# Handling the asymmetric spectral line profile

D. Solovyev1,2 A. Anikin1,3<EMAIL_ADDRESS>T. Zalialiutdinov1,2
1 Department of Physics, St. Petersburg State University, Petrodvorets,
Oulianovskaya 1, 198504, St. Petersburg, Russia
2 Petersburg Nuclear Physics Institute named by B.P. Konstantinov of National
Research Centre ’Kurchatov Institut’, St. Petersburg, Gatchina, 188300, Russia
3D.I. Mendeleev Institute for Metrology, St. Petersburg, 190005, Russia

###### Abstract

This paper discusses some features of the spectral line profile theory used in
the treatment of measured atomic transitions. It is shown that going beyond
the established linear approximation for the spectral line contour in the case
of its nonresonant extension, the potential for a more accurate extraction of
atomic characteristics from experimental data arises. Using the example of the
Lyman-$\alpha$ (Lyα) transition in hydrogen, a simple analysis of the observed
spectral line distorted by a possible interfering transitions is given. In
particular, the results obtained in the present work clearly demonstrate that
the processing of the same experimental data at different settings can provide
an accurate determination of the transition frequency, the centre of gravity
as well as the hyperfine splitting of the ground state in hydrogen-like atomic
systems. The latter is especially important for setting up precision
spectroscopic experiments on the antihydrogen atom.

##  I Introduction

A detailed theoretical description of the observed spectral line profile has
become an inherent part of the precision determination of the transition
frequency in the last decade. The first rigorous derivation of the line
profile in the framework of quantum electrodynamics (QED) was given in [1]. It
was shown that the line profile is determined by the Lorentz contour in the
resonance approximation. Until recently, such approach was sufficient to
accurately extract the transition frequency from experimental data, although
theoretical studies of the phenomena affecting the line profile have been
carried out for quite some time, having emerged as a separate field of
research, see, e.g., [2, 3] and references therein.

In particular, theoretical works dealing with particularities of the line
shape have shown the importance of taking into account the asymmetry arising
when measuring transition frequencies [4, 5]. Theoretical examination of the
spectral line asymmetry unveils a notable distinction in determining the
transition frequency for an atom unaffected by the measurement process
compared to one influenced by experimental observation. Namely, the frequency
obtained in an experiment should be correlated to the Bohr model of the atom
(underlying theoretical studies) in order to extract, for example, physical
constants or to test fundamental interactions and symmetries.

The most significant influence comes from the effect resulting from
interfering pathways for neighboring states, which is known as the quantum
interference effect (QIE) [5]. Depending on the experimental conditions and
method of measuring the transition frequency, the asymmetry of the spectral
line profile should be taken into account for each specific experiment [6].
Giving rise to a frequency shift that can significantly exceed, for example,
radiative corrections, the corresponding asymmetry is, as usual, calculated
theoretically [7, 8, 9, 10, 11, 12, 13].

The most significant advance was made recently in the [14]. First of all, the
experiment [14] demonstrated that the spectral line profile has become a
precisely measurable quantity. As another important consequence, it is
necessary to point out the processing of the experimental data, which enabled
to significantly improve the accuracy of the transition frequency extraction.
It is the line profile that represents the physical quantity, while the
transition frequency, level widths, etc. are determined as its parameters. The
accuracy of determining these characteristics strongly depends on the used
fitting contour.

The experimental data treatment performed in [14] was based on the procedure
described in [5] and the use of the Fano contour (Fano-Voigt) to fit the
observed line profile, see also [15]. Once an indispensable part of the
experiment [16, 17], the fit of the observed line beyond the resonance
approximation is performed by an asymmetric profile derived in the linear
approximation, see [5] for details. In the lowest order this corresponds
exactly to the description of the QIE. With the same result, the QIE can be
obtained as the frequency shift via an extremum condition for the line shape
given by the corresponding cross section [7, 8, 9, 10, 11, 12, 13].

Adhering to the analysis presented in [5, 14], we discuss the principles of
extracting the transition frequency from the experiment [18, 19] and the
possibility of accurately determining the hyperfine splitting intervals in the
hydrogen atom for the Lyman-$\alpha$ (Lyα) transition. In the framework of the
linear approximation, it is shown that a thorough treatment of the
experimental data should significantly improve the accuracy of extracting the
values of hyperfine intervals in such experiments. The latter may have a key
role in the detailed comparison with the antihydrogen atom [20, 21, 22].

##  II Theory of spectral line asymmetry

In the Furry picture and $S$-matrix formalism [23, 24, 25] for the one-
electron atom the standard Lorentz line profile can be obtained by considering
the elastic photon scattering by an atomic electron in the state $n$. In the
resonant approximation, the amplitude of the process can be reduced to the
form [1, 2]:

$\displaystyle U_{nn}^{\rm sc}=\frac{\langle
n|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|r\rangle\langle
r|\bm{\alpha}\bm{A}_{\bm{k}_{2},\bm{e}_{2}}|n\rangle}{E_{r}+L_{r}^{\rm
SE}-E_{n}-\omega-\frac{\mathrm{i}}{2}\Gamma_{r}},$ (1)

where $L_{r}^{\rm SE}$ and $\mathrm{i}\Gamma_{r}/2$ represent, respectively,
the real and imaginary parts of the one-loop self-energy correction (the
leading-order Lamb shift and the level width, respectively), the inserts of
which were used to regularize the divergent resonance contribution, see for
more details [1, 2]. Notations $\bm{A}_{\bm{k}_{2},\bm{e}_{2}}$ and
$\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}$ are introduced for the wave functions in
the coordinate space representation of absorbed and emitted photons, with the
corresponding wave, $\bm{k}$, and polarization, $\bm{e}$, vectors,
$\bm{\alpha}$ represents the Dirac matrix, $E_{r}$ denotes the energy of the
resonant state $r$ and $\omega\equiv|\bm{k}|$ is the frequency of the
emitted/absorbed photon (the latter can be redefined by the energy
conservation law). The Dirac binding energies are $E_{n}$ for an arbitrary
state of atom $n$, which we characterize by a set of quantum numbers $nljF$
($n$ is the principal quantum number, $l$ is the orbital momentum, $j$
represents the total angular momentum of the bound electron, and $F$
represents the total atomic momentum taking into account the nuclear spin).
Unless otherwise stated, relativistic units $\hbar=c=m=1$ are used ($\hbar$ is
the reduced Planck constant, $c$ is the speed of light, and $m$ is the
electron mass).

Then the squared modules of the scattering amplitude (1) leads to the standard
expression for the Lorentz profile. In the following calculations, we will
turn to the dominant effect represented by quantum interference. The QI effect
can be obtained by considering neighboring states $r_{1}$ and $r_{2}$. Then,
the scattering amplitude is

$\displaystyle U_{nn}^{\rm sc}=\frac{\langle
n|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|r_{1}\rangle\langle
r_{1}|\bm{\alpha}\bm{A}_{\bm{k}_{2},\bm{e}_{2}}|n\rangle}{\omega_{0}-\omega-\frac{\mathrm{i}}{2}\Gamma_{r_{1}}}+$
(2) $\displaystyle\frac{\langle
n|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|r_{2}\rangle\langle
r_{2}|\bm{\alpha}\bm{A}_{\bm{k}_{2},\bm{e}_{2}}|n\rangle}{\omega_{0}+\Delta-\omega-\frac{\mathrm{i}}{2}\Gamma_{r_{2}}},$

where with high accuracy one can set
$\Gamma_{r_{1}}=\Gamma_{r_{2}}\equiv\Gamma_{r}$ (as for the Lyα transition),
$\omega_{0}\equiv E_{r}-E_{n}$ (all possible radiative corrections can be
included in the definition of $E_{n}$) and $\Delta=E_{r_{2}}-E_{r_{1}}$. The
expression (3) is written for the case when both resonant lines are observed.
It can be considered as excitation by two independent laser beams or a single
broadband laser overlapping the two lines.

By introducing a shortcut notation for matrix elements $\langle
a|\bm{\alpha}\bm{A}_{\bm{k}_{2},\bm{e}_{2}}|r_{1(2)}\rangle\langle
r_{1(2)}|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|a\rangle=A_{1(2)}$, for
the squared modulus of Eq. (2) we arrive at

$\displaystyle\phi(x)\sim\frac{\left|A_{1}\right|^{2}}{x^{2}+\frac{1}{4}\Gamma^{2}_{r}}+\frac{\left|A_{2}\right|^{2}}{(x+\Delta)^{2}+\frac{1}{4}\Gamma^{2}_{r}}\qquad$
(3)
$\displaystyle+\frac{2Re\left[A_{1}A^{*}_{2}\right]\left[x(x+\Delta)+\frac{1}{4}\Gamma^{2}_{r}\right]}{\left[x^{2}+\frac{1}{4}\Gamma^{2}_{r}\right]\left[(x+\Delta)^{2}+\frac{1}{4}\Gamma^{2}_{r}\right]},\qquad$

where $x\equiv\omega_{0}-\omega$ and $Re[\dots]$ denotes the real part of
magnitude in brackets. The dependence on the angles between the polarization
and propagation vectors (or their combinations) for the absorbed and emitted
photons is contained in the amplitudes $A_{1}$, $A_{2}$. The factor
$Re\left[A_{1}A^{*}_{2}\right]$ determines the angular dependence of the line
profile along with the squares of the corresponding amplitudes.

By applying the extremum condition to Eq. (3), where $d\phi(\omega)/dx=0$, it
is possible to derive a fifth-degree equation with respect to $x$. For obvious
reasons, the solution of this equation should be performed numerically.
However, assuming the smallness of $x$, the conventional manner corresponds to
the linear approximation, which immediately leads to an expression for the
frequency shift caused by the QIE. The details of such a description are
extensively covered in the literature, see, e.g., [26] and references therein.

Here we only note that the three real roots of the emerging equation can be
interpreted, obviously, as values of two maxima and one minimum of the line
profile. The latter is of particular interest for determining the ”gravity
centre” of two lines. According to the results of calculations taking into
account the fine structure of the Lyman-$\alpha$ transition, i.e., for the
interfering $1s_{1/2}\rightarrow 2p_{1/2},2p_{3/2}$ transitions [5], this
centre of gravity can be shifted by a nonresonant correction.

To demonstrate this statement, it is enough to consider the case when the
interference is zero. Then one can find the magic angle for electric dipole
transitions with vanishing amplitude $Re[A_{1}A^{*}_{2}]$ in Eq. (3). By
performing the calculations (see Appendix), the roots determined by the
extremum condition of Eq. (3) at the magic angle can be found as follows

$\displaystyle x^{(1)}$ $\displaystyle=$ $\displaystyle 0,x^{(2)}=\Delta_{\rm
fs},$ $\displaystyle x^{(3)}$ $\displaystyle=$
$\displaystyle\left(1-2^{1/3}+2^{2/3}\right)\frac{\Delta_{\rm
fs}}{3}-\frac{\left(2-2^{1/3}\right)}{2^{2/3}}\frac{\Gamma^{2}}{6\Delta_{\rm
fs}},$ (4) $\displaystyle x^{(4,5)}$ $\displaystyle=$ $\displaystyle
0.278753\Delta_{\rm fs}+0.0388517\frac{\Gamma^{2}}{\Delta_{\rm fs}}$
$\displaystyle\pm$
$\displaystyle\mathrm{i}\left(0.296415\frac{\Gamma^{2}}{\Delta_{\rm
fs}}+0.821951\Delta_{\rm fs}\right).$

Here $\Delta_{\rm fs}$ denotes the fine splitting interval
$E_{2p_{3/2}}-E_{2p_{1/2}}$, and $\Gamma$ is the width of the $2p$ state in
the hydrogen atom. The expressions above were obtained for the case of the
correlation between the two polarization vectors. The angle between the
polarization vector of the absorbed photon and the direction vector of the
emitted one will be treated in later sections.

As can be seen from Eqs. (II), the first and second roots correspond to the
transition frequencies $E_{1s_{1/2}}-E_{2p_{1/2}}$ and
$E_{1s_{1/2}}-E_{2p_{3/2}}$, respectively. Both are shifted by a fine
structure interval with respect to each other, and both are not subject to QIE
at a magic angle. The root $x^{(3)}$ represents the position of the minimum of
the line profile Eq. (3) and can be understood as the centre of gravity of the
two lines, whose position is determined by the coefficient at $\Delta_{\rm
fs}$ and is subject to the effect of quantum interference even at a magic
angle.

Finally, the remaining roots $x^{(4,5)}$ yield inflection points (the second
derivative of Eq. (3) is zero at these points). We give their numerical
coefficients because of the unwieldiness of the expressions, although we can
state that, as in all other roots, the coefficients are determined by the
amplitudes of $A_{1}$ and $A_{2}$. Keeping in mind that $x=\omega_{0}-\omega$,
the real part of these roots can be interpreted as a shift, and their
imaginary part as the value of the width at inflection points (like the
natural width in Eqs. (1), (2)). At first glance, the roots of $x^{(4,5)}$ are
unphysical, but depending on the angle, they can become real. Calculations for
the angle $\pi/2$ at lowest order give: $x^{(1)}=\Gamma^{2}/(2\Delta_{\rm
fs})$, $x^{(2)}=\Delta_{\rm fs}-\Gamma^{2}/(2\Delta_{\rm fs})$,
$x^{(3)}=\Delta_{\rm fs}/2$ and $x^{(4,5)}=\Delta_{\rm
fs}/2\pm\sqrt{5}\Delta_{\rm fs}/2$. The first three roots, as before,
represent the maxima (shifted by nonresonant correction) and the unshifted
centre of gravity. The centre of gravity is located exactly in the middle of
two lines corresponding to fine sublevels at angle $\pi/2$. In turn, the
expressions for $x^{(4,5)}$ give additional positions of minima located on the
left and right of the two peaks, see Fig. 1.

Figure 1: Graph illustrating the spectral profile of two lines,
$1s_{1/2}\rightarrow 2p_{1/2}$ and $1s_{1/2}\rightarrow 2p_{3/2}$, separated
by the fine structure interval $\Delta_{\rm fs}$. Roots corresponding to the
extremum condition are marked with red squares. The graph is plotted at an
angle of $\pi/2$ between the polarization vectors of the emitted and absorbed
photons, and an enlarged region of minima to the left and right of both lines
is inserted for demonstration (black arrows connect the same roots). Spectral
profile values are multiplied by the square of the natural width for clarity
and for convenience, the graph is shifted so that the x-coordinate of the
centre of gravity ($\Delta_{\rm fs}/2$) is zero.

Thus, measuring the centre of gravity requires different data processing with
respect to determining the transition frequency. In the case of a larger
number of lines, which may overlap significantly, the appropriate analysis
becomes much more complicated (the extremum condition will lead to polynomials
of higher powers on $x$).

##  III Determination of the HFS of the ground state of hydrogen by measuring
the Lyα transition

We now turn to the linear approximation, within which the line profile fitting
was presented theoretically in [5] and later used in [14] to treat the
experimental results. Considering the Lyman-$\alpha$ transition, we discuss
the possibility of accurately determining the hyperfine structure (HFS)
interval of the ground state in the hydrogen atom. Referring to the experiment
[18], the level scheme is shown in Fig. 2.

Figure 2: Scheme of transitions in the experiment [18]. The hyperfine
splittings of levels $1s_{1/2}$, $2p_{3/2}$, as well as the $2p_{1/2}$ state
splitting, are denoted by the corresponding values of the total atomic
momentum $F$. The transitions are shown by lines with an arrow (single line
for absorption and double line for emission). Since there is no direct
transition to the $2p_{1/2}$ state, but interference exists, the corresponding
pathways are shown by dashed and dotted lines.

The experiment [18] showed that the observation of Lyman-$\alpha$ emission
lines can be used to accurately measure the hyperfine structure (HFS) of the
ground state. A value differing by $\approx 20$ MHz from the generally
accepted result (see, for example, [27, 28]) was found in order to further
apply a similar scheme to measurements in antimatter [18]. Possessing
particular importance, a detailed comparison of the hyperfine ground state of
hydrogen and antihydrogen atoms was recently performed in [29], where the HFS
interval for the latter was determined as $1420.4\pm 0.5$ MHz. It should be
noted that hydrogen HFS interval experiments are orders of magnitude more
accurate than antihydrogen experiments, see for example [27, 28]. Although
experiments with the antihydrogen atom are more challenging (see also [22]),
we turn to the experimental setup of the [18], suggesting their future
adaptation to antimatter. The main purpose of the following discussion is to
demonstrate that the HFS interval determination of the ground state through
Lyα emission can be determined more accurately using the same experiment as
[18].

To realize this, one should consider the emission lines (depicted by the
double solid line in Fig. 2) without distinguishing the hyperfine sublevels of
the initial state, but for different hyperfine sublevels of the final state,
i.e., with total atomic momenta $F_{f}=1$ and $F_{f}=0$. In consistency with
the experiment [18], one should also sum over the total momentum, $F$, of the
$2p_{3/2}$ resonance state atom, since excitation is possible to any hyperfine
sublevel. Then, the scattering amplitude can be presented in form:

$\displaystyle U_{1s_{1/2}\rightarrow 2p_{3/2}\rightarrow
1s_{1/2}^{F=0}}+U_{1s_{1/2}\rightarrow 2p_{3/2}\rightarrow 1s_{1/2}^{F=1}}=$
$\displaystyle\frac{A_{1}}{E_{2p_{3/2}}-\omega-
E_{1s_{1/2}^{F=1}}-\mathrm{i}\frac{\Gamma}{2}}+\frac{A_{2}}{E_{2p_{1/2}}-\omega-
E_{1s_{1/2}^{F=1}}}+$ (5) $\displaystyle\frac{A_{3}}{E_{2p_{3/2}}-\omega-
E_{1s_{1/2}^{F=0}}-\mathrm{i}\frac{\Gamma}{2}}+\frac{A_{4}}{E_{2p_{1/2}}-\omega-
E_{1s_{1/2}^{F=0}}}.$

Here the divergent resonance contributions have been regularized according to
the procedure presented in [1]. The terms with amplitudes $A_{2}$ and $A_{4}$
were introduced as interfering paths to the $2p_{1/2}$ fine splitting state.
The notations for amplitudes in Eq. III are

$\displaystyle A_{1}=\sum\limits_{F_{i}}(2F_{i}+1)\sum\limits_{F}\langle
1s_{1/2}^{F_{i}}|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|2p_{3/2}^{F}\rangle\times$
(6) $\displaystyle\langle
2p_{3/2}^{F}|\bm{\alpha}\bm{A}_{\bm{k}_{2},\bm{e}_{2}}|1s_{1/2}^{F_{f}=1}\rangle,$
$\displaystyle A_{2}=\sum\limits_{F_{i}}(2F_{i}+1)\sum\limits_{F}\langle
1s_{1/2}^{F_{i}}|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|2p_{1/2}^{F}\rangle\times$
$\displaystyle\langle
2p_{1/2}^{F}|\bm{\alpha}\bm{A}_{\bm{k}_{2},\bm{e}_{2}}|1s_{1/2}^{F_{f}=1}\rangle,$
$\displaystyle A_{3}=\sum\limits_{F_{i}}(2F_{i}+1)\sum\limits_{F}\langle
1s_{1/2}^{F_{i}}|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|2p_{3/2}^{F}\rangle\times$
$\displaystyle\langle
2p_{3/2}^{F}|\bm{\alpha}\bm{A}_{\bm{k}_{2},\bm{e}_{2}}|1s_{1/2}^{F_{f}=0}\rangle,$
$\displaystyle A_{4}=\sum\limits_{F_{i}}(2F_{i}+1)\sum\limits_{F}\langle
1s_{1/2}^{F_{i}}|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|2p_{1/2}^{F}\rangle\times$
$\displaystyle\langle
2p_{1/2}^{F}|\bm{\alpha}\bm{A}_{\bm{k}_{2},\bm{e}_{2}}|1s_{1/2}^{F_{f}=0}\rangle.$

The $(2F_{i}+1)$ coefficient is introduced so that the excitation process
proceeds in accordance with the experiment [18], from the $1s_{1/2}$ state
with an unfixed hyperfine sublevel.

By performing the calculations, see [6] for details, we find in the
nonrelativistic limit

$\displaystyle\left|A_{1}\right|^{2}=-\frac{1}{54}\left(7-3\cos{2\vartheta}\right),$
$\displaystyle\left|A_{2}\right|^{2}=-\frac{2}{27},$ (7) $\displaystyle
A_{1}A_{2}=\frac{1}{54}\left(1+3\cos{2\vartheta}\right)$

for the decay to the $1s_{1/2}^{F_{f}=1}$ state, and for the
$1s_{1/2}^{F_{f}=0}$ state:

$\displaystyle\left|A_{3}\right|^{2}=-\frac{1}{162}\left(7-3\cos{2\vartheta}\right),$
$\displaystyle\left|A_{4}\right|^{2}=-\frac{2}{81},$ (8) $\displaystyle
A_{3}A_{4}=\frac{1}{162}\left(1+3\cos{2\vartheta}\right).$

Here we assume that there is no interference between channels with different
total atomic momentum of the final state, i.e. between transitions with
$F_{f}=1$ and $F_{f}=0$.

Presence of interference in Eqs. (III) and (III) results in asymmetric line
profiles for two scattering channels $1s_{1/2}\rightarrow 2p_{3/2}\rightarrow
1s_{1/2}^{F_{f}=1}$ and $1s_{1/2}\rightarrow 2p_{3/2}\rightarrow
1s_{1/2}^{F_{f}=0}$. The corresponding transition frequencies can be
determined accurately in two ways. The first is given by subtracting the
nonresonant correction from the transition frequency, determined as the
maximum of the line profile. This correction in the linear approximation on
$x$ is easily calculated and is expressed through $\Gamma^{2}/(4\Delta_{\rm
fs})\approx-227.4$ kHz, where $\Delta_{\rm fs}$ is the fine structure interval
between states $2p_{3/2}$ and $2p_{1/2}$. At the same time, as before, it
follows from Eqs. (III) and (III) that the angle at which the low-order
asymmetry vanishes is the magic angle, $\vartheta_{\rm
m}=\arccos{(1/\sqrt{3})}=\arccos{(-1/3)}/2$. This is the result of
distinguishable transitions into hyperfine sublevels of the ground state.

Another scenario refers to the use of an asymmetric profiles to fit the
observed spectral line shapes. Following the theory stated in [5], within a
linear approximation the fitting contour can be given by a sum of two
profiles:

$\displaystyle\frac{C_{1}}{(x-\Delta_{1}(x))^{2}+\frac{1}{4}\Gamma^{2}}+\frac{C_{2}}{(x+\Delta^{(1s)}_{\rm
HFS}-\Delta_{2}(x))^{2}+\frac{1}{4}\Gamma^{2}}.\qquad$ (9)

In Eq. (9) the first function is written for the scattering channel to the
$1s_{1/2}^{F_{f}=1}$ state and the second function corresponds to
$1s_{1/2}^{F_{f}=0}$. The introduced coefficients are

$\displaystyle
C_{1}=\left|A_{1}\right|^{2},\,a_{1}=2\frac{\left|A_{2}\right|^{2}}{\Delta_{\rm
fs}^{3}},\,b_{1}=-2\frac{A_{1}A_{2}}{\Delta_{\rm fs}}$ (10) $\displaystyle
C_{2}=\left|A_{3}\right|^{2},\,a_{2}=2\frac{\left|A_{4}\right|^{2}}{\Delta_{\rm
fs}^{3}},\,b_{2}=-2\frac{A_{3}A_{4}}{\Delta_{\rm fs}}.$

The frequency shifts are defined as

$\displaystyle\Delta_{1}(x)=\frac{a_{1}}{2C_{1}}\left[x^{2}+\frac{1}{4}\Gamma^{2}\right]^{2}+\frac{b_{1}}{2C_{1}}\left[x^{2}+\frac{1}{4}\Gamma^{2}\right],$
(11)
$\displaystyle\Delta_{2}(x)=\frac{a_{2}}{2C_{2}}\left[\left(x+\Delta^{(1s)}_{\rm
HFS}\right)^{2}+\frac{1}{4}\Gamma^{2}\right]^{2}$
$\displaystyle+\frac{b_{2}}{2C_{2}}\left[\left(x+\Delta^{(1s)}_{\rm
HFS}\right)^{2}+\frac{1}{4}\Gamma^{2}\right].$

Thus, the contour given by the expression (9) becomes defined as a function of
the angle $\vartheta$ and the atomic characteristics $\Delta_{\rm fs}$,
$\Delta^{(1s)}_{\rm HFS}$, $\Gamma$. The corresponding profile is shown in
Fig. 3 as a function of frequency and angle. The difference between the model
shape (9) and the one based on (III) is invisible to the naked eye.

Figure 3: The spectral profile arising for two scattering channels
$1s_{1/2}\rightarrow 2p_{3/2}\rightarrow 1s_{1/2}^{F=1}$ and
$1s_{1/2}\rightarrow 2p_{3/2}\rightarrow 1s_{1/2}^{F=0}$ as a function of
frequency and angle in arbitrary units for visualisation. The axis with
notations in gamma refers to the variable $x\equiv
E_{2p_{3/2}^{F=1}}-E_{1s_{1/2}^{F=1}}-\omega$, the axis in $\pi$ scale refers
to the dependence of line profile on $\vartheta$ angle, and the line contour
values are plotted along the height. Two resonant lines occur at $x=0$ and
$x=\Delta^{(1s)}_{\rm HFS}$.

Solving equation $\Delta_{1}(x=0)=0$ (here $x=\omega_{0}-\omega\equiv
E_{2p_{3/2}}-E_{1s_{1/2}^{F_{f}=1}}-\omega$) and
$\Delta_{2}\left(x=-\Delta^{(1s)}_{\rm HFS}\right)=0$, one can find the angle
at which the unshifted frequencies can be determined. The result is

$\displaystyle\vartheta_{0}=\pm\frac{1}{2}\arccos{\left[\frac{-2\Gamma^{2}-\Delta_{\rm
fs}^{2}}{3\Delta_{\rm fs}^{2}}\right]},$ (12)

which gives the ”magic angle” in approximation of small width $\Gamma$. The
addition to the magic angle at lowest order is
$\pm\Gamma^{2}/\left(2\sqrt{2}\Delta_{\rm fs}^{2}\right)\approx 2.9\times
10^{-5}$ in radians.

So far we have considered the case of determining the hyperfine splitting
interval by finding the two frequencies corresponding to the
$1s_{1/2}\rightarrow 2p_{3/2}\rightarrow 1s_{1/2}^{F=1}$ and
$1s_{1/2}\rightarrow 2p_{3/2}\rightarrow 1s_{1/2}^{F=0}$ spectral lines.
Accordingly, there are two possibilities: i) to determine the frequencies and
angle from experimental data using the asymmetric profile (9), ii) to utilize
the symmetric line profile obtained by fitting parameter equal to the angle
$\vartheta=\vartheta_{0}$ calculated theoretically. Such a symmetric contour
will not neatly fit the observed line shape, but the maxima will be unshifted,
which in turn can be used to define $\Delta^{(1s)}_{\rm HFS}$ as the frequency
difference at their maxima (symmetrization procedure [14]).

However, according to the previous section, the hyperfine splitting of the
ground state can be found in another way. The centre of gravity can be used
for this purpose. Solving the set of equations $\Delta_{1}(x)=0$,
$\Delta_{2}(x)=0$, one can find ten pairs $\\{x,\vartheta\\}$, two of which
are real. Dropping out the solutions corresponding to
$x=\pm\mathrm{i}\Gamma/2$ and $x=-\Delta^{(1s)}_{\rm
HFS}\pm\mathrm{i}\Gamma/2$, we found

$\displaystyle x=-\frac{\Delta^{(1s)}_{\rm HFS}}{2}$ (13)

at the angle

$\displaystyle\vartheta_{\rm
gc}=\pm\frac{1}{2}\arccos{\left(\frac{-\Delta^{2}_{\rm
fs}-2\left(\Delta^{(1s)}_{\rm HFS}\right)^{2}-2\Gamma^{2}}{3\Delta_{\rm
fs}^{2}}\right)}.$ (14)

The angle $\vartheta_{\rm gc}\approx 0.961319$ and differs on the magic angle,
$\vartheta_{0}$: $\delta\vartheta=\vartheta_{\rm
gc}-\vartheta_{0}\approx\left(\Delta^{(1s)}_{\rm
HFS}\right)^{2}/\left(2\sqrt{2}\Delta_{\rm fs}^{2}\right)\approx 6\times
10^{-3}$. However, at angle $\vartheta_{\rm gc}$ the centre of gravity is
unshifted and is located exactly (in lowest order) in the middle of the two
peaks, thus providing an accurate determination of the hyperfine splitting of
the ground state.

##  IV $2p_{3/2}$ HFS in Lyα transition measurements

So far we have described the Lyα transition without emphasizing the hyperfine
structure of the $2p_{3/2}$ state, $\Delta_{\rm HFS}^{(2p_{3/2})}$. Meanwhile,
the value of the appropriate energy interval is about $23.6516$ MHz, see [30],
which admits the influence of the $2p_{3/2}$ state hyperfine structure on the
determination of the HFS of the ground state carried out in [18]. Note that
the value of the $\Delta_{\rm HFS}^{(2p_{3/2})}$ splitting is about four times
smaller than the natural level width $\Gamma\approx 99.7624$ MHz, which leads
to the failure to observe it in experiments of the type. Hence, investigating
the impact of such a blurring on the determination of the $\Delta^{(1s)}_{\rm
HFS}$, as well as the possibility of extracting the $\Delta_{\rm
HFS}^{(2p_{3/2})}$ interval, is the main goal of this section.

To account for the $2p_{3/2}$ level hyperfine splitting of atomic hydrogen, we
use the experimental setup of [18], see Fig. 2, where two emission lines
corresponding to $F_{f}=0$ and $F_{f}=1$ of the $1s_{1/2}$ ground state were
observed. Then, evidently, the line corresponding to the final momentum
$F_{f}=0$ is not responsible for the $\Delta_{\rm HFS}^{(2p_{3/2})}$, and
moreover does not interfere with the channel scattering to $F_{f}=0$. So, we
start by considering the scattering channel $1s_{1/2}\rightarrow
2p_{3/2}^{F=1}/2p_{3/2}^{F=2}\rightarrow 1s_{1/2}^{F_{f}=1}$, which can be
further generalized to the case described in the previous section III (by
adding the remaining line profile in Eq. (9)).

According to the expression (2), we have

$\displaystyle U^{\rm sc}_{1s_{1/2}\rightarrow
1s_{1/2}^{F_{f}=1}}\sim\frac{A_{1}}{x-\frac{\mathrm{i}}{2}\Gamma}+\frac{A_{2}}{x+\Delta_{\rm
HFS}^{(2p_{3/2})}-\frac{\mathrm{i}}{2}\Gamma}\qquad$ (15)
$\displaystyle+\frac{A_{3}}{x-\Delta_{\rm
fs}^{(11)}}+\frac{A_{4}}{x-\Delta_{\rm fs}^{(10)}}+\frac{A_{5}}{x-\Delta_{\rm
fs}^{(21)}}+\frac{A_{6}}{x-\Delta_{\rm fs}^{(20)}}.$

Here the first two terms represent transitions to $2p_{3/2}^{F=1}$ and
$2p_{3/2}^{F=2}$, respectively, while the last four terms correspond to the
interferening paths with $2p_{1/2}^{F=1}$ and $2p_{1/2}^{F=0}$ separated by
the fine splitting interval. The corresponding splitting energies are
$\Delta_{\rm fs}^{(11)}=E_{2p_{3/2}^{F=1}}-E_{2p_{1/2}^{F=1}}$, $\Delta_{\rm
fs}^{(10)}=E_{2p_{3/2}^{F=1}}-E_{2p_{1/2}^{F=0}}$, $\Delta_{\rm
fs}^{(21)}=E_{2p_{3/2}^{F=2}}-E_{2p_{1/2}^{F=1}}$, $\Delta_{\rm
fs}^{(20)}=E_{2p_{3/2}^{F=2}}-E_{2p_{1/2}^{F=0}}$, and $x\equiv
E_{2p_{3/2}^{F=1}}-E_{1s_{1/2}^{F_{f}=1}}-\omega$.

At first, it can be found that the resonant terms give rise to the overlapping
profiles (indistinguishable due to the level width). The interference
represented by the product $A_{1}A_{2}$ cannot be attributed to either one of
them. However, using the approximation of $\Delta_{\rm HFS}^{(2p_{3/2})}$
smallness, we can write

$\displaystyle\phi^{(0)}(x)\approx\frac{\left|A_{1}\right|^{2}+A_{1}A_{2}}{x^{2}+\frac{1}{4}\Gamma^{2}}+\frac{\left|A_{2}\right|^{2}}{\left(x+\Delta_{\rm
HFS}^{(2p_{3/2})}\right)^{2}+\frac{1}{4}\Gamma^{2}},\qquad$ (16)

where the contribution $\sim-\Delta_{\rm HFS}^{(2p_{3/2})}\left(x+\Delta_{\rm
HFS}^{(2p_{3/2})}\right)$ is discarded. Performing calculations as before, we
obtain

$\displaystyle\left|A_{1}\right|^{2}=-\frac{1}{648}(17+3\cos{2\vartheta}),$
$\displaystyle\left|A_{2}\right|^{2}=\frac{1}{648}(21\cos{2\vartheta}-73),$
(17) $\displaystyle A_{1}A_{2}=\frac{1}{216}(1+3\cos{2\vartheta}).$

The result (IV) shows that albeit $\left|A_{1}\right|^{2}$ and
$\left|A_{2}\right|^{2}$ cannot be zero, the first term vanishes when
$\left|A_{1}\right|^{2}+2A_{1}A_{2}=0$, i.e.
$\left|A_{1}\right|^{2}+2A_{1}A_{2}=(15\cos(2\vartheta)-11)/648=0$ and hence
$\vartheta=\pm(1/2)\arccos(11/15)\approx 21.4^{\circ}$.

The shift of the maximum as a function of angle is shown in Fig. 4.

Figure 4: The spectral line profile given by Eq. (16). To illustrate the shift
of the profile maximum, the graph is plotted in logarithmic scale for the
values of angle $\vartheta=(1/2)\arccos(11/15)$, magic angle
$\vartheta=\arccos(1/\sqrt{3})$, and $\vartheta=\pi/2$.

In spite of obvious shift of the line profile maximum from Fig. 4, this
representation does not take into account the asymmetry caused by interference
transitions due to the fine structure of levels. The latter can be considered
with the remaining terms in Eq. (15) independently.

According to the theory of [5], generalized in the previous section to the two
observed lines, one should evaluate the amplitudes and determine the asymmetry
parameters. Then the spectral profile of the emission line corresponding to
the final state $1s_{1/2}^{F_{f}=1}$ is given by

$\displaystyle\phi(x)\approx\frac{\left|A_{1}\right|^{2}+A_{1}A_{2}}{(x-\tilde{\Delta}_{1}(x))^{2}+\frac{1}{4}\Gamma^{2}}$
(18) $\displaystyle+\frac{\left|A_{2}\right|^{2}}{\left(x+\Delta_{\rm
HFS}^{(2p_{3/2})}-\tilde{\Delta}_{2}(x)\right)^{2}+\frac{1}{4}\Gamma^{2}}.$

Here $\tilde{\Delta}_{1}(x)$ and $\tilde{\Delta}_{2}(x)$ are expressed as
before, see Eq. (11), with the substitution $\Delta^{(1s)}_{\rm
HFS}\rightarrow\Delta_{\rm HFS}^{(2p_{3/2})}$ and coefficients

$\displaystyle\tilde{C}_{1}=\left|A_{1}\right|^{2}+A_{1}A_{2},\,\tilde{a}_{1}$
$\displaystyle=$
$\displaystyle\frac{2\left|A_{3}\right|^{2}}{\left(\Delta_{\rm
fs}^{(11)}\right)^{3}}+\frac{2\left|A_{4}\right|^{2}}{\left(\Delta_{\rm
fs}^{(10)}\right)^{3}},$ $\displaystyle\tilde{b}_{1}$ $\displaystyle=$
$\displaystyle-\frac{2A_{1}A_{3}}{\Delta_{\rm
fs}^{(11)}}-\frac{2A_{1}A_{4}}{\Delta_{\rm fs}^{(10)}},$ (19)
$\displaystyle\tilde{C}_{2}=\left|A_{2}\right|^{2},\,\tilde{a}_{2}$
$\displaystyle=$
$\displaystyle\frac{2\left|A_{5}\right|^{2}}{\left(\Delta_{\rm
fs}^{(21)}\right)^{3}}+\frac{2\left|A_{6}\right|^{2}}{\left(\Delta_{\rm
fs}^{(20)}\right)^{3}},$ $\displaystyle\tilde{b}_{2}$ $\displaystyle=$
$\displaystyle-\frac{2A_{2}A_{5}}{\Delta_{\rm
fs}^{(21)}}-\frac{2A_{2}A_{6}}{\Delta_{\rm fs}^{(20)}},$

and together with the expressions (IV) we found

$\displaystyle\left|A_{3}\right|^{2}=-\frac{4}{81},\,A_{1}A_{3}=\frac{1+3\cos{2\vartheta}}{324},$
$\displaystyle\left|A_{4}\right|^{2}=-\frac{2}{81},\,A_{1}A_{4}=0,$ (20)
$\displaystyle\left|A_{5}\right|^{2}=-\frac{4}{81},\,A_{2}A_{5}=\frac{1+3\cos{2\vartheta}}{108},$
$\displaystyle\left|A_{6}\right|^{2}=-\frac{2}{81},\,A_{2}A_{6}=\frac{1+3\cos{2\vartheta}}{162}.$

Taking into account the result (IV), we find that the emission line profile to
the $1s_{1/2}^{F_{f}=1}$ state can be approximated by the second summand in
Eq. (18), where $\left|A_{2}\right|^{2}$ is defined at angle
$(1/2)\arccos(11/15)$. As a next step to accurately consider the hyperfine
splitting of the $2p_{3/2}$ state, an angle close to the magic angle should
obviously be chosen. Numerical calculation gives $\tilde{\Delta}_{2}(x=0)=0$
at $\vartheta=\pm 0.955338$ ($\vartheta_{\rm m}=0.955317$).

The schematic behavior of the profile constructed as the second summands in
Eqs. (9) and (18) is illustrated in Fig. 5.

Figure 5: The line profile constructed as the second summands in Eqs. (9),
(18). The graphs are plotted in logarithmic scale at different angles
$\vartheta=0$, $\vartheta=\vartheta_{\rm m}$ and $\vartheta=\pi/2$ to clearly
demonstrate the difference in decay emission line to the $1s_{1/2}^{F=0}$
state, while the second emission line remains the same.

In particular, from Fig. 5 it is clear that with the chosen asymmetry
parameters, the emission line to the $1s_{1/2}^{F=1}$ state remains unchanged.
In turn, the line profile corresponding to the state $1s_{1/2}^{F=0}$ changes
significantly. Moreover, a shift in the local minimum (centre of gravity) is
also obvious. Finally, in Fig. 6 we show in detail the emission line to the
$1s_{1/2}^{F=1}$ state.

Figure 6: The emission line to the $1s_{1/2}^{F=1}$ state at different angles
$\vartheta=0$, $\vartheta=\vartheta_{\rm m}$ and $\vartheta=\pi/2$. At
arbitrary angles, the profiles coincide due to the smallness of the parameters
$\tilde{a}_{2}$ and $\tilde{b}_{2}$. Dashed line is plotted at $x=-\Delta_{\rm
HFS}^{(2p_{3/2})}$.

The corresponding line profile is shifted by the value $\Delta_{\rm
HFS}^{(2p_{3/2})}$.

##  V Conclusions

In this work, we analyzed the solutions of the equation arising from the
extremum condition determined for the spectral line profile. Here, for
simplicity, the extremum condition has been used to determine the transition
frequency of the observed spectral line (the full-width-half-maximum
condition, i.e., $x=\Gamma/2$, can just as well be used). In the case of an
’isolated’ line, there is a single solution corresponding to the line profile
parameter representing the transition frequency. However, the presence of a
neighboring lines leads to significant complications. The asymmetry of the
line profile arising from nonresonant contributions to the scattering cross
section should be taken into account.

The most considerable contribution to the line asymmetry comes from the
interference effect occurring for the two pathways between neighboring
resonance states. As a rule for the description of QIE, the linear
approximation for the line profile is employed (Fano contour). The QI effect
has been widely discussed in the literature and has several important
properties. One of them is that the angular dependence is determined by the
set of quantum numbers for the described transitions.

Another important aspect of the QIE is that the dominant nonresonant
contribution can be reduced to zero by choosing the angle between the absorbed
and emitted photons. For the hydrogen atom, a theoretical description of the
one-photon scattering process was given in [4, 5] and later adapted in [14] to
accurately determine the $2s-4p$ transition frequency. In particular, it turns
out that the magic angle $\vartheta_{\rm m}=\arccos(1/\sqrt{3})$ should be
used for electric dipole transitions. At the same time, the model profile of
the line (Fano contour, see [5]) taking into account the asymmetry parameter
demonstrates an improved extrapolation of the experimental data to beyond the
resonance approximation, as in [14].

Using the Lyα transition as an example, we find that the choice of the magic
angle is very relevant for determining the unshifted transition frequency and
fine splitting interval (and hence the second transition frequency). However,
the centre of gravity of these lines cannot be defined at this angle through
an integer relation, such as a centroid for example. Determination of the
gravity centre is subjected to nonresonant effects in another way, the angle
dependence is determined not by the ratio of resonant and nonresonant
amplitudes but by their permissible combinations. The physical interpretation
of all roots for the equation arising from the extremum condition clearly
demonstrates this. The two lines corresponding to fine sublevels of the
Lyman-$\alpha$ transition (discarding the hyperfine structure of levels) are
equally distant from the centre of gravity at an angle of $\pi/2$, although
each is distorted due to nonresonant effects.

Despite the standalone interest in the gravity centre of lines in atomic
spectroscopy, its measurement can also be used to accurately determine the
hyperfine splitting. According to the analysis of section III, we at first
considered the possibility of processing experimental data to determine the
transition frequencies to selected hyperfine sublevels of the ground state in
the hydrogen atom, discarding the hyperfine structure of the $2p_{3/2}$ state.
Then in section IV the HFS of the resonant state was included into analysis.
The main conclusion of this consideration is that the hyperfine splitting
interval can be found exactly as the difference of the transition frequencies
representing the maxima of these symmetrized lines.

For this purpose, the symmetrization procedure should be performed using Eqs.
(9), (18) in accordance with the theory [5]. It can be expected that the HFS
interval of the ground state can be determined more accurately than it was
found in [18] from exactly the same experiment. In principle, determining the
HFS of the $2p_{3/2}$ state is also possible (although it is four times
smaller than the natural level width) by noticing the shift of the maximum at
different angles and calculating the frequency shift in accordance with the
extremum condition. Focusing on $\Delta^{(1s)}_{\rm HFS}$, according to the
conclusions drawn in sections II, III and IV, we find that first the emission
line corresponding to the transition to $1s_{1/2}^{F=1}$ state should be
processed to obtain a symmetrized profile shifted by the value of $\Delta_{\rm
HFS}^{(2p_{3/2})}$. The angle determining the corresponding asymmetry
parameters is about $21.4^{\circ}$. Then other asymmetry parameters
(determined at the magic angle) should be used to avoid QIE due to the fine
structure of the levels.

The importance of the above analysis can be emphasized by recent experiments
[22] aimed at a detailed comparison of the spectra of hydrogen and
antihydrogen. It can be expected that the results obtained in this work can
serve for the experimental improvement of such measurements.

## Acknowledgements

The work of T. Z. and A. A. was supported by the foundation for the
advancement of theoretical physics and mathematics ”BASIS” (grants No.
23-1-3-31-1 and 22-1-5-9-1, respectively).

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##  Appendix A Evaluation of transition matrix elements

To describe the one-photon scattering process, we use the $S$-matrix formalism
[24, 31]. According to the Feynman diagram depicted in Fig. A1,

Figure A1: Feynman diagram representing the one-photon scattering process by
an atomic electron. Wavy lines with arrows denote absorption (down arrow) and
emission (up arrow) with corresponding frequencies $\omega_{2}$ and
$\omega_{1}$, respectively. The double solid line represents the bound
electron state (Furry picture). The designations $f$, $n$ and $i$ denote the
final, intermediate, and initial states.

the $S$-matrix element is

$\displaystyle S_{fi}=(-\mathrm{i}e)^{2}\int
d^{4}x_{1}d^{4}x_{2}\overline{\psi}_{f}(x_{1})\gamma^{\mu}A_{\mu}^{*}(x_{1})S(x_{1},x_{2})\gamma^{\nu}A_{\nu}(x_{2})\psi_{i}(x_{2}).$
(A1)

The graph in Fig. A1 should additionally include a part with permuted photons.
However, this part is not essential for our purposes, and we omit its
consideration (the corresponding discussion can be found, for example, in [32,
4]).

For an arbitrary atomic state $a$

$\displaystyle\psi_{a}(x)=\psi_{a}(\bm{r})e^{-\mathrm{i}E_{a}t},$ (A2)

where $\psi_{a}(\bm{r})$ is the solution of the Dirac equation for the atomic
electron, $E_{a}$ is the Dirac energy,
$\overline{\psi}_{a}=\psi^{+}_{a}\gamma_{0}$ is the Dirac conjugated wave
function, $\gamma_{\mu}\equiv(\gamma_{0},\bm{\gamma})$ are the Dirac matrices
and $x\equiv(t,\bm{r})$ (vectors are in bold) is the four-dimensional space-
time coordinate. The photon field or the photon wave function $A_{\mu}(x)$ is
defined by

$\displaystyle
A_{\mu}(x)=\sqrt{\frac{2\pi}{\omega}}e_{\mu}e^{\mathrm{i}(\bm{k}\bm{r}-\omega
t)}=e^{-\mathrm{i}\omega t}A_{\mu}(\bm{r}),$ (A3)

where $e_{\mu}$ are the components of the photon polarization four-vector
($\bm{e}$ is 3-dimensional polarization vector for real photons),
$k\equiv(\omega,\bm{k})$ is the photon momentum four-vector, $\bm{k}$ is the
wave vector, $\omega=|\bm{k}|$ is the photon frequency. Eq. (A3) corresponds
to the absorbed photon and $A^{*}_{\mu}(x)$ corresponds to the emitted photon.
Finally, the electron propagator for the bound electron can be presented in
the form of the eigenmode decomposition with respect to one-electron
eigenstates:

$\displaystyle
S(x_{1},x_{2})=\frac{\mathrm{i}}{2\pi}\int\limits^{\infty}_{-\infty}d\Omega\,e^{-i\Omega(t_{1}-t_{2})}\sum\limits_{n}\frac{\psi_{n}(\bm{r}_{1})\overline{\psi}_{n}(\bm{r}_{2})}{\Omega-
E_{n}(1-\mathrm{i}0)}.$ (A4)

Integration over time variables yields

$\displaystyle S_{fi}=-2\pi\mathrm{i}e^{2}\int
d\bm{r}_{1}d\bm{r}_{2}\psi_{f}^{*}(\bm{r}_{1})\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}\int\limits^{\infty}_{-\infty}d\Omega\,\sum\limits_{n}\frac{\psi_{n}(\bm{r}_{1})\overline{\psi}_{n}(\bm{r}_{2})}{\Omega-
E_{n}(1-\mathrm{i}0)}\bm{\alpha}\bm{A}_{\bm{k}_{1},\bm{e}_{1}}\psi_{i}(\bm{r}_{2})\delta(E_{f}+\omega_{1}-\Omega)\delta(\Omega-\omega_{2}-E_{i}),\qquad$
(A5)

where $\bm{\alpha}$ is the Dirac matrix and $\bm{A}_{\bm{k},\bm{e}}$
represents the vector part of the photon wave function characterized by the
wave vector $\bm{k}$ and transversal polarization $\bm{e}$. Then, after the
integration over $\Omega$, we have

$\displaystyle
S_{fi}=-2\pi\mathrm{i}e^{2}\delta(E_{f}+\omega_{1}-\omega_{2}-E_{i})\sum\limits_{n}\frac{\langle
f|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|n\rangle\langle
n|\bm{\alpha}\bm{A}_{\bm{k}_{1},\bm{e}_{1}}|i\rangle}{E_{i}+\omega_{2}-E_{n}(1-\mathrm{i}0)}.$
(A6)

For the equal initial and final states (elastic scattering), $f=i=a$ and
$\omega_{1}=\omega_{2}\equiv\omega$, using the relation
$S_{fi}=-2\pi\mathrm{i}\,U_{fi}$, the expression (A6) gives the amplitude of
the scattering process:

$\displaystyle U_{aa}^{\rm sc}=e^{2}\sum\limits_{n}\frac{\langle
a|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|n\rangle\langle
n|\bm{\alpha}\bm{A}_{\bm{k}_{1},\bm{e}_{1}}|a\rangle}{E_{a}+\omega-
E_{n}(1-\mathrm{i}0)},$ (A7)

which in the resonance approximation (the resonant state is fixed in the sum
over $n$), $n=r$, reduces to

$\displaystyle U_{aa}^{\rm sc}=e^{2}\frac{\langle
a|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|r\rangle\langle
r|\bm{\alpha}\bm{A}_{\bm{k}_{1},\bm{e}_{1}}|a\rangle}{E_{a}+\omega-
E_{r}(1-\mathrm{i}0)}.$ (A8)

To obtain the spectral line profile, it is necessary to regularize the
expression (A8), since it diverges at $\omega=E_{r}-E_{a}$. This can be done
using the procedure described in [1], see also [2] for details, according to
which an infinite number of one-loop self-energy corrections should be
inserted sequentially into the graph in Fig. A1. The regularization procedure
is shown in Fig A2.

Figure A2: Feynman diagram representing the one-loop self-energy insertions
into the photon scattering amplitude. All designations are the same as in Fig.
A1. The parts given by the photon loop (closed wavy lines) are the
corresponding lower order self-energy correction. The dots denote an infinite
number of such inserts. The summation of the diagrams with a different number
of loops is assumed.

The sum of the diagrams in Fig. A2 is a geometric progression, which leads to
the appearance of the real and imaginary parts of the lower-order self-energy
correction in the energy denominator of Eq. (A8), see [2]:

$\displaystyle U_{aa}^{\rm sc}=e^{2}\frac{\langle
a|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|r\rangle\langle
r|\bm{\alpha}\bm{A}_{\bm{k}_{1},\bm{e}_{1}}|a\rangle}{E_{a}+\omega-
E_{r}-L_{r}^{\rm SE}+\mathrm{i}\frac{\Gamma_{r}}{2}},$ (A9)

where we have used that the imaginary part of the self-energy correction is
minus half of the $\Gamma_{r}$ level width, and the real part is the lower-
order Lamb shift of the $r$ resonance state. In turn, choosing the nearest
states in the sum over $n$, one can write

$\displaystyle U_{aa}^{\rm sc}=e^{2}\frac{\langle
a|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|r_{1}\rangle\langle
r_{1}|\bm{\alpha}\bm{A}_{\bm{k}_{1},\bm{e}_{1}}|a\rangle}{E_{a}+\omega-
E_{r_{1}}-L_{r_{1}}^{\rm
SE}+\mathrm{i}\frac{\Gamma_{r_{1}}}{2}}+e^{2}\frac{\langle
a|\bm{\alpha}\bm{A}^{*}_{\bm{k}_{1},\bm{e}_{1}}|r_{2}\rangle\langle
r_{2}|\bm{\alpha}\bm{A}_{\bm{k}_{1},\bm{e}_{1}}|a\rangle}{E_{a}+\omega-
E_{r_{2}}-L_{r_{2}}^{\rm SE}+\mathrm{i}\frac{\Gamma_{r_{2}}}{2}},$ (A10)

where we used the regularization procedure depicted in Fig. A2. Introducing
the notation $\omega_{0}=E_{r_{1}}-E_{a}$, we find that the second energy
denominator is shifted by $\Delta=E_{r_{2}}-E_{r_{1}}$ with respect to the
first one.

The cross section of the elastic one-photon scattering process is obtained by
the formula:

$\displaystyle d\sigma^{\rm
sc}_{a}=2\pi\sum\limits_{\bm{e}_{1}\bm{e}_{1}}\left|U_{aa}^{sc}\right|^{2}\delta(\omega_{1}-\omega_{2})\frac{d\bm{k}}{(2\pi)^{3}},$
(A11)

which should be summed over the projection of the angular momentum of the
final state and averaged over the projection of the initial state. The
summation over the projections of the intermediate states in Eqs. (A9) and
(A10) is also assumed. The $\delta$-function is written for clarity in Eq.
(A11) and was taken into account in the expressions (A7)-(A10).

Further evaluation concerns the matrix elements in the numerators of Eq.
(A10). For the hydrogen atom within the framework of the dipole approximation
$e^{i\bm{k}\bm{r}}\approx 1$ and
$\bm{\alpha}\bm{A}_{\bm{k},\bm{e}}=\sqrt{2\pi/\omega}\,\bm{e}\bm{\alpha}$.
Then, in the nonrelativistic limit $\bm{\alpha}$ is replaced by $\bm{p}$
(electron momentum operator). The electron momentum operator can be replaced
by the coordinate, $\bm{r}$, according to the commutation relation
$\bm{p}=\mathrm{i}[\hat{H},\bm{r}]$, which for the matrix element
$(\bm{p})_{ab}$ reduces to $\mathrm{i}(E_{a}-E_{b})(\bm{r})_{ab}=\Delta
E_{ab}(\bm{r})_{ab}$. According to this, we arrive at

$\displaystyle U_{aa}^{\rm sc}=e^{2}2\pi\omega_{0}^{2}\left[\frac{\langle
a|\bm{e}_{2}\bm{r}|r_{1}\rangle\langle
r_{1}|\bm{e}^{*}_{1}\bm{r}|a\rangle}{\omega_{0}-\omega-\mathrm{i}\frac{\Gamma_{r_{1}}}{2}}+e^{2}\frac{\langle
a|\bm{e}^{*}_{1}\bm{r}|r_{2}\rangle\langle
r_{2}|\bm{e}_{2}\bm{r}|a\rangle}{\omega_{0}-\omega+\Delta-\mathrm{i}\frac{\Gamma_{r_{2}}}{2}}\right].$
(A12)

Here the wave functions are given by the solution of the Schrödinger-Pauli
equation. The common pre-factor arising before the squared amplitude is not
essential for our purposes and, in further, we focus on the matrix elements in
brackets of Eq. (A12).

Considering the states shifted relative to each other by the fine structure,
the resonance states $r_{1}$ and $r_{2}$ are described as the atomic energy
levels with different angular momentum of the bound electron, $j$. Then, the
states can be characterized by the set of quantum numbers $njlsm_{j}$, where
$n$ is the principal quantum number, $l$ is the orbital momentum, $s$
represents the spin of the bound electron and $j$, $m_{j}$ represents the
total angular momentum, $\bm{j}=\bm{l}+\bm{s}$ and its projection,
respectively. Using the definition of the scalar product
$(\bm{e}\bm{r})=\sum_{q}(-1)^{q}\bm{e}_{q}\bm{r}_{-q}$, the matrix elements in
Eq. (A12) can be evaluated via the Wigner-Eckart theorem:

$\displaystyle\langle
n^{\prime}j^{\prime}m^{\prime}_{j}|\hat{T}_{k\kappa}|njm_{j}\rangle=(-1)^{2k}C_{jm_{j}\,k\kappa}^{j^{\prime}m^{\prime}_{j}}\frac{\langle
n^{\prime}j^{\prime}\|\hat{T}_{k}\|nj\rangle}{\sqrt{2j^{\prime}+1}},$ (A13)

where the coefficient $C_{j_{1}m_{1}\,j_{2}m_{2}}^{jm}$ gives the Clebsch-
Gordan coefficient for the decomposition of $|jm\rangle$ in terms of
$|j_{1}m_{1}\rangle$, $|j_{2}m_{2}\rangle$ [33]. The reduced matrix element of
the spin-independent operator for the set of quantum numbers $njlsm_{j}$ is
defined by

$\displaystyle\langle
n^{\prime}l^{\prime}s^{\prime}j^{\prime}\|\hat{T}_{k}\|nlsj\rangle=\delta_{ss^{\prime}}(-1)^{j+l^{\prime}+s+k}\Pi_{jj^{\prime}}\begin{Bmatrix}l&s&j\\\
j^{\prime}&k&l^{\prime}\end{Bmatrix}\langle
n^{\prime}l^{\prime}\|\hat{T}_{k}\|nl\rangle.$ (A14)

Then, for the coordinate operator

$\displaystyle\langle
n^{\prime}l^{\prime}\|r\|nl\rangle=(-1)^{l^{\prime}}\Pi_{ll^{\prime}}\begin{pmatrix}l^{\prime}&1&l\\\
0&0&0\end{pmatrix}\int\limits_{0}^{\infty}dr\,r^{3}\,R_{n^{\prime}l^{\prime}}(r)R_{nl}(r),$
(A15)

where $R_{nl}(r)$ is the radial part of the hydrogen wave function.

The summation over projections can be performed as follows. For the squared
amplitude (A12) we should consider

$\displaystyle\left|A\right|^{2}=\sum\limits_{\begin{subarray}{c}q_{1}q_{2}\\\
q^{\prime}_{1}q^{\prime}_{2}\end{subarray}}(-1)^{q_{1}+q_{2}+q^{\prime}_{1}+q^{\prime}_{2}}\bm{e}^{*}_{1_{q_{1}}}\bm{e}_{2_{q_{2}}}\bm{e}_{1_{q^{\prime}_{1}}}\bm{e}^{*}_{2_{q^{\prime}_{2}}}\sum\limits_{\begin{subarray}{c}m_{j_{f}}m_{j_{i}}\\\
m_{j_{r}}m^{\prime}_{j_{r}}\end{subarray}}\langle
n_{a}l_{a}sj_{a}m_{j_{f}}|r_{-q_{1}}|n_{r}l_{r}sj_{r}m_{j_{r}}\rangle\langle
n_{r}l_{r}sj_{r}m_{j_{r}}|r_{-q_{2}}|n_{a}l_{a}sj_{a}m_{j_{i}}\rangle\qquad$
$\displaystyle\times\langle
n_{a}l_{a}sj_{a}m_{j_{i}}|r_{-q^{\prime}_{2}}|n_{r}l_{r}sj^{\prime}_{r}m^{\prime}_{j_{r}}\rangle\langle
n_{r}l_{r}sj^{\prime}_{r}m^{\prime}_{j_{r}}|r_{-q^{\prime}_{1}}|n_{a}l_{a}sj_{a}m_{j_{i}}\rangle,\qquad\qquad$
(A16)

where it is taken into account that for the process of elastic scattering, the
initial and final states can differ in the projection of the angular momentum
and for a one-electron atom $s_{a}=s_{r}\equiv s=1/2$ with fixed orbital
momenta of the initial, final, and resonant states. Denoting all coefficients
which are independent of projections as $\\{{\rm red}\\}$, we find that the
angular correlations between the polarization vectors are determined by

$\displaystyle\left|A\right|^{2}=\sum\limits_{\begin{subarray}{c}q_{1}q_{2}\\\
q^{\prime}_{1}q^{\prime}_{2}\end{subarray}}(-1)^{q_{1}+q_{2}+q^{\prime}_{1}+q^{\prime}_{2}}\bm{e}^{*}_{1_{q_{1}}}\bm{e}_{2_{q_{2}}}\bm{e}_{1_{q^{\prime}_{1}}}\bm{e}^{*}_{2_{q^{\prime}_{2}}}\sum\limits_{\begin{subarray}{c}m_{j_{f}}m_{j_{i}}\\\
m_{j_{r}}m^{\prime}_{j_{r}}\end{subarray}}\\{{\rm
red}\\}\Pi^{-1}_{j_{a}j_{r}j_{a}j^{\prime}_{r}}C^{j_{a}m_{j_{f}}}_{j_{r}m_{r}\,1-q_{1}}\,C^{j_{r}m_{j_{r}}}_{j_{a}m_{j_{i}}\,1-q_{2}}\,C^{j_{a}m_{j_{i}}}_{j^{\prime}_{r}m^{\prime}_{r}\,1-q^{\prime}_{2}}\,C^{j^{\prime}_{r}m^{\prime}_{j_{r}}}_{j_{a}m_{j_{f}}\,1-q^{\prime}_{1}}.\qquad$
(A17)

By rearranging the indices of the Clebsch-Gordan coefficients according to the
symmetry property [33], we have employed the following relation:

$\displaystyle\sum\limits_{\delta}C^{e\epsilon}_{b\beta\,d\delta}C^{d\delta}_{a\alpha\,f\varphi}=\sum\limits_{c\gamma}(-1)^{2e}\Pi_{cd}C^{c\gamma}_{a\alpha\,b\beta}C^{e\epsilon}_{f\varphi\,c\gamma}\begin{Bmatrix}a&b&c\\\
e&f&d\end{Bmatrix}.$ (A18)

Then, summing over $m_{j_{r}}$ and $m^{\prime}_{j_{r}}$, one can obtain

$\displaystyle\left|A\right|^{2}\sim\sum\limits_{\begin{subarray}{c}q_{1}q_{2}\\\
q^{\prime}_{1}q^{\prime}_{2}\end{subarray}}(-1)^{q_{1}+q_{2}+q^{\prime}_{1}+q^{\prime}_{2}}\bm{e}^{*}_{1_{q_{1}}}\bm{e}_{2_{q_{2}}}\bm{e}_{1_{q^{\prime}_{1}}}\bm{e}^{*}_{2_{q^{\prime}_{2}}}\sum\limits_{m_{j_{f}}m_{j_{i}}}\sum\limits_{c\gamma\,d\delta}C^{c\gamma}_{1-q_{2}\,1-q_{1}}C^{j_{a}m_{j_{f}}}_{j_{a}m_{j_{i}}\,c\gamma}C^{d\delta}_{1-q^{\prime}_{1}\,1-q^{\prime}_{1}}C^{j_{a}m_{j_{i}}}_{j_{a}m_{j_{f}}\,d\delta}.$
(A19)

Summing over $m_{j_{f}}$ and $m^{\prime}_{j_{i}}$, we can get

$\displaystyle
C^{j_{a}m_{j_{f}}}_{j_{a}m_{j_{i}}\,c\gamma}C^{j_{a}m_{j_{i}}}_{j_{a}m_{j_{f}}\,d\delta}=(-1)^{-\gamma}\frac{2j_{a}+1}{2c+1}\delta_{c\,d}\delta_{\gamma\,-\delta}.$
(A20)

Then, taking into account the symmetry property of the Clebsch-Gordan
coefficients and $q_{1}+q_{2}+q^{\prime}_{1}+q^{\prime}_{2}=-\gamma-\delta$,
by the definition of the tensor product

$\displaystyle\left|A\right|^{2}\sim\sum\limits_{c\gamma}(-1)^{c-\gamma}\\{\bm{e}_{2}\otimes\bm{e}^{*}_{1}\\}_{c-\gamma}\\{\bm{e}^{*}_{2}\otimes\bm{e}_{1}\\}_{c\gamma}.$
(A21)

The latter is the scalar product of two tensors.

Performing the numerical calculations for Eq. (A21), one can find

$\displaystyle\left|A_{j_{r}=j^{\prime}_{r}=\frac{1}{2}}\right|^{2}$
$\displaystyle\equiv$
$\displaystyle\left|A_{1}\right|^{2}=\frac{2R}{81}\left(3\\{\bm{e}_{2}\otimes\bm{e}^{*}_{1}\\}_{0}\\{\bm{e}^{*}_{2}\otimes\bm{e}_{1}\\}_{0}+2\\{\bm{e}_{2}\otimes\bm{e}^{*}_{1}\\}_{1}\\{\bm{e}^{*}_{2}\otimes\bm{e}_{1}\\}_{1}\right)=\frac{2R}{81}\left(\cos^{2}\vartheta+\sin^{2}\vartheta\right)=\frac{2R}{81},\qquad$
$\displaystyle A_{j_{r}=\frac{1}{2}}A_{j^{\prime}_{r}=\frac{3}{2}}$
$\displaystyle\equiv$ $\displaystyle
A_{1}A_{2}=\frac{4R}{81}\left(3\\{\bm{e}_{2}\otimes\bm{e}^{*}_{1}\\}_{0}\\{\bm{e}^{*}_{2}\otimes\bm{e}_{1}\\}_{0}-\\{\bm{e}_{2}\otimes\bm{e}^{*}_{1}\\}_{1}\\{\bm{e}^{*}_{2}\otimes\bm{e}_{1}\\}_{1}\right)=\frac{2R}{81}\left(2\cos^{2}\vartheta-\sin^{2}\vartheta\right),$
(A22) $\displaystyle\left|A_{j_{r}=j^{\prime}_{r}=\frac{3}{2}}\right|^{2}$
$\displaystyle\equiv$
$\displaystyle\left|A_{2}\right|^{2}=\frac{4R}{81}\left(6\\{\bm{e}_{2}\otimes\bm{e}^{*}_{1}\\}_{0}\\{\bm{e}^{*}_{2}\otimes\bm{e}_{1}\\}_{0}+\\{\bm{e}_{2}\otimes\bm{e}^{*}_{1}\\}_{1}\\{\bm{e}^{*}_{2}\otimes\bm{e}_{1}\\}_{1}\right)=\frac{2R}{81}\left(4\cos^{2}\vartheta+\sin^{2}\vartheta\right).$

Here $R$ denotes all the remaining radial integrals, and we have used
relations for the tensor components of ranks $0$, $1$ with scalar and vector
products of polarization vectors [33]. The second equation can be solved with
respect to $\vartheta$:
$2\cos^{2}\vartheta-\sin^{2}\vartheta=3\cos^{2}\vartheta-1=0$ leading to the
magic angle $\vartheta_{\rm m}=\arccos(1/\sqrt{3})$ at which the interference
of two pathways $j_{a}=1/2\rightarrow j_{r}=1/2$ and $j_{a}=1/2\rightarrow
j^{\prime}_{r}=3/2$ is equal to zero.


$\tilde{\Gamma}=-\tilde{\mathfrak{D}}^{\dagger,x}K_{xy}\tilde{\nabla}_{t}\tilde{\mathfrak{D}}^{y}=\tilde{\nabla}_{t}\tilde{\mathfrak{D}}^{\dagger,x}K_{xy}\tilde{\mathfrak{D}}^{y}+\tilde{\mathfrak{D}}^{\dagger,x}\frac{d{K}_{xy}}{dt}\tilde{\mathfrak{D}}^{y}$
(156)

The explicit derivative $\tilde{\nabla}_{t}\tilde{\mathfrak{D}}^{y}$ is
explicitly written in (188). The last equality comes from the fact that
$\tilde{\Gamma}=-\tilde{\Gamma}^{\dagger_{H}}+\phi^{x}\frac{d{K}_{xy}}{dt}\Delta^{xy}\partial_{\phi^{x}}$
as a direct consequence of
$(\tilde{\nabla}_{t}{\mathcal{O}})^{\dagger_{H}}=\tilde{\nabla}_{t}({\mathcal{O}}^{\dagger_{H}})$.
We derive the explicit expression and compute the derivative of creation and
annihilation operator in the Appendix A. Recall the inhomogeneous
transformation law of the connection that under a change of coordinates
$\Psi\rightarrow\mathcal{C}\Psi$ transforms as

$\Gamma\rightarrow\mathcal{C}\Gamma\mathcal{C}^{-1}+\mathcal{C}\frac{d\mathcal{C}^{-1}}{dt}$

###  6.2 Dynamics: The Modified Schrödinger equation

We want the evolution to preserve the symplectic structure. For that matter we
opt for a Hamiltonian flow generated by an Hermitian operator $\hat{H}$ to
postulate evolution. Then, for any function $F(t,\Psi,\bar{\Psi})$, its
evolution is generated by some $f_{\hat{H}}$

$\frac{d}{dt}F=\partial_{t}F+\\{F,f_{\hat{H}}\\}_{\mathscr{P}}$ (157)

where $\hat{H}$ is a self adjoint operator. The form of $f_{\hat{H}}$ is
(130). With the Poisson Bracket $\\{\cdot,\cdot\\}_{\mathscr{P}}$, we can
readily write down the modified Schrödinger in any set of real coordinates

$\nabla_{t}\left(\begin{array}[]{c}\Phi^{\phi}\\\
\Pi^{\phi}\end{array}\right)=-\mathcal{J}\hat{H}_{\sigma}^{\phi}\left(\begin{array}[]{c}\Phi^{\sigma}\\\
\Pi^{\sigma}\end{array}\right).$ (158)

It is clear that this equation does not have a solution on
$(\mathcal{N})\times(\mathcal{N})$. In Quantum theory it is desirable to solve
this equation over a Hilbert space. Unfortunately for Quantum Field theory
this is not possible either, to solve this equation we must look for solutions
on $(\mathcal{N})^{\prime}\times(\mathcal{N})^{\prime}$, this is our way out
to deal with different instantaneous Hilbert spaces.

##  7 Application: Klein Gordon theory on curved spacetimes

We have seen above how the choice of a complex structure on the phase space of
classical fields is a crucial ingredient of the quantization procedure.
Nonetheless, from a physical point of view, given a relativistic classical
field theory, it is an open question how to identify the correct complex
structure to define the quantization procedure. There exists a successful
answer for stationary space times [18, 8, 19], but the solution for arbitrary
models is, to the best of our knowledge, still missing. In this section, we
review this treatment form the geometrical perspective developed above, in the
simple case of a Klein-Gordon model.

###  7.1 The classical theory and the choice of the complex structure on the
classical phase space

To start with, we consider a $d+1$ globally hyperbolic spacetime
$(\mathcal{M},\mathbf{g},\nabla_{g})$ endowed with a pseudo-Riemannian
structure $\mathbf{g}$ of Lorentzian signature $(-,+,\cdots,+)$ and the Levi-
Civita connection $\nabla_{g}$. $\mathcal{M}$ is diffeomorphic to
$\mathbb{R}\times\Sigma$ where $\Sigma$ is a $d-$dimensional manifold
diffeomorphic to every space-like Cauchy hypersurface of $\mathcal{M}$. A one
parameter family of embeddings $\Sigma\to\mathcal{M}$ is determined by a
scalar function $\hat{t}:\mathcal{M}\to\mathbb{R}$ such that
$\Sigma_{t}=\\{\hat{t}(x)=t\textrm{ with }x\in\mathcal{M}\\}$ i.e.
$\Sigma_{t}$ are the level sets of $\hat{t}$. The structure of the whole
spacetime project into $(\Sigma_{t},\mathbf{h}_{t},\mathbf{D}:t)$ where
$\mathbf{h}$ is a Riemannian structure and $\mathbf{D}$ the corresponding
Levi-Cività connection $\mathbf{D}\mathbf{h}=0$.

We can also define the dynamics directly on $(\Sigma,\mathbf{h},\mathbf{D})$.
To this end we take a coordinate system $\mathbf{x}=(x^{i})$ $i=1,\cdots,d$
over $\Sigma$ that induces a coordinate system over $\mathcal{M}$ as of
$(t,\mathbf{x})$. In this coordinate system we obtain the relation

$\partial_{t}=N\mathbf{n}+N^{i}\frac{\partial}{\partial{x^{i}}},$ (159)

where $N\mathbf{n}=\vec{\nabla}\hat{t}$ and
$N=[-\mathbf{g}(\vec{\nabla}\hat{t},\vec{\nabla}\hat{t})]^{-\frac{1}{2}}$ is
the lapse function while $\vec{N}=N^{i}\frac{\partial}{\partial{x^{i}}}$ is
the shift vector. Thus, if we provide as data $(N,\vec{N})$ over $\Sigma$, the
embedding is fully determined.

In the Klein-Gordon classical theory the dynamics of a scalar field can be
defined as a parametric curve $\mathbb{R}\to\mathcal{N}$, where $\mathcal{N}$
is a nuclear space of test functions. We denote this solution by
$\varphi_{\mathbf{x}}(t)$. Associated to the space of solutions, we build a
Gel’fand triple with respect to the Hilbert space
$L^{2}(\mathcal{N},d\mathrm{Vol})$, where $d\mathrm{Vol}$ represents the
measure associated with the 3-metric tensor $h$.

With these ingredients we can introduce a Lagrangian formalism, as it is
usually done in QFT. From the usual Lagrangian density in four dimensions, we
can write the following bilinear form:

$\mathcal{L}(t)=\int_{\Sigma}d\mathrm{Vol}\frac{1}{2}\Bigl{[}\frac{(\dot{\varphi})^{2}}{N}-\frac{N^{i}}{N}2\dot{\varphi}D_{i}\varphi+\\\
\Bigl{(}\frac{N^{i}N^{j}}{N}-Nh^{ij}\Bigr{)}(D_{i}\varphi)(D_{j}\varphi)-m^{2}(\varphi)^{2}\Bigr{]}$
(160)

Where $N,N^{i}$ and $h^{ij}$ are functions on $\Sigma$.

In order to define the canonical momentum we must choose between different
options. One possible path is to define $\pi^{\mathbf{x}}$ as a density and
therefore to build it as an element of $\mathcal{N}^{\prime}$, which is the
choice made in [8]. Another option is to consider $\varphi_{\mathbf{x}}$ as an
element of a Rigged Hilbert space and profit from this structure to let
$\pi_{\mathbf{x}}$ be an element of the same space as the field
$\varphi_{\mathbf{x}}$. With respect to the Gel’fand triple and the injection
$\mathcal{N}\subset L^{2}(\Sigma,dVol)$ we obtain as canonical momentum

$\pi_{x}=\delta_{xy}\frac{\partial{\mathcal{L}}}{\partial\dot{\varphi}_{y}}=\frac{1}{N}\Big{[}\dot{\varphi}_{x}-N^{i}D_{i}\varphi_{x}\Big{]}$
(161)

The Hamiltonian is defined using the same rigging structure
$\mathcal{H}=\delta^{xy}\pi_{x}\dot{\varphi}_{y}-\mathcal{L}$ and can be
written as

$H=\int_{\Sigma}d^{d}x\sqrt{h}[N\mathcal{H}+N^{i}\mathcal{H}_{i}]$ (162)

Where $\mathcal{H}$ is the so called superHamiltonian and $\mathcal{H}_{i}$
the supermomenta given by

$\displaystyle\mathcal{H}$
$\displaystyle=\frac{1}{2}[\pi^{2}+h^{ij}D_{i}\varphi
D_{j}\varphi+m^{2}\varphi^{2}]$ (163) $\displaystyle\mathcal{H}_{i}$
$\displaystyle=\pi D_{i}\varphi$ (164)

As we saw in [1], with this definition we can endow $\mathcal{M}_{F}$ with a
Poisson bracket, which in our notation reads

$\\{P,Q\\}=\delta_{xy}\Big{(}\frac{\partial
P}{\partial\varphi_{x}}\frac{\partial Q}{\partial\pi_{y}}-\frac{\partial
P}{\partial\pi_{x}}\frac{\partial Q}{\partial\varphi_{y}}\Big{)}.$ (165)

Notice that the definition of the Hilbert space, and hence of the rigging,
depends on the complex structure and thus on time. Therefore, if we write
Liouville equation with the Hamiltonian and the Poisson bracket, it is also
necessary to introduce a covariant derivative for time already at the
classical level. The usual prescription for classical theory is to consider
$(\varphi_{x},\pi^{x})$ as time independent. In such a case, we get a
correction term for $\pi_{x}$ and the evolution of the classical theory is
written in our coordinates as:

$\left(\frac{d}{dt}+\Gamma_{c}\right)\left(\begin{array}[]{c}\varphi_{x}\\\
\pi_{x}\end{array}\right)=\Big{\\{}\left(\begin{array}[]{c}\varphi_{x}\\\
\pi_{x}\end{array}\right),H\Big{\\}}=F\left(\begin{array}[]{c}\varphi_{x}\\\
\pi_{x}\end{array}\right)$ (166)

with

$\Gamma_{c}=\left(\begin{array}[]{cc}0&0\\\
0&\dot{\delta}_{xz}\delta^{zy}\end{array}\right)\textrm{ and
}F=\left(\begin{array}[]{cc}N^{i}D_{i}&N\\\ -\Theta&\Lambda\end{array}\right)$

where $\dot{\delta}_{xz}\delta^{zy}\varphi_{y}=-\frac{\dot{h}}{2h}\varphi(x)$,
$-\Theta=ND^{i}D_{i}+(D^{i}N)D_{i}-Nm^{2}$ and
$\Lambda=N^{i}D_{i}+D_{i}N^{i}$. This implies that $\Theta^{x}_{y}$ is a
symmetric operator.

We can connect this classical dynamics with the quantum one considering one
particle states, which are those that can be written in holomorphic second-
quantized coordinates for the holomorphic quantization as
$\Psi^{\phi}_{1PS}=\tilde{\Phi}_{x}\phi^{x}+i\tilde{\Pi}_{x}\phi^{x}$. The
main problem with this representation is that it relies on the rigging
provided by $L^{2}_{Hol}(\mathcal{N}^{\prime}_{\mathbb{C}},D\mu_{c})$ that, in
turn, depends on the complex structure (30) which is, for now, unknown. Notice
that, as the complex structure is built on the phase space of fields defined
on the submanifold $\Sigma$, which in general evolves in time, it will be, in
general, time dependent. In [18] it is proved that, for static and stationary
spacetimes, there exists a unique complex structure $J$ once the submanifold
$\Sigma$, the 3-metric $h_{jk}$ (and hence the covariant derivative $D$) and
operators $N_{0}$ and $\vec{N}$ have been fixed. The expression of such a
complex structure in our notation reads

$J=\lvert F\rvert^{-1}F,$ (167)

for $F$ the classical linear evolution defined by Equation (166).

For simplicity we will take this prescription as well in the general case.
Nonetheless, this prescription requires further investigation. In the first
place the connection $\Gamma_{c}$ appearing in (167) does not coincide with
the the connection that must appear in the one particle state theory $\Gamma$
derived from the second quantized framework . To solve this issue one is
tempted to try and add the connection that preserves the Kähler structure
already at the classical level. However, this connection $\Gamma(J)$ stems
from the complex structure that we are deriving. Thus this procedure is
difficult to implement in the general case because both ingredients have to be
determined at the same time. Besides, this procedure would modify the
classical equations of motion and it is unclear its interpretation in physical
terms.

We can write down an explicit expression for $J$ using the prescription (167)
studying first two limiting cases where $J$ can be written down
straightforwardly .

##### Null Shift

In this case we set $N^{i}=0$ and we get

$J_{0}=\left(\begin{array}[]{cc}0&\Theta^{-\frac{1}{2}}N^{\frac{1}{2}}\\\
-N^{-\frac{1}{2}}\Theta^{\frac{1}{2}}&0\end{array}\right)$ (168)

##### Huge shift

In this case we consider $N^{i}N_{i}\gg N^{2}$ and therefore we set $N\simeq
0$. This is, of course, a limiting approximation because the lapse function
can never vanish. Then we get

$J_{\infty}=\left(\begin{array}[]{cc}(-\alpha^{2})^{-\frac{1}{2}}N^{i}D_{i}&0\\\
0&(-\Lambda^{2})^{-\frac{1}{2}}\Lambda\end{array}\right)$ (169)

with $\alpha=N^{i}D_{i}$.

##### Interpolating cases

The other cases can be obtained as

$J=A_{0}J_{0}+A_{\infty}J_{\infty}$ (170)

Where $A_{0}=\lvert F\rvert^{-1}\lvert F_{0}\rvert$ and $A_{\infty}=\lvert
F\rvert^{-1}\lvert F_{\infty}\rvert$ are operators being

$\displaystyle\lvert
F_{0}\rvert=\left(\begin{array}[]{cc}(-N\Theta)^{\frac{1}{2}}&0\\\ 0&(-\Theta
N)^{\frac{1}{2}}\end{array}\right),\hskip 10.00002pt\lvert
F_{\infty}\rvert=\left(\begin{array}[]{cc}(-\alpha^{2})^{\frac{1}{2}}&0\\\
0&(-\Lambda^{2})^{\frac{1}{2}}\end{array}\right),$ (175) $\displaystyle\lvert
F\rvert^{2}=\lvert F_{0}\rvert^{2}+\lvert F_{\infty}\rvert^{2}-\big{(}\lvert
F_{0}\rvert J_{0}J_{\infty}\lvert F_{\infty}\rvert+\lvert F_{\infty}\rvert
J_{\infty}J_{0}\lvert F_{0}\rvert\big{)}$ (176)

In general this expression is difficult to work out explicitly because of the
need to compute $\lvert F\rvert^{-1}$. The derivation of the general complex
structure can be found in [8] with the corresponding changes due to $\pi$
being considered a density instead of a scalar.

Remember that what we refer to as the time-dependence of these structures
reflects a dependence on the geometric Cauchy data ($h_{ij},\pi^{ij}$) and on
the lapse and shift functions ($N,N^{i}$). This dependence is necessary in
order to preserve the physical interpretation of our QFT throughout the
foliation.

Indeed, for a given background spacetime the geometric content on each
hypersurface is the one given by the choice of the foliation and thus, we have
a description of the spacetime in terms of a curve
$\gamma:\mathbb{R}\rightarrow\mathcal{M}_{G}\times\mathcal{M}_{N}$ such that
$\gamma(t)=(h_{ij}(t),\pi^{ij}(t),N(t),N^{i}(t))$ is the geometrical content
at the hypersurface $\Sigma_{t}$, which is considered known $\forall
t\in[\tau_{0},\tau_{f}]$. Thus, the dependence on these geometric variables of
the Kähler structure described above manifest itself as an additional
dependence on the Hamiltonian time.

On the other hand, in a theory where, instead of being a background, the
metric evolves coupled to the quantum matter and therefore exhibits
backreaction from the matter evolution, this geometric curve is unknown
beforehand and one must make the Kähler structure depend on the gravitational
degrees of freedom, instead of time. This is the subject of study of [2].

###  7.2 The quantum model

####  7.2.1 Obtaining the quantum evolution

Having chosen a complex structure on $\mathcal{M}_{F}$, we can define
$\mathcal{M}_{C}$ and proceed with the quantization of the theory.
Nonetheless, the quantization cannot be carried out directly in the limit
$N\simeq 0$ because $\Delta^{\mathbf{xy}}=0$ but we can study it in the limit
$N^{i}=0$. In this limit we obtain

$\varphi_{x}\Delta^{xy}\xi_{y}=\int_{\Sigma}d^{d}x\sqrt{h}\varphi(x)\Theta^{-\frac{1}{2}}N^{\frac{1}{2}}\xi(x)$
(177)

Also, the Hamiltonian in holomorphic coordinates is written
${H}=\bar{\phi}^{x}\Theta_{xy}\phi^{y}$ and the quantum Hamiltonian acting
over this space is obtained with Wick quantization as

$\hat{H}=\Theta_{xy}(a^{\dagger})^{x}a^{y}$ (178)

In holomorphic s.q. coordinates this expression is just $\phi^{y}(\sqrt{\Theta
N})_{y}^{x}\partial_{\phi^{x}}$ where $N^{x}_{y}\varphi_{x}=N\varphi_{y}$

The diagonal elements of the complex structure are $A=0$, then
$D^{-1}=-\Delta$ and the Fourier transform (113) is particularly simple to
compute. With this expression it follows that the connection (156) is given by

$\tilde{\Gamma}=-\frac{1-\epsilon}{2}\phi^{y}K_{yz}\dot{\Delta}^{zx}\partial_{\phi^{x}}+\epsilon\phi^{x}\dot{\delta}_{xz}\delta^{zy}\partial_{\phi^{y}}$

It is important to notice that (152) does not hold with this prescription of
complex structure. In the holomorphic setting the Schrodinger equation is
particularly simple

$i\left[\frac{\partial}{\partial
t}-\frac{1}{2}\phi^{y}K_{yz}\dot{\Delta}^{zx}\partial_{\phi^{x}}\right]\Psi=\\\
[\phi^{y}(\sqrt{\Theta
N})_{y}^{x}\partial_{\phi^{x}}+\frac{1}{2}\phi^{y}K_{yz}\dot{\Delta}^{zx}\partial_{\phi^{x}}+\phi^{x}\dot{\delta}_{xz}\delta^{zy}\partial_{\phi^{y}}]\Psi$
(179)

There are new terms in the equation that involves the time derivative
$\dot{\Delta}$ or $\dot{\delta}_{\mathbf{xy}}$. The left hand side term is not
self adjoint but is needed in the evolution to preserve the probabilistic
nature of quantum mechanics. In the right hand side we get terms that modify
the self adjoint part and can be interpreted as modifications of the
Hamiltonian. The phenomenology of these equations must be addressed in future
works.

####  7.2.2 Stationary spacetimes

In the particular case of an stationary spacetime there is a time-like Killing
vector field and a foliation that in a local region has constant shift
function $N$ and null shift. Therefore the non self-adjoint term disappears
recovering the usual Schrödinger equation. In this particular framework we can
solve Schrödinger equation with the integral kernel expression

$\Psi(\phi,t)=\int
D\beta(\sigma)\exp[\phi^{x}(\delta_{xy}-itE)_{xy}\bar{\sigma}^{y}]\Psi(\sigma,0)$
(180)

Where $E=\Theta^{-\frac{1}{2}}N^{\frac{1}{2}}\Theta$. In the particular case
Minkowsky space-time we can split space and time with a foliation in which
$N=1$, in this case $E=\sqrt{-\nabla^{2}+m^{2}}$ which is the energy operator.

##  8 Conclusions

In this paper we have made extensive use of the mathematical tools presented
in the first part of this series [1] to study the mathematical formalism
underlying different approaches to quantization of a system in quantum field
theory. The focus of this work is the geometric interpretation of the tools of
Gaussian analysis to describe the Hamiltonian picture of a quantum field
theory of a scalar field over the space of Cauchy data.

Following this idea, we started our analysis with geometric quantization. The
first step of this program is prequantization. In this context we used Minlos
theorem to define the measure of the prequantum Hilbert space and the
Malliavin derivative as the central tool to define a connection on such space.

The next step is the choice of a polarization. Each choice of polarization
amounts to a particular picture of the QFT to be described. We discussed the
(anti)holomorphic, Schrödinger and Momentum-Field pictures. In this case the
tools of Gaussian analysis suited for the discussion are the Wiener-Ito
decomposition theorem that provides the particle interpretation of the theory
through the Segal isomorphism with the Bosonic Fock space. But this setting is
also suited for the discussion of more general relations among these spaces.
In particular we defined a new notion of Fourier transform among infinite
dimensional spaces that relates the Schrödinger and momentum-field
representations.

General operators other than constant or linear operators fail to be
consistently quantized by the procedure of geometric quantization. For this
reason we had to study problems of ordering in this picture. We found that
trigonometric exponentials and coherent states are algebras of functions
specially well suited for the discussion of Weyl and Wick quantization and the
study of the star products that arise in these quantization procedures.

In accordance with our geometric approach to this problem, we abandoned the
description of the spaces of pure states in terms of Hilbert spaces and
treated the as Kähler manifolds. The space of Hida test functions is specially
suited for this matter. This point of view led us to a deeper understanding of
the different tools described so far, in particular we described the
ingredients of Fock quantization in the holomorphic and antiholomorphic
representation and its relation through the Fourier transform.

From this geometric perspective we have been able to propose a Hamiltonian
evolution that stems from the Kähler structure of the problem. In this case we
introduce a quantum connection to preserve it. In turn, this modifies the
Schrödinger equation in such a way that losses of norm, found in other
approaches to quantization, are not allowed in our prescription.

However, there is an infinite family of connections that achieves this
purpose. In our case we found a unique choice based on the
indistinguishability of the Holomorphic and Antiholomorphic representations.
We illustrated the choice of this connection with an example of Klein Gordon
theory in a generic curved spacetime. However, this construction is ambiguous
in the election of a complex structure that suits the need of the theory. For
simplicity we choose an example that matches the stationary cases but it is
known that this prescription leads to some unwanted features. This matter will
be the subject of future investigation.

Acknowledgments The authors would like to thank Carlos Escudero Liébana for
pointing out to very useful bibliography on the connection of Malliavin
calculus with QFT. The authors acknowledge partial financial support of Grant
PID2021- 123251NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by the
European Union, and of Grant E48-20R funded by Government of Aragón. C.B-M and
D.M-C acknowledge financial support by Gobierno de Aragón through the grants
defined in ORDEN IIU/1408/2018 and ORDEN CUS/581/2020 respectively.

##  Appendix A Explicit expressions for the connection and the quantized
operators in s.q. coordinates.

In this appendix we will compute explicitly the expressions of the connection
in (155) in holomorphic s.q. coordinates. Let $\mathcal{O}^{\phi}_{\sigma}$ be
an operator then the expression of the covariant derivative is

$\displaystyle\tilde{\nabla}_{t}{\mathcal{O}}=\frac{1}{2}\left[\frac{\partial{\mathcal{O}}}{\partial
t}+\left(\frac{\partial{\mathcal{O}}^{\dagger_{H}}}{\partial
t}\right)^{\dagger_{H}}\right]+\epsilon\frac{1}{2}\mathcal{F}^{-1}\left[\frac{\partial\hat{\mathcal{O}}}{\partial
t}-\left(\frac{\partial{\hat{\mathcal{O}}}^{\dagger_{\bar{H}}}}{\partial
t}\right)^{\dagger_{\bar{H}}}\right]\mathcal{F}$ (181)

To compute this term recall the kernels with respect to white noise of the
Fourier transform (113) and its inverse. We will start by the linear operators
(135). For simplicity consider a real direction $\xi_{x}$ then we see that

$\displaystyle\tilde{\mathcal{F}}\xi_{x}\tilde{\mathfrak{D}}^{x}\tilde{\mathcal{F}}^{-1}$
$\displaystyle=-\xi_{x}\left(\begin{array}[]{cc}A&\delta\\\
-\delta&A\end{array}\right)^{xy}\partial_{\phi^{y}},$ (184)
$\displaystyle\tilde{\mathcal{F}}\
\xi_{x}\tilde{\mathfrak{D}}^{\dagger_{H},x}\ \tilde{\mathcal{F}}^{-1}$
$\displaystyle=\xi_{x}\left(\begin{array}[]{cc}AD^{-1}&D^{-1}\\\
-D^{-1}&AD^{-1}\end{array}\right)_{y}^{x}\phi^{y}.$ (187)

Let the time derivative be represented by a $\cdot$, consider also $\xi_{x}$
time independent. We will consider that the coordinates
$(\phi^{x},\bar{\phi}^{x})$ are just placeholders for integration and, as such
do not depend on the time parameter.

Lets consider the derivative of the operator
$O=\xi_{x}\tilde{\mathfrak{D}}^{x}$. We can compute every term of the
derivative using the rule (115) and the definitions of the $\dagger$ operators
given by (133). We get
$\dot{O}=\xi_{x}{\dot{\Delta}^{xy}\partial_{\phi^{y}}}$,
$\dot{O}^{\dagger_{H}}=0$. For the other piece of the derivative we get

$\displaystyle\mathcal{F}^{-1}\frac{d\mathcal{F}\mathcal{O}\mathcal{F}^{-1}}{dt}\mathcal{F}=$
$\displaystyle-{\xi_{x}}\left(\begin{array}[]{cc}\dot{A}&\dot{\delta}\\\
-\dot{\delta}&\dot{A}\end{array}\right)^{xy}\left(\begin{array}[]{cc}{A}D^{-1}&-D^{-1}\\\
D^{-1}&{A}D^{-1}\end{array}\right)^{z}_{y}\partial_{\phi^{z}},$
$\displaystyle\mathcal{F}^{-1}\left(\frac{d{\mathcal{F}\mathcal{O}^{\dagger_{{H}}}\mathcal{F}^{-1}}}{dt}\right)^{\dagger_{\bar{H}}}\mathcal{F}=$
$\displaystyle-{\xi_{x}}\left[\frac{d}{dt}\left(\begin{array}[]{cc}{A}D^{-1}&D^{-1}\\\
-D^{-1}&{A}D^{-1}\end{array}\right)^{x}_{y}\right]\left(\begin{array}[]{cc}A^{t}&-\delta\\\
\delta&A^{t}\end{array}\right)^{yz}\partial_{\phi^{z}}.$

The hardest term to compute is the second one. First notice that

$\frac{d{\mathcal{F}\mathcal{O}^{\dagger_{{H}}}\mathcal{F}^{-1}}}{dt}=\xi_{x}\frac{d}{dt}\left(\begin{array}[]{cc}AD^{-1}&-D^{-1}\\\
D^{-1}&AD^{-1}\end{array}\right)_{y}^{x}\phi^{y}$

And the $\dagger_{\bar{H}}$ operator is given by substituting $\phi^{x}$ by
$-D^{xy}\partial_{\phi^{y}}$ and transposing the matrix (in this particular
case in which $\epsilon$ commutes with the operator). Applying the
transformations leads to the desired result.

Once this is computed, simply by using the definitions of (30) it follows that

$\Delta^{xy}=-\left(\begin{array}[]{cc}{A}&\delta\\\
-\delta&{A}\end{array}\right)^{xu}D^{-1}_{uv}\left(\begin{array}[]{cc}A^{t}&-\delta\\\
\delta&A^{t}\end{array}\right)^{vy}$

Then

$\nabla_{t}{\mathfrak{D}}^{x}=\frac{1}{2}{\dot{\Delta}^{xy}\partial_{\phi^{y}}}+\frac{\epsilon}{2}\left(\begin{array}[]{cc}{A}&\delta\\\
-\delta&{A}\end{array}\right)^{xu}\dot{D}^{-1}_{uv}\left(\begin{array}[]{cc}A^{t}&-\delta\\\
\delta&A^{t}\end{array}\right)^{vy}\partial_{\phi^{y}}$ (188)

For the creation operator we use the fact
$(\nabla_{t}O)^{\dagger}=\nabla_{t}(O^{\dagger})$ and simply substitute
$\partial_{\phi^{x}}$ by $K_{xy}\phi^{y}$ and transpose the matrices.

Now we will express the connection $\Gamma$. In this fashion see that because
of (133) we can express

$\left(\frac{d{\mathcal{O}}^{\dagger_{H}}}{dt}\right)^{\dagger_{H}}=\dot{\mathcal{O}}+[\dot{\mathcal{K}}^{\phi\bar{\gamma}}\varDelta_{\bar{\gamma}\sigma},\mathcal{O}]$
(189)

We use here the identity
$\dot{\mathcal{K}}^{\phi\bar{\gamma}}\varDelta_{\bar{\gamma}\sigma}=-{\mathcal{K}}^{\phi\bar{\gamma}}\dot{\varDelta}_{\bar{\gamma}\sigma}$
that is a consequence of being one the inverse of the other and thus its
contraction is $\delta^{\phi}_{\sigma}$ which is time independent. Similarly
we get
$\mathcal{F}^{-1}\dot{\mathcal{F}}=-\dot{\mathcal{F}}^{-1}{\mathcal{F}}$ and

$\mathcal{F}^{-1}\frac{d\mathcal{F}\mathcal{O}\mathcal{F}^{-1}}{dt}\mathcal{F}=\dot{\mathcal{O}}+[\mathcal{F}^{-1}\dot{\mathcal{F}},\mathcal{O}]$
(190)

The last term is a combination of the two above.

$(\mathcal{F}^{-1})\left(\frac{\partial{\mathcal{F}\mathcal{O}^{\dagger_{H}}\mathcal{F}^{-1}}}{\partial
t}\right)^{\dagger_{\bar{H}}}\mathcal{F}=\dot{\mathcal{O}}+[(\mathcal{F}^{-1}\mathcal{D})_{\phi\bar{\gamma}}\frac{d(\mathcal{D}^{-1}{\mathcal{F}})^{\bar{\gamma}\sigma}}{dt},\mathcal{O}]$
(191)

Then, explicitly we get that in holomorphic second quantized coordinates

$\tilde{\Gamma}=\frac{1}{2}\left[\dot{\mathcal{K}}^{\phi\bar{\gamma}}\varDelta_{\bar{\gamma}\sigma}+\epsilon\mathcal{F}^{-1}\dot{\mathcal{F}}-\epsilon(\mathcal{F}^{-1}\mathcal{D})_{\phi\bar{\gamma}}\frac{d(\mathcal{D}^{-1}{\mathcal{F}})^{\bar{\gamma}\sigma}}{dt}\right]=-\tilde{\mathfrak{D}}^{\dagger,x}K_{xy}\tilde{\nabla}_{t}\tilde{\mathfrak{D}}^{y}$

$\int D\beta(\sigma)A(\phi,\bar{\sigma})B(\sigma,\bar{\gamma}):$

$A$ $B$ | $\sqrt{2}\cosh(N)\cos(V)$ | $\sqrt{2}\cosh(N)\sin(V)$ | $\sqrt{2}\sinh(N)\cos(V)$ | $\sqrt{2}\sinh(N)\sin(V)$
---|---|---|---|---
$\sqrt{2}\cosh(M)\cos(U)$ | $\cosh\boxplus\cos\circleddash+\cosh\boxminus\cos\oplus$ | 0 | 0 | $-\sinh\boxplus\sin\circleddash+\sinh\boxminus\sin\oplus$
$\sqrt{2}\cosh(M)\sin(U)$ | 0 | $\sinh\boxplus\cos\circleddash-\sinh\boxminus\cos\oplus$ | $\cosh\boxplus\sin\circleddash+\cosh\boxminus\sin\oplus$ | 0
$\sqrt{2}\sinh(M)\cos(U)$ | 0 | $-\cosh\boxplus\sin\circleddash+\cosh\boxminus\sin\oplus$ | $\sinh\boxplus\cos\circleddash+\sinh\boxminus\cos\oplus$ | 0
$\sqrt{2}\sinh(M)\sin(U)$ | $\sinh\boxplus\sin\circleddash+\sinh\boxminus\sin\oplus$ | 0 | 0 | $\cosh\boxplus\cos\circleddash-\cosh\boxminus\cos\oplus$
Table 1: Result of the integral $\int
D\beta(\sigma)A(\phi,\bar{\sigma})B(\sigma,\bar{\gamma})$ where the kernels
$A$ and $B$ are specified in the first column and row respectively

Where
$M=\phi^{x}M_{xy}\bar{\sigma}^{y},U=\phi^{x}U_{xy}\bar{\sigma}^{y},N=\sigma^{x}N_{xy}\bar{\gamma}^{y}$
and $V=\sigma^{x}V_{xy}\bar{\gamma}^{y}$.

Also $\boxplus=\phi^{x}(MN+UV)_{xy}\bar{\gamma}^{y},$ $\boxminus=\phi^{x}(MN-
UV)_{xy}\bar{\gamma}^{y},$ $\oplus=\phi^{x}(UN+MV)_{xy}\bar{\gamma}^{y}$ and
$\circleddash=\phi^{x}(UN-MV)_{xy}\bar{\gamma}^{y}$

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