta)}{{\rm
d}\eta}\;=\;-\,\tilde{t}\,\lim_{\alpha\to+\infty}\,\Xi_{\alpha}(c\,t_{c})\;=\;\frac{(-1)}{\eta\,c}\hskip
14.22636pt.$ (76)

In (76) we have exploited the definition (58) and the asymptotic behavior
(62).
Relation (76) reveals having simply

$\zeta(\eta)\;=\;\zeta\;-\;\frac{1}{c}\,\ln\eta\hskip 14.22636pt;$ (77)

with

$\zeta\;\equiv\;\zeta(1)\;=\;\lim_{\alpha\to+\infty}\,\left\\{\int_{0}^{+\infty}{\rm
d}k\;\,e^{-\frac{k^{2}}{2\,\alpha}}\;\frac{\Bigl{(}\cos\left(\frac{\hbar
k}{mc}\right)\,-\,1\Bigr{)}}{kc}\;+\;\frac{1}{2\,c}\,\ln\alpha\right\\}\hskip
14.22636pt.$ (78)

For our present purposes, it is not necessary to work out a more explicit
prescription for $\zeta=(\ref{zeta-def})$.
It is sufficient to regard $\zeta$ as some well defined finite number
depending upon $(m,c,\hbar)$.
After plugging (77) into (74) one arrives towards a finalized
$\alpha$-expansion of ${\cal F}_{\alpha}(0)$, namely,

${\cal
F}_{\alpha}(0)\;=\;-\,\frac{1}{2\,c}\,\ln\Bigl{(}\eta^{2}\alpha\Bigr{)}\;+\;\zeta\;+\;\frac{1}{2\,c^{3}\,\eta^{2}\,\tilde{t}^{2}\,\alpha}\;+\;{\cal
O}(\alpha^{-2})\hskip 14.22636pt.$ (79)

Our sought RR force $\hat{\bm{F}}_{\rm RR}^{\rm el}(t)=(\ref{F-RR-analysis-
take-4})$ equals then to

$\displaystyle\hat{\bm{F}}_{\rm RR}^{\rm el}(t)$ $\displaystyle=$
$\displaystyle-\,\frac{q_{t}^{2}}{4\,\pi\,c^{2}}\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\,\ln\alpha\;-\;\frac{q_{t}^{2}}{2\,\pi\,c^{2}}\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\,\ln\eta\;+\;\frac{q_{t}^{2}}{2\,\pi\,c}\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\,\zeta$
$\displaystyle-$
$\displaystyle\frac{q_{t}}{2\,\pi\,c^{2}}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;\ln\left(\frac{t-t^{\prime}}{\eta\,\tilde{t}}\right){\rm
d}t^{\prime}$ $\displaystyle+$ $\displaystyle{\rm an}\;\,\alpha{\rm-
dependent\;\,correction\;\,vanishing\;\,as}\;\,\alpha\to+\infty\hskip
14.22636pt.$

Additionally, an identity

$\displaystyle-\;\frac{q_{t}}{2\,\pi\,c^{2}}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;\ln\left(\frac{t-t^{\prime}}{\eta\,\tilde{t}}\right){\rm
d}t^{\prime}\;=$ (81) $\displaystyle=$
$\displaystyle-\;\frac{q_{t}}{2\,\pi\,c^{2}}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;\ln\left(\frac{t-t^{\prime}}{\tilde{t}}\right){\rm
d}t^{\prime}\;+\;\frac{q_{t}^{2}}{2\,\pi\,c^{2}}\,\bm{\ddot{}}{\hat{\bm{x}}}(t^{\prime})\,\ln\eta$

shows that the two terms proportional to $\ln\eta$ cancel each other in (•).
Thereby our electric RR force $\hat{\bm{F}}_{\rm RR}^{\rm el}(t)$ turns out to
be independent upon $\eta$,
and takes the final, simple, compelling appearance

$\displaystyle{\color[rgb]{0,0,1}\hat{\bm{F}}_{\rm RR}^{\rm el}(t)}$
$\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}-\,\frac{q_{t}^{2}}{4\,\pi\,c^{2}}\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\,\ln\left(\frac{\alpha}{\alpha_{0}}\right)\;-\;\frac{q_{t}}{2\,\pi\,c^{2}}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;\ln\left(\frac{t-t^{\prime}}{\tilde{t}}\right){\rm
d}t^{\prime}}$ $\displaystyle{\color[rgb]{0,0,1}+}$
$\displaystyle{\color[rgb]{0,0,1}{\rm an}\;\,\alpha{\rm-
dependent\;\,correction\;\,vanishing\;\,as}\;\,\alpha\to+\infty\hskip
56.9055pt.}$

We have conveniently set here

$\zeta\;=\;\frac{1}{2\,c}\,\ln\alpha_{0}\hskip 14.22636pt,\hskip
14.22636pt{\rm i.e.}\hskip 14.22636pt\alpha_{0}\;=\;e^{2\,c\,\zeta}\hskip
14.22636pt.$ (83)

The passage from (57) to (•) was painfully technical and tedious, but
certainly worthy of an effort.
One may observe that the just derived electric RR force $\hat{\bm{F}}_{\rm
RR}^{\rm el}(t)=(\ref{F-RR-analysis-take-6})$ is given as a sum over:
i)ii a local term proportional to $\bm{\ddot{}}{\hat{\bm{x}}}(t)$,  whose
prefactor diverges logarithmically with $\alpha\to+\infty$;
ii)i an $\alpha$-independent nonlocal term, contanining
$\dddot{\hat{\bm{x}}}(t^{\prime})$ $[\,t^{\prime}<t\,]$ and a memory kernel
$\ln\left(\frac{t-t^{\prime}}{\tilde{t}}\right)$;
iii) other contributions which however fall off to zero as $\alpha\to+\infty$.
Due to i), our just evaluated electric RR force $\hat{\bm{F}}_{{\rm RR}}^{\rm
el}(t)$
possesses an infinite magnitude in the limit of $\alpha\to+\infty$.
This highly unsettling conclusion
is bringing us already to the essence of renormalization, much as in C.4.
We will carefully analyze the implications of formula (•) later on in Section
D.

  * •

C.6 Equation of motion for the electron revisited again
Let us return to the Newton-Heisenberg equation of motion (39) for the
electron,
and incorporate our above derived expressions (43) and (55) or (•)
which determine the electric forces $\hat{\bm{F}}_{0}^{\rm el}(t)$ and
$\hat{\bm{F}}_{\rm RR}^{\rm el}(t)$.
In the case of $d=2$, one gets explicitly

$\displaystyle{\color[rgb]{0,0,1}m_{o}^{t}(\alpha)\;\frac{{\rm d}^{2}}{{\rm
d}t^{2}}\,\hat{\bm{x}}(t)}$ $\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}-\,\bm{\nabla}V(\hat{\bm{x}}(t))\;+\;q_{t}\;\hat{\bm{E}}_{0}(t,{\bm{0}})}$
$\displaystyle{\color[rgb]{0,0,1}-}$
$\displaystyle{\color[rgb]{0,0,1}\frac{q_{t}^{2}}{4\,\pi\,c^{2}}\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\,\ln\left(\frac{\alpha}{\alpha_{0}}\right)\;-\;\frac{q_{t}}{2\,\pi\,c^{2}}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;\ln\left(\frac{t-t^{\prime}}{\tilde{t}}\right){\rm
d}t^{\prime}}$ $\displaystyle{\color[rgb]{0,0,1}+}$
$\displaystyle{\color[rgb]{0,0,1}{\rm an}\;\,\alpha{\rm-
dependent\;\,correction\;\,vanishing\;\,as}\;\,\alpha\to+\infty\hskip
42.67912pt.\hskip 14.22636pt[\,d=2\,]}$

In the case of $d=3$, one gets explicitly

$\displaystyle{\color[rgb]{0,0,1}m_{o}^{t}(\alpha)\;\frac{{\rm d}^{2}}{{\rm
d}t^{2}}\,\hat{\bm{x}}(t)}$ $\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}-\,\bm{\nabla}V(\hat{\bm{x}}(t))\;+\;q_{t}\;\hat{\bm{E}}_{0}(t,{\bm{0}})}$
$\displaystyle{\color[rgb]{0,0,1}-}$
$\displaystyle{\color[rgb]{0,0,1}\frac{4}{3}\,\frac{q_{t}^{2}}{c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;\bm{\ddot{}}{\hat{\bm{x}}}(t)\;+\;\frac{2}{3}\,\frac{q_{t}^{2}}{c^{3}}\,\dddot{\hat{\bm{x}}}(t)\;+\;{\cal
O}(\alpha^{-1/2})\hskip 14.22636pt.\hskip 14.22636pt[\,d=3\,]}$

Formulas (•)-(•) represent final outcome of the somewhat technical Section C.
The electric RR force of equation (•) takes a qualitatively different
appearance
than the electric RR force of equation (•).
Both in (•) and (•), the electric RR force term diverges for
$\alpha\to+\infty$.
We are done with all the technical preparations,
and ready to jump into the matters of mass renormalization now!

D. Mass renormalization

  * •

D.1 Preliminaries
The concept or mass renormalization will be introduced here via examining the
$\alpha\to+\infty$ limit
of the Newton-Heisenberg equations of motion (•) and (•) for
$\hat{\bm{x}}(t)$.
Recall that for $\alpha\to+\infty$ the charge density
$\varrho({\bm{y}})=(\ref{varrho-def})$ approaches $\delta^{d}({\bm{y}})$.
This is equivalent to saying that our electron approaches the limit of a
charged mass point.
Pushing $\alpha\to+\infty$ elevates towards infinity
the UV cutoff in our ”electron – radiation field” coupling (13), see A.2.
In other words, for $\alpha\to+\infty$, all the radiation field modes
become affected by the electronic motion and vice versa.
As we have just explicitly seen above in C.4 and C.5,
coupling the electron to all the UV field modes
causes an UV divergence of the corresponding electric RR force,
see again formulas (55), (•) or the equations of motion (•), (•) themselves.
Our Newton-Heisenberg equations of motion (•) and (•)
become inevitably unphysical in the $\alpha\to+\infty$ limit, unless
the so far unspecified bare mass term $m_{o}^{t}(\alpha)$ is adjusted in such
an ingenuous manner
as to overcompensate the UV divergent part of the electric RR force.
We shall implement exactly this kind of trick below, separately for $d=2$ and
$d=3$.

  * •

D.2 Mass renormalization in the case of $\bm{d}\bm{=}\bm{2}$
The Newton-Heisenberg equation of motion (•) can be equivalently redisplayed
as follows:

$\displaystyle\left\\{\,m_{o}^{t}(\alpha)\;+\;\frac{q_{t}^{2}}{4\,\pi\,c^{2}}\;\ln\left(\frac{\alpha}{\alpha_{0}}\right)\,\right\\}\frac{{\rm
d}^{2}}{{\rm d}t^{2}}\,\hat{\bm{x}}(t)$ $\displaystyle=$
$\displaystyle-\,\bm{\nabla}V(\hat{\bm{x}}(t))\;+\;q_{t}\;\hat{\bm{E}}_{0}(t,{\bm{0}})$
$\displaystyle-$
$\displaystyle\frac{q_{t}}{2\,\pi\,c^{2}}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;\ln\left(\frac{t-t^{\prime}}{\tilde{t}}\right){\rm
d}t^{\prime}$ $\displaystyle+$ $\displaystyle{\rm an}\;\,\alpha{\rm-
dependent\;\,correction\;\,vanishing\;\,as}\;\,\alpha\to+\infty\hskip
7.11317pt.$

Showing that, for large values of $\alpha$,
a physically meaningful dynamical time evolution of $\hat{\bm{x}}(t)$ can take
place only if
one adjusts the bare mass $m_{o}^{t}(\alpha)$ according to the  mass
renormalization prescription

${\color[rgb]{0,0,1}m_{o}^{t}(\alpha)\;+\;\frac{q_{t}^{2}}{4\,\pi\,c^{2}}\;\ln\left(\frac{\alpha}{\alpha_{0}}\right)\;=\;m\hskip
14.22636pt.\hskip 14.22636pt[d=2]}$ (87)

Here $m$ represents of course the physical mass of the electron
(which enters also into the atomic Hamiltonian $\hat{H}_{{\rm A}}=(\ref{H-A-
def})$).
Note that $m_{o}^{t}(\alpha)$ diverges to $-\infty$ as $\alpha\to+\infty$.
$[\,$Thus $m_{o}^{t}(\alpha)$ should be regarded as mere mathematical tool
lacking any physical significance.$\,]$
Having accepted the above peculiar arrangement (87), and having subsequently
pushed $\alpha\to+\infty$,
we are able to simplify the Newton-Heisenberg equation of motion (•)
into its final and compelling appearance

$\displaystyle{\color[rgb]{0,0,1}m\,\frac{{\rm d}^{2}}{{\rm
d}t^{2}}\,\hat{\bm{x}}(t)}$ $\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}-\,\bm{\nabla}V(\hat{\bm{x}}(t))\;+\;q_{t}\;\hat{\bm{E}}_{0}(t,{\bm{0}})}$
$\displaystyle{\color[rgb]{0,0,1}-}$
$\displaystyle{\color[rgb]{0,0,1}\frac{q_{t}}{2\,\pi\,c^{2}}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;\ln\left(\frac{t-t^{\prime}}{\tilde{t}}\right){\rm
d}t^{\prime}\hskip 14.22636pt.\hskip 14.22636pt[d=2]}$

Success! Mass renormalization has just been performed!
The integro-differential equation (•) should be solved for $\hat{\bm{x}}(t)$
with the initial conditions (28), (29), (31).
Recall that $q_{t}$ equals to the physical charge $q$ of the electron at
finite times.
The first line on the r.h.s. of (•) displays
standard scalar potential contribution to the force acting on the electron,
as well as the interaction picture electric force contribution acting on the
electron,
which we have already discussed before $[\,$see equation (43) and the
accompanying remarks$\,]$.
The second line on the r.h.s. of (•) displays
a finite contribution of the electric RR force which has survived the mass
renormalization.
This is the so called Abraham-Lorentz force,

${\color[rgb]{0,0,1}\hat{\bm{F}}_{\rm
ALF}(t)\;=\;-\;\frac{q_{t}}{2\,\pi\,c^{2}}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;\ln\left(\frac{t-t^{\prime}}{\tilde{t}}\right){\rm
d}t^{\prime}\hskip 14.22636pt.\hskip 14.22636pt[d=2]}$ (89)

Most importantly, for $d=2$, the resulting Abraham-Lorentz force is nonlocal
in time,
contanining $\dddot{\hat{\bm{x}}}(t^{\prime})$ $[\,t^{\prime}<t\,]$ and a
logarithmic memory kernel $\ln\left(\frac{t-t^{\prime}}{\tilde{t}}\right)$.
Hence the quantum dynamical evolution of $\hat{\bm{x}}(t)$ is non-Markovian.
$[\,$The term ”non-Markovian” means that our equation of motion (•) has a
memory, i.e., that
$[\,$previous history of the system (including all the time instants
$t^{\prime}$ finitely separated from $t$)
$[\,$matters when propagating $\hat{\bm{x}}(t)$ from the time instant $t$
forward.$\,]$
In Appendix A we analyze in more detail
mathematical well definedness of the just mentioned Abraham-Lorentz force term
(89),
and suggest an iterative (perturbative) solution of equation (•), starting
from
the zeroth order interaction picture approximation
$\hat{\bm{x}}(t)\doteq\hat{\bm{x}}_{I}(t)\;=\;e^{+\frac{i}{\hbar}\hat{H}_{{\rm
A}}t}\;\hat{\bm{x}}\;e^{-\frac{i}{\hbar}\hat{H}_{{\rm A}}t}$.

  * •

D.3 Mass renormalization in the case of $\bm{d}\bm{=}\bm{3}$
The Newton-Heisenberg equation of motion (•) can be equivalently redisplayed
as follows:

$\displaystyle\left\\{m_{o}^{t}(\alpha)\,+\,\frac{4}{3}\,\frac{q_{t}^{2}}{c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\,\right\\}\frac{{\rm
d}^{2}}{{\rm d}t^{2}}\,\hat{\bm{x}}(t)$ $\displaystyle=$
$\displaystyle-\,\bm{\nabla}V(\hat{\bm{x}}(t))\;+\;q_{t}\;\hat{\bm{E}}_{0}(t,{\bm{0}})\;+\;\frac{2}{3}\,\frac{q_{t}^{2}}{c^{3}}\,\dddot{\hat{\bm{x}}}(t)\;+\;{\cal
O}(\alpha^{-1/2})\hskip 7.11317pt.$ (90)

Showing that, for large values of $\alpha$,
a physically meaningful dynamical time evolution of $\hat{\bm{x}}(t)$ can take
place only if
one adjusts the bare mass $m_{o}^{t}(\alpha)$ according to the  mass
renormalization prescription

${\color[rgb]{0,0,1}m_{o}^{t}(\alpha)\;+\;\frac{4}{3}\,\frac{q_{t}^{2}}{c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;=\;m\hskip
14.22636pt.\hskip 14.22636pt[d=3]}$ (91)

Here $m$ represents of course the physical mass of the electron
(which enters also into the atomic Hamiltonian $\hat{H}_{{\rm A}}=(\ref{H-A-
def})$).
Note that $m_{o}^{t}(\alpha)$ diverges to $-\infty$ as $\alpha\to+\infty$.
$[\,$Thus $m_{o}^{t}(\alpha)$ should be regarded as mere mathematical tool
lacking any physical significance.$\,]$
Having accepted the above peculiar arrangement (91), and having subsequently
pushed $\alpha\to+\infty$,
we are able to simplify the Newton-Heisenberg equation of motion (90)
into its final and compelling appearance

$\displaystyle{\color[rgb]{0,0,1}m\,\frac{{\rm d}^{2}}{{\rm
d}t^{2}}\,\hat{\bm{x}}(t)}$ $\displaystyle{\color[rgb]{0,0,1}=}$
$\displaystyle{\color[rgb]{0,0,1}-\,\bm{\nabla}V(\hat{\bm{x}}(t))\;+\;q_{t}\;\hat{\bm{E}}_{0}(t,{\bm{0}})\;+\;\frac{2}{3}\,\frac{q_{t}^{2}}{c^{3}}\,\dddot{\hat{\bm{x}}}(t)\hskip
14.22636pt.\hskip 14.22636pt[d=3]}$ (92)

Success! Mass renormalization has just been performed!
The differential equation (92) should be solved for $\hat{\bm{x}}(t)$
with the initial conditions (28), (29), (31).
Recall that $q_{t}$ equals to the physical charge $q$ of the electron at
finite times.
The first term on the r.h.s. of (92) displays
standard scalar potential contribution to the force acting on the electron.
The second term on the r.h.s. of (92) corresponds
to the interaction picture electric force contribution acting on the electron,
which we have already discussed before $[\,$see equation (43) and the
accompanying remarks$\,]$.
The third term on the r.h.s. of (92) displays
a finite contribution of the electric RR force which has survived the mass
renormalization.
This is the so called Abraham-Lorentz force,

${\color[rgb]{0,0,1}\hat{\bm{F}}_{\rm
ALF}(t)\;=\;\frac{2}{3}\,\frac{q_{t}^{2}}{c^{3}}\,\dddot{\hat{\bm{x}}}(t)\hskip
14.22636pt.\hskip 14.22636pt[d=3]}$ (93)

Most importantly, for $d=3$, the resulting Abraham-Lorentz force
is local in time and proportional to the third time derivative
$\dddot{\hat{\bm{x}}}(t)$.
Hence the quantum dynamical evolution of $\hat{\bm{x}}(t)$ is Markovian.
$[\,$The term ”Markovian” means that our equation of motion (92) has no
memory, i.e., that
$[\,$previous history of the system (including the time instants $t^{\prime}$
finitely separated from $t$)
$[\,$does not matter when propagating $\hat{\bm{x}}(t)$ from the time instant
$t$ forward.$\,]$
Physics intuition suggests that equation (92) can be solved iteratively
(perturbatively),
starting from the zeroth order interaction picture approximation
$\hat{\bm{x}}(t)\doteq\hat{\bm{x}}_{I}(t)\;=\;e^{+\frac{i}{\hbar}\hat{H}_{{\rm
A}}t}\;\hat{\bm{x}}\;e^{-\frac{i}{\hbar}\hat{H}_{{\rm A}}t}$.
We have accomplished the main goal of this course: To introduce the concept of
mass renormalization.
Now we shall proceed further and elaborate on important applications.

E. Renormalized perturbation theory for the atomic level shifts

  * •

Perturbative treatment of the atomic energy levels dressed by the radiation
field ($d=2,3$).
Vacuum fluctuations in the radiation field modes invoke level shifts
as well as spontaneous emission of photons from the excited atomic levels.

  * •

Worked out in Appendices B.1, B.2, B.3, B.4.
Appendix B.1: Introducing the game.
Appendix B.2: Naive perturbation theory, breaking down for $d=2,3$ due to UV
divergences.
Appendix B.3: Renormalized perturbation theory for $d=2$.
Appendix B.4: Renormalized perturbation theory for $d=3$.

F. Renormalized mean field theory

  * •

We show explicitly how is the time dependent mean field (Hartree) method
properly renormalized.
Such a renormalized mean field approach is even suitable for numerical
implementation.

  * •

Worked out in Appendix C.

Appendix A: More on the electric RR force for $\bm{d}\bm{=}\bm{2}$

  * •

The purpose of this Appendix A is to explore in more detail mathematical well
definedness
of the non-Markovian Abraham-Lorentz force term (89), that is, of an entity

$\hat{\bm{F}}_{\rm
ALF}(t)\;=\;-\;\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\,\ln\left(\frac{t-t^{\prime}}{t_{c}}\right){\rm
d}t^{\prime}\hskip 14.22636pt;$ (94)

which appears in the main text first inside equation (•),
and which enters later on into the final Newton-Heisenberg equation of motion
(•) for the electron.

  * •

Let us look at a convolution

$I_{t}\;=\;\int_{-\infty}^{t}f(t^{\prime})\;\ln\left(\frac{t-t^{\prime}}{t_{c}}\right)\,{\rm
d}t^{\prime}\hskip 14.22636pt;$ (95)

with $f(t^{\prime})$ representing an as yet unspecified $t^{\prime}$-dependent
real valued entity or signal
(or possibly a hermitian operator).
Assume for a while that the integral of equation (95) converges for the chosen
$f(t^{\prime})$,
and write conveniently

$I_{t}\;=\;\lim_{\varepsilon\to+0}\;\int_{-\infty}^{t}f(t^{\prime})\;e^{-\varepsilon(t-t^{\prime})}\ln\left(\frac{t-t^{\prime}}{t_{c}}\right)\,{\rm
d}t^{\prime}\hskip 14.22636pt.$ (96)

Assume for a while also that  $f(t^{\prime})$ is Fourier transformable,

${\color[rgb]{0,0,1}f(t^{\prime})\;=\;\int_{-\infty}^{+\infty}\tilde{f}(\Omega)\;e^{+i\Omega(t-t^{\prime})}\,{\rm
d}\Omega\hskip 14.22636pt.}$ (97)

Real valuedness (or hermiticity) of $f(t^{\prime})$ implies having

$f(t^{\prime})\;=\;f^{*}(t^{\prime})\hskip 14.22636pt,\hskip
14.22636pt\tilde{f}(\Omega)\;=\;\tilde{f}^{*}(-\Omega)\hskip 14.22636pt.$ (98)

Then of course

$f(t^{\prime})\;=\;\int_{0}^{+\infty}\tilde{f}(\Omega)\;e^{+i\Omega(t-t^{\prime})}\,{\rm
d}\Omega\;\,+\;\,{\rm c.c.}\hskip 14.22636pt;$ (99)

and our convolution formula $I_{t}=(\ref{I-t-take-2})$ can be rewritten into

$\displaystyle I_{t}$ $\displaystyle=$
$\displaystyle\lim_{\varepsilon\to+0}\;\int_{0}^{+\infty}{\rm
d}\Omega\;\,\tilde{f}(\Omega)\;\int_{-\infty}^{t}e^{+i\Omega(t-t^{\prime})}\;e^{-\varepsilon(t-t^{\prime})}\ln\left(\frac{t-t^{\prime}}{t_{c}}\right)\,{\rm
d}t^{\prime}\;+\;{\rm c.c.}\;=$ (100) $\displaystyle=$
$\displaystyle\lim_{\varepsilon\to+0}\;\int_{0}^{+\infty}{\rm
d}\Omega\;\,\tilde{f}(\Omega)\,\left\\{\,\int_{0}^{+\infty}e^{+i\Omega\tau}\;e^{-\varepsilon\tau}\ln\left(\frac{\tau}{t_{c}}\right)\,{\rm
d}\tau\right\\}\;+\;{\rm c.c.}\;=$ $\displaystyle=$
$\displaystyle\lim_{\varepsilon\to+0}\;\int_{0}^{+\infty}{\rm
d}\Omega\;\,\tilde{f}(\Omega)\;\left\\{\frac{\ln\Bigl{(}(\Omega+i\varepsilon)\,t_{c}\Bigr{)}+\gamma-i\,\frac{\pi}{2}}{i\,(\Omega+i\varepsilon)}\right\\}\;+\;{\rm
c.c.}\hskip 14.22636pt.$

Here $\gamma=0.577\cdots$ is the Euler-Mascheroni constant.
The $\\{\bm{\cdots}\\}$ term of (100) diverges to infinity both for
$\Omega\to+0$ and for $\Omega\to+\infty$,
this demands an appropriate analysis of convergence conditions for the
integral $I_{t}=(\ref{I-t-take-3})$:

    * $\star$

Let $\Omega_{\rm cut}$ be an arbitrary auxiliary parameter, $0<\Omega_{\rm
cut}<\infty$.

    * $\star$

The $\Omega\to+0$ singularity of the $\\{\bm{\cdots}\\}$ term is actually weak
enough to be integrable,
provided only that

${\color[rgb]{0,0,1}\tilde{f}(\Omega\to+0)\;{\rm
admits\;a\;power\;series\;expansion}}$
${\color[rgb]{0,0,1}\tilde{f}(\Omega)\;=\;\tilde{f}(0)\;+\;\Omega\,\tilde{f}^{\prime}(0)\;+\;{\cal
O}(\Omega^{2})}$ (101) ${\color[rgb]{0,0,1}{\rm
with\;finite\;coefficients}\;\tilde{f}(0)\;{\rm
and}\;\tilde{f}^{\prime}(0)\;.}$

Valid since the entity

$\lim_{\varepsilon\to+0}\;\int_{0}^{\Omega_{\rm cut}}{\rm
d}\Omega\;\,\tilde{f}(0)\;\left\\{\frac{\ln\Bigl{(}(\Omega+i\varepsilon)\,t_{c}\Bigr{)}+\gamma-i\,\frac{\pi}{2}}{i\,(\Omega+i\varepsilon)}\right\\}\;+\;{\rm
c.c.}$ (102)

is mathematically well defined,
as one may easily check by exploiting the property
$\tilde{f}(0)=\tilde{f}^{*}(0)$ coming from (98).
Indeed, direct calculation yields

$\displaystyle\int_{0}^{\Omega_{\rm cut}}{\rm
d}\Omega\;\,\tilde{f}(0)\;\left\\{\frac{\ln\Bigl{(}(\Omega+i\varepsilon)\,t_{c}\Bigr{)}+\gamma-i\,\frac{\pi}{2}}{i\,(\Omega+i\varepsilon)}\right\\}\;+\;{\rm
c.c.}\;=$ (103) $\displaystyle=$ $\displaystyle\left\\{\,2\,{\rm
arctan}\left(\frac{\varepsilon}{\Omega_{\rm
cut}}\right)-\pi\,\right\\}\left\\{\,\ln\left(\sqrt{\Omega_{\rm
cut}^{2}+\varepsilon^{2}}\,t_{c}\right)\,+\,\gamma\,\right\\}\hskip
14.22636pt;$

and hence

$(\ref{tilde-f-0-entity})\;=\;-\,\pi\,\tilde{f}(0)\,\Bigl{(}\ln(\Omega_{\rm
cut}\,t_{c})+\gamma\Bigr{)}\hskip 14.22636pt.$ (104)

    * $\star$

The $\Omega\to+\infty$ singularity of the $\\{\bm{\cdots}\\}$ term is
encountered when dealing with an integral

$\lim_{\varepsilon\to+0}\;\int_{\Omega_{\rm cut}}^{+\infty}{\rm
d}\Omega\;\,\tilde{f}(\Omega)\;\left\\{\frac{\ln\Bigl{(}(\Omega+i\varepsilon)\,t_{c}\Bigr{)}+\gamma-i\,\frac{\pi}{2}}{i\,(\Omega+i\varepsilon)}\right\\}\hskip
14.22636pt.$ (105)

Here the limit of $\varepsilon\to+0$ can be taken inside the
$\\{\bm{\cdots}\\}$ term, converting (105) into

$\int_{\Omega_{\rm cut}}^{+\infty}{\rm
d}\Omega\;\,\tilde{f}(\Omega)\;\left\\{\frac{\ln\Bigl{(}\Omega\,t_{c}\Bigr{)}+\gamma-i\,\frac{\pi}{2}}{i\,\Omega}\right\\}\hskip
14.22636pt.$ (106)

The integral (106) is granted to converge provided only that

${\color[rgb]{0,0,1}\tilde{f}(\Omega\to+\infty)\;{\rm
is\;falling\;off\;to\;zero\;as}\;{\cal O}(\Omega^{-1})\;{\rm or\;faster.}}$
(107)

We may proceed now in reverse order, starting from (100) & assumptions (97),
(101), (107),
and going stepwise up to (95). Such an argument enables us to conclude
that the integral $I_{t}=(\ref{I-t-take-1})$ is a well defined entity
provided only that the conditions (97), (101), (107) are satisfied by
$f(t^{\prime})$.
$[\,$Note that most signals $f(t^{\prime})$ encountered in physics
$[\,$would satisfy conditions as weak as required in (97), (101), (107).$\,]$

  * •

Let us identify $f(t^{\prime})$ with
$q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\,\Theta(t-t^{\prime})$, as
appropriate for our starting equation (94).
The basic question is now whether or not
$f(t^{\prime})=q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\,\Theta(t-t^{\prime})$
matches the above conditions (97), (101), (107).

  * •

Assume that
the Newton-Heisenberg equation of motion (•)
supplemented by the initial conditions (28), (29), (31)
has been solved for the electron up to some given time instant $t$.
Assume that the solution $\hat{\bm{x}}(t^{\prime})$ (with $t^{\prime}\leq t$)
behaves in a physically sensible manner,
as appropriate for an electron trapped inside a bound potential well
whose motion is slightly affected by the electric RR force.
$[\,$To apply at least in some subspace ${\mathfrak{S}}$ of quantum state
vectors
$[\,$which correspond to non-relativistic motions of the electron.$\,]$
If so, then the FTs

$\hat{\bm{\chi}}_{\nu}(\Omega)\;=\;\frac{1}{2\,\pi}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\frac{{\rm
d}^{\nu}\hat{\bm{x}}(t^{\prime})}{{\rm
d}t^{\prime\nu}}\;e^{-i\Omega(t-t^{\prime})}\,{\rm d}t^{\prime}$ (108)

do exist for all $\Omega\in(-\infty,+\infty)$ as well as for all
$\nu=\\{1,2,3,4,\bm{\cdots}\\}$, and moreover

$\hat{\bm{\chi}}_{\nu}(\Omega\to\pm\infty)\;=\;0\hskip 14.22636pt.\hskip
14.22636pt[\,\nu\geq 1\,]$ (109)

Valid since:

    * $\circ$

The charge $q_{t^{\prime}}$ possesses the properties (8) and (9);

    * $\circ$

The entity $\frac{{\rm d}^{\nu}\hat{\bm{x}}(t^{\prime})}{{\rm
d}t^{\prime\nu}}$ is bounded in magnitude, and
oscillates with $t^{\prime}$ due to motion of the electron in our strictly
bound potential $V({\bm{x}})$, see A.1.
Such that the average of $\frac{{\rm d}^{\nu}\hat{\bm{x}}(t^{\prime})}{{\rm
d}t^{\prime\nu}}$ over a sufficiently long time interval is zero.
Moreover, the oscillations of $\frac{{\rm
d}^{\nu}\hat{\bm{x}}(t^{\prime})}{{\rm d}t^{\prime\nu}}$ do not contain ultra
high frequencies.

After integrating by parts in (108) with the help of (8)-(9) one finds a neat
recursive formula

$\hat{\bm{\chi}}_{\nu}(\Omega)\;=\;\frac{q_{t}}{2\,\pi}\,\frac{{\rm
d}^{(\nu-1)}\hat{\bm{x}}(t)}{{\rm
d}t^{(\nu-1)}}\;-\;i\,\Omega\,\hat{\bm{\chi}}_{\nu-1}(\Omega)\hskip
14.22636pt.\hskip 14.22636pt[\,\nu\geq 2\,]$ (110)

  * •

Note that $\tilde{f}(\Omega)=\hat{\bm{\chi}}_{3}(\Omega)$.
Condition (97) is thus fulfilled by
$f(t^{\prime})=q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\,\Theta(t-t^{\prime})$.

  * •

Consider now the regime of $\Omega\to 0$. Straightforward calculation
exploiting (8) and (9) yields

$\displaystyle\tilde{f}(\Omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2\,\pi}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\;e^{-i\Omega(t-t^{\prime})}\,{\rm
d}t^{\prime}\;=$ (111) $\displaystyle=$
$\displaystyle\frac{1}{2\,\pi}\,\int_{-\infty}^{t}q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\,\Bigl{\\{}\,1\,-\,i\,\Omega\,(t-t^{\prime})\,+\,{\cal
O}(\Omega^{2})\,\Bigr{\\}}\,{\rm d}t^{\prime}\;=$ $\displaystyle=$
$\displaystyle\frac{1}{2\,\pi}\,q_{t}\,\bm{\ddot{}}{\hat{\bm{x}}}(t)\;-\;\frac{1}{2\,\pi}\;i\,\Omega\;q_{t}\,\bm{\dot{}}{\hat{\bm{x}}}(t)\;+\;{\cal
O}(\Omega^{2})\hskip 14.22636pt.$

Showing thus that the condition (101) is matched by
$f(t^{\prime})=q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\,\Theta(t-t^{\prime})$.

  * •

Consider now the regime of $\Omega\to\pm\infty$. Formula (110) implies having

$\tilde{f}(\Omega)\;=\;\hat{\bm{\chi}}_{3}(\Omega)\;=\;\frac{1}{i\,\Omega}\,\frac{q_{t}}{2\,\pi}\,\dddot{\hat{\bm{x}}}(t)\;-\;\frac{\hat{\bm{\chi}}_{4}(\Omega)}{i\,\Omega}\hskip
14.22636pt.$ (112)

Insights (112) and (109) enable us to conclude that
the condition (107) is satisfied by
$f(t^{\prime})=q_{t^{\prime}}\,\dddot{\hat{\bm{x}}}(t^{\prime})\,\Theta(t-t^{\prime})$.

  * •

Summarizing the contents of the present Appendix A,
we have shown that the non-Markovian electric RR force term (94) is well
defined,
in spite of singularities of the logarithmic memory kernel
$\ln\left(\frac{t-t^{\prime}}{t_{c}}\right)$.
This conclusion holds provided only that
the corresponding Heisenberg picture position operator
$\hat{\bm{x}}(t^{\prime})$ $(t^{\prime}\leq t)$
behaves in a physically sensible manner,
as appropriate for non-relativistic motions of an electron
trapped inside an external bound potential well $V({\bm{x}})$
whose motion is only slightly affected by the electric RR force.
In particular, our non-Markovian electric RR force term (94) is well defined
when one approximates $\hat{\bm{x}}(t^{\prime})$ by its interaction picture
counterpart,

$\hat{\bm{x}}_{I}(t^{\prime})\;=\;e^{+\frac{i}{\hbar}\hat{H}_{{\rm
A}}t^{\prime}}\;\hat{\bm{x}}\;e^{-\frac{i}{\hbar}\hat{H}_{{\rm
A}}t^{\prime}}\hskip 14.22636pt;$ (113)

cf. equation (28).
Hence the Newton-Heisenberg equation of motion (•)
supplemented by the initial conditions (28), (29), (31)
can be solved iteratively (perturbatively),
starting from the initial zeroth order approximation
$\hat{\bm{x}}_{I}(t^{\prime})=(\ref{hat-bm-x-I})$.

Appendix B: Perturbation theory for the atomic level shifts

Appendix B.1: Introducing the game

  * •

Consider first our model atom alone.
The corresponding atomic Hamiltonian takes the form $\hat{H}_{{\rm
A}}=(\ref{H-A-def})$.
Since the potential $V(\hat{\bm{x}})$ is strictly bound, only discrete energy
levels are permitted.
We write as usual

$\hat{H}_{{\rm
A}}\,|\psi_{j}\rangle_{A}\;=\;E_{j}^{A}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt;$ (114)

with $j=0,1,2,\bm{\cdots}$.

  * •

Secondly, consider the quantum radiation field alone.
The corresponding Hamiltonian takes the form $\hat{H}_{{\rm R}}=(\ref{H-R-
def})$.
Recall that $\hat{H}_{{\rm R}}$ describes an infinite ensemble of mutually
uncoupled harmonic oscillators,
these are labeled by modal indices $({\bm{k}}\wp)$. An oscillator
$({\bm{k}}\wp)$ possesses the circular frequency $\omega_{k}$.
The ground state $|{\rm vac}\rangle_{\rm R}$ of $\hat{H}_{{\rm R}}$ is
annihilated by all the $\hat{a}_{{\bm{k}}\wp}$’s.
All the other eigenstates of $\hat{H}_{{\rm R}}$ are built up via repeated
actions of the $\hat{a}_{{\bm{k}}\wp}^{\dagger}$’s. Such that

$|({\bm{k}_{1}}\wp_{1})\bm{\cdots}({\bm{k}_{n}}\wp_{n})\rangle_{R}\;=\;\frac{1}{\sqrt{n!}}\;\hat{a}_{{\bm{k}}_{1}\wp_{1}}^{\dagger}\bm{\cdots}\,\hat{a}_{{\bm{k}}_{n}\wp_{n}}^{\dagger}\,|{\rm
vac}\rangle_{\rm R}\hskip 14.22636pt.\hskip 14.22636pt[n\geq 0]$ (115)

Clearly,

$\hat{H}_{{\rm
R}}\,|({\bm{k}_{1}}\wp_{1})\bm{\cdots}({\bm{k}_{n}}\wp_{n})\rangle_{R}\;=\;\Bigl{(}\hbar\omega_{k_{1}}\,+\,\bm{\cdots}\,+\,\hbar\omega_{k_{n}}\Bigr{)}\,|({\bm{k}_{1}}\wp_{1})\bm{\cdots}({\bm{k}_{n}}\wp_{n})\rangle_{R}\hskip
14.22636pt.\hskip 14.22636pt[n\geq 0]$ (116)

  * •

Thirdly, consider both our model atom (subsystem #1)
and the quantized radiation field (subsystem #2).
When the subsystems #1 and #2 are mutually uncoupled (non-interacting),
the total Hamiltonian of our problem takes of course the form $\hat{H}_{\rm
free}=(\ref{H-free-def})$,
and the associated stationary eigenstates result trivially. One has

$\hat{H}_{\rm
free}\,|\psi_{j}\rangle_{A}\,|({\bm{k}_{1}}\wp_{1})\bm{\cdots}({\bm{k}_{n}}\wp_{n})\rangle_{R}\;=\;\Bigl{(}E_{j}^{A}\,+\,\hbar\omega_{k_{1}}\,+\,\bm{\cdots}\,+\,\hbar\omega_{k_{n}}\Bigr{)}\,|\psi_{j}\rangle_{A}\,|({\bm{k}_{1}}\wp_{1})\bm{\cdots}({\bm{k}_{n}}\wp_{n})\rangle_{R}\hskip
14.22636pt.$ (117)

  * •

Fourthly, consider both our model atom (subsystem #1)
and the quantized radiation field (subsystem #2),
coupled together as specified in the Hamiltonian (7).
Recall that the Hamiltonian (7) is parameterized by a finite (yet arbitrarily
large) value of $\alpha>0$.
Moreover, the mass renormalization prescriptions [(89) or (93)] supplement the
formula (7).
Hereafter we shall be concerned only with physical properties
of our system $[\,$$\equiv$ mutually interacting subsystems #1 and #2$\,]$
pertaining to finite time instants $t$.
At finite time instants, the charge $q_{t}$ of our electron equals to its
physical value $q$.
This enables us to update (7) & [(89) or (93)] into

$\hat{H}\;=\;\frac{1}{2\,m_{o}(\alpha)}\left(\hat{\bm{p}}\,-\,\frac{q}{c}\,\hat{\bm{A}}[\alpha]\right)^{2}+\;V(\hat{\bm{x}})\;+\;\hat{H}_{{\rm
R}}\hskip 14.22636pt;$ (118)

with

$m_{o}(\alpha)\;+\;\frac{q^{2}}{4\,\pi\,c^{2}}\;\ln\left(\frac{\alpha}{\alpha_{0}}\right)\;=\;m\hskip
14.22636pt;\hskip 14.22636pt[d=2]$ (119)

or

$m_{o}(\alpha)\;+\;\frac{4}{3}\,\frac{q^{2}}{c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;=\;m\hskip
14.22636pt.\hskip 14.22636pt[d=3]$ (120)

In the Hamiltonian formula (118), we have adopted for the sake of notational
convenience
an auxiliary shorthand symbol

$\displaystyle\hat{\bm{A}}[\alpha]$ $\displaystyle\equiv$
$\displaystyle\int_{{\mathbb{R}}^{d}}\varrho({\bm{y}})\;\hat{\bm{A}}({\bm{y}})\;{\rm
d}^{d}y$ (121) $\displaystyle=$ $\displaystyle\int_{{\mathbb{R}}^{2}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\frac{\hbar\,c^{2}}{\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\,\hat{a}_{{\bm{k}}\wp}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\;+\;{\rm
c.c.}\hskip 14.22636pt.$

The second line of (121) comes from (13).
An eigenvalue problem of the Hamiltonian $\hat{H}=(\ref{H-no-t})$ is formally
written down as

$\hat{H}\,|\Psi\rangle\;=\;E\,|\Psi\rangle\hskip 14.22636pt.$ (122)

In what follows, we shall investigate the eigenproblem (122)
in the weak coupling regime
(such that the charge $q$ of the electron is set to be arbitrarily small but
nonzero),
using appropriate perturbation expansions in $q$.
$[\,$The case of $q=0$ is resolved trivially, since
$\hat{H}|_{q=0}=\hat{H}_{\rm free}$ and $(\ref{hat-H-eigenproblem})|_{q=0}$
reduces to $(\ref{hat-H-free-eigenproblem})$.$\,]$
Our interest actually consists in finding only those stationary eigenstates of
$\hat{H}=(\ref{H-no-t})$
which reduce in the limit of $q\to 0$ just to

$|\Psi^{(0)}\rangle\;=\;|\psi_{j}\rangle_{A}\,|{\rm vac}\rangle_{R}\hskip
14.22636pt;$ (123)

cf. equation $(\ref{hat-H-free-eigenproblem})|_{n=0}$. Here $j$ represents an
arbitrary chosen atomic stationary state.
Loosely speaking, our goal is to investigate those stationary eigenstates of
$\hat{H}=(\ref{H-no-t})$
which describe a given stationary state $|\psi_{j}\rangle_{A}$ of our model
atom
dressed by zero temperature bath of the radiation field modes
(i.e., dressed by the electromagnetic vacuum).
Vacuum fluctuations in the radiation field modes invoke atomic level shifts,
as well as spontaneous emission of photons from the excited atomic levels
$j\geq 1$.

Appendix B.2: Naive perturbation theory

  * •

Before implementing explicitly the just outlined program
of renormalized perturbation theory for the atomic level shifts,
it is instructive to try out a naive treatment which ignores the matters of
mass renormalization.
Such a naive formulation would replace our starting Hamiltonian
$\hat{H}=(\ref{H-no-t})$
by a heuristically plausible (yet deceptive, treacherous) prescription

$\hat{H}\;=\;\frac{1}{2\,m}\left(\hat{\bm{p}}\,-\,\frac{q}{c}\,\hat{\bm{A}}({\bm{0}})\right)^{2}+\;V(\hat{\bm{x}})\;+\;\hat{H}_{{\rm
R}}\hskip 14.22636pt.$ (124)

Here $m$ and $q$ stand for the physical mass and charge of our electron, and
$\hat{\bm{A}}({\bm{0}})$ comes from (4).
Clearly, the Hamiltonian (124) is obtainable from (118)
via identifying hastily $m_{o}(\alpha)$ with $m$ and taking hastily
$\alpha\to+\infty$.
$[\,$Such a hasty attitude would hardly seem suspicious to those who never
heard of renormalization.$\,]$
As we shall see shortly, the naive Hamiltonian prescription (124) leads to an
epic failure!
This is not surprising at all for everyone who passed through Sections C and D
of the main text.

  * •

We wish to solve the eigenvalue problem (122)
by employing the standard Rayleigh-Schrödinger perturbation theory (RSPT or
just PT)
in the coupling parameter $q$.
It is thus convenient to express the Hamiltonian $\hat{H}=(\ref{H-naive})$ in
the form of a sum

$\hat{H}\;=\;\hat{H}_{\rm free}\;+\;\hat{W}\hskip 14.22636pt;$ (125)

where $\hat{H}_{\rm free}=(\ref{H-free-def})$ is understood as the unperturbed
reference part, and

$\hat{W}\;=\;-\,\frac{q}{mc}\,\hat{\bm{A}}({\bm{0}})\bm{\cdot}\hat{\bm{p}}\;+\;\frac{q^{2}}{2\,m\,c^{2}}\,\hat{\bm{A}}^{2}({\bm{0}})$
(126)

determines the perturbation.
Our unperturbed reference state
$|\Psi^{(0)}\rangle=(\ref{Psi-(0)-unperturbed})$
corresponds to an unperturbed energy eigenvalue $E^{(0)}=E_{j}^{A}$,
consistently with relation $(\ref{hat-H-free-eigenproblem})|_{n=0}$.

  * •

An application of the standard RSPT is now straightforward.
Our PT1 energy correction equals to

$E^{(1)}\;=\;\langle\Psi^{(0)}|\,-\,\frac{q}{mc}\,\hat{\bm{A}}({\bm{0}})\bm{\cdot}\hat{\bm{p}}\,|\Psi^{(0)}\rangle\;=\;0\hskip
14.22636pt.$ (127)

Our PT1 eigenvector correction reads as

$|\Psi^{(1)}\rangle\;=\;\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{{}_{A}\langle\psi_{j^{\prime}}|\,_{R}\langle({\bm{k}}\wp)|\,\left(-\frac{q}{mc}\,\hat{\bm{A}}({\bm{0}})\bm{\cdot}\hat{\bm{p}}\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}\;|\psi_{j^{\prime}}\rangle_{A}\,|({\bm{k}}\wp)\rangle_{R}\hskip
14.22636pt.$ (128)

Accordingly, our PT2 energy correction takes the form

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{\left|\,{}_{A}\langle\psi_{j^{\prime}}|\,_{R}\langle({\bm{k}}\wp)|\,\left(-\frac{q}{mc}\,\hat{\bm{A}}({\bm{0}})\bm{\cdot}\hat{\bm{p}}\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\,\right|^{2}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}$
$\displaystyle+$ ${}_{A}\langle\psi_{j}|\,_{R}\langle{\rm
vac}|\,\left(\frac{q^{2}}{2\,m\,c^{2}}\,\hat{\bm{A}}^{2}({\bm{0}})\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\hskip 14.22636pt.$

Denominators of the formulas (128) and (•) contain a small imaginary factor
$(+i\,\varepsilon)$ where $\varepsilon\to+0$.
Clearly, the mentioned $(+i\,\varepsilon)$ factor cures singularities
which otherwise would occur in (128) and (•) whenever $j\geq 1$,
at the resonant modal frequencies
$\hbar\omega_{k}=E_{j}^{A}\,-\,E_{j^{\prime}<j}^{A}$.
It is important to recall at this point that all the $j\geq 1$ atomic levels
are metastable
(decaying due to spontaneous emission).
Metastable states (resonances) are studied by the nonhermitian quantum
mechanics, see Ref. [2].
The just discussed insertion of the $(+i\,\varepsilon)$ factor into the
denominators of (128) and (•)
represents a common practice of the nonhermitian RSPT.
Note that the $(+i\,\varepsilon)$ factor makes $E^{(2)}$ complex valued for
$j\geq 1$.
$[\,$Negative imaginary part $\sim$ decay rate of the $j$-th level due to
spontaneous emission.$\,]$
Detailed justification for including the $(+i\,\varepsilon)$ term
in the present context of spontaneous emission can be found in my recent work,
see Ref. [3].

  * •

Proceeding further, we wish to evaluate the PT corrections (128) and (•) more
explicitly.
For this purpose we introduce shorthand notation

${\bm{p}}_{j^{\prime}j}\;=\;_{A}\langle\psi_{j^{\prime}}|\,\hat{\bm{p}}\,|\psi_{j}\rangle_{A}\,\hskip
14.22636pt.$ (130)

We also evaluate an action of $\hat{\bm{A}}({\bm{0}})$ and
$\hat{\bm{A}}^{2}({\bm{0}})$ on the vacuum state $|{\rm vac}\rangle_{R}$. One
has

$\hat{\bm{A}}({\bm{0}})\,|{\rm vac}\rangle_{R}\;=\;\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\frac{\hbar\,c^{2}}{\omega_{k}}}\;|({\bm{k}}\wp)\rangle_{R}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\hskip
14.22636pt;$ (131)

and

$\hat{\bm{A}}^{2}({\bm{0}})\,|{\rm
vac}\rangle_{R}\;=\;\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{4\,\pi^{2}}\,\frac{\hbar\,c^{2}}{\omega_{k}}\;|{\rm
vac}\rangle_{R}\;+\;{\rm
a\phantom{i}two\phantom{i}photon\phantom{i}term}\hskip 14.22636pt.$ (132)

Subsequently one may work out the relevant matrix elements

${}_{A}\langle\psi_{j^{\prime}}|\,_{R}\langle({\bm{k}}\wp)|\,\left(-\frac{q}{mc}\,\hat{\bm{A}}({\bm{0}})\bm{\cdot}\hat{\bm{p}}\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\;=\;-\frac{q}{2\,\pi\,m}\,\sqrt{\frac{\hbar}{\omega_{k}}}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\hskip
14.22636pt;$ (133)

and

${}_{A}\langle\psi_{j}|\,_{R}\langle{\rm
vac}|\,\left(\frac{q^{2}}{2\,m\,c^{2}}\,\hat{\bm{A}}^{2}({\bm{0}})\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\;=\;\frac{q^{2}}{2\,m\,c^{2}}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{4\,\pi^{2}}\,\frac{\hbar\,c^{2}}{\omega_{k}}\hskip
14.22636pt.$ (134)

  * •

Returning back to the PT corrections (128) and (•), we find that

$|\Psi^{(1)}\rangle\;=\;-\frac{q}{2\,\pi\,m}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\sqrt{\frac{\hbar}{\omega_{k}}}\,\frac{{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}}{\left(E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon\right)}\;|\psi_{j^{\prime}}\rangle_{A}\,|({\bm{k}}\wp)\rangle_{R}\hskip
14.22636pt;$ (135)

and

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\;\,\omega_{k}^{-1}\sum_{\wp}\,\frac{\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}$
$\displaystyle+$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\;\,\omega_{k}^{-1}\hskip 14.22636pt.$

These are the final explicit results of our naive RSPT.

  * •

From now on, we shall focus on examining more closely the PT2 energy
correction $E^{(2)}=(\ref{E-PT2-take-2-naive})$.
The term

$\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\;\,\omega_{k}^{-1}$ (137)

is obviously UV divergent, yet it does not depend upon the atomic label $j$.
If so, it can be discarded as a physically unimportant constant energy shift
affecting uniformly all the atomic levels.
$[\,$Exactly the same reasoning was implicitly encountered before in A.1,
$[\,$when we eliminated the divergent ZPE part of $\hat{H}_{{\rm
R}}=(\ref{H-R-def})$.$\,]$
After discarding the contribution (137), we are left with

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\;\,\frac{1}{\omega_{k}}\,\frac{1}{\Bigl{(}E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon\Bigr{)}}\;\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\hskip
14.22636pt.$ (138)

Since $\int_{{\mathbb{R}}^{d}}{\rm d}^{d}k=\int_{0}^{\infty}k^{d-1}\,{\rm
d}k\int_{{\mathbb{S}}^{d-1}}{\rm d}^{d-1}\Omega_{\bm{k}}$, one may even write

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\lim_{K\to+\infty}\,\int_{0}^{K}k^{d-1}\,{\rm
d}k\;\,\frac{1}{\omega_{k}}\,\frac{1}{\Bigl{(}E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon\Bigr{)}}$
(139) $\displaystyle\int_{{\mathbb{S}}^{d-1}}{\rm
d}^{d-1}\Omega_{\bm{k}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\hskip
14.22636pt.$

Observe now that the $k$-integral of equation (139)
refers to an integrand which behaves as $k^{d-3}$ for $k\to+\infty$. Hence:
$\circ$ For $d=3$, our energy correction $E^{(2)}=(\ref{E-PT2-take-4-naive})$
is UV divergent as $K^{+1}$.
$\circ$ $[\,$In passing we note that $\Im E^{(2)}$ actually does not diverge,
only $\Re E^{(2)}$ is a troublemaker.$\,]$
$\circ$ For $d=2$, our energy correction $E^{(2)}=(\ref{E-PT2-take-4-naive})$
is UV divergent as $\ln K$.
$\circ$ $[\,$In passing we note that $\Im E^{(2)}$ actually does not diverge,
only $\Re E^{(2)}$ is a troublemaker.$\,]$
Our naively calculated leading order RSPT energy correction
$E^{(2)}=(\ref{E-PT2-take-4-naive})$
turns out to be UV divergent!
Epic failure of the naive RSPT formulation which ignores mass renormalization!
The UV divergences just encountered for $d=2,3$ can be cured only
by applying the renormalized perturbation theory, see the Appendices B.3 and
B.4 below.

Appendix B.3: Renormalized perturbation theory for $\bm{d}\bm{=}\bm{2}$

  * •

We wish to implement now
the renormalized Rayleigh-Schrödinger perturbation theory (RSPT)
for the atomic level shifts, assuming $d=2$.
Consider our starting Hamiltonian $\hat{H}=(\ref{H-no-t})|_{d=2}$
supplemented by the mass renormalization prescription (119).
Our goal is to solve an eigenvalue problem (122)
by employing the standard RSPT in the coupling parameter $q$.
It is thus necessary to express $\hat{H}=(\ref{H-no-t})|_{d=2}$ as a power
series in $q$.
This is accomplished through writing

$\displaystyle\frac{1}{2\,m_{o}(\alpha)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\,m}\,\frac{m}{m_{o}(\alpha)}\;=\;\frac{1}{2\,m}\;\frac{1}{1\,-\,\frac{q^{2}}{4\,\pi\,mc^{2}}\ln\left(\frac{\alpha}{\alpha_{0}}\right)}\;=$
(140) $\displaystyle=$
$\displaystyle\frac{1}{2\,m}\;+\;\frac{1}{2\,m}\,\sum_{\ell=1}^{\infty}\left(\frac{q^{2}}{4\,\pi\,mc^{2}}\ln\left(\frac{\alpha}{\alpha_{0}}\right)\right)^{\ell}\hskip
14.22636pt.$

Entity
$\frac{1}{2\,m}\,\sum_{\ell=1}^{\infty}\left(\frac{q^{2}}{4\,\pi\,mc^{2}}\ln\left(\frac{\alpha}{\alpha_{0}}\right)\right)^{\ell}$
is the so called mass counterterm.
Note that the just displayed infinite series (140)
is not summable in an ordinary sense for $\alpha\to+\infty$, thus some readers
may feel uneasy here.
In these introductory lecture notes, we avoid delving into
mathematical peculiarities accompanying the summation (140).
Such a light hearted attitude is common for vast majority of the present day
QFT theorists,
who keep on using weird expansions of the form similar to (140) daily,
in their studies of high energy physics within (and beyond) the Standard
Model!
The Hamiltonian $\hat{H}=(\ref{H-no-t})|_{d=2}$ is then split into a sum

$\hat{H}\;=\;\hat{H}_{\rm free}\;+\;\hat{W}\hskip 14.22636pt;$ (141)

where $\hat{H}_{\rm free}=(\ref{H-free-def})$ is understood as the unperturbed
reference part, and

$\hat{W}\;=\;-\,\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\;+\;\frac{q^{2}}{2\,m\,c^{2}}\,\hat{\bm{A}}^{2}[\alpha]\;+\;\frac{q^{2}}{8\,\pi\,m^{2}c^{2}}\ln\left(\frac{\alpha}{\alpha_{0}}\right)\hat{\bm{p}}^{2}\;+\;{\cal
O}(q^{3})$ (142)

determines the (adequately renormalized) perturbation.
Note that the perturbation $\hat{W}=(\ref{W-Appendix-B.3})$ contains all the
mass counterterms!
This is what makes $\hat{W}=(\ref{W-Appendix-B.3})$ small compared to
$\hat{H}_{\rm free}=(\ref{H-free-def})$ for $q$ small enough.
$[\,$Our subsequent calculation of the atomic level shifts will confirm the
just presented anticipation.$\,]$
Note that the naive perturbation term $\hat{W}=(\ref{W-naive})$ from B.2 does
not contain any mass counterterms!
This is why $\hat{W}=(\ref{W-naive})$ represents an infinite (and in fact ill-
defined) contribution to $\hat{H}=(\ref{H-naive})$.
Much as in B.2, our unperturbed reference state
$|\Psi^{(0)}\rangle=(\ref{Psi-(0)-unperturbed})$
corresponds to an unperturbed energy eigenvalue $E^{(0)}=E_{j}^{A}$,
consistently with relation $(\ref{hat-H-free-eigenproblem})|_{n=0}$.

  * •

An application of the standard RSPT is now straightforward.
Our PT1 energy correction equals to

$E^{(1)}\;=\;\langle\Psi^{(0)}|\,-\,\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\;|\Psi^{(0)}\rangle\;=\;0\hskip
14.22636pt.$ (143)

Our PT1 eigenvector correction reads as

$|\Psi^{(1)}\rangle\;=\;\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\,\sum_{\wp}\,\frac{{}_{A}\langle\psi_{j^{\prime}}|\,_{R}\langle({\bm{k}}\wp)|\,\left(-\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}\;|\psi_{j^{\prime}}\rangle_{A}\,|({\bm{k}}\wp)\rangle_{R}\hskip
14.22636pt.$ (144)

Accordingly, our PT2 energy correction takes the form

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\,\sum_{\wp}\,\frac{\left|\,{}_{A}\langle\psi_{j^{\prime}}|\,_{R}\langle({\bm{k}}\wp)|\,\left(-\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\,\right|^{2}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}$
$\displaystyle+$ ${}_{A}\langle\psi_{j}|\,_{R}\langle{\rm
vac}|\,\left(\frac{q^{2}}{2\,m\,c^{2}}\,\hat{\bm{A}}^{2}[\alpha]\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}$ $\displaystyle+$
$\displaystyle\frac{q^{2}}{8\,\pi\,m^{2}c^{2}}\,\ln\left(\frac{\alpha}{\alpha_{0}}\right)\,_{A}\langle\psi_{j}|\,_{R}\langle{\rm
vac}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\,|{\rm vac}\rangle_{R}\hskip
14.22636pt.$

Denominators of the formulas (144) and (•) contain a small imaginary factor
$(+i\,\varepsilon)$ where $\varepsilon\to+0$.
Clearly, the mentioned $(+i\,\varepsilon)$ factor cures singularities
which otherwise would occur in (144) and (•) whenever $j\geq 1$,
at the resonant modal frequencies
$\hbar\omega_{k}=E_{j}^{A}\,-\,E_{j^{\prime}<j}^{A}$.
It is important to recall at this point that all the $j\geq 1$ atomic levels
are metastable
(decaying due to spontaneous emission).
Metastable states (resonances) are studied by the nonhermitian quantum
mechanics, see Ref. [2].
The just discussed insertion of the $(+i\,\varepsilon)$ factor into the
denominators of (144) and (•)
represents a common practice of the nonhermitian RSPT.
Note that the $(+i\,\varepsilon)$ factor makes $E^{(2)}$ complex valued for
$j\geq 1$.
$[\,$Negative imaginary part $\sim$ decay rate of the $j$-th level due to
spontaneous emission.$\,]$
Detailed justification for including the $(+i\,\varepsilon)$ term
in the present context of spontaneous emission can be found in my recent work,
see Ref. [3].

  * •

Proceeding further, we wish to evaluate the PT corrections (144) and (•) more
explicitly.
For this purpose we introduce shorthand notation

${\bm{p}}_{j^{\prime}j}\;=\;_{A}\langle\psi_{j^{\prime}}|\,\hat{\bm{p}}\,|\psi_{j}\rangle_{A}\,\hskip
14.22636pt.$ (146)

We also evaluate an action of $\hat{\bm{A}}[\alpha]$ and
$\hat{\bm{A}}^{2}[\alpha]$ on the vacuum state $|{\rm vac}\rangle_{R}$. One
has

$\hat{\bm{A}}[\alpha]\,|{\rm vac}\rangle_{R}\;=\;\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\frac{\hbar\,c^{2}}{\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\;|({\bm{k}}\wp)\rangle_{R}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\hskip
14.22636pt;$ (147)

and

$\hat{\bm{A}}^{2}[\alpha]\,|{\rm
vac}\rangle_{R}\;=\;\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\,\sum_{\wp}\,\frac{1}{4\,\pi^{2}}\,\frac{\hbar\,c^{2}}{\omega_{k}}\;e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\;|{\rm
vac}\rangle_{R}\;+\;{\rm
a\phantom{i}two\phantom{i}photon\phantom{i}term}\hskip 14.22636pt.$ (148)

Subsequently one may work out the relevant matrix elements

${}_{A}\langle\psi_{j^{\prime}}|\,_{R}\langle({\bm{k}}\wp)|\,\left(-\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\;=\;-\frac{q}{2\,\pi\,m}\,\sqrt{\frac{\hbar}{\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\hskip
14.22636pt;$ (149)

and

${}_{A}\langle\psi_{j}|\,_{R}\langle{\rm
vac}|\,\left(\frac{q^{2}}{2\,m\,c^{2}}\,\hat{\bm{A}}^{2}[\alpha]\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\;=\;\frac{q^{2}}{2\,m\,c^{2}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\,\sum_{\wp}\,\frac{1}{4\,\pi^{2}}\,\frac{\hbar\,c^{2}}{\omega_{k}}\;e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\hskip
14.22636pt.$ (150)

  * •

Returning back to the PT corrections (144)-(•), we find that

$|\Psi^{(1)}\rangle\;=\;-\frac{q}{2\,\pi\,m}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\,\sum_{\wp}\,\sqrt{\frac{\hbar}{\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\;\frac{{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}}{\left(E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon\right)}\;|\psi_{j^{\prime}}\rangle_{A}\,|({\bm{k}}\wp)\rangle_{R}\hskip
14.22636pt;$ (151)

and

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\frac{\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}$
$\displaystyle+$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\;+\;\frac{q^{2}}{8\,\pi\,m^{2}c^{2}}\,\ln\left(\frac{\alpha}{\alpha_{0}}\right)\,_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$

Writing down these explicit results
serves as an important intermediate step of our renormalized RSPT.

  * •

From now on, we shall focus on examining more closely the PT2 energy
correction $E^{(2)}=(\ref{E-PT2-take-2-d=2})$.
The term

$\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\;=\;\frac{q^{2}\,\hbar}{2\,\pi\,m\,c}\,\int_{0}^{\infty}{\rm
d}k\;\,e^{-\frac{k^{2}}{2\,\alpha}}\;=\;\frac{q^{2}\,\hbar}{2\,m\,c}\;\sqrt{\frac{\alpha}{2\,\pi}}$
(153)

is obviously UV divergent for $\alpha\to+\infty$, yet it does not depend upon
the atomic label $j$.
If so, it can be discarded as a physically unimportant constant energy shift
affecting uniformly all the atomic levels.
$[\,$Exactly the same reasoning was implicitly encountered before in A.1,
$[\,$when we eliminated the divergent ZPE part of $\hat{H}_{{\rm
R}}=(\ref{H-R-def})$.$\,]$
After discarding the contribution (153), we are left with

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\frac{\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}$
$\displaystyle+$
$\displaystyle\frac{q^{2}}{8\,\pi\,m^{2}c^{2}}\,\ln\left(\frac{\alpha}{\alpha_{0}}\right)\,_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$

Formula (•) can be redisplayed in an equivalent but more convenient appearance

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\,\left\\{\,\frac{1}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}\,+\,\frac{\chi(k,{\cal
K})}{\hbar\omega_{k}}\,\right\\}$ $\displaystyle-$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\,\frac{\chi(k,{\cal
K})}{\hbar\omega_{k}}$ $\displaystyle+$
$\displaystyle\frac{q^{2}}{8\,\pi\,m^{2}c^{2}}\,\ln\left(\frac{\alpha}{\alpha_{0}}\right)\,_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$

Here ${\cal K}>0$ stands for an as yet unspecified parameter,
the associated factor $\chi(k,{\cal K})$ equals to unity for $|{\bm{k}}|>{\cal
K}$ and vanishes otherwise.
We set also $\tilde{\chi}(k,{\cal K})=1-\chi(k,{\cal K})$.
Instead of (•) one may write even more conveniently

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\,\frac{\hbar\omega_{jj^{\prime}}^{A}\,\chi(k,{\cal
K})+\hbar\omega_{k}\,\tilde{\chi}(k,{\cal
K})}{\hbar\omega_{k}\,(\hbar\omega_{jj^{\prime}}^{A}-\hbar\omega_{k}+i\varepsilon)}$
$\displaystyle-$
$\displaystyle\frac{q^{2}}{4\,\pi^{2}\,m^{2}\,c^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\chi(k,{\cal
K})\;k^{-2}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}$
$\displaystyle+$
$\displaystyle\frac{q^{2}}{8\,\pi\,m^{2}c^{2}}\,\ln\left(\frac{\alpha}{\alpha_{0}}\right)\,_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$

We have adopted here a shorthand symbol

$\hbar\omega_{jj^{\prime}}^{A}\;=\;E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\hskip
14.22636pt.$ (157)

  * •

Our next effort is invested into simplifying further the second line of (•).
We have

$\sum_{j^{\prime}}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\;=\;\sum_{j^{\prime}}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{jj^{\prime}}\;{\bm{p}}_{j^{\prime}j}\bm{\cdot}{\bm{\varepsilon}}_{{\bm{k}}\wp}\;=\;_{A}\langle\psi_{j}|\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}\hat{\bm{p}}\;\hat{\bm{p}}\bm{\cdot}{\bm{\varepsilon}}_{{\bm{k}}\wp}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt;$ (158)

and subsequently

$\displaystyle\sum_{\wp}\sum_{j^{\prime}}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}$
$\displaystyle=$
${}_{A}\langle\psi_{j}|\,\sum_{\wp}\,\hat{\bm{p}}^{T}{\bm{\varepsilon}}_{{\bm{k}}\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}^{T}\,\hat{\bm{p}}\,|\psi_{j}\rangle_{A}\;=\;_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{T}\Bigl{(}{\mathbb{I}}-{\bm{\kappa}}{\bm{\kappa}}^{T}\Bigr{)}\,\hat{\bm{p}}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$ (159)

The previously established property (56) implies in turn

$\displaystyle\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\chi(k,{\cal
K})\;k^{-2}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\;=\;\pi\;_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\,\int_{{\cal
K}}^{\infty}k^{-1}\,e^{-\frac{k^{2}}{2\,\alpha}}\;{\rm d}k\;=$ (160)
$\displaystyle=$
$\displaystyle\pi\,_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\;\frac{1}{2}\,\ln\left(\frac{2\,\alpha}{{\cal
K}^{2}\,e^{\gamma}}\right)\;+\;{\cal O}(\alpha^{-1})\hskip 14.22636pt.$

Here $\gamma=0.577\cdots$ is the Euler-Mascheroni constant encountered already
in Appendix A.
After setting most conveniently

${\cal K}\;=\;\sqrt{2\;\alpha_{0}\;e^{-\gamma}}\hskip 14.22636pt;$ (161)

the second line of (•) exactly overcompensates the third line,
up to a nonvanishing ${\cal O}(\alpha^{-1})$ contribution.
This is how the logarithmically UV divergent parts of
$E^{(2)}=(\ref{E-PT2-take-5-d=2})$ overkill each other.
Logarithmic UV divergence of the naive RSPT outcome
is overcompensated by the logarithmically UV divergent mass counterterm.
We are thus left with the PT2 energy correction formula

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{2}}{\rm
d}^{2}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\,\frac{\hbar\omega_{jj^{\prime}}^{A}\,\chi(k,{\cal
K})+\hbar\omega_{k}\,\tilde{\chi}(k,{\cal
K})}{\hbar\omega_{k}\,(\hbar\omega_{jj^{\prime}}^{A}-\hbar\omega_{k}+i\varepsilon)}$
$\displaystyle+$ $\displaystyle{\cal O}(\alpha^{-1})\hskip 14.22636pt.$

  * •

Expression (•) can still be simplified, by using again the property (56).
Indeed, one writes

$\displaystyle\int_{0}^{2\pi}{\rm
d}\varphi_{\bm{k}}\;\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\;=\;\int_{0}^{2\pi}{\rm
d}\varphi_{\bm{k}}\;\,{\bm{p}}_{jj^{\prime}}\sum_{\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}^{T}\;{\bm{p}}_{j^{\prime}j}\;=$
(163) $\displaystyle=$ $\displaystyle\int_{0}^{2\pi}{\rm
d}\varphi_{\bm{k}}\;\,{\bm{p}}_{jj^{\prime}}\Bigl{(}{\mathbb{I}}-{\bm{\kappa}}{\bm{\kappa}}^{T}\Bigr{)}\,{\bm{p}}_{j^{\prime}j}\;=\;\phantom{2}\,\pi\;{\bm{p}}_{jj^{\prime}}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\;=\;\phantom{2}\,\pi\;\Bigl{|}\,{\bm{p}}_{jj^{\prime}}\Bigr{|}^{2}\hskip
14.22636pt.$

Correspondingly, $E^{(2)}=(\ref{E-PT2-take-6-d=2})$ boils down into

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi\,m^{2}\,c}\,\sum_{j^{\prime}\neq
j}\;\Bigl{|}\,{\bm{p}}_{jj^{\prime}}\Bigr{|}^{2}\int_{0}^{\infty}{\rm
d}k\;\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\;\,\frac{\hbar\omega_{jj^{\prime}}^{A}\,\chi(k,{\cal
K})+\hbar\omega_{k}\,\tilde{\chi}(k,{\cal
K})}{\hbar\omega_{k}\,(\hbar\omega_{jj^{\prime}}^{A}-\hbar\omega_{k}+i\varepsilon)}$
$\displaystyle+$ $\displaystyle{\cal O}(\alpha^{-1})\hskip 14.22636pt.$

Most importantly, moment of reflection reveals
that the r.h.s. of (•) remains finite when the limit of $\alpha\to+\infty$ is
taken.
Our finalized form of the resulting finite PT2 energy correction reads thus as
follows:

${\color[rgb]{0,0,1}E^{(2)}\;=\;\frac{q^{2}\,\hbar}{4\,\pi\,m^{2}\,c}\,\sum_{j^{\prime}\neq
j}\;\Bigl{|}\,{\bm{p}}_{jj^{\prime}}\Bigr{|}^{2}\int_{0}^{\infty}{\rm
d}k\;\,\frac{\hbar\omega_{jj^{\prime}}^{A}\,\chi(k,{\cal
K})+\hbar\omega_{k}\,\tilde{\chi}(k,{\cal
K})}{\hbar\omega_{k}\,(\hbar\omega_{jj^{\prime}}^{A}-\hbar\omega_{k}+i\varepsilon)}\hskip
14.22636pt;\hskip 14.22636pt[d=2]}$ (165)

with ${\cal K}=(\ref{cal-K-fixing-Appendix-4})$.
We may conclude Appendix B.3 by saying that the renormalized second order RSPT
has provided us finite and physically meaningful atomic level shifts for the
case of $d=2$.
Some parts of Appendix B.3 were a bit technical, but the output (165) is
worthy of an effort!

Appendix B.4: Renormalized perturbation theory for $\bm{d}\bm{=}\bm{3}$

  * •

We wish to implement now
the renormalized Rayleigh-Schrödinger perturbation theory (RSPT)
for the atomic level shifts, assuming $d=3$.
Consider our starting Hamiltonian $\hat{H}=(\ref{H-no-t})|_{d=3}$
supplemented by the mass renormalization prescription (120).
Our goal is to solve an eigenvalue problem (122)
by employing the standard RSPT in the coupling parameter $q$.
It is thus necessary to express $\hat{H}=(\ref{H-no-t})|_{d=3}$ as a power
series in $q$.
This is accomplished through writing

$\displaystyle\frac{1}{2\,m_{o}(\alpha)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\,m}\,\frac{m}{m_{o}(\alpha)}\;=\;\frac{1}{2\,m}\;\frac{1}{1\,-\,\frac{4}{3}\,\frac{q^{2}}{mc^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}}\;=\;\frac{1}{2\,m}\;+\;\frac{1}{2\,m}\,\sum_{\ell=1}^{\infty}\left(\frac{4}{3}\,\frac{q^{2}}{mc^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;\right)^{\ell}\;.\hskip
7.11317pt$ (166)

Entity
$\frac{1}{2\,m}\,\sum_{\ell=1}^{\infty}\left(\frac{4}{3}\,\frac{q^{2}}{mc^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\right)^{\ell}$
is the so called mass counterterm.
Note that the just displayed infinite series (166)
is not summable in an ordinary sense for $\alpha\to+\infty$, thus some readers
may feel uneasy here.
In these introductory lecture notes, we avoid delving into
mathematical peculiarities accompanying the summation (166).
Such a light hearted attitude is common for vast majority of the present day
QFT theorists,
who keep on using weird expansions of the form similar to (166) daily,
in their studies of high energy physics within (and beyond) the Standard
Model!
The Hamiltonian $\hat{H}=(\ref{H-no-t})|_{d=3}$ is then split into a sum

$\hat{H}\;=\;\hat{H}_{\rm free}\;+\;\hat{W}\hskip 14.22636pt;$ (167)

where $\hat{H}_{\rm free}=(\ref{H-free-def})$ is understood as the unperturbed
reference part, and

$\hat{W}\;=\;-\,\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\;+\;\frac{q^{2}}{2\,m\,c^{2}}\,\hat{\bm{A}}^{2}[\alpha]\;+\;\frac{2}{3}\,\frac{q^{2}}{m^{2}c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;\hat{\bm{p}}^{2}\;+\;{\cal
O}(q^{3})$ (168)

determines the (adequately renormalized) perturbation.
Note that the perturbation $\hat{W}=(\ref{W-Appendix-B.4})$ contains all the
mass counterterms!
This is what should make $\hat{W}=(\ref{W-Appendix-B.4})$ small compared to
$\hat{H}_{\rm free}=(\ref{H-free-def})$ for $q$ small enough.
$[\,$Our subsequent calculation of the atomic level shifts will test the just
presented anticipation.$\,]$
Note that the naive perturbation term $\hat{W}=(\ref{W-naive})$ from B.2 does
not contain any mass counterterms!
This is why $\hat{W}=(\ref{W-naive})$ represents an infinite (and in fact ill-
defined) contribution to $\hat{H}=(\ref{H-naive})$.
Much as in B.2 and B.3, our unperturbed reference state
$|\Psi^{(0)}\rangle=(\ref{Psi-(0)-unperturbed})$
corresponds to an unperturbed energy eigenvalue $E^{(0)}=E_{j}^{A}$,
consistently with relation $(\ref{hat-H-free-eigenproblem})|_{n=0}$.

  * •

An application of the standard RSPT is now straightforward.
Our PT1 energy correction equals to

$E^{(1)}\;=\;\langle\Psi^{(0)}|\,-\,\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\;|\Psi^{(0)}\rangle\;=\;0\hskip
14.22636pt.$ (169)

Our PT1 eigenvector correction reads as

$|\Psi^{(1)}\rangle\;=\;\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\,\sum_{\wp}\,\frac{{}_{A}\langle\psi_{j^{\prime}}|\,_{R}\langle({\bm{k}}\wp)|\,\left(-\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}\;|\psi_{j^{\prime}}\rangle_{A}\,|({\bm{k}}\wp)\rangle_{R}\hskip
14.22636pt.$ (170)

Accordingly, our PT2 energy correction takes the form

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\,\sum_{\wp}\,\frac{\left|\,{}_{A}\langle\psi_{j^{\prime}}|\,_{R}\langle({\bm{k}}\wp)|\,\left(-\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\,\right|^{2}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}$
$\displaystyle+$ ${}_{A}\langle\psi_{j}|\,_{R}\langle{\rm
vac}|\,\left(\frac{q^{2}}{2\,m\,c^{2}}\,\hat{\bm{A}}^{2}[\alpha]\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}$ $\displaystyle+$
$\displaystyle\frac{2}{3}\,\frac{q^{2}}{m^{2}c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;\;_{A}\langle\psi_{j}|\,_{R}\langle{\rm
vac}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\,|{\rm vac}\rangle_{R}\hskip
14.22636pt.$

Denominators of the formulas (170) and (•) contain a small imaginary factor
$(+i\,\varepsilon)$ where $\varepsilon\to+0$.
Clearly, the mentioned $(+i\,\varepsilon)$ factor cures singularities
which otherwise would occur in (170) and (•) whenever $j\geq 1$,
at the resonant modal frequencies
$\hbar\omega_{k}=E_{j}^{A}\,-\,E_{j^{\prime}<j}^{A}$.
It is important to recall at this point that all the $j\geq 1$ atomic levels
are metastable
(decaying due to spontaneous emission).
Metastable states (resonances) are studied by the nonhermitian quantum
mechanics, see Ref. [2].
The just discussed insertion of the $(+i\,\varepsilon)$ factor into the
denominators of (170) and (•)
represents a common practice of the nonhermitian RSPT.
Note that the $(+i\,\varepsilon)$ factor makes $E^{(2)}$ complex valued for
$j\geq 1$.
$[\,$Negative imaginary part $\sim$ decay rate of the $j$-th level due to
spontaneous emission.$\,]$
Detailed justification for including the $(+i\,\varepsilon)$ term
in the present context of spontaneous emission can be found in my recent work,
see Ref. [3].

  * •

Proceeding further, we wish to evaluate the PT corrections (170) and (•) more
explicitly.
For this purpose we introduce shorthand notation

${\bm{p}}_{j^{\prime}j}\;=\;_{A}\langle\psi_{j^{\prime}}|\,\hat{\bm{p}}\,|\psi_{j}\rangle_{A}\,\hskip
14.22636pt.$ (172)

We also evaluate an action of $\hat{\bm{A}}[\alpha]$ and
$\hat{\bm{A}}^{2}[\alpha]$ on the vacuum state $|{\rm vac}\rangle_{R}$. One
has

$\hat{\bm{A}}[\alpha]\,|{\rm vac}\rangle_{R}\;=\;\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\frac{\hbar\,c^{2}}{\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\;|({\bm{k}}\wp)\rangle_{R}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\hskip
14.22636pt;$ (173)

and

$\hat{\bm{A}}^{2}[\alpha]\,|{\rm
vac}\rangle_{R}\;=\;\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\,\sum_{\wp}\,\frac{1}{4\,\pi^{2}}\,\frac{\hbar\,c^{2}}{\omega_{k}}\;e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\;|{\rm
vac}\rangle_{R}\;+\;{\rm
a\phantom{i}two\phantom{i}photon\phantom{i}term}\hskip 14.22636pt.$ (174)

Subsequently one may work out the relevant matrix elements

${}_{A}\langle\psi_{j^{\prime}}|\,_{R}\langle({\bm{k}}\wp)|\,\left(-\frac{q}{mc}\,\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\;=\;-\frac{q}{2\,\pi\,m}\,\sqrt{\frac{\hbar}{\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\hskip
14.22636pt;$ (175)

and

${}_{A}\langle\psi_{j}|\,_{R}\langle{\rm
vac}|\,\left(\frac{q^{2}}{2\,m\,c^{2}}\,\hat{\bm{A}}^{2}[\alpha]\right)\,|\psi_{j}\rangle_{A}\,|{\rm
vac}\rangle_{R}\;=\;\frac{q^{2}}{2\,m\,c^{2}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\,\sum_{\wp}\,\frac{1}{4\,\pi^{2}}\,\frac{\hbar\,c^{2}}{\omega_{k}}\;e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\hskip
14.22636pt.$ (176)

  * •

Returning back to the PT corrections (170)-(•), we find that

$|\Psi^{(1)}\rangle\;=\;-\frac{q}{2\,\pi\,m}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\,\sum_{\wp}\,\sqrt{\frac{\hbar}{\omega_{k}}}\;e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\;\frac{{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}}{\left(E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon\right)}\;|\psi_{j^{\prime}}\rangle_{A}\,|({\bm{k}}\wp)\rangle_{R}\hskip
14.22636pt;$ (177)

and

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\frac{\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}$
$\displaystyle+$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\;+\;\frac{2}{3}\,\frac{q^{2}}{m^{2}c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;\;_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$

Writing down these explicit results
serves as an important intermediate step of our renormalized RSPT.

  * •

From now on, we shall focus on examining more closely the PT2 energy
correction $E^{(2)}=(\ref{E-PT2-take-2})$.
The term

$\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\;=\;\frac{q^{2}\,\hbar}{\pi\,m\,c}\,\int_{0}^{\infty}{\rm
d}k\;\,k\,e^{-\frac{k^{2}}{2\,\alpha}}\;=\;\frac{q^{2}\,\hbar}{\pi\,m\,c}\;\alpha$
(179)

is obviously UV divergent for $\alpha\to+\infty$, yet it does not depend upon
the atomic label $j$.
If so, it can be discarded as a physically unimportant constant energy shift
affecting uniformly all the atomic levels.
$[\,$Exactly the same reasoning was implicitly encountered before in A.1,
$[\,$when we eliminated the divergent ZPE part of $\hat{H}_{{\rm
R}}=(\ref{H-R-def})$.$\,]$
After discarding the contribution (179), we are left with

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\frac{\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}$
$\displaystyle+$
$\displaystyle\frac{2}{3}\,\frac{q^{2}}{m^{2}c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;\;_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$

Formula (•) can be redisplayed in an equivalent but more convenient appearance

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\,\left\\{\,\frac{1}{E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\,-\,\hbar\omega_{k}\,+\,i\,\varepsilon}\,+\,\frac{1}{\hbar\omega_{k}}\,\right\\}$
$\displaystyle-$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\,\frac{1}{\hbar\omega_{k}}$
$\displaystyle+$
$\displaystyle\frac{2}{3}\,\frac{q^{2}}{m^{2}c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;\;_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt;$

or even more conveniently as

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}\,\hbar}{4\,\pi^{2}\,m^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,\omega_{k}^{-1}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\,\frac{\hbar\omega_{jj^{\prime}}^{A}}{\hbar\omega_{k}\,(\hbar\omega_{jj^{\prime}}^{A}-\hbar\omega_{k}+i\varepsilon)}$
$\displaystyle-$
$\displaystyle\frac{q^{2}}{4\,\pi^{2}\,m^{2}\,c^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,k^{-2}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}$
$\displaystyle+$
$\displaystyle\frac{2}{3}\,\frac{q^{2}}{m^{2}c^{2}}\,\sqrt{\frac{\alpha}{2\,\pi}}\;\;_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$

We have adopted here a shorthand symbol

$\hbar\omega_{jj^{\prime}}^{A}\;=\;E_{j}^{A}\,-\,E_{j^{\prime}}^{A}\hskip
14.22636pt.$ (183)

  * •

Our next effort is invested into simplifying further the second line of (•).
We have

$\sum_{j^{\prime}}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\;=\;\sum_{j^{\prime}}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{jj^{\prime}}\;{\bm{p}}_{j^{\prime}j}\bm{\cdot}{\bm{\varepsilon}}_{{\bm{k}}\wp}\;=\;_{A}\langle\psi_{j}|\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}\hat{\bm{p}}\;\hat{\bm{p}}\bm{\cdot}{\bm{\varepsilon}}_{{\bm{k}}\wp}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt;$ (184)

and subsequently

$\displaystyle\sum_{\wp}\sum_{j^{\prime}}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}$
$\displaystyle=$
${}_{A}\langle\psi_{j}|\,\sum_{\wp}\,\hat{\bm{p}}^{T}{\bm{\varepsilon}}_{{\bm{k}}\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}^{T}\,\hat{\bm{p}}\,|\psi_{j}\rangle_{A}\;=\;_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{T}\Bigl{(}{\mathbb{I}}-{\bm{\kappa}}{\bm{\kappa}}^{T}\Bigr{)}\,\hat{\bm{p}}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$ (185)

The previously established property (45) implies in turn

$\displaystyle\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,k^{-2}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\;=\;\frac{8\,\pi}{3}\,_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\,\int_{0}^{\infty}e^{-\frac{k^{2}}{2\,\alpha}}\;{\rm
d}k\;=$ (186) $\displaystyle=$
$\displaystyle\frac{8\,\pi}{3}\,\sqrt{\frac{\pi\,\alpha}{2}}\,_{A}\langle\psi_{j}|\,\hat{\bm{p}}^{2}\,|\psi_{j}\rangle_{A}\hskip
14.22636pt.$

If so, then the second line of (•) exactly overcompensates the third line.
This is how the most abruptly UV divergent parts of
$E^{(2)}=(\ref{E-PT2-take-5})$ overkill each other.
The most abrupt UV divergence of our naive RSPT outcome (of order ${\cal
O}(\alpha^{1/2})$)
is overcompensated by the accordingly UV divergent mass counterterm.
We are thus left with the PT2 energy correction formula

$\displaystyle E^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{4\,\pi^{2}\,m^{2}\,c^{2}}\,\sum_{j^{\prime}}\,\int_{{\mathbb{R}}^{3}}{\rm
d}^{3}k\;\,k^{-2}\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\,\frac{\hbar\omega_{jj^{\prime}}^{A}}{(\hbar\omega_{jj^{\prime}}^{A}-\hbar\omega_{k}+i\varepsilon)}\hskip
14.22636pt.$ (187)

  * •

Expression (187) can still be simplified, by using again the property (45).
Indeed, one writes

$\displaystyle\int_{0}^{2\pi}{\rm
d}\varphi_{\bm{k}}\,\int_{0}^{\pi}\sin\vartheta_{\bm{k}}\;{\rm
d}\vartheta_{\bm{k}}\;\,\sum_{\wp}\,\Bigl{|}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\,\Bigr{|}^{2}\;=\;\int_{0}^{2\pi}{\rm
d}\varphi_{\bm{k}}\,\int_{0}^{\pi}\sin\vartheta_{\bm{k}}\;{\rm
d}\vartheta_{\bm{k}}\;\,{\bm{p}}_{jj^{\prime}}\sum_{\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}\,{\bm{\varepsilon}}_{{\bm{k}}\wp}^{T}\;{\bm{p}}_{j^{\prime}j}\;=$
(188) $\displaystyle=$ $\displaystyle\int_{0}^{2\pi}{\rm
d}\varphi_{\bm{k}}\,\int_{0}^{\pi}\sin\vartheta_{\bm{k}}\;{\rm
d}\vartheta_{\bm{k}}\;\,{\bm{p}}_{jj^{\prime}}\Bigl{(}{\mathbb{I}}-{\bm{\kappa}}{\bm{\kappa}}^{T}\Bigr{)}\,{\bm{p}}_{j^{\prime}j}\;=\;\frac{8\,\pi}{3}\;{\bm{p}}_{jj^{\prime}}\bm{\cdot}{\bm{p}}_{j^{\prime}j}\;=\;\frac{8\,\pi}{3}\;\Bigl{|}\,{\bm{p}}_{jj^{\prime}}\Bigr{|}^{2}\hskip
14.22636pt.$

Our finalized form of the resulting PT2 energy correction reads thus as
follows:

${\color[rgb]{0,0,1}E^{(2)}\;=\;\frac{2}{3\,\pi}\,\frac{q^{2}}{m^{2}c^{2}}\,\sum_{j^{\prime}}\,\Bigl{|}\,{\bm{p}}_{jj^{\prime}}\Bigr{|}^{2}\,\int_{0}^{\infty}{\rm
d}k\;\,e^{-\frac{k^{2}}{2\,\alpha}}\,\frac{\hbar\omega_{jj^{\prime}}^{A}}{(\hbar\omega_{jj^{\prime}}^{A}-\hbar\omega_{k}+i\varepsilon)}\hskip
14.22636pt.\hskip 14.22636pt[d=3]}$ (189)

The $k$-integral of equation (189) diverges for $\alpha\to+\infty$ as
$\ln\alpha$.
Showing that our PT2 energy correction $E^{(2)}=(\ref{E-PT2-take-7})$
turns out to be logarithmically UV divergent even after the mass
renormalization.
Note however that $E^{(2)}=(\ref{E-PT2-take-7})$ is robustly insensitive with
respect to changing $\ln\alpha$
on the scales $\alpha\sim\lambda^{-2}$ where $\lambda=(\ref{lambda-Compton})$
is the Compton wavelength of the electron.
In fact, our above derived formula $E^{(2)}=(\ref{E-PT2-take-7})$ does provide
physically meaningful results
for the atomic level shifts, provided only that one employs any finite cutoff
value of $\alpha\sim\lambda^{-2}$.
The just discussed logarithmic divergence of $E^{(2)}=(\ref{E-PT2-take-7})$
represents a characteristic feature/artefact of the non-relativistic QED
theory.
As a matter of fact, the studied atomic level shifts (more precisely, their
relative differences)
come out finite for $d=3$ only within the fully relativistic QED treatment,
where the vacuum polarization effects are properly accounted for,
and where also the charge $q$ of the electron is renormalized.
$[\,$Such a treatment is presented e.g. in Ref. [4].
$[\,$The underlying technical details are painfully tedious,
$[\,$much as in any other calculation involving relativistic QFT.
$[\,$Yet computational complexity is the price one needs to pay.$\,]$
We may conclude Appendix B.4 by saying that the renormalized second order RSPT
has provided us a logarithmically divergent (yet still useful) result
for the atomic level shifts in the case of $d=3$.
We are done with introducing the renormalized RSPT,
and ready to proceed into Appendix C which deals with the renormalized mean
field theory!

Appendix C: Renormalized mean field theory

Appendix C.1: Introducing the game

  * •

Suppose that we wish to explore in more detail the quantum dynamics of our
model problem
by calculating explicitly the time evolution of some chosen physical
observables.
$[\,$Like e.g. the time dependent velocity $\bm{\dot{}}{\hat{\bm{x}}}(t)$ or
acceleration $\bm{\ddot{}}{\hat{\bm{x}}}(t)$ of the electron,
$[\,$and the electric field $\hat{\bm{E}}(t,{\bm{y}})$ affected by motion of
the electron.$\,]$
One possible strategy would be to solve explicitly
the renormalized Newton-Heisenberg equation of motion (•) or (92),
starting from the initial conditions (28), (29), (31).
Such an approach, based upon the Heisenberg picture, seems however to be
impractical or even untractable from the computational (numerical) point of
view.
$[\,$Unless one decides to apply perturbation expansion in the electronic
charge $q_{t}$.
$[\,$Yet the power of perturbation methods is not unlimited.
$[\,$It is important to explore also nonperturbative aspects of the
theory.$\,]$

  * •

Another possibility is to switch into the Schrödinger picture,
and aim at resolving explicitly the time dependent Schrödinger equation
(TDSCHE)

$i\hbar\,\partial_{t}\,|\Psi_{t}\rangle\;=\;\hat{H}_{t}\,|\Psi_{t}\rangle\hskip
14.22636pt;$ (190)

starting from an appropriate initial condition $|\Psi_{t_{0}}\rangle$ of our
interest.
Here $\hat{H}_{t}=(\ref{H})$ represents of course the full Hamiltonian
defining our model system in Section A.
Recall that formula (7) is supplemented by the mass renormalization
prescription (87) or (91).
Recall also that equations (7) & (87) or (91) contain an UV regulator
$\alpha$, which needs to be pushed
towards $+\infty$ at the very end of every computation of a physically
relevant observable quantity.

  * •

The Hamiltonian $\hat{H}_{t}=(\ref{H})$ can be redisplayed in an equivalent
form

$\displaystyle\hat{H}_{t}$ $\displaystyle=$
$\displaystyle\frac{\hat{\bm{p}}^{2}}{2\,m_{o}^{t}(\alpha)}\;-\;\frac{q_{t}}{m_{o}^{t}(\alpha)\,c}\;\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\;+\;\frac{q_{t}^{2}}{2\,m_{o}^{t}(\alpha)\,c^{2}}\;\hat{\bm{A}}^{2}[\alpha]\;+\;V(\hat{\bm{x}})\;+\;\hat{H}_{{\rm
R}}\hskip 14.22636pt.$ (191)

Here

$\hat{\bm{A}}[\alpha]\;=\;\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{2\,\pi}\,\sqrt{\frac{\hbar\,c^{2}}{\omega_{k}}}\;\,e^{-\frac{{\bm{k}}^{2}}{4\,\alpha}}\,\hat{a}_{{\bm{k}}\wp}\;{\bm{\varepsilon}}_{{\bm{k}}\wp}\;+\;{\rm
c.c.}\hskip 14.22636pt;$ (192)

exactly as in equation (121) of Appendix B, and $\hat{H}_{{\rm R}}=(\ref{H-R-
def})$.
The squared vector potential term $\hat{\bm{A}}^{2}[\alpha]$ can conveniently
be normal ordered, one gets

$\hat{\bm{A}}^{2}[\alpha]\;=\;\bm{:}\hat{\bm{A}}^{2}[\alpha]\bm{:}\;+\;\,\varsigma\,\hat{1}\hskip
14.22636pt;$ (193)

where by definition

$\varsigma\;=\;\int_{{\mathbb{R}}^{d}}{\rm
d}^{d}k\,\sum_{\wp}\,\frac{1}{4\,\pi^{2}}\,\frac{\hbar\,c^{2}}{\omega_{k}}\;\,e^{-\frac{{\bm{k}}^{2}}{2\,\alpha}}\hskip
14.22636pt.$ (194)

The constant contribution $\varsigma\,\hat{1}$ (which is UV divergent for
$\alpha\to+\infty$)
is discarded consonantly with our common practice,
cf. our elimination of the divergent ZPE part of $\hat{H}_{{\rm R}}=(\ref{H-R-
def})$ in A.1,
and similar steps taken in Appendix B $[\,$equations (137), (153), (179)$\,]$.
Correspondingly, our finalized Hamiltonian $[\,$entering into the TDSCHE
(190)$\,]$ reads as follows:

$\displaystyle\hat{H}_{t}$ $\displaystyle=$
$\displaystyle\frac{\hat{\bm{p}}^{2}}{2\,m_{o}^{t}(\alpha)}\;-\;\frac{q_{t}}{m_{o}^{t}(\alpha)\,c}\;\hat{\bm{A}}[\alpha]\bm{\cdot}\hat{\bm{p}}\;+\;\frac{q_{t}^{2}}{2\,m_{o}^{t}(\alpha)\,c^{2}}\;\bm{:}\hat{\bm{A}}^{2}[\alpha]\bm{:}\;+\;V(\hat{\bm{x}})\;+\;\hat{H}_{{\rm
R}}\hskip 14.22636pt.$ (195)

  * •

An explicit numerical solution of the TDSCHE (190) represents a worthy
challenge.
Difficulties do arise not only due to an  enormous number of the degrees of
freedom involved
(in fact, the radiation field possesses an infinite number of degrees of
freedom),
but also due to the  UV divergencies
entering into the TDSCHE (190) via (7) & (87) or (91) in the limit of
$\alpha\to+\infty$.
In the present lecture notes, we prefer not to pursue an ambitious (yet
feasible) program
of resolving the TDSCHE (190) in a numerically exact manner without
approximations.
Instead, we attack equation (190) by employing the  approximative mean field
(Hartree) method. The mean field formalism must of course be properly adapted,
in order to cope with the UV divergencies just mentioned above.
This is explained in a self contained fashion below, see Appendix C.2 and
Appendix C.3.
As we shall see shortly,
the resulting renormalized mean field method (in which the limit of
$\alpha\to+\infty$ is already taken)
does not contain any UV divergencies,
and the corresponding algorithm is suitable for numerical implementation.
See the summary given in Appendix C.4.
$[\,$It is not the purpose of Appendix C to comment on an (in)capability of
the mean field approach
$[\,$to describe various physical phenomena accommodated by our studied model.
$[\,$Our goal is mainly to  illustrate how is the concept of mass
renormalization implemented
$[\,$into a nonperturbative (and approximative) numerical method.$\,]$

Appendix C.2: Standard developments of the mean field theory

  * •

Within the mean field (Hartree) method,
the sought solution $|\Psi_{t}\rangle$ of the TDSCHE (190) is approximated by
a product ansatz

$|\Psi_{t}\rangle\;=\;|\psi_{t}\rangle_{A}\;|\beta_{t}\rangle_{R}\hskip
14.22636pt.$ (196)

Here $|\psi_{t}\rangle_{A}$ stands for the pertinent unit normalized dynamical
atomic state vector,
and $|\beta_{t}\rangle_{R}$ represents an (unit normalized) coherent state of
the radiation field.
Recall that $|\beta_{t}\rangle_{R}$ is an eigenstate of all the modal
annihilation operators, one has

$\hat{a}_{{\bm{k}}\wp}\,|\beta_{t}\rangle_{R}\;=\;\beta_{{\bm{k}}\wp}(t)\,|\beta_{t}\rangle_{R}\hskip
14.22636pt.$ (197)

The corresponding modal eigenvalues $\beta_{{\bm{k}}\wp}(t)$ characterize
completely $|\beta_{t}\rangle_{R}$.
More information on the concept of coherent states can be found e.g. in Ref.
[5].
$[\,$In passing we note that an ansatz (196) is almost always well justified
in the (semi)classical regime,
$[\,$where our model atom undergoes driving by a laser pulse. Yet important
exceptions do occur either.
$[\,$See Ref. [5] for details.$\,]$
Appropriate equations of motion for $|\psi_{t}\rangle_{A}$ and
$|\beta_{t}\rangle_{R}$ will be determined below
by employing the Dirac–Frenkel variational principle [6], which demands having

$\delta\,\int_{t_{1}}^{t_{2}}\langle\Psi_{t}|\,\hat{H}_{t}\,-\,i\hbar\,\partial_{t}\,|\Psi_{t}\rangle\;{\rm
d}t\;=\;0\hskip 14.22636pt.$ (198)

Recall that in (198) one tacitly assumes
$\delta\,|\Psi_{t_{1}}\rangle\,=\,|\emptyset\rangle\,=\,\delta\,|\Psi_{t_{2}}\rangle$
and similarly for the bras.

  * •

It is a straightforward matter to write down an explicit formula for our
Dirac–Frenkel functional

${\cal
S}\;=\;\int_{t_{1}}^{t_{2}}\langle\Psi_{t}|\,\hat{H}_{t}\,-\,i\hbar\,\partial_{t}\,|\Psi_{t}\rangle\;{\rm
d}t\hskip 14.22636pt.$ (199)

One just combines (195), (196), (197), (198)
and takes advantage of the well known textbook properties of the coherent
states
listed e.g. in Ref. [5]. This yields