###### Abstract

In an atom, the interaction of a bound electron with the vacuum fluctuations
of the electromagnetic field leads to complex shifts in the energy levels of
the electron, with the real part of the shift corresponding to a shift in the
energy level and the imaginary part to the width of the energy level. The most
celebrated radiative shift is the Lamb shift between the $2S_{1/2}$ and the
$2P_{1/2}$ levels of the hydrogen atom. The measurement of this shift in 1947
by Willis Lamb Jr. proved that the prediction by Dirac theory that the energy
levels were degenerate was incorrect. Hans Bethe’s calculation of the shift
demonstrated the renormalization process required to deal with the divergences
plaguing the existing theories and led to the understanding that it was
essential for theory to include interactions with the zero-point quantum
vacuum field. This was the birth of modern quantum electrodynamics (QED).
Other calculations of the Lamb shift followed by Welton and Power in an effort
to clarify the physical mechanisms leading to the shift. We have done a
calculation of the shift using a group theoretical approach which gives the
shift as an integral over frequency of a function, which we call the spectral
density of the shift. The spectral density reveals how different frequencies
contribute to the total energy shift. We find, for example, that half the
radiative shift for the ground state 1S level in H comes from photon energies
below 9700 eV, and that the expressions by Power and Welton do not have the
correct low frequency behavior, although they do give approximately the
correct value for the total shift.

###### keywords:

Bethe; radiative shift; shift spectral density; spectral density; vacuum
fluctuations; vacuum field; mass renormalization; Lamb shift; QED; radiative
reaction; zero point fluctuations; hydrogen atom

00 0 0 Received: 22 April 2022; Published: 19 October 2022; Updated: 30 May,
2023 yes New Insights into the Lamb Shift: The Spectral Density of the Shift
G. Jordan MaclayG. Jordan Maclay Correspondence:
<EMAIL_ADDRESS>
0000-0003-4901-1942

##  1 Introduction

In astronomy, in quantum theory, in quantum electrodynamics (QED), there have
been periods of great progress in which solutions to challenging problems have
been obtained, and the fields have moved forward. However, in some cases
getting the right answers can still leave fundamental questions unanswered.
The Big Bang explained the origin of the cosmic background radiation, but left
the problem of why the universe appears to be made of matter and not equal
amounts of matter and antimatterastro . In quantum theory, we can compute the
behavior of atoms yet we cannot describe a measurement in a self-consistent
way, or make sense of the collapse of a photon wavefunction from a near
infinite volume to a pointespeg . In quantum electrodynamics we can compute
the Lamb shift of the H atom to 15 decimal placesbeyer , yet we are left with
the paradox of using perturbation theory to remove infinite terms, or to
understand a quantum vacuum with infinite energy. In this paper, we examine
different approaches to the computation of the non-relativistic Lamb shift.
For all these approaches, the Lamb shift can be expressed in different ways as
an integral over frequency of a spectral density. We analyze the differences
in the spectral densities for the different approaches as a function of
frequency and compare the spectral densities to those obtained by using a
group theoretical analysis. The integral of the spectral density over all
frequencies gives the corresponding value of the Lamb shift.

Feynman called the the three page long 1947 non-relativistic Lamb shift
calculation by Hans Bethe the most important calculation in quantum
electrodynamics because it tamed the infinities plaguing earlier attempts.
When the sum over all states is evaluated numerically, it gives a finite
prediction that agreed with experimentlam1 bethe . Dirac said it
"fundamentally changed the nature of theoretical physics." Yet when this
calculation is explored more deeply, questions arise about it and about other
calculations of the Lamb shift, for example those by Welton welton  and
Powerpower , that employ different methods that have different low frequency
behavior from Bethe’s result yet give approximately the same value for the
level shift mil . These three approaches to the Lamb shift and the
corresponding vacuum energy densities have also been considered in passante .

There is an intimate relationship between radiative shifts and vacuum
fluctuations. The shift can be interpreted as arising from virtual transitions
induced by the quantum fluctuations of the electromagnetic field. Since the
vacuum field contains all frequencies, virtual transitions to all states,
bound and scattering, are possible. These short lived virtual transition
result in a slight shift in the average energy of the atom, a shift which we
call the Lamb shift gjmradiative . We note that the Lamb shift can also be
described as an interaction of the electron with its own radiation field,
yielding the same results as the vacuum fieldmil .

Bethe’s calculation was based on second order perturbation theory applied to
the minimal coupling of the atom with the vacuum field $(e/mc)\bm{A\cdot p}$
and a dipole approximation. This interaction leads to the emission and
absorption of virtual photons corresponding to virtual transitions. The shift
is expressed as a sum over the intermediate states reached by virtual
transitions. The predicted shift is divergent, but Bethe subtracted the term
that corresponded to the linearly divergent vacuum energy shift for a free
bare electron, essentially doing a mass renormalization to remove this higher
order divergence in the spectral density for the shift. For S states, the
resulting spectral density has a 1/frequency behavior for high frequencies
giving a logarithmic divergence in the shift.

Welton’s model for computing the Lamb shift was based on the perturbation of
the motion of a bound electron in the H atom due to the quantum vacuum
fluctuations altering the location of the electron, which resulted in a slight
shift of the bound state energy welton mil gjmradiative . This simplified
intuitive model predicts a spectral density proportional to 1/frequency for
all frequencies and a shift only for S states. The approach of Feynmanfeyn ,
interpreted by Power power , considers a large box containing H atoms and is
based on the shift in the energy in the quantum vacuum field due to the change
in the index of refraction arising from the presence of H atoms. This approach
predicts that the shift in the energy in the vacuum field around the H atoms
exactly equals the radiative shift predicted by Bethe for all energy mil
passante . It gives a spectral density with the same high frequency dependence
as Bethe, but a different low frequency dependence. A similar calculation to
Power’s models the Lamb shift as a Stark shift mil .

The Lamb shift has been previously computed using O(4) symmetry lieber  and by
a different approach from ours using SO(4,2) symmetry huff . We present the
results of a calculation of the Lamb shift that is based on a SO(4,2) group
theoretical analysis of the H atom that allows us to determine the dependence
of the shift on frequency with no sum over statesgjmdynam . The degeneracy
group of the non-relativistic H atom is O(4), with generators angular momentum
operator $\bm{L}$ and Runge-Lenz vector $\bm{A}$. A representation of O(4) of
dimension $n^{2}$ exists for each value of the principal quantum number $n$,
where the angular momentum $L$ has values from 0 to $n-1$, and there are
$2L+1$ possible values of $L_{z}=m$. If we extend this group by adding a 4
vector of generators we get the non-invariance group SO(4,1) which has
representations that include all states of different $n$ and $L$ and operators
that connect states with different principal quantum numbers. Adding a 5
vector of additional generators gives the group SO(4,2) and allows us to
express Schrodinger’s equation in terms of the new generators, and to make
effective group theoretical calculations gjmdynam . We use basis states that
allow us to include both bound and scattering states seamlessly brow  and no
sum over states appears in the final expression for the spectral density. One
advantage of this approach is that for each energy level we can easily compute
a spectral density for the shift whose integral over frequency from 0 to
$mc^{2}/\hbar$ is the radiative shift that includes transitions to all
possible states. Thus we can see how different frequencies of the vacuum field
contribute to the radiative shift.

We compare the different approaches of Bethe, Welton and Power to the group
theoretical spectral density of the non-relativistic Lamb shift for the 1S
ground state, the 2S and 2P levels. With this new picture of the Lamb shift,
we have found differences between the various approaches. Knowing the spectral
density of the shift provides new insights into understanding the Lamb shift.

##  2 Background of Radiative Shift Calculations

The first calculation of the Lamb shift of a hydrogen atom was done by Bethe
in 1947, who assumed the shift was do the interaction of the atom with the
vacuum field. He calculated the shift using second order perturbation theory,
assuming that there was minimal coupling in the Hamiltonian:

$H_{int}=-\frac{e}{mc}\textbf{A}\cdot\textbf{p}+\frac{e^{2}}{2mc^{2}}\textbf{A}^{2}$
(1)

where $m$, $e$, and p are the mass, charge and momentum of the electron, $c$
is the speed of light in vacuum, and A is the vector potential in the dipole
approximation for the vacuum field in a large quantization volume $V$

$\textbf{A}=\sum_{\textbf{k},\lambda}(\frac{2\pi\hbar
c^{2}}{\omega_{k}V})^{1/2}(a_{\textbf{k},\lambda}+a^{\dagger}_{\textbf{k},\lambda})\textbf{e}_{\textbf{k}\lambda}$
(2)

where the sum is over the virtual photon wave number $\mathbf{k}$, where
$kc=\hbar\omega_{k}$, the energy of the virtual photon, and the polarization
$\lambda$; $a_{\textbf{k},\lambda}$ and $a^{\dagger}_{\textbf{k},\lambda}$ are
the annihilation and creation operators, and $\textbf{e}_{\textbf{k}\lambda}$
is a unit vector in the direction of polarization of the electric field. The
shift from the $\textbf{A}^{2}$ term is independent of the state of the atom
and is therefore neglected. The total shift $\Delta E_{nTot}$ for energy level
n of the atom in state $|n\rangle$ is given by second order perturbation
theory as mil

$\Delta
E_{nTot}=-\frac{2}{3\pi}\frac{\alpha}{m^{2}c^{2}}\sum_{m}|\textbf{p}_{mn}|^{2}\int\frac{EdE}{E_{m}-E_{n}+E}$
(3)

where the integral is over the quantum vacuum field energy $E=\hbar\omega$ and
the momentum matrix elements are $|\textbf{p}_{mn}|=|\langle
m|\textbf{p}|n\rangle|$. The sum is over all intermediate states $|m\rangle$,
scattering and bound, where $m\neq n$. The fine structure constant is
$\alpha=e^{2}/\hbar c$.

The integrand in Eq. 3 has a linear divergence. Bethe observed that this
divergence in Eq. 3 corresponded to the integral that occurs when the binding
energy vanishes $(E_{m}-E_{n})\rightarrow 0$ and the electrons are free:

$\Delta
E_{free}=-\frac{2}{3\pi}\frac{\alpha}{m^{2}c^{2}}\sum_{m}|\textbf{p}_{mn}|^{2}\int
dE.$ (4)

He subtracted this divergent term $\Delta E_{free}$ from the total shift
$\Delta E_{nTot}$

$\Delta E_{nL}=\Delta E_{nTot}-\Delta E_{free}$ (5)

to obtain a finite observable shift $\Delta E_{nL}$ for the state $|nL\rangle$

$\Delta
E_{nL}=\frac{2\alpha}{3\pi(mc)^{2}}\sum_{m}^{s}|\textbf{p}_{mn}|^{2}\int_{0}^{\hbar\omega_{c}}dE\frac{(E_{m}-E_{n})}{E_{m}-E_{n}+E-i\epsilon},$
(6)

where $\omega_{C}$ is a cutoff frequency for the integration that Bethe took
as $\hbar\omega_{c}=mc^{2}$. Using an idea from Kramers, Bethe did this
renormalization, taking the difference between the terms with a potential
present and without a potential present, essentially performing the free
electron mass renormalization. He reasoned that relativistic retardation could
be neglected and the radiative shift could be reasonable approximated using a
non-relativistic approach and he cut the integration off at an energy
corresponding to the mass of the electron. He obtained a finite result that
required a numerical calculation over all states, bound and scattering, that
gave good agreement with measurements lam1 bethe bandsbook .

The spectral density in the Bethe formalism, which we will analyse, is the
quantity in Eq. 6 being integrated over $E$. It includes the sum over states
$m$. The term for $m$ represents the contribution to the Lamb shift for the
virtual transition from state $n$ to state $m$. Note since the ground state is
the lowest state, all intermediate states have higher energies so the ground
state shift has to be positive.

For the purposes of comparison to the other calculations of the Lamb shift it
is helpful to show the next steps Bethe took to evaluate the shift $\Delta
E_{n}$ for S states, which have the largest shifts. Note that the spectral
density we will analyse in Eq. 6 is not affected by the subsequent
approximations Bethe made to evaluate the integral. First the E integration is
done:

$\Delta
E_{n}^{Bethe}=\frac{2\alpha}{3\pi}(\frac{1}{mc})^{2}\sum_{m}|\textbf{p}_{nm}|^{2}(E_{m}-E_{n})ln\frac{(mc^{2}+E_{m}-E_{n})}{|E_{m}-E_{n}|}.$
(7)

To simplify the evaluation Bethe assumed $|E_{m}-E_{n}|<<mc^{2}$ in the
logarithm and that the logarithm would vary slowly with $m$ so it could be
replaced by an average value

$\widehat{\Delta{E}}_{n}^{Bethe}=\frac{2\alpha}{3\pi}(\frac{1}{mc})^{2}ln\frac{mc^{2}}{|E_{m}-E_{n}|_{Ave}}\sum_{m}|\textbf{p}_{nm}|^{2}(E_{m}-E_{n})$
(8)

where the hat over the $\Delta E$ indicates this is an approximation to Eq. 7.
Now that the E integration is done, the spectral density is no longer
manifest. The summation can be evaluated using the dipole sum rule

$2\sum_{m}^{s}|\textbf{p}_{nm}|^{2}\left(E_{m}-E_{n}\right)=\hbar^{2}\left\langle
n\left|\nabla^{2}V\right|n\right\rangle.$ (9)

The value of the Laplacian with a Coulomb potential V=$-Ze^{2}/r$ is
$\nabla^{2}V(r)=4\pi Ze^{2}\delta(\textbf{r})$ so we have

$\left\langle n\left|\nabla^{2}V\right|n\right\rangle=4\pi
Ze^{2}|\psi_{n}(0)|^{2},$ (10)

where $\psi(r)$ is the wave function for a Coulomb potential and
$|\psi_{n}(0)|^{2}$ is zero except for $S$ states

$|\psi_{n}(0)|^{2}=\frac{1}{\pi}\big{(}\frac{Z\alpha mc}{n\hbar}\big{)}^{3}.$
(11)

For S states, this gives an energy shift equal to mil :

$\widehat{\Delta{E}}_{n}^{Bethe}=\frac{4mc^{2}}{3\pi}\alpha(Z\alpha)^{4}\frac{1}{n^{3}}ln\frac{mc^{2}}{|E_{m}-E_{n}|_{Ave}}.$
(12)

where the so called Bethe log for an S states with principal quantum number n
is

$ln\frac{mc^{2}}{|E_{m}-E_{n}|_{Ave}}=\frac{\sum_{m}|\textbf{p}_{nm}|^{2}(E_{m}-E_{n})ln\frac{mc^{2}}{|E_{m}-E_{n}|}}{\sum_{m}|\textbf{p}_{nm}|^{2}\left(E_{m}-E_{n}\right)}$
(13)

where the sum is over all states, bound and scattering. Bethe also has
extended the formalism for shifts for states that are not S states bandsbook .

Regarding the approximations Bethe made to obtain Eq. 8 from Eq. 7 and the use
of the Bethe log Eq. 13, he commented: "The important values of
$|E_{m}-E_{n}|$ will be of order of the ground state binding energy for a
hydrogenic atom. This energy is very small compared to $mc^{2}$ so the log [in
our Eq. 7] is very large and not sensitive to the exact value of
$(E_{m}-E_{n})$. In the numerator we neglect $(E_{m}-E_{n})$ altogether and
replace it by an average energy bandsbook ." Our work shows that Bethe was
correct that the relative contribution from energies of the order of the
ground state is very important, but we find the contribution from higher
energy scattering states is very significant, and therefore that the
approximation $|E_{m}-E_{n}|<<mc^{2}$ is not valid for higher energy
scattering states for which $E_{m}$ increases to the value $mc^{2}$. We are
not aware of any quantitative estimates of the error in the approximation. The
difference, $0.3\%$, between our value for the total 1S shift and that of
Bethe may be due to this approximation, although we have not verified this. On
the other hand Bethe’s approximation may have made his non-relativistic
approach viable.

To provide a more intuitive physical picture of the shift, Welton considered
the effect of a zero-point vacuum field on the motion of an electron bound in
a coulomb potential $V(\textbf{r})$ at a location r. The perturbation
$\xi$=$(\xi_{x},\xi_{y},\xi_{z})$ in the position of the bound electron due to
the random zero-point vacuum field $\textbf{E}_{0}$ causes a variation in the
potential energy

$V(\textbf{r}+\textbf{$\xi$})=V(\textbf{r})+\textbf{$\xi$}\cdot\textbf{$\nabla$}V(\textbf{r})+\frac{1}{2}\left({\textbf{$\xi$}\cdot\textbf{$\nabla$}}\right)^{2}V(\textbf{r})...$
(14)

Because of the harmonic time dependence of the vacuum field,
$\langle\textbf{$\xi$}\rangle$ vanishes and the radiative shift is given
approximately by the vacuum expectation value of the last term:

$\Delta
E^{Welton}_{n}=\frac{\langle\textbf{$\xi$}^{2}\rangle}{6}\left\langle\nabla^{2}\mathrm{V}(\vec{\mathrm{r}})\right\rangle_{n}$
(15)

where we assume the potential has spherical symmetry, thus
$\langle\xi_{x}^{2}\rangle=\langle\xi_{y}^{2}\rangle=\langle\xi_{z}^{2}\rangle=\langle{\textbf{$\xi$}}^{2}/{3}\rangle$.
Eq. 15 gives $\Delta E_{n}^{Welton}$ as the product of two factors, the first
depending on the nature of the fluctuations in the position of the bound
electron due to the vacuum field and the second depending on the structure of
the system. $\xi$ is determined by $m$$\ddot{\xi}$=$e\textbf{E}_{0}$. With a
Fourier decomposition of $E_{0}$ and $\xi$, and integrating over the frequency
distribution of the vacuum field, we obtain the vacuum expectation valuemil
gjmradiative

$\langle(\vec{\xi})^{2}\rangle=\frac{2\alpha}{\pi}(\frac{\hbar}{mc})^{2}\int_{0}^{mc^{2}}\frac{dE}{E}.$
(16)

Using the results in Eqs. 10 and 11 we can evaluate the Laplacian in Eq. 15
and obtain a shift for S states equal to mil :

$\Delta
E_{n}^{Welton}=\frac{4mc^{2}}{3\pi}\alpha(Z\alpha)^{4}\frac{1}{n^{3}}\int_{0}^{mc^{2}}\frac{dE}{E}.$
(17)

Eq. 17 shows that the spectral density for the Welton approach is proportional
to 1/E. For the upper limit of integration, we take $mc^{2}$ as Bethe did. The
lower limit of 0 gives a divergent shift. Sometimes a lower limit of the
ground state energy is taken. On the other hand, if we happen to compare Eq.
17 to Eq. 12, we see that if we take for the lower limit the Bethe log Eq. 13,
we get exactly the same total S state shift as in the approximate Bethe
formalism Eq. 12. With these limits, the RMS amplitude of oscillation of the
electron bound in the Coulomb potential $\sqrt{\langle(\vec{\xi})^{2}\rangle}$
is about 72 fermis, about 1/740 of the mean radius of the 1S electron orbit.

Feynman proposed another approach for computing the Lamb shift based on a
fundamental observation about the interaction of matter and the vacuum
fieldfeyn . He considered a large box containing a low density of atoms in the
quantum vacuum. The atoms cause a change in the index of refraction, which
leads to changes in the frequencies of the vacuum field. The wavelengths
remain the same. He maintained that the change in the energy of the zero point
vacuum field in the box due to the frequency changes resulting from a weak
perturbing background of atoms acting as a refracting medium would correspond
to the self energy of the atoms, which is precisely the Lamb shift.

Power, based on the suggestion by Feynman, considered the change in vacuum
energy when N hydrogen atoms are placed in a volume V, using the Kramers-
Heisenberg expression for the index of refraction $n(\omega_{k})$power mil .
The H atoms cause a change in the index of refraction and therefore a change
in the frequencies of the vacuum fluctuations present. The corresponding
change in vacuum energy $\Delta E$ is

$\Delta
E=\sum_{k}\frac{1}{n(\omega_{k})}\frac{1}{2}\hbar\omega_{k}-\frac{1}{2}\hbar\omega_{k}$
(18)

where the sum is over all frequencies $\omega_{k}$ present. For a dilute gas
of atoms in a level n, the index of refraction is mil

$n(\omega_{k})=1+\frac{4\pi
N}{3\hbar}\sum_{m}\frac{\omega_{mn}|\textbf{d}|_{mn}^{2}}{\omega^{2}_{mn}-\omega_{k}^{2}}$
(19)

where $\omega_{mn}=(E_{m}-E_{n})/\hbar$ and
$\textbf{d}_{mn}=e\textbf{x}_{mn}$, the transition dipole moment. After
substituting $n(\omega_{k})$ into Eq. 18, we get a divergent result for the
energy shift. Following Bethe’s approach, Power subtracted from $\Delta E$ the
energy shift for the N free electrons, which equals the shift when
$\omega_{mn}\rightarrow 0$, with no binding energy. After making this
subtraction and converting the sum over $\omega_{k}$ to an integral over
$\omega$, and letting $NV\rightarrow 1$ the observable shift in energy is
obtainedmil :

$\Delta E_{n}^{Power}=-\frac{2}{3\pi
c^{3}}\sum_{m}\omega_{mn}^{3}|\textbf{d}_{mn}|^{2}\int_{0}^{mc^{2}/\hbar}\frac{d\omega\omega}{\omega_{mn}^{2}-\omega^{2}}.$
(20)

Noting that

$\langle m|\frac{\textbf{p}}{m}|n\rangle=\frac{i}{\hbar}\langle
m|[H,\textbf{x}]|n\rangle=\frac{i}{\hbar}(E_{m}-E_{n})\langle
m|\textbf{x}|n\rangle$ (21)

we can show

$|\textbf{p}_{mn}|^{2}=m^{2}\omega_{mn}^{2}|\textbf{x}_{mn}|^{2}=\frac{m^{2}\omega_{mn}^{2}}{e^{2}}|\textbf{d}_{mn}|^{2}.$
(22)

This allows us to write Power’s result Eq. 20 as

$\Delta E_{n}^{Power}=-\frac{2e^{2}}{3\pi
m^{2}c^{3}}\sum_{m}\omega_{mn}|\textbf{p}_{mn}|^{2}\int_{0}^{mc^{2}/\hbar}\frac{d\omega\omega}{\omega_{mn}^{2}-\omega^{2}}.$
(23)

Writing this equation in terms of $E=\hbar\omega$ instead of $\omega$ yields

$\Delta
E_{n}^{Power}=-\frac{2\alpha}{3\pi}(\frac{1}{mc})^{2}\sum_{m}|\textbf{p}_{mn}|^{2}(E_{m}-E_{n})\int_{0}^{mc^{2}}\frac{EdE}{(E_{m}-E_{n})^{2}-E^{2}}$
(24)

We will use this equation to analyze the spectral density for Power’s method,
showing the spectral density is different from Bethe’s at low frequencies but
the same at high frequencies. When Eq. 24 is integrated with respect to E,
taking the principal value, we obtain

$\Delta
E_{n}^{Power}=\frac{2\alpha}{3\pi}(\frac{1}{mc})^{2}\sum_{m}|\textbf{p}_{mn}|^{2}(E_{m}-E_{n})ln[{\frac{mc^{2}+(E_{m}-E_{n})}{E_{m}-E_{n}}\times\frac{mc^{2}-(E_{m}-E_{n})}{E_{m}-E_{n}}]^{1/2}}.$
(25)

Except for the argument in the $ln$ function, which corresponds to the upper
limit of integration, this is the same as Bethe’s expression Eq. 7 for the
shift. If we assume $mc^{2}>>E_{m}-E_{n}$, as Bethe did, then both expressions
for the total shift are identical. It is clear, however, that this
approximation is not valid at high energies for the second factor in the $ln$
function in Eq. 25, which may even become less than one making the $ln$ term
negative. Feynman’s approach highlights the changes in the vacuum field energy
due to the interactions with the H atoms.

One assumption in the computation by Power is that the index of refraction in
the box containing the atoms is spatially uniform. We will return to this
assumptions and suggest a model that predicts, for a single atom, the changes
in the vacuum field energy as a function of position for each spectral
component of the radiative shift.

##  3 Spectral Density of the Lamb Shift

Our goal is to develop an expression for the energy shift of a level, in terms
of the generators of the group SO(4,2), that is an integral over frequency.
Then the integrand will be the spectral density of the shift, and group
theoretical techniques can be used to evaluate it gjmdynam . We derive a
generating function for the shifts for all levels. We first focus on the
ground state 1S level as an illustration of the results. At ordinary
temperatures and pressures, most atoms are in the ground state. The radiative
shift for the 1S level is gjmdynam

$\Delta{E}_{1}=\frac{4mc^{2}\alpha(Z\alpha)^{4}}{3\pi}\int_{0}^{\phi_{c}}{d}\phi{e}^{\phi}\sinh\phi\int_{0}^{\infty}{dse^{s{e^{-\phi}}}}\frac{d}{ds}\frac{1}{\left(\coth\frac{s}{2}+\cosh\phi\right)^{2}}$
(26)

where the dimensionless normalized frequency variable $\phi$ is defined as

$\phi=\frac{1}{2}ln[1+\frac{\hbar\omega}{|E_{1}|}]$ (27)

where $E_{1}$ is the ground state energy -13.6 eV. The cutoff $\phi_{c}$
corresponds to $E=\hbar\omega_{c}=mc^{2}$, 511 keV corresponding to the
electron mass.

The group theoretical expression for the Lamb shift Eq. 26 is directly derived
from the Klein-Gordon equations of motion using a non-relativistic dipole
approximation, assuming infinite proton mass, and minimal coupling with the
vacuum field. Basis states of $(1/Z\alpha)$ are used since they have no
scattering states and have the same quantum numbers as the usual bound energy
eigenstates gjmdynam . The level shift is obtained as the difference between
the mass renormalization for a spinless meson bound in the desired state and
the mass renormalization for a free meson. Second order perturbation theory is
not used. Near the end of the derivation an equation which is equivalent to
Bethe’s result Eq. 6 for the radiative shift can be derived by inserting a
complete set of Schrodinger energy eigenstates. Thus we expect the fundamental
results from Bethe’s spectral density (with no approximations) and the group
theoretical spectral density to be in agreement gjmradiative gjmdynam . For
convenience an explanation of the basis states used to derive Eq. 26 is given
in Appendix A, and the derivation of Eq. 26 is given in Appendix B since the
derivation in gjmdynam  is spread in steps throughout the paper as the group
theory methods are developed.

We can write Eq. 26 as an integral over $E=\hbar\omega$, which is the energy
of the vacuum field in eV, and evaluate the definite integral over $s$
analytically for different values of $E$. We measure the ground state Lamb
shift $\Delta E_{1}$ in eV so the spectral density of the shift $d\Delta
E_{1}/dE$ is measured in eV/eV which is dimensionless:

$\Delta E_{1}=\int_{0}^{mc^{2}}\frac{d\Delta E_{1}}{dE}dE$ (28)

where the ground state spectral density from Eq. 26 is

$\frac{d\Delta
E_{1}}{dE}=\frac{4\alpha^{3}}{3\pi}{e}^{-2\phi}\sinh\phi\int_{0}^{\infty}{dse^{s{e^{-\phi}}}}\frac{1}{\sinh(\frac{s}{2})^{2}}\frac{1}{\left(\coth\frac{s}{2}+\cosh\phi\right)^{3}}.$
(29)

Fig. 1 shows a logarithmic plot (ordinate is a log, abscissa is linear) of the
spectral density $\frac{d\Delta E_{1}}{dE}$ of the ground state Lamb shift
with Z=1 over the entire range of energy $E$ computed from Eq. 29 using
Mathematica. The spectral density is largest at the lowest energies, and
decreases monotonically by about 4 orders of magnitude as the energy increases
to 511 eV. The ground state shift is the integral of the spectral density from
energy 0 to 511 keV. Fig. 2 is a loglog plot (both ordinate and abscissa are
log) of the same information. The use of the loglog plot expands the energy
range for each decade, revealing that for energy above about 1000 eV the slope
is approximately -1, indicating that the spectral density is nearly
proportional to $1/E$. For energy below about 10 eV, the spectral density in
Fig. 2 is almost flat, corresponding to a linear decrease as energy increases,
with a maximum spectral density at the lowest energy computed, as shown in
Fig. 3.

Fig. 2 shows that there are essentially two different behaviors of the
spectral density. For values of the energy E of the vacuum field that are
about 10 eV and below, in the range of the changes in energy for bound state
transitions, the spectral density corresponds to the near horizontal portion
of the spectral density in Fig. 2, and when E is much larger than the bound
state energies, the spectral density goes as 1/E.

Fig. 3 shows linear plots (linear in ordinate and abscissa) of the spectral
density of the shift for the ground state computed from Eq. 29 for several
lower energy regions. Fig 3a shows a linear decrease in the spectral density
as the energy increases over the small energy interval plotted. Fig 3b show a
linear decrease of about 15% as the energy increases from 0 eV to 3 eV. Fig.
3c shows that the spectral density decreases by a factor of about 4 as the
energy increases from 0 eV to 100 eV. In the low frequency limit, the spectral
density decreases linearly from the asymptotic constant value as the energy
increases.

Figure 1: Plot of the log of the spectral density of the ground state Lamb
shift from the group theoretical expression Eq. 29 on the vertical axis versus
the energy in eV from 0 to 510 keV on the horizontal axis. Figure 2: This
loglog plot shows the log of the spectral density of the ground state shift
from the group theoretical expression Eq. 29 on the vertical axis versus the
log of the energy in eV. From about 0 eV to 10 eV, there is a slow linear
decrease in the spectral density. For energies above about 100 eV, the
behavior is dominated by a 1/energy dependence.

(a) Linear decrease in ground state spectral density at very low energies.
Note ordinate changes very little over small energy region plotted.

(b) Near linear change in ground state spectral density for visible and near
IR energies. The contribution to the total shift for energies below 3 eV is
about 0.7% .

(c) Ground state spectral density calculated for energies below 80 eV, which
contribute about $8.6\%$ to the total shift.

Figure 3: Linear plot of the ground state spectral density as a function of eV
calculated from group theory, plotted as a function of energy for low and mid
energies. From about 0 eV to 10 eV, the spectral density decreases linearly
from its maximum value at the origin which corresponds to 0 eV for all graphs.

From explicit evaluations, we will show in Section 4 that for shifts in S
states with principal quantum number n, the asymptotic spectral density for
large $E$ is proportional to $\alpha(Z\alpha)^{4}(1/n^{3})$, and show in
Section 5 that as the energy E goes to zero, the spectral density increases
linearly, reaching a maximum value that is proportional to
$\alpha(Z\alpha)^{2}(1/n^{2})$. An approximate fit to the ground state data in
Fig. 1 is

$\frac{d\Delta E_{1}^{Fit}}{dE}=A\frac{(1+e^{-BE})}{(E+C)}.$ (30)

where $A=4.4008\times 10^{-6}$, $B=0.08445$, $C=106.79$. The fit is quite good
at the asymptotes and within 10% over the entire energy range.

We can use the spectral density shown in Fig. 1 or 2 in order to determine the
contribution to the total ground state shift from different energy regions. If
we integrate the spectral density from 0 eV to energy $E$, we obtain the value
of the partial shift $\Delta_{1}(E)$ that these energies (0 eV to $E$ eV)
contribute to the total shift $\Delta E_{1}$ for the ground state. In Fig. 4
we have plotted $\Delta_{1}(E)/\Delta E_{1}$, which is the fraction of the
total shift $\Delta E_{1}$ due to the contributions from energies below $E$,
as a function of $E$. Fig. 4a shows that almost 80% of the shift comes from
energies below about 100,000 eV. Fig. 4b shows that about half the total shift
is from energies below 9050 eV. Fig. 4c shows that energies below 100 eV
contribute about 10% of the total shift. Energies below 13.6 eV contribute
about 2.5% while energies below 1 eV contribute about 1/4% of the total. As
Fig. 4c shows, the fraction of the total shift increases linearly for E<10 eV,
corresponding to the nearly horizontal portion of the shift density for E<10
eV, as shown in Fig. 2. The contribution to the total 1S shift for the visible
spectral interval 400-700 nm (1.770 eV to 3.10 eV) is about $1.00342\times
10^{-7}$ eV or about 3/10 % of the total shift.

The relative contribution to the total shift per eV is much greater for lower
energies. For example, half the 1S shift corresponds to energies 0 to 9000 eV,
but only about 0.2% corresponds to 500,000 to 509,000 eV. The largest
contribution to the shift per eV is at the lowest energies, which have the
steepest slope of the spectral density curve in Fig. 1, about 1000 times
greater than the slope for the largest values of the energy. But the total
range for the large energies, from 9050 to 510,000 is so large that the
absolute contribution to the total shift for large energies is significant.

(a) Fraction of the 1S shift due to energies from 0 to $E$ plotted versus $E$
on the abscissa, for $0<E<510$ keV.

(b) Fraction of the 1S shift due to energies below $E$ plotted versus $E$, for
$0<E<9000$ eV.

(c) Fraction of the 1S shift due to energies from 0 to $E$ plotted versus $E$
on the abscissa, for $0<E<100$ eV. Energies below 30 eV account for about 0.05
of the total shift. The variation is linear for E<10 eV.

Figure 4: The ordinate is the fraction of the ground state shift $\Delta
E_{1}$ due to vacuum field energies between 0 and E, plotted as a function of
E on the abscissa. This plot is obtained by integration of the spectral
density from Eq. 29, shown in Fig. 1. The plot is linear in the ordinate and
abscissa. The origin corresponds to (0,0) for all plots.

For the ground state Fig. 5 shows how the dominant terms for different $m$ in
the Bethe sum over states in Eq. 6 contribute to the full spectral density
obtained from group theory Eq. 29. Each such term in the Bethe sum could be
interpreted as corresponding to the shift resulting from virtual transitions
from state $n$ to state $m$ occurring due to the vacuum field. Each term shown
has a behavior similar to that of the full spectral density, but the
magnitudes decrease as the transition probabilities decrease.

Figure 5: This loglog plot shows the 1S spectral density from group theory Eq.
29 in black, and the contributions to this shift in the Bethe formalism for
the transition $1S\rightarrow 2P$ (blue), $1S\rightarrow 4P$ (red),
$1S\rightarrow 8P$ (green). The dashed blue line shows the high frequency
$1/E$ asymptote. The black line is the complete spectral density which is the
summation of the contributions from all transitions.

Fig. 6 shows the spectral densities for 1S (black) and 2S (orange) shifts. The
shapes are similar but the spectral density for the 1S shift is about eight
times as large at high frequencies and about four times as large at low
frequencies, factors that we will derive explicitly by considering the
asymptotic forms of the spectra density for S states with different principal
quantum numbers. Both have a $1/E$ high frequency behavior. The s integration
in the group theoretical calculation for the 2S state diverges for energies
below 10.2 eV due to a non-relativistic approximation, but the spectral
density of the shift can be obtained from a low energy approximation, Eq. 47,
to the group theory result, which we derive in Section 5.

Figure 6: This loglog plot shows the log of the group theoretical spectral
density for the 1S (black) and 2S (orange) shifts on the vertical axis versus
the log of the frequency in eV. The dashed orange curve below 1 eV is a 2S low
energy approximation Eq. 47 from group theory or the Bethe formula. The blue
is the largest single contribution in the Bethe formalism to the 2S shift for
the transition $2S\rightarrow 3P$. Figure 7: This loglog plot shows the log of
the absolute value of the spectral density on the vertical axis versus the log
of the frequency in eV for the 2S shift (orange), which goes as $1/E$ for
large $E$, and for the 2P shift (green), which goes as $1/E^{2}$ for large
$E$. At 511 keV, the 2P spectral density is about 5 orders of magnitude
smaller than the 2S spectral density. Below 20 eV, the absolute value of the
2P spectral density is greater than the 2S spectral density. Note that the 2P
spectral density is actually negative and the 2S spectral density is positive.
Figure 8: This loglog plot shows the log of the spectral density for the 2S
shift (orange) and the 2S-2P Lamb shift (blue) versus the log of the energy.
The solid black line is the $1/E$ asymptote.

We can define the spectral density $\frac{d\Delta E_{n}}{dE}$ for a state $n$
in a convenient form suggested by Eq. 29,

$\frac{d\Delta
E_{n}}{dE}=\frac{4\alpha^{3}}{3\pi}\int_{0}^{\infty}dsW_{n}(s,\phi_{n})\hskip
20.0pt\text{where}\hskip 10.0pt\phi_{n}=\ln{[1+\frac{E}{|E_{n}|}]}$ (31)

where the energy for state $n$ is $E_{n}=-mc^{2}(Z\alpha)^{2}/2n^{2}$. From
our group theoretical results, we have for the 2S-2P Lamb shift gjmdynam

$W_{2S-2P}(s,\phi_{2})=\frac{4e^{(2se^{-\phi_{2}}+\phi_{2})}\sinh^{3}(\phi_{2})\text{csch}^{2}\left(\frac{s}{2}\right)}{\left(\cosh(\phi_{2})+\coth\left(\frac{s}{2}\right)\right)^{5}}$
(32)

and for the 2P shift gjmdynam :

$W_{2P}(s,\phi_{2})=-\frac{e^{(2se^{-\phi_{2}}+\phi_{2})}\sinh(\phi_{2})\text{csch}^{4}\left(\frac{s}{2}\right)(\cosh(\phi_{2})\sinh(s)+\cosh(s)-3)}{2\left(\cosh(\phi_{2})+\coth\left(\frac{s}{2}\right)\right)^{5}}$
(33)

The spectral density of the 2P shift has a very different behavior from the
spectral density of the 2S shift (Fig. 7). It is negative and and it falls off
as $1/E^{2}$. The shift is negative because the dominant contribution to the
shift is from virtual transitions from the 2P state to the lower 1S state,
with an energy difference of about 10.2 eV. For frequencies below about 20 eV,
the absolute value of the spectral density of the 2P shift increases rapidly
in magnitude as the energy is reduced and is much bigger than the spectral
density for the 2S shift. The 2S shift cannot have a negative contribution
from the lower 1S state since the transition 2S->1S is forbidden by the
conservation of angular momentum. The classic Lamb shift arises from the
difference between the two spectral densities, so the negative 2P spectral
density actually increases the 2S-2P Lamb shift as the energy decreases (Fig.
8). The total 2P shift is about 0.3% percent of the 2S shift. Bethe also
computed a negative contribution for the shift from the 2P statebandsbook .

###  3.1 Comparing the Ground State Group Theoretical Lamb Shift Calculations
to Those of Bethe, Welton, and Feynman

Integrating the group theoretical spectral density Eq. 29 from near zero
energy $(5.4x10^{-7}$ eV) to 511 keV, about the rest mass energy of the
electron, gives the 1S shift of $3.4027x10^{-}5$ eV, in agreement with the
numerical result of Bethe and Salpeter summing over states and using the Bethe
log approximation, $3.392x10^{-}5$ eV, to about $0.3\%$ bethe .

Bethe and Salpeter reported that the ground state Bethe log Eq. 13, which is a
logarithmically weighted average value of the excitation of the energy levels
contributing to the radiative 1S shift, was 19.77 Ry or 269 eV bandsbook .
Because of the weighting, it is not clear how one should interpret this value,
other than it indicates that high energy photons and scattering states
contribute significantly to the shift. As we have noted, our group theoretical
method does not provide an equivalent weighted average value for direct
comparison.

Although the methods of Bethe, Welton, and Power as defined all give
approximately the same value for the 1S shift, which equals the integral of
the spectral density in our approach, they differ significantly in their
frequency dependence, which we will now examine.

##  4 The Spectral Density of The Lamb shift at High Frequency

The form for $d\Delta E_{n}/dE$, which is the Lamb shift spectral density for
level $n$, can be obtained at high energies from 1) the classic calculation by
Bethe using second order perturbation theory; 2) the calculation by Welton of
the Lamb shift; 3) the calculation of Power of the Lamb shift based on
Feynman’s approach; and 4) our group theoretical calculation.

The spectral density for level $n$ can be written from Bethe’s expression Eq.
6

$\frac{\Delta E_{n}^{Bethe}}{\Delta
E}=\frac{2\alpha}{3\pi}(\frac{1}{mc})^{2}\sum_{m}|\textbf{p}_{mn}|^{2}(E_{n}-E_{m})\frac{1}{E_{n}-E_{m}-E}.$
(34)

If we are evaluating the spectral density for the ground state $n=1$, $Z=1$,
then $E_{1}=-13.613$eV, and for the bound states $E_{m}=-13.613eV/m^{2}$. For
scattering states $E_{m}$ is positive. Hence the denominator is negative for
all terms in the sum over $m$ and never vanishes, and the spectral density is
positive, and the ground state shift is positive as it must be. For large
values of $E$, we can make the approximation

$\frac{\Delta E_{n}^{Bethe}}{\Delta
E}|_{E->\infty}=\frac{2\alpha}{3\pi}(\frac{1}{mc})^{2}\sum_{m}|\textbf{p}_{mn}|^{2}(E_{m}-E_{n})\frac{1}{E}.$
(35)

The summation can be evaluated using the dipole sum rule Eq. 9, and Eqs. 10
and 11 for the Coulomb S state wavefunction, obtaining the final result for
the high frequency spectral density for S states with principal quantum number
$n$

$\frac{d\Delta
E_{n}^{Bethe}}{dE}|_{E->\infty}=\frac{4mc^{2}}{3\pi}\alpha(Z\alpha)^{4}\frac{1}{n^{3}}\frac{1}{E}.$
(36)

The result highlights the $1/E$ divergence at high frequencies, and shows the
presence of a coefficient proportional to $1/n{{}^{3}}$. To put a scale on the
coefficient, we note that the high frequency spectral density can be written
as $(8/3\pi)(\alpha(Z\alpha)^{2}/n)(E_{n}/E)$.

The spectral density for all frequencies from Welton’s model, Eq. 17 , is
identical to this high frequency limit of Bethe’s calculation. Thus at low
frequencies, the spectral density for Welton’s calculation diverges as $1/E.$
Because of the expectation value of the Laplacian, Welton’s approach predicts
a shift only for S states. Its appeal is that it gives a clear physical
picture of the primary role of vacuum fluctuations in the Lamb shift and shows
the presence of the $1/E$ characteristic behavior. To obtain a level shift, it
requires providing a low energy limit for the integration. As we have noted,
if the lower limit is the Bethe’s log average excitation energy, 269 eV for
n=1, and the upper limit $mc^{2}$ then Welton’s total 1S shift agrees with
Bethe’s. A choice of this type works since 1) it does not include any
contributions from energies below 269 eV and 2) it gives a compensating
contribution for energies from 269 eV to about 1000 eV that is larger than the
actual spectral density, as shown in Fig. 4, and 3) above about 1000 eV,
Welton’s model gives the same $1/E$ spectral density as Bethe.

The spectral density for Power’s model can be obtained from Eq. 24

$\frac{\Delta
E_{n}^{Power}}{dE}=-\frac{2\alpha}{3\pi}(\frac{1}{mc})^{2}\sum_{m}|\textbf{p}_{mn}|^{2}(E_{m}-E_{n})\frac{E}{(E_{m}-E_{n})^{2}-E^{2}}$
(37)

Letting E become large, we see the result is identical to the high frequency
limit Eq. 35 for the Bethe formalism and the Welton model so we have

$\frac{\Delta
E_{n}^{Power}}{dE}|_{E->\infty}=\frac{4mc^{2}}{3\pi}\frac{\alpha(Z\alpha)^{4}}{n^{3}}\frac{1}{E}.$
(38)

Thus we find for S states a $1/E$ dependence of the high frequency spectral
density, corresponding to the logarithmic divergence at high frequency. We can
write this high energy theoretical result in a form allowing easy comparison
to the calculated group theoretical spectral density eV/eV:

$\frac{d\Delta
E_{n}^{Bethe}}{dE}|_{E->\infty}=\frac{4mc^{2}}{3\pi}\frac{\alpha(Z\alpha)^{4}}{n^{3}}\frac{1}{E}.$
(39)

The spectral density goes as $1/n^{3}$ for S states. For the ground state
$n=1,Z=1$ we have

$\frac{d\Delta E_{1}^{Bethe}}{dE}|_{E->\infty}=4.488\times 10^{-6}\frac{1}{E}$
(40)

A fit to the last two data points near 510 KeV in the group theoretical
calculations gives:

$\frac{d\Delta E_{1}^{GTcalc}}{dE}|_{E->\infty}=4.4008\times
10^{-6}\frac{1}{E}.$ (41)

The coefficients differ by about $2\%$. Fig. 9 is a plot of the ground state
group theoretical calculated spectral density (red) from Eq. 29 and the
theoretical high energy $1/E$ function from Bethe, Power and Welton, Eq. 40
(black), and the difference times a factor of 10. The asymptotic theoretical
result agrees with the full group theoretical calculation from Eq. 29 to
within about $2\%$ at 511 keV, and to about $6\%$ at 50 KeV. It is notable
that the high frequency form is a reasonable approximation down to 50 keV.
Indeed, the Welton approach is based on this observation; it has the same
$1/E$ energy dependence at all energies.

Figure 9: Top red curve is the 1S group theoretical calculated spectral
density Eq. 29, slightly lower black curve is the $1/E$ asymptotic model Eq.
39, and the bottom green curve is the difference times 10, plotted for the
interval 50-510keV. Both axes are linear.

##  5 Spectral Density of the Lamb Shift at Low Frequency

We can obtain a low frequency limit of the spectral density of the Lamb shift
from the Bethe spectral density Eq. 34. For small values of $E$, the spectral
density can be expanded to first order in E, giving

$\frac{\Delta
E_{n}^{Bethe}}{dE}|_{E->0}=\frac{2\alpha}{3\pi}(\frac{1}{mc})^{2}\sum_{m}|\textbf{p}_{mn}|^{2}(1-\frac{E}{E_{m}-E_{n}}).$
(42)

Since the sum is over a complete set of states $m$ including scattering states
we can evaluate the first term in parenthesis using the sum rule

$\sum_{m}|\textbf{p}_{mn}|^{2}=-2mE_{n}=(mc)^{2}\frac{(Z\alpha)^{2}}{n^{2}}.$
(43)

For the second term we use Eq. 22 and the Thomas-Reiche-Kuhn sum rule sakurai

$\sum_{m}\omega_{mn}|\textbf{d}_{mn}|^{2}=\frac{3e^{2}\hbar}{2m}$ (44)

to evaluate the resulting summation. The final result for $E\rightarrow 0$ is

$\frac{\Delta
E_{n}^{Bethe}}{dE}|_{E->0}=\frac{2\alpha}{3\pi}\frac{(Z\alpha)^{2}}{n^{2}}-\frac{\alpha}{\pi
mc^{2}}E.$ (45)

The corresponding spectral density for $n=1,Z=1$ is

$\frac{d\Delta E_{1}^{Bethe}}{dE}|_{E->0}=\frac{4\alpha\times 13.6}{3\pi
mc^{2}}(1-\frac{3E}{4\times 13.6})=8.253\times 10^{-8}(1-0.0551E).$ (46)

As E decreases to zero, the spectral density increases linearly to a constant
value $\frac{4\alpha}{3\pi}\frac{|E_{n}|}{mc^{2}}=2\alpha^{3}Z^{2}/3\pi
n^{2}=8.253\times 10^{-8}/n^{2}$. The intercept goes as $1/n^{2}$, but the
slope $\alpha/\pi mc^{2}$, which has a remarkable simple form, is independent
of $n$.

If we take the low frequency limit of the group theoretical result
analytically, we obtain exactly the same result as in Eq. 45 from the Bethe
formulation

$\frac{d\Delta E_{1}^{GTheory}}{dE}|_{E->0}=\frac{d\Delta
E_{1}^{Bethe}}{dE}|_{E->0}=\frac{2\alpha}{3\pi}\frac{(Z\alpha)^{2}}{n^{2}}-\frac{\alpha}{\pi
mc^{2}}E.$ (47)

Fig. 3 shows the results of group theoretical calculations of the spectral
density of the ground state Lamb shift for different energy regions, showing
the near linear increase in the spectral density as the frequency decreases
from 80 eV to $10^{-5}$ eV. For low values of E, the slopes and intercept
agree within about two tenth of a percent with the theoretical values from Eq.
47.

To explore Power’s approach at low frequency, we can let $E$ become very small
in the spectral density Eq. 37, giving

$\frac{\Delta E_{n}^{Power}}{dE}|_{E->0}=-\frac{2\alpha}{3\pi
mc^{3}}\sum_{m}|\textbf{p}_{mn}|^{2}\frac{E}{E_{m}-E_{n}}$ (48)

which is identical to the second term in the low E approximation to the Bethe
result Eq. 42 so we have:

$\frac{\Delta E_{n}^{Power}}{dE}=-\frac{1}{\pi}\frac{\alpha}{mc^{2}}E.$ (49)

This result Eq. 49 is identical to the frequency dependent term in Eq. 47,
which is the low frequency spectral density from the Bethe approach and from
the group theoretical expression. However, in the low frequency limit based on
Power’s expression for the spectral density, the constant term that is present
in the other approaches does not appear. This a consequence of the form used
for the index of refraction, which assumes that real photons are present that
can excite the atom with resonant transitions. More sophisticated
implementations of Feynman’s proposal may avoid this issue.

##  6 Comparison of the Spectral Energy Density of the Vacuum Field and the
Spectral Density of the Radiative Shift

The theory of Feynman proposes that the vacuum energy density in a large box
containing H atoms, which we assume are all in the 1S ground state, increases
uniformly with the addition of the atoms. He maintains that the total vacuum
energy in the box increases by the Lamb shift times the number of atoms
present. If we had one atom in a very large box, we would not expect the
change in energy density to be uniform but more concentrated near the atom. To
develop a model of the spatial dependence of the change in energy density for
one atom, we can use the close relationship between the vacuum field and the
radiative shift. The spectral densities of the ground state shift and of the
quantum vacuum with no H atoms present are both know. In the box the vacuum
field density must increase so that the integral gives the 1S Lamb shift. The
spectral energy density of the vacuum field with no H atom present is equal to
mil

$\rho_{0}(\omega)=\frac{\hbar\omega^{3}}{2\pi^{2}c^{3}}$ (50)

where c is the speed of light in cm/sec and $\omega$ is in $sec^{-1}$. If we
measure frequency in eV so $\hbar\omega=E$ then the vacuum spectral energy
density in $1/cc$ is

$\rho_{0}(E)=\frac{E^{3}}{2\pi^{2}\hbar^{3}c^{3}}.$ (51)

and $\int_{E_{1}}^{E_{2}}\rho_{0}(E)dE$ would be the energy density eV/cc in
the energy interval $E_{1}$ to $E_{2}$. The question we are addressing is:
what volume of vacuum energy of density $\rho_{0}(E)$ is required to supply
the amount of energy needed for the radiative shift? We can express the total
radiative shift $\Delta E_{1}$ as the integral of the vacuum energy density
$\rho_{0}(E)$ over an effective volume $V_{1}(E)$

$\Delta E_{1}=\int_{0}^{mc^{2}}dE\rho_{0}(E)V_{1}(E)$ (52)

where we use the same upper limit for $E$ as in all of our calculations.
Recall our definition of the spectral shift Eq. 28:

$\Delta E_{1}=\int_{0}^{mc^{2}}dE\frac{\Delta E_{1}}{dE}.$ (53)

By comparison of Eq. 52 and Eq. 53 we determine that to insure energy balance
at each energy E, the effective spectral volume $V_{1}(E)$ is

$V_{1}(E)=\frac{d\Delta E_{1}}{dE}\frac{1}{\rho_{0}(E)}.$ (54)

The spectral volume $V_{1}(E)$ has the dimensions of $cc$ and contains the
amount of vacuum energy at energy value $E$ that corresponds to the ground
state spectral density at the same energy $E$. In Fig. 10, for the 1S ground
state radiative shift, we plot the log of the spectral volume $V_{1}(E)$ on
the y-axis in units of cubic Angstroms versus the log of the energy $E$ in eV
on the x-axis.

Figure 10: This loglog plot shows the spectral volume $V_{1}(E)$ as a function
of $E$. The spectral volume $V_{1}(E)$ contains the free field vacuum energy
at energy value $E$ that corresponds to the ground state shift spectral
density at the same energy $E$.

For energies above about 100 eV, the spectral volume is less than 1 cubic
Angstrom, approximately the volume of the ground state wavefunction. For an
energy of 1 eV, the spectral volume is $11850A^{3}$, corresponding to a sphere
of radius about 14 A. This calculation predicts that there is a sphere of
positive vacuum energy of radius 14 A around the atom corresponding to the 1
eV shift spectral density. Fig. 11 shows the radius of the spherical spectral
volume $V_{1}(E)$ for energies from 0.05 eV, with spectral radius of 278 A, to
23 eV, with radius 0.49 A.

Figure 11: This plot shows the log of the radius in Angstroms of the spherical
spectral volume $V_{1}(E)$ as a function of the vacuum field energy E from
0.05 eV to 23 eV.

##  7 Conclusion

The non-relativistic Lamb shift can be interpreted as due to the interaction
of an atom with the fluctuating electromagnetic field of the quantum vacuum.
We introduce the concept of a spectral shift density which is a function of
frequency $\omega$ or energy $E=\hbar\omega$ of the vacuum field. The integral
of the spectral density from E=0 to the rest mass energy of an electron, 511
keV, gives the radiative shift. We report on calculations of the spectral
density of the level shifts for 1S, 2S and 2P states based on a group
theoretical analysis and compare the results to the spectral densities
implicit in previous calculations of the Lamb shift. The group theoretical
calculation provides an explicit form for the spectral density over the entire
spectral range. Bethe’s approach requires a summation over an infinite number
of states, all bound and all scattering, to obtain a comparable spectral
density. We compare all approaches for asymptotic cases, for very large and
very small energies E.

The calculations of the shift spectral density provide a new perspective on
radiative shifts. The group theory approach as well as the approaches of
Bethe, Power, and Welton all show the same $1/E$ high frequency behavior for S
states above about $E=\hbar\omega$= 1000 eV to E=511 keV, namely a spectral
density for S states equal to $(4/3\pi)(\alpha(Z\alpha)^{4}mc^{2}/n^{3})(1/E)$
for states with principal quantum number $n$. Since our group theory
calculation shows that about 76% of the ground state 1S shift is contributed
by E above 1000 eV, this is essentially why all the approaches give
approximately the same result for the 1S Lamb shift.

Only the Bethe and group theory calculations have the correct low frequency
behavior. We find that for S states the spectral density increases linearly as
E approaches zero. Its maximum value is at E=0 and for S states equals
$(2\alpha/3\pi)(Z\alpha)^{2}/n^{2}$. This maximum value is about
$1/(Z\alpha)^{2}$ or about $2\times 10^{4}$ larger than the high frequency
spectral density at E=510 keV. Thus low energies contribute much more to the
shift for a given spectral interval than the high energies. Energies below
13.6 eV contribute about 2.5 %. Because of the huge spectral range
contributing to the shift, contributions to the shift from high energies are
very important. Half the contribution to the 1S shift is from energies above
9050 eV.

The 2P shift has a very different spectral density from an S state: it is
negative and has an asymptotic behavior that goes as $1/E^{2}$ rather than as
$1/E$. Below about 20 eV, the absolute value of the 2P spectral density is
much larger than the 2S spectral density and it dominates the 2S-2P shift
spectral density, yet the total 2P shift is only about 0.3% of the total 2S
shift.

##  Appendix A : Eigenstates $|nlm;a)$ of $1/Z\alpha$

To obtain an equation for these basis states $|nlm;a)$ we write Schrodinger’s
equation for a charged non-relativistic particle with energy
$E=-\frac{a^{2}}{2m}$gjmdynam brow  in a Coulomb potential

$\left[p^{2}+a^{2}-\frac{2m\hbar cZ\alpha}{r}\right]|a\rangle=0.$ (A1)

There are solutions for $|a\rangle$ for certain critical values of the energy
$E_{n}=-\frac{a_{n}^{2}}{2m}$ or equivalently when $a=a_{n}$ where
$\frac{a_{n}}{mcZ\alpha}=\frac{1}{n}$. These are the usual energy eigenstates
which we label as $|nlm;a_{n}\rangle$. Conversely we can let $a$ be fixed in
value and let $Z\alpha$ have different values. If it has certain eigenvalues
$Z\alpha_{n}$ then for any value of $a$ we can have another set of
eigenvectors corresponding to eigenvalues
$\frac{a}{mcZ\alpha_{n}}=\frac{1}{n}$. To demonstrate this we start by
inserting factors of $1=\sqrt{ar}\frac{1}{\sqrt{ar}}$ in Schrodinger’s
equation Eq. A1 obtaining

$\left(\sqrt{ar}(p^{2}+a^{2})\sqrt{ar}-2amZ\alpha\right)\frac{1}{\sqrt{ar}}|a\rangle=0.$
(A2)

We can rewrite this equation, multiplying successively from the left by
$\frac{1}{\sqrt{ar}}$,$\frac{1}{p^{2}+a^{2}}$, and $\frac{1}{\sqrt{ar}}$, and
then multiplying by $a^{2}$, and dividing by $mcZ\alpha$, multiplying by
$\sqrt{n\hbar}$ obtaining

$\left(\frac{a}{mcZ\alpha}-K_{1}(a)\right)\sqrt{\frac{n\hbar}{ar}}|a\rangle=0$
(A3)

where

$K_{1}(a)=\frac{1}{\sqrt{ar}}\frac{2a^{2}\hbar}{p^{2}+a^{2}}\frac{1}{\sqrt{ar}}$
(A4)

There are solutions to this equation for eigenvalues of $1/Z\alpha$ such that
$\frac{a}{mcZ\alpha_{n}}=\frac{1}{n}$:

$\left(\frac{1}{n}-K_{1}(a)\right)|nlm;a)=0$ (A5)

where

$\sqrt{\frac{n\hbar}{ar}}|nlm;a\rangle=|nlm;a)$

The $n\hbar$ in the square root insures the new states are also normalized to
1. The kernel $K_{1}(a)$ is bounded and Hermetian with respect to the
eigenstates $|nlm;a)$ of $1/Z\alpha$, therefore these eigenstates of
$1/Z\alpha$ form a complete orthonormal basis for the hydrogen atom. Because
the kernel is bounded, there are no continuum states in this representation.
To show they have the same quantum numbers as the usual states, we note when
$a=a_{n}$ then the eigenstates of $K_{1}(a_{n})$ becomes $|nlm;a_{n})$ and
these corresponds to the usual energy eigenstates $|nlm;a_{n}\rangle$. We can
change the value of $a$ in Eq. A5 to obtain these eigenstates using the
dilation operator $D(\lambda)=e^{iS\lambda}$ where the dimensionless operator
S, which is also a generator of transformations of SO(4,2), is

$S=\frac{1}{2\hbar}(\bm{p}\cdot\bm{r}+\bm{r}\cdot\bm{p}).$ (A6)

When S operates on the canonical variables we obtain

$D(\lambda)\textbf{p}D^{-1}(\lambda)=e^{-\lambda}\textbf{p}$
$D(\lambda)\textbf{r}D^{-1}(\lambda)=e^{\lambda}\textbf{r}.$

Operating on $K_{1}(a)$ with $D(\lambda)$ we find

$D(\lambda)K_{1}(a)D^{-1}(\lambda)=K_{1}(ae^{\lambda}).$

We can pick $\lambda$ as

$\lambda_{n}=ln(a_{n}/a)$

so that $ae^{\lambda_{n}}=a_{n}$. Thus operating with $D(\lambda_{n})$ on Eq.
A5 we obtain

$\left(\frac{1}{n}-K_{1}(a_{n})\right)D(\lambda_{n})|nlm;a)=0.$ (A7)

This is the equation for the usual Schrodinger energy eigenstates so

$D(\lambda_{n})|nlm;a)=|nlm;a_{n})=\sqrt{\frac{n\hbar}{a_{n}r}}|nlm;a_{n}\rangle.$
(A8)

Thus the usual Schrodinger energy eigenstates $|nlm;a_{n}\rangle$ can be
expressed in terms of the eigenstates of $1/Z\alpha$ as

$|nlm;a_{n}\rangle=\sqrt{\frac{a_{n}r}{n\hbar}}D(\lambda_{n})|nlm;a).$ (A9)

The relationship shows that complete basis functions $|nlm;a)$ of $1/Z\alpha$
are proportional to the ordinary bound state energy wavefunctions and
therefore have the same quantum numbers as the ordinary bound statesgjmdynam
brow . A comparable set of $1/Z\alpha$ eigenstates useful for momentum space
calculations is derived in gjmdynam .

##  Appendix B : Derivation of Group Theoretical Formula for the Shift
Spectral Density

The group theoretical approach is based solely on the Schrodinger and Klein-
Gordon equations of motion in the non-relativistic dipole approximation. We
obtain a result gjmdynam

$\Delta E_{NL}=\frac{2\alpha}{3\pi(mc)^{2}}\int_{0}^{\hbar\omega_{c}}dE\langle
NL|p_{i}\frac{H-E_{N}}{H-(E_{N}-E)-i\epsilon}p_{i}|NL\rangle.$ (B1)

where $E=\hbar\omega$, $H=\frac{p^{2}}{2m}-\frac{Z\alpha\hbar c}{r}$ and the
states $|NL\rangle$ are the usual H atom energy eigenstates. $\omega_{C}$ is a
cutoff frequency for the integration that we will take as
$\hbar\omega_{c}=mc^{2}$. If we insert a complete set of states in this
expression we obtain Bethe’s result Eq. 6, a step we avoid with the group
theoretical approach. If we add and subtract $E$ from the numerator in Eq. B1,
we find the real part of the shift is

$\Delta
E_{NL}=\frac{2\alpha}{3\pi(mc)^{2}}Re\int_{0}^{\hbar\omega_{c}}dE[\langle
NL|p^{2}|NL\rangle-E\Omega_{NL}]$ (B2)

where

$\Omega_{NL}=\langle
NL|p_{i}\frac{1}{H-E_{N}+\hbar\omega-i\epsilon}p_{i}|NL\rangle.$ (B3)

We want to convert the matrix element $\Omega_{NL}$ to a matrix element of a
function of SO(4,2) generators taken between a new set of basis states
$|nlm;a)$, which are complete with no scattering states, where
$a=\sqrt{2m|E|}$, and n,l,m have their usual meaning and values. The new basis
states $|nlm;a)$ are eigenstates of $(Z\alpha)^{-1}$gjmdynam  brow . Sometimes
we simply write them as $|nlm)$ with the $a$ implicit.

We define a generator of SO(4,1) as
$\Gamma_{0}=1/K_{1}(a)=(1/2)(\frac{\sqrt{r}p^{2}\sqrt{r}}{a}+ar)$ so

$(\Gamma_{0}-n)|nlm;a)=0.$ (B4)

This is Schrodinger’s equation in the language of SO(4,2).

We need to define several more generators. Since the algebra of SO(4,2)
generators closes, commutators of generators must also be generators. To find
$\Gamma_{4}$, we calculate $\Gamma_{4}=-i[S,\Gamma_{0}]$, obtaining

$\Gamma_{4}=\frac{1}{2\hbar}\left(\frac{\sqrt{r}p^{2}\sqrt{r}}{a}-ar\right)\hskip
20.0pt\Gamma_{0}=\frac{1}{2\hbar}\left(\frac{\sqrt{r}p^{2}\sqrt{r}}{a}+ar\right)$
(B5)

where the generator $S$ is defined in Appendix A. The generators
$({\Gamma_{4},S,\Gamma_{0}})=(j_{1},j_{2},j_{3})$ form a O(2,1) subgroup of
SO(4,2) and $S=i[\Gamma_{4},\Gamma_{0}],\Gamma_{0}=-i[S,\Gamma_{4}]$ and for
our representations $\Gamma_{0}^{2}-\Gamma_{4}^{2}-S^{2}=\bm{L^{2}}=l(l+1)$.
The scale change S transforms $\Gamma_{0}\equiv\Gamma_{0}(a)$ according to the
equation

$e^{i\lambda S}\Gamma_{0}(a)e^{-i\lambda
S}=\Gamma_{0}(e^{\lambda}a)=\Gamma_{0}\cosh{\lambda}-\Gamma_{4}\sinh{\lambda}$
(B6)

and similarly

$e^{i\lambda S}\Gamma_{4}(a)e^{-i\lambda
S}=\Gamma_{4}(e^{\lambda}a)=\Gamma_{4}\cosh{\lambda}-\Gamma_{0}\sinh{\lambda}.$
(B7)

Finally we define a three vector of generators proportional to the momentum

$\Gamma_{i}=\frac{1}{\hbar}\sqrt{r}p_{i}\sqrt{r}.$ (B8)

The quantity
$\bm{\Gamma}=(\Gamma_{0},\Gamma_{1},\Gamma_{2},\Gamma_{3},\Gamma_{4})$ is a
five vector of generators under transformations generated by SO(4,2). For the
representation of SO(4,2) based on the states $|nlm)$, all generators are
Hermetian, and
$\bm{\Gamma}^{2}=\Gamma_{A}\Gamma^{A}=-\Gamma_{0}^{2}+\Gamma_{1}^{2}+\Gamma_{2}^{2}+\Gamma_{3}^{2}+\Gamma_{4}^{2}=1$
for our representation, and $g_{AB}=(-1,1,1,1,1)$ for $A,B=0,1,2,3,4$. The
commutators of the components of the five vector are also generators of
$SO(4,2)$ transformations.

Inserting factors of $1=\sqrt{ar}\frac{1}{\sqrt{ar}}$ and using the
definitions of the generators we can transform Eq. B3 to

$\Omega_{NL}=\frac{m\nu}{N^{2}}(NL|\Gamma_{i}\frac{}{\Gamma
n(\xi)-\nu}\Gamma_{i}|NL)$ (B9)

where

$n^{0}(\xi)=\frac{2+\xi}{2\sqrt{1+\xi}}=\cosh\phi\hskip 20.0ptn^{i}=0\hskip
20.0ptn^{4}(\xi)=-\frac{\xi}{2\sqrt{1+\xi}}=-\sinh\phi$ (B10)

and

$\xi=\frac{\hbar\omega}{|E_{N}|}\hskip
20.0pt\nu=\frac{N}{\sqrt{1+\xi}}=Ne^{-\phi}.$ (B11)

From the definitions we see $\phi=\frac{1}{2}ln(1+\xi)>0$ and
$n_{A}(\xi)n^{A}(\xi)=-1$. The contraction over $i$ in $\Omega_{NL}$ may be
evaluated using the group theoretical formula gjmdynam :

$\sum_{B}\Gamma_{B}f(n\Gamma)\Gamma^{B}=\frac{1}{2}(n\Gamma+1)^{2}f(n\Gamma+1)+\frac{1}{2}(n\Gamma-1)^{2}f(n\Gamma-1)-(n\Gamma)^{2}f(n\Gamma).$
(B12)

We apply the contraction formula to the the integral representation

$f(n\Gamma)=\frac{1}{\Gamma n-\nu}=\int_{0}^{\infty}dse^{\nu
s}e^{-n\Gamma\hskip 1.0pts}$ (B13)

and obtain the result

$\Gamma_{A}\frac{1}{\Gamma n-\nu}\Gamma^{A}=-2\nu\int_{0}^{\infty}ds\hskip
2.0pte^{\nu s}\frac{d}{ds}(\sinh^{2}\frac{s}{2}\hskip 2.0pte^{-n\Gamma\hskip
1.0pts}).$ (B14)

Applying this to our expression Eq. B9 for $\Omega_{NL}$ gives

$\displaystyle\Omega_{NL}$
$\displaystyle=-2\frac{m\nu^{2}}{N^{2}}\int_{0}^{\infty}dse^{\nu
s}\frac{d}{ds}\left(\sinh^{2}\frac{s}{2}M_{NL}(s)\right)$ (B15)
$\displaystyle\vspace{5pt}-m\frac{\nu}{N^{2}}(NL|\Gamma_{4}\frac{1}{\Gamma
n(\xi)-\nu}\Gamma_{4}|NL)+m\frac{\nu}{N^{2}}(NL|\Gamma_{0}\frac{1}{\Gamma
n(\xi)-\nu}\Gamma_{0}|NL)$

where

$M_{NL}(s)=(NL|e^{-\Gamma n(\xi)\hskip 1.0pts}|NL).$ (B16)

In order to evaluate the last two terms in Eq. B15 we use
$\Gamma_{0}|NL)=N|NL)$ and express the action of $\Gamma_{4}$ on our states as
$\Gamma_{4}=N-(1/\sinh{\phi})(\Gamma n(\xi)-\nu)$. This expression for
$\Gamma_{4}$ is derived from Eqs. B9 and B10: $\Gamma
n(\xi)-\nu=\Gamma_{0}\cosh{\phi}-\Gamma_{4}\sinh{\phi}-\nu$, and then
substituting Eq.B11, $\nu=Ne^{-\phi}$. Using the virial theorem
$(NLM|p^{2}|NLM)=a_{N}^{2}$, we find that the term in $p^{2}$ in Eq. B15
exactly cancels the last two terms in $\Omega_{NL}$, yielding the result for
the level shift

$\Delta E_{NL}=\frac{4mc^{2}\alpha(Z\alpha)^{4}}{3\pi
N^{4}}\int_{0}^{\phi_{c}}d\phi\sinh\phi e^{\phi}\int_{0}^{\infty}ds\hskip
2.0pte^{\nu s}\frac{d}{ds}\left(\sinh^{2}\frac{s}{2}M_{NL}(s)\right)$ (B17)

where

$\phi_{c}=\frac{1}{2}ln\left(1+\frac{\hbar\omega_{c}}{|E_{N}|}\right)=\frac{1}{2}ln\left(1+\frac{2N^{2}}{(Z\alpha)^{2}}\right).$
(B18)

We can derive a generating function for the shifts for any eigenstate
characterized by $N$ and $L$ if we multiply Eq.B17 by $N^{4}e^{-\beta N}$ and
sum over all $N,N\geq L+1$. To simplify the right side of the resulting
equation, we use the definition Eq. B16 and the fact that $\Gamma_{4}$, $S$,
and $\Gamma_{0}$ form an O(2,1) algebra so we have:

$\sum_{N={L}+1}^{\infty}e^{-\beta{N}}M_{NL}=\sum_{{N}={L}+1}^{\infty}({NL}|e^{-\bm{j}\cdot\bm{\psi}}|{NL}),$
(B19)

where

$e^{\hskip 1.0pt-\bm{j}\cdot\bm{\psi}}\equiv e^{-\beta\Gamma_{0}}e^{-s\Gamma
n(\xi)}.$ (B20)

We perform a $\bm{j}$ transformation generated by $e^{i\phi S}$, such that
$e^{-\bm{j}\cdot\psi}\rightarrow e^{-j_{3}\psi}=e^{-\Gamma_{0}\psi}$. The
trace is invariant with respect to this transformation so we have

$\sum_{N={L}+1}^{\infty}e^{-\beta{N}}M_{NL}=\sum_{{N}={L}+1}^{\infty}({NL}|e^{-j_{3}\psi}|{NL})=\sum_{N={L}+1}^{\infty}e^{-N\psi}=\frac{e^{-\psi(L+1)}}{1-e^{-\psi}},$
(B21)

where we have used $(NL|\Gamma_{0})|NL)=N$.

In order to find a particular $M_{NL}$, we must expand the right hand side of
the equation in powers of $e^{-\beta}$ and equate the coefficients to those on
the left hand side. Using the isomorphism between $\bm{j}$ and the Pauli
$\bm{\sigma}$ matrices
$(\Gamma_{4},S,\Gamma_{0})\rightarrow(j_{1},j_{2},j_{3})\rightarrow(\frac{i}{2}\sigma_{1},\hskip
2.0pt\frac{i}{2}\sigma_{2},\hskip 2.0pt\frac{1}{2}\sigma_{3})$ gives the
result

$\cosh{\frac{\psi}{2}}=\cosh{\frac{\beta}{2}}\cosh{\frac{s}{2}}+\sinh{\frac{\beta}{2}}\sinh{\frac{s}{2}}\cosh{\phi}.$
(B22)

Rewriting this equation gives

$e^{+\frac{1}{2}\psi}=de^{\frac{1}{2}\beta}+be^{-\frac{1}{2}\beta}-e^{-\frac{1}{2}\psi}$
(B23)

where

$\begin{array}[]{l}d=\cosh\frac{s}{2}+\sinh\frac{s}{2}\cosh\phi\\\
b=\cosh\frac{s}{2}-\sinh\frac{s}{2}\cosh\phi\end{array}.$ (B24)

Let $\beta$ become very large, which implies large $\psi$, and iterate the
equation for $e^{-\frac{1}{2}\psi}$ to obtain the result

${e}^{-\psi}={Ae}^{-\beta}\left[1+{A}_{1}{e}^{-\beta}+{A}_{2}{e}^{-2\beta}+\ldots\right]$
(B25)

where $A=1/d^{2}$ and $A_{1}=-(2/d)(b-d^{-1})$. To obtain $M_{NL}$, we expand
the right side of Eq. B21 in powers of $\psi$

$\frac{e^{-\psi(L+1)}}{1-e^{-\psi}}=\sum_{m=1}^{\infty}e^{-\psi(m+L)}.$ (B26)

For large $\beta$ it follows from Eqs. B21, B25, and B26 that

$\sum_{N=L+1}^{\infty}e^{-\beta
N}M_{NL}=\sum_{m=1}^{\infty}\left[e^{-\beta}A(1+A_{1}e^{-\beta}+A_{2}e^{-2\beta}+...\right]^{m+L}.$
(B27)

Using the multinomial theorem morse  the right side of Eq. B27 becomes

$\sum_{m=1}^{\infty}A^{m+L}\sum_{r,s,t,...}\frac{(m+L)!}{r!s!t!...}A_{1}^{s}A_{2}^{t}...e^{-\beta(m+L+s+2t+...)}$
(B28)

where $r+s+t+...=m+L$. To obtain the expression for $M_{NL}$, we note $N$ is
the coefficient of $\beta$ so $N=m+L+s+2t+...=r+2s+3t+...$ Accordingly we find

$M_{NL}=\sum_{r,s,t,}A^{(r+s+t+...)}\frac{(r+s+t+...)!}{r!s!t!}A_{1}^{s}A_{2}^{t}...$
(B29)

where r+s+t+…=N and r+s+t+…>L. For the 1S shift, as Eq. B17 indicates, we want
the matrix element $M_{10}$ which corresponds to $e^{-\beta}$ so $M_{10}=A$.
For the 2S shift we have $M_{20}=A^{2}+AA_{1}$, and for the 2P shift
$M_{21}=A^{2}$. Therefore the radiative shift for the $1S$ ground state is

$Re\Delta{E}_{10}=\frac{4mc^{2}\alpha(Z\alpha)^{4}}{3\pi}\int_{0}^{\phi_{c}}{d}\phi{e}^{\phi}\sinh\phi\int_{0}^{\infty}{dse^{s{e^{-\phi}}}}\frac{d}{ds}\frac{1}{\left(\coth\frac{s}{2}+\cosh\phi\right)^{2}}.$
(B30)

The shift for the 2S-2P level is

$Re(\Delta E_{20}-\Delta
E_{21})=\frac{m\alpha(Z\alpha)^{4}}{6\pi}\int_{0}^{\phi_{c}}d\phi
e^{\phi}\sinh^{3}\phi\int_{0}^{\infty}dse^{2se^{-\phi}}\frac{d}{ds}\frac{1}{\left(\coth\frac{s}{2}+\cosh\phi\right)^{4}}.$
(B31)

## Funding

This research received no external funding. The author has no conflicts of
interest.

## Acknowledgements

I thank Prof. Peter Milonni for many insightful and enjoyable discussions,
particularly about the resonant behavior of the index of refraction and the
volume of vacuum energy corresponding the the spectral density , and I thank
Prof. Lowell S. Brown for his observations, especially about the 1/ frequency
asymptotic behavior.

## References

## References

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  * (8) Milonni, P., _The Quantum Vacuum_ ; Academic Press: San Diego, CA, USA, 1994.
  * (9) Campagno, G., Passante, R., and Persico.F., _Atom-field Interactions and dressed Atoms_ ; Cambridge University Press:Cambridge, England, 1995.
  * (10) Maclay, G. Jordan, History and Some Aspects of the Lamb Shift, Physics 2(2), 105-149(2020).
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11institutetext:  Hikvision Research Institute, China 11email:
<EMAIL_ADDRESS>
22institutetext: Fudan University, Shanghai, China
22email<EMAIL_ADDRESS>
# REAPS: Towards Better Recognition of Fine-grained Images by Region Attending
and Part Sequencing

Peng Zhang 11 Xinyu Zhu 22 Zhanzhan Cheng 11 Shuigeng Zhou 22 Yi Niu 11

###### Abstract

Fine-grained image recognition has been a hot research topic in computer
vision due to its various applications. The-state-of-the-art is the
part/region-based approaches that first localize discriminative parts/regions,
and then learn their fine-grained features. However, these approaches have
some inherent drawbacks: 1) the discriminative feature representation of an
object is prone to be disturbed by complicated background; 2) it is
unreasonable and inflexible to fix the number of salient parts, because the
intended parts may be unavailable under certain circumstances due to occlusion
or incompleteness, and 3) the spatial correlation among different salient
parts has not been thoroughly exploited (if not completely neglected). To
overcome these drawbacks, in this paper we propose a new, simple yet robust
method by building part sequence model on the attended object region.
Concretely, we first try to alleviate the background effect by using a region
attention mechanism to generate the attended region from the original image.
Then, instead of localizing different salient parts and extracting their
features separately, we learn the part representation implicitly by applying a
mapping function on the serialized features of the object. Finally, we combine
the region attending network and the part sequence learning network into a
unified framework that can be trained end-to-end with only image-level labels.
Our extensive experiments on three fine-grained benchmarks show that the
proposed method achieves the state of the art performance.

###### Keywords:

Fine Grained Region Attending Part Sequencing.

##  1 Introduction

Fine-grained image recognition has attracted much research interest of the
computer vision community [1, 2, 3, 35], which tries to distinguish sub-
ordinate categories such as car models [18], bird species [4, 14, 29], dog
breeds [16] and flower categories [26] etc. Though much effort [9, 14, 15, 23,
30, 39] has been devoted to solving this problem, recognizing fine-grained
images is still a challenging task due to their relatively small inter-class
difference and large intra-class variation.

Roughly speaking, there are two kinds of popular frameworks for handling fine-
grained categorization: _key region localization and amplification_ (_abbr_.
RLA) and _discriminative part learning_ (_abbr_. PL). In general, RLA tries to
attend and amplify the key region for capturing detailed visual representation
while avoiding background disturbance. On the other hand, PL usually first
localizes discriminative parts via some sophisticated part selection
mechanisms such as part attentions [9, 32, 39] and convolutional responses
[33, 38], and then extracts the visual representations of the selected parts
by using multiple independent feature extractors. Fig. 1 (a) and (b)
illustrate these two frameworks. Though previous studies proved their
effectiveness, they have several inherent drawbacks. For example, RLA-based
methods may miss some salient parts when progressively attending the key
region, as shown in Fig. 2 (a). PL-based methods usually fix the number of
salient parts to be extracted, which is unreasonable and inflexible because in
certain scenarios some of the intended parts may be unavailable due to image
occlusion and incompleteness, as shown in Fig. 2 (b). Furthermore, learning
independent extractor for each salient part neglects the spatial correlation
among these different parts, which should be useful for image recognition if
properly exploited.

Figure 1: An illustrative comparison between our framework and two popular
existing fine-grained recognition frameworks. (a), (b) and (c) represent RLA,
PL and our proposed framework respectively. $O$, $P_{i}$, $R_{i}$ and $B_{i}$
correspond to the original image, a selected part, an attended region and a
backbone network respectively. +⃝ represents the operation of crop and zoom
in. Figure 2: Drawbacks of the RLA and PL frameworks. (a) RLA focuses on a
detailed region progressively, while neglecting some other salient parts (the
feet and wings of the bird disappear in scale1 and scale2 images). (b) PL
detects a fixed number of pre-set parts and may get into trouble when some
intended parts miss or be occluded (the back of the bird cannot be seen).
[Best viewed in color]

These drawbacks mentioned above motivate us to develop a new method for fine-
grained image recognition by simultaneously taking region attending and part
sequence learning into consideration. We call the new method _REAPS_ —— an
abbreviation of REgion Attending and Part Sequence learning. The framework of
REAPS is shown in Fig. 1 (c), and its detailed architecture is illustrated in
Fig. 3, which consists of two major components:

_Region attending network_ (RAN): Inspired by previous works [9, 12], we apply
the class activation mapping (CAM) mechanism [40] to constructing the region
attending network for generating the region attention. The attended region
cropped and amplified from the origin image are fed into the part sequence-
learning network (PSN). RAN can effectively depress the impact of the
background noise, while PSN can depict the detailed visual features well.

_Part sequence-learning network_ (PSN): As mentioned above, the traditional
part-based methods usually fix the number of parts and the representation of
each part is explicitly learned by independent extractors. So we call them
‘hard-part’-based methods. In addition to the drawbacks we previously
mentioned, ‘hard-part’-based methods adopt independent feature extractor
(_e.g._ VGG19) for each part, which incurs high computational cost. In order
to overcome the above drawbacks, we propose the ‘soft-part’ concept, which is
implemented by mapping the serialized visual features into a group of implicit
discriminative part representation and capturing the spatial correlation among
different salient parts simultaneously.

In REAPS, we integrate the region attending network and the part sequence
learning network into a unified framework, which can be trained end-to-end
with only image-level annotations.

Our contributions are as follows: 1) We propose the novel ‘soft-part’ concept,
and implement this concept by designing a part sequence learning network
(PSN), which learns implicit discriminative part representation and captures
the spatial context simultaneously. 2) We apply the region attending network
to localizing the object region and alleviating the interference of
complicated background to fine feature representation. 3) We integrate the
region attending network and the part sequence learning network into a unified
framework, and train it end-to-end without any part-level annotation. 4) We
conduct extensive experiments on three challenging datasets (Stanford Cars,
FGVC-Aircraft and CUB Birds), which demonstrate the superiority of our method
over the existing ones.

##  2 Related Work

Fine-grained recognition (or categorization) is a challenging problem that has
been extensively studied. Related works can be grouped in two dimensions:
representation learning and part localization.

###  2.1 Representation Learning

Discriminative representation learning is crucial for fine-grained
recognition. Thanks to their strong encoding capability, most existing fine-
grained recognition algorithms [8, 9, 25, 39] employ deep convolutional
networks for feature representation, which have achieved much better
performance than traditional descriptors and hand-crafted features [27, 36].

To better handle the subtle inter-class difference and large intra-class
variation in fine-grained recognition tasks, [21] proposes a bilinear
structure to model second-order interactions of local convolutional features
in a translationally invariant manner. This idea was later extended by [20]
and other variants [10, 17]for better recognition performance. Recently, [5,
8] further exploit higher-order integration of convolutional activations that
can yield more discriminative representation, and achieve impressive
performance.

Besides, some approaches (e.g. [28], [37]) try to learn more robust
representations via distance metric learning. [38] unifies deep CNN features
with spatially weighted Fisher vectors to capture important details and
eliminate background disturbance. [25] incorporates deep CNNs into a generic
boosting framework to combine the strength of multiple weaker learners, which
improves the classification accuracy of a single model and simplifies the
network design.

###  2.2 Part Localization

Previous works have studied the localization impact of discriminative parts on
capturing subtle visual difference. Early part-based approaches rely on extra
annotations of bounding boxes or part landmarks to localize pre-defined
semantic parts. [11, 22] assume that annotations are available in both
training and testing. Some later works [14, 18, 19] use annotations only in
training. However, the cost-prohibitive manually-labeled annotations prevent
the application of these algorithms to large-scale real problems. Therefore,
most of recent works focus on weakly-supervised task-driven part localization
with only category labels. Attention-based models have been widely used to
automatically discover salient parts. [33] proposes a two-level attention
model, where one object-level filter-net selects relevant patches for a
certain object, another part-level domain-net localizes discriminative parts.
Since deep filter responses from CNNs are able to significantly and
consistently respond to specific visual patterns, [38] proposes to learn part
detectors by picking distinctive filters, while [30] identifies discriminative
regions according to channel activations. [15] takes one step further and
proposes a spatial transform module based on the differentiable attention
mechanism, which enables CNN to learn better invariance to classification and
all kinds of warping.

The latest works use hidden filter responses of deep CNNs as part detectors.
[23] proposes a fully convolutional attention network to optimally glimpse
local discriminative regions by reinforcement learning. [9] introduces a
recurrent attention convolutional neural network (RA-CNN) that recursively
learns discriminative region attention from coarse to fine by an attention
proposal network. [39] develops a multi-attention convolutional neural network
(MA-CNN) that generates multiple part attentions by clustering, weighting and
pooling from spatially-correlated channels, and achieves the state-of-art
performance.

##  3 The REAPS Method

###  3.1 Overview

The architecture of our REAPS method is shown in Fig. 3, which consists of two
major components: a _region attending network_ (RAN) and a _part sequence
learning network_ (PSN).

Figure 3: The REAPS architecture. The region attending network (RAN) takes an
original image as input and produces the region attention by weighting the
last convolutional feature maps with the parameters of softmax layer. The
attended region is cropped out and zoomed in, and fed into the part sequence
learning network (PSN) where part representation is learned in an implicit way
by applying the mapping function on the serialized features. The whole network
can be trained end-to-end under the supervision of three softmax loss
functions in Eq. (5). [Best viewed in color]

RAN leverages deep convolutional responses to generate the discriminative
region attention. Then the attended region is cropped out and zoomed in as the
input of PSN. While in PSN, the deep visual features are extracted by a
backbone network and further serialized to a sequence of vectors, each of
which describes a rectangle region in the raw image. A mapping function is
learned to map the sequence of vectors to discriminative part representation.

###  3.2 Region Attending Network

The RAN is based on a standard classification network, where global average
pooling is applied on the last convolutional feature maps, followed by a
SoftmaxLoss layer. Given an input image $\mathcal{I}$, we can get its last
convolutional feature maps, denoted as $\mathcal{F}$. In order to eliminate
the background disturbance, we apply the CAM [40] mechanism to generating the
region attention in a “self-guided” way. Concretely, let $f_{(x,y)}^{k}$
denote the activation value of unit $k$ at $(x,y)$ in $\mathcal{F}$, the
global average pooling (_abbr._ GAP) can be represented as
$\mathcal{P}_{k}=\sum_{x,y}f^{k}_{(x,y)}$. For a specific class $c$, the
probability yielding $c$ is $\sum_{k}W_{k,c}\mathcal{P}_{k}$, where $W_{k,c}$
is the weight of the last inner-product layer and can be learned under the
supervision of SoftmaxLoss $\mathcal{L}_{A}$. $W_{k,c}$ also acts as the
weight indicating the importance of unit $k$ for class $c$. Therefore, we can
compute the region attention of $c$ at $(x,y)$ by

$RA^{c}_{x,y}=\sum_{k}W_{k}^{c}f^{k}_{(x,y)}.$ (1)

Since each unit $k$ responds to a certain type visual pattern (_e.g._ circle),
the region attention can represent all the discriminative visual patterns at
different locations by conducting weighted-sum over all units. Furthermore, we
locate the attended region of class $c$ by

$AR^{c}=f_{+}(BB_{\tau}(RA^{c}),\mathcal{I}),$ (2)

where $BB_{\tau}$ is the operation of calculating rectangular bounding box
over the binary mask based on the preset threshold $\tau$, and $f_{+}$
represents the operation of crop and zoom in.

###  3.3 Part Sequence Modeling

Let $X\in\mathbb{R}^{(H\times W\times C)}$ denote the deep representation
through the deep convolutional neural network (the backbone network in Fig.
3), where $H$, $W$ and $C$ respectively refer to the height, width and the
number of channels of $X$. Instead of localizing a fixed number of parts and
extracting their representation separately, we learn the part representation
in a soft way. Concretely, we evenly decompose $X$ into a sequence of $N$
vectors by

$Y=[Y_{1},Y_{2},...,Y_{N}]=seq(X),$ (3)

where $Y_{i}\in\mathbb{R}^{(1\times C)}$ describes a rectangle region in the
raw image and $seq$ is the pooling operation with kernel size of
$[H\times\frac{W}{N}]$. We abstract the sequence of visual feature vectors
into $M$ implicit parts with the learned mapping function

$P_{P}=[P_{1},P_{2},...,P_{M}]\simeq mapping(Y),$ (4)

where $M\leq N$ refers to the rough number of learnt discriminative parts.

The mapping function here should keep the discriminative features and depress
the useless ones. Several sequence learning techniques meet this requirement,
_e.g._ the recurrent neural networks (GRU [6] and LSTM [13]) and attention-
based sequence models [7, 34]. Attention-based models can fulfill the mapping
from a source sequence of length $N$ to a target sequence of length $M$. With
sequential labels, they can effectively learn the alignment between labels and
their corresponding salient representation of features [7, 34]. Recurrent
neural networks (GRU [6] and LSTM [13] can strengthen the sequence
representation with the sequence length being unchanged. In our case,
classification is performed in a weakly-supervised fashion and the only
available supervision information is the image-level category, so we apply a
bi-directional LSTM on the sequenced vectors and concatenate all the hidden
states as the part representation $P_{P}\in\mathbb{R}^{(N\times U)}$, followed
by a fully connected layer and a SoftmaxLoss $\mathcal{L}_{P}^{p}$.

To accelerate the training process, another SoftmaxLoss $\mathcal{L}_{P}^{g}$
is attached to the global representation $P_{g}\in\mathbb{R}^{C}$ of the
backbone network of PSN.

###  3.4 Training and Joint Representation

Instead of alternative optimization, REAPS can be trained end-to-end
straightforwardly by

$\mathcal{L}=\lambda_{1}\mathcal{L}_{A}+\lambda_{2}\mathcal{L}_{P}^{g}+\lambda_{3}\mathcal{L}_{P}^{p},$
(5)

where $\lambda_{j}(j=1,2,3)$ is the corresponding loss weight. Once the
training process converges, the joint representation $F$ of the input image
$\mathcal{I}$ can be represented by a set of descriptors, followed by a fully-
connected layer with softmax function for final classification:

$F=\\{A_{g},P_{g},P_{p}\\},$ (6)

where $A_{g}\in\mathbb{R}^{C}$ denotes the global representations of the
backbone network of RAN.

##  4 Performance Evaluation

###  4.1 Datasets and Implementation Details

We conduct extensive experiments on three benchmark datasets, including
Stanford Cars[18], FGVC-aircraft[24] and CUB-200-2011[29], which are widely
used to evaluate fine-grained image recognition. Table 1 shows the detailed
statistics of the three datasets.

Table 1: Statistics of the fine-grained benchmark datasets used in this paper. Dataset | Category | Training | Testing
---|---|---|---
Stanford Cars[18] | 196 | 8,144 | 8,041
FGVC-aircraft[24] | 100 | 6,667 | 3,333
CUB-200-2011[29] | 200 | 5,994 | 5,794

For fair comparison, all compared methods employ similar backbone network.
Specifically, we start with the 19-layer VGGNets pre-trained on ImageNet and
fine-tune it on the three fine-grained datasets. The parameters of RAN and PSN
are initialized with the same pre-trained model. Input images and the cropped
attended regions are both resized to $448\times 448$, where high resolution
highlights details and benefits recognition. We use SGD with momentum 0.9 to
minimize the loss function $\mathcal{L}$ in Eq. (5), where $\lambda_{1}$,
$\lambda_{2}$, and $\lambda_{3}$ are all set to 1. The threshold $\tau$ in Eq.
(2) is set to $0.1$. Following the common practice of learning rate decaying
schedule, the initial learning rate is set to 0.001 and multiplied by 0.1
every 60 epoches.

###  4.2 Experiment and Analysis

#### Effectiveness of Region Attending Network.

We first visualize some results of our CAM-based region attending network in
Fig. 4 for qualitative analysis. We can see that the discriminative regions of
the input images are highlighted. The attended regions that are cropped from
the raw images and then amplified preserve the object-level structure,
eliminate background interference and enrich local visual details. We evaluate
the effectiveness of RAN in terms of the single scale classification accuracy.
In Table 2, we compare the result of our PSN _without_ part modeling branch
with that of two other attention based approaches. To be fair, we select the
_single-attention_ based performance from FCAN [23], and _the second scale_
result from RA-CNN [9]. RA-CNN[9] is the most relevant work to ours
considering the region attention concept and the way to use it. We can see
that our method outperforms FCAN [23] with a clear margin (7.1% relative gain)
and RA-CNN[9] with 1.3% accuracy improvement

#### Effectiveness of part branch in PSN.

As pointed out in Section 3.3, we take a bi-directional LSTM as the mapping
function to map the serialized features to discriminative part
representation.To evaluate its effectiveness, we present the classification
results of two models: _PSN wo/w part_ , which are the classification accuracy
based on the $P_{g}$ _without/ with_ the part modeling branch. The results are
shown in Table 2. We can see that the model adopting part sequence modeling
branch achieves a relative performance gain of 1.0%. Some illustrations are
given in Figure 5. It can be observed that the part branch encourages a more
compact distribution on the feature maps, further enhances the crucial part
areas and depresses the unless ones.

Table 2: Performance comparison of attention localization on the Stanford Cars dataset. Approach | Accuracy
---|---
FCAN (single-attention) [23] | 84.2
RA-CNN (scale 2) [9] | 90.0
PSN _wo part_ | 91.3
PSN | 92.3
Figure 4: Region attention localization results of RAN for some examples from
(a) Stanford Cars, (b) FGVC-aircraft, and (c) CUB-200-2011. Pictures from left
to right in (a-c) are the raw image, the attention mask with bounding box
indicating the area of top attention response, and the cropped object-level
region respectively. Figure 5: Visualization of feature maps for some examples
from (a) Stanford Cars, (b) FGVC Aircraft, (c) CUB-200-2011. Pictures from
left to right in (a-c) are the raw image, the feature map generated by PSN
_without_ part branch and the feature map generated by PSN _with_ part branch,
respectively.

#### Fine-grained Categorization.

We compare our REAPS framework with several existing methods and the results
are summarized in Table 3. Our method achieves better performance than those
[4, 18, 30, 31, 32] using ground-truth bounding boxes or part annotations
during training or testing time on three datasets. Compared with BCNN-based
methods [8, 20, 25] our method obtains comparable or even better performance
due to accurate attention localization and part sequence modeling. To further
enhance the capacity of REAPS, as RA-CNN [9] does, we incorporate one more PSN
into our framework, and the 2nd PSN is based on the attended region of the 1st
PSN. We call the resulting network REAPS+. Note that, REAPS+ obtains the best
performance on three datasets. Especially on FGVC Aircraft dataset, our
proposed REAPS+ obtains the best accuracy of 92.6%, surpassing state-of-the-
art MA-CNN [39] by a relative 2.7% gain. The significant improvement suggests
that the proposed part sequence modeling network works as expected to leverage
spatial information of parts, and does even better when the objects to be
recognized have strong sequential structures.

Table 3: Performance comparison on the Stanford Cars, FGVC Aircraft and CUB-200-2011 datasets. _(*)_ indicates whether bounding box or part annotation is used in training. Approach | Stanford Cars | FGVC Aircraft | CUB200-2011
---|---|---|---
PA-CNN [18] | 92.8 (*) | $-$ | 82.8 (*)
MDTP [31] | 92.5 (*) | 88.4 (*) | $-$
MG-CNN [30] | $-$ | 86.6 (*) | 83.0 (*)
PN-CNN [4] | $-$ | $-$ | 85.4 (*)
Mask-CNN [32] | $-$ | $-$ | 85.4 (*)
STNs [15] | $-$ | $-$ | 84.1
FCAN [23] | 91.5 | $-$ | 84.3
PDFR [38] | $-$ | $-$ | 84.5
Improved B-CNN [20] | 92.0 | 88.5 | 85.8
BoostCNN [25] | 92.1 | 88.5 | 86.2
KP [8] | 92.4 | 86.9 | 86.2
RA-CNN(scale 1+2+3) [9] | 92.5 | $-$ | 85.3
MA-CNN [39] | 92.8 | 89.9 | 86.5
REAPS _wo PSN_ | 92.0 | 89.8 | 81.3
REAPS | 93.1 | 91.8 | 86.0
REAPS+ | 93.5 | 92.6 | 86.8

##  5 Conclusion

In this paper, we propose a novel framework REAPS for fine-grained
recognition, which consists of a region attending network and a part sequence-
learning network. The proposed framework does not need bounding box/ part
annotations for training and can be trained end-to-end. We conduct extensive
experiments on three fine-grained benchmark datasets, and the experimental
results show that REAPS outperforms the existing methods.

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