# Phenomenology from Dirac equation
with Euclidean-Minkowskian “gravity phase”

Jens Köplinger  105 E Avondale Dr, Greensboro, NC 27403, USA
<EMAIL_ADDRESS>preprints, personal versions at
http://jenskoeplinger.com/P

(Date: 16 Jan 2023. This preprint has not undergone peer review or any post-
submission improvements or corrections. The Version of Record of this article
is published in Int. J. Theor. Phys., and is available online at
https://doi.org/10.1007/s10773-023-05283-2)

###### Abstract.

Over the past decades, many authors advertised models on complexified
spacetime algebras for use in describing gravity. This work aims at providing
phenomenological support to such claims, by introducing a one-parameter real
phase $\alpha$ to the conventional Dirac equation with $\frac{1}{r}$-type
potential. This phase allows to transition between Euclidean
($\alpha=0,\pm\pi,\pm 2\pi,\ldots$) and Minkowskian
($\alpha=\pm\frac{\pi}{2},\pm\frac{3\pi}{2},\ldots$) geometry, as two distinct
cases that one may expect from some complexified spacetime. The configuration
space is modeled on $4\times 4$ matrix algebra over the bicomplex numbers,
$\mathbb{C}\oplus\mathbb{C}$. Spin-$\frac{1}{2}$ Coulomb scattering
(Rutherford scattering) in Born approximation is then executed. All
calculations are done “from scratch”, as they could have been done some 85
years ago. By removing elegance from field theory that has since become
customary, this paper aims at remaining as generally applicable as possible,
for a wide range of candidate models that contain such a phase $\alpha$ in one
way or another. Results for backscattering and cross section at high energies
are compared with results from General Relativity calculations. Effects on
intergalactic gas distribution and momentum transfer from scattering high-
energy leptons are sketched.

###### Contents

  1. 1 Context
  2. 2 Introduce a phase $\alpha$ to the Dirac equation
    1. 2.1 Define phase $\alpha$
    2. 2.2 Introduce $\alpha$ to Dirac equation
    3. 2.3 Properties of the generalized Dirac-$\gamma$
    4. 2.4 Energy, mass, momentum, relativity
      1. 2.4.1 Physical implications
      2. 2.4.2 Mathematical implications
    5. 2.5 Green’s function in energy-momentum space
  3. 3 Rutherford scattering of a spin-$\frac{1}{2}$ particle for general $\alpha$
    1. 3.1 Adjoint wave function and conservation of probability
    2. 3.2 Solutions of the free equation of motion
    3. 3.3 Normalizing the plane-wave eigenfunctions
    4. 3.4 Normalizing to invariant volume
    5. 3.5 Coulomb-type field of a point charge
    6. 3.6 Lowest-order transition matrix element in Born approximation
    7. 3.7 Spin contribution
      1. 3.7.1 Particle / antiparticle transition, spin flip
      2. 3.7.2 Spin and particle type remains unchanged
    8. 3.8 Result and comparison with the classical case
    9. 3.9 Approximation for slow moving incident particle
    10. 3.10 Approximation for fast incident particle
    11. 3.11 Approximation for purely gravitational interaction
    12. 3.12 Estimate for momentum transfer of a fast fermion
    13. 3.13 Qualitative prediction for intergalactic medium distribution

##  1\. Context

Numerous models have been proposed in mathematical physics over the past
decades, which provide ways that one could interpret as “complexifying
spacetime” algebraically. Out of those, several claim to be modelling effects
from gravity this way. For a certainly not complete list see e.g. [1, 2, 3, 4,
5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26,
27]. These works don’t necessarily assume locally-Lorentzian spacetime as
foundational. They are therefore distinct from canonical “spin-$2$” type
gravity, as well as, use of four-dimensional Euclidean spacetime as
mathematical tool in contemporary field theory (keywords “$ict$”, “Wick
rotation”).

In order to support taking such a stance a priori, this paper provides a
simple phenomenological calculation: In the spirit of retracing early quantum
electrodynamics [18], mathematical elegance is removed as much as possible,
leaving just enough structure to be able to arrive at qualitative and
quantitative results. The hope is that these results could then be reproduced
in the respective algebraic frameworks of a wide range of models, if so
desired.

Complexifying a conventional $4\times 4$ complex matrix representation of the
Dirac equation can be understood as matrix algebra over the bicomplex numbers
$\mathbb{C}\oplus\mathbb{C}$, which have rich historical and contemporary
interest in physics (for pointers see e.g. [28]).

##  2\. Introduce a phase $\alpha$ to the Dirac equation

We start with the conventional Dirac equation and a $\frac{1}{r}$-type
potential, and introduce a one-parameter real phase $\alpha$ that transitions
between Euclidean ($\alpha=0,\pm\pi,\pm 2\pi,\ldots$) and Minkowskian
($\alpha=\pm\frac{\pi}{2},\pm\frac{3\pi}{2},\ldots$) geometry. This phase is
proposed to serve as the minimal assumption that justifies utility of the
calculation in “complexified spacetime”, which contains Euclidean and
Lorentzian subspaces, respectively.

###  2.1. Define phase $\alpha$

Let $\alpha\in\mathbb{R}$ be the phase that transitions between Euclidean
($\alpha=0,\pm\pi,\pm 2\pi,\ldots$) and Minkowskian
($\alpha=\pm\frac{\pi}{2},\pm\frac{3\pi}{2},\ldots$) geometry. Define two
copies of the complex numbers, $\mathbb{C}$ and $\mathbb{C}_{0}$, to basis
elements $\left\\{1,i\right\\}$ and $\left\\{1,i_{0}\right\\}$, respectively.
Dedicate $\mathbb{C}_{0}$ to modeling the phase from $\alpha$, and
$\mathbb{C}$ for modeling conventional Dirac equation and operators. Elements
from $\mathbb{C}$ and $\mathbb{C}_{0}$ commute, associate, and distribute just
like real coefficients. Given a variable $x$, complex conjugation in
$\mathbb{C}$ is written as $\overline{x}$ and complex conjugation in
$\mathbb{C}_{0}$ as $\underline{x}$.

With this, a transformation $\phi$ and its conjugate $\underline{\phi}$ are
then defined as:

(2.1)
$\phi\,:=\,e^{i_{0}\alpha},\qquad\underline{\phi}\,:=\,e^{-i_{0}\alpha},\qquad\phi\underline{\phi}=\underline{\phi}\phi=1.$

###  2.2. Introduce $\alpha$ to Dirac equation

Using $4\times 4$ matrices over the complexes, $\gamma_{\mu}$, with
$\mu=0\ldots 3$, linear derivatives
$\partial_{\mu}:=\frac{\partial_{\mu}}{\partial x_{\mu}}$, a property
$\widetilde{m}$ that is invariant under $\partial_{\mu}$ (with
$\left.\widetilde{m}\right|_{\mathrm{QED}}\equiv m\in\mathbb{R}$ mass in the
classical case), and functions $\psi:\mathbb{R}^{4}\rightarrow\mathbb{C}^{4}$,
the Dirac equation is written as the eigenvalue relation:

(2.2) $\sum_{\mu=0}^{3}i\gamma_{\mu}\partial_{\mu}\psi=\widetilde{m}\psi.$

In contrast to notation convention in physics today, all indices are now
written as lower indices, and summation is written explicitly (i.e. without
implicit Minkowski tensor). Spelling out summations and metric explicitly
avoids confusion going forward, when Minkowski metric is considered an edge
case in a generalized geometry.

Generalized Dirac matrices, using the same symbol $\gamma_{\mu}$, are now
defined as a function of $\alpha$, to model the conventional Dirac equation in
the $\alpha=\pm\frac{\pi}{2},\pm\frac{3\pi}{2},\ldots$ cases and a counterpart
on Euclidean metric in the $\alpha=0,\pm\pi,\pm 2\pi,\ldots$ case (per [13,
14]):

(2.3) $\gamma_{0}:=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&-1&0\\\
0&0&0&-1\end{array}\right),\,\gamma_{1}:=\left(\begin{array}[]{cccc}0&0&0&\phi^{2}\\\
0&0&\phi^{2}&0\\\ 0&1&0&0\\\
1&0&0&0\end{array}\right),\,\gamma_{2}:=\left(\begin{array}[]{cccc}0&0&0&-i\phi^{2}\\\
0&0&i\phi^{2}&0\\\ 0&-i&0&0\\\
i&0&0&0\end{array}\right),\,\gamma_{3}:=\left(\begin{array}[]{cccc}0&0&\phi^{2}&0\\\
0&0&0&-\phi^{2}\\\ 1&0&0&0\\\ 0&-1&0&0\end{array}\right).$

With $\gamma_{\mu}\in\left(\mathbb{C}\times\mathbb{C}_{0}\right)^{4}$ the wave
functions $\psi$ are now generally
$\psi:\mathbb{R}^{4}\rightarrow\left(\mathbb{C}\times\mathbb{C}_{0}\right)^{4}$.

Using Pauli spinors $\sigma_{j}$ with $j=1,2,3$

(2.4) $\sigma_{1}:=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\qquad\sigma_{2}:=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\qquad\sigma_{3}:=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$

and identifying the unit matrix with

(2.7) $\displaystyle I_{2}\equiv\sigma_{0}$ $\displaystyle:=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),$

the generalized $\gamma_{\mu}$ can be written as:

(2.8) $\gamma_{0}=\left(\begin{array}[]{cc}\sigma_{0}&0\\\
0&-\sigma_{0}\end{array}\right)=\left(\begin{array}[]{cc}I_{2}&0\\\
0&-I_{2}\end{array}\right),\qquad\gamma_{j}=\left(\begin{array}[]{cc}0&\phi^{2}\sigma_{j}\\\
\sigma_{j}&0\end{array}\right).$

It is left open for now whether this requires the parameter $\widetilde{m}$ to
become complex in $\mathbb{C}_{0}$ or not.

###  2.3. Properties of the generalized Dirac-$\gamma$

The Euclidean and Minkowskian edge cases from the referenced papers ([13, 14])
are satisfied by inspection, for $\phi^{2}=1$ and $\phi^{2}=-1$, respectively.

Writing $I_{4}$ for the identity $4\times 4$ matrix, the generalized Dirac
matrices have the property:

(2.9)
$\displaystyle\frac{1}{2}\left(\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}\right)$
$\displaystyle=I_{4}*\begin{cases}0&\textrm{for }\mu\neq\nu,\\\ 1&\textrm{for
}\mu=\nu=0,\\\ \phi^{2}&\textrm{otherwise
}\left(\mu=\nu\in\left\\{1,2,3\right\\}\right).\end{cases}$

This property reflects the choice of metric.

###  2.4. Energy, mass, momentum, relativity

Understanding the derivatives $i\partial_{\mu}$ as quantum mechanical
operators for energy $E$ and momentum
$\vec{p}:=\left(p_{1},p_{2},p_{3}\right)$,
$\left|\vec{p}\right|^{2}:=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}$, multiplying the
Dirac equation (2.2) with its conjugate in $\mathbb{C}$ recovers the
Minkowskian
$\left|E\right|^{2}-\left|\vec{p}\right|^{2}=m_{\mathrm{Mink}}^{2}$ and
Euclidean $\left|E\right|^{2}+\left|\vec{p}\right|^{2}=m_{\mathrm{Eucl}}^{2}$
edge cases for $\phi^{2}=1$ and $\phi^{2}=-1$, respectively.

In general, the sum over four momentum $p$,

(2.10)
$p:=\left(p_{\mu}\right)=\left(E,\vec{p}\right)=\left(E,p_{1},p_{2},p_{3}\right),\qquad\textrm{with
}E,\vec{p}\in\mathbb{R}.$

becomes:

(2.11) $\displaystyle m_{0}^{2}$ $\displaystyle:=$
$\displaystyle\sum_{\mu=0}^{3}\sum_{\nu=0}^{3}\gamma_{\mu}\gamma_{\nu}p_{\mu}p_{\nu}=E^{2}+\phi^{2}\left|\vec{p}\right|^{2}.$

This makes $m_{0}^{2}$ complex in $\mathbb{C}_{0}$ in the general case. In
order to keep the classical mass parameter $m_{\mathbb{R}}\in\mathbb{R}$ real,
it has to be redefined as compared to the conventional case. The simplest way
is to take the absolute of $m_{0}^{2}$, which makes $m_{\mathbb{R}}$ a fourth-
order expression in $E$ and $\vec{p}$:

(2.12) $\displaystyle\phi^{2}$ $\displaystyle=$ $\displaystyle
e^{2i_{0}\alpha},\textrm{ therefore}$ (2.13) $\displaystyle
m_{\mathbb{R}}^{4}:$ $\displaystyle=$
$\displaystyle\left|m_{0}^{2}\right|^{2}=E^{4}+2E^{2}\left|\vec{p}\right|^{2}\cos\left(2\alpha\right)+\left|\vec{p}\right|^{4},$
(2.14) $\displaystyle m_{\mathbb{R}}$ $\displaystyle=$
$\displaystyle\sqrt{\left|m_{0}^{2}\right|}=\sqrt[4]{E^{4}+2E^{2}\left|\vec{p}\right|^{2}\cos\left(2\alpha\right)+\left|\vec{p}\right|^{4}}.$

There are both physical and mathematical implications to this.

####  2.4.1. Physical implications

From the physical side, it raises the question on what makes physical lab
frames equivalent. Next to energy and momentum of test bodies in unaccelerated
frames of reference, the parameter $\alpha$ now factors into the equivalence
condition as well. Equation 2.14 therefore becomes the new definition of
relativity111A semi-classical approximation allows to reconstruct linearized
General Relativity, per “proposition 4” in [15] (“ _NatAliE_ equations”).
Ignoring the complex-octonion setting in that paper, this “proposition 4” can
be interpreted as stand-alone phenomenology in the large body, non-quantum
limit. This requires clarification of the meaning of $\alpha$, which is
subject of this current research..

The relation between speed $\vec{v}$, energy $E$, and momentum $\vec{p}$
remains unchanged for any $\alpha$ by definition:

(2.15) $\displaystyle\vec{v}$ $\displaystyle:=$
$\displaystyle\frac{\vec{p}}{E}.$

Conserved properties between equivalent frames of reference are generalized
per equation (2.11) and therefore complex in $\mathbb{C}_{0}$ in the general
case. For example, invariant length elements $\ell_{0}$, volumes $V_{0}$, or
time intervals $d\tau$:

(2.16)
$\ell_{0}:=\ell\frac{E}{m_{0}}=\frac{\ell}{\sqrt{1+\phi^{2}\left|\vec{v}\right|^{2}}},\qquad
V_{0}:=V\frac{E}{m_{0}}=\frac{V}{\sqrt{1+\phi^{2}\left|\vec{v}\right|^{2}}},\qquad
d\tau:=dt\frac{m_{0}}{E}=dt\sqrt{1+\phi^{2}\left|\vec{v}\right|^{2}}.$

####  2.4.2. Mathematical implications

From the mathematical side, fourth-order expressions are a departure from the
pure (Dirac-)spinor expression of the equation of motion of a
spin-$\frac{1}{2}$ particle. It requires clarification on what mathematical
construct we’re looking at exactly - or alternatively, find a more natural
mathematical representation of the phase $\alpha$. This may ultimately lead to
a more natural mathematical description of relativity, which might not be
apparent here due to the focus on a special case.

For now it is still left open whether the placeholder $\widetilde{m}$ in the
generalized Dirac equation is the classical, real-valued mass parameter
$m_{\mathbb{R}}$, or whether it has to become $m_{0}$ which is complex-valued
in $\mathbb{C}_{0}$.

###  2.5. Green’s function in energy-momentum space

In momentum space, Green’s function $G\left(p\right)$ solves

(2.17)
$\left(\sum_{\mu=0}^{3}\gamma_{\mu}p_{\mu}-\widetilde{m}\right)G\left(p\right)=1$

and is:

(2.18) $\displaystyle G\left(p\right)$ $\displaystyle=$
$\displaystyle\frac{\sum_{\mu=0}^{3}\gamma_{\mu}p_{\mu}+\widetilde{m}}{E^{2}+\phi^{2}\left|\vec{p}\right|^{2}-\widetilde{m}^{2}}=\frac{\sum_{\mu=0}^{3}\gamma_{\mu}p_{\mu}+\widetilde{m}}{m_{0}^{2}\left(p\right)-\widetilde{m}^{2}}.$

The factor $m_{0}^{2}\left(p\right)\in\mathbb{C}_{0}$ varies not only in
$\alpha$ but also in energy and momentum. If $\widetilde{m}$ would be assumed
real, it would mean that there is no pole in $G\left(p\right)$ on the real
$p_{\mu}$ parameter space except for the Euclidean ($\phi^{2}=1$,
$\alpha=0,\pm\pi,\pm 2\pi,\ldots$) and Minkowskian ($\phi^{2}=-1$,
$\alpha=\pm\frac{\pi}{2},\pm\frac{3\pi}{2},\ldots$) edge cases.

This raises questions, coming from conventional theory where the propagator of
a plane-wave particle would be expected to have a pole at the particle’s
invariant mass $m_{\mathbb{R}}$. There it would mark the exact momentum and
energy associated with the infinite plane wave. Invariant mass
$m_{\mathbb{R}}$ is a particle property, just as is $\alpha$. As such, it
seems from Green’s function that it is not correct to model the generalized
Dirac equation using the real mass property
$\widetilde{m}\overset{?}{\equiv}m_{\mathbb{R}}$, but instead the complex mass
property $\widetilde{m}\overset{?}{\equiv}\sqrt{m_{0}^{2}}$ should be used.
Nevertheless, for now the placeholder $\widetilde{m}$ continues to be used
until more evidence is gathered.

##  3\. Rutherford scattering of a spin-$\frac{1}{2}$ particle for general
$\alpha$

In order to check for plausibility of the overall approach, and to get a
feeling on what challenges may arise when generalizing the calculation,
execute a simple quantum calculation in a special case. Observe how the phase
$\alpha$ plays into that special case, and learn how this may later need to be
handled generally.

Spin-$\frac{1}{2}$ Coulomb scattering (Rutherford scattering) in Born
approximation is a simple and well understood calculation that can be done
from the Dirac equation with minimal prerequisites. The calculation will
correct and clean up the $\alpha=0$ case in [16], but then also provide the
result for general alpha:

  * •

Fix a representation of the Dirac equation and introduce the phase $\alpha$ in
the simplest possible way.

  * •

Define some basics (adjoint wave function, conservation of probability,
particle propagator).

  * •

Use Born approximation to calculate cross section.

Interpret the result:

  * •

Backscattering, cross section for high energies,

  * •

compare with cross section of "scattering a rock on a Black Hole",

  * •

estimate effects on intergalactic gas distribution,

  * •

estimate momentum transfer $\triangle p$ from scattering high-energy neutrinos
on atomic nuclei when traveling through matter.

###  3.1. Adjoint wave function and conservation of probability

For a given volume and time interval without sources, probability must be
conserved in the general case. This is a prerequisite for calculating
transition properties using perturbation theory methods, but is also a
physical principle underlying quantum mechanics in general.

Define a matrix $\gamma_{\phi}$ as:

(3.7) $\displaystyle\gamma_{\phi}$ $\displaystyle:=$
$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&\phi&0\\\
0&0&0&\phi\end{array}\right)=\left(\begin{array}[]{cc}\sigma_{0}&0\\\
0&\phi\sigma_{0}\end{array}\right),\qquad\gamma_{\phi}^{2}=\gamma_{\phi^{2}},$

with

(3.14) $\displaystyle\gamma_{\underline{\phi}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&\underline{\phi}&0\\\
0&0&0&\underline{\phi}\end{array}\right)=\left(\begin{array}[]{cc}\sigma_{0}&0\\\
0&\underline{\phi}\sigma_{0}\end{array}\right),\qquad\gamma_{\underline{\phi}}^{2}=\gamma_{\underline{\phi}^{2}},$

accordingly. Therefore

(3.15)
$\displaystyle\gamma_{\underline{\phi}}\gamma_{\phi}=\gamma_{\phi}\gamma_{\underline{\phi}}=\gamma_{\underline{\phi}^{2}}\gamma_{\phi^{2}}=\gamma_{\phi^{2}}\gamma_{\underline{\phi}^{2}}$
$\displaystyle=$ $\displaystyle I_{4}.$

It relates a $\gamma_{\mu}$ matrix with its $\mathbb{C}$-hermitian transpose,
$\overline{\gamma}_{\mu}^{T}$:

(3.16)
$\gamma_{\mu}=\gamma_{\underline{\phi}^{2}}\overline{\gamma}_{\mu}^{T}\gamma_{\phi^{2}},\qquad\overline{\gamma}_{\mu}^{T}=\gamma_{\phi^{2}}\gamma_{\mu}\gamma_{\underline{\phi}^{2}}.$

Writing $\psi^{\dagger}$ for adjoint and $\psi^{T}$ for transpose of $\psi$,
all
$\mathbb{R}^{4}\rightarrow\left(\mathbb{C}\times\mathbb{C}_{0}\right)^{4}$,
the probability density four-vector
$j:\mathbb{R}^{4}\rightarrow\mathbb{R}^{4}$ is defined as:

(3.17) $\displaystyle j_{\mu}$ $\displaystyle:=$
$\displaystyle\psi^{\dagger}\gamma_{\mu}\psi\textrm{ with }\mu=0\ldots 3.$

In the absence of sources (charges and fields) probability must be conserved
globally:

(3.18) $\displaystyle\sum_{\mu=0}^{3}\partial_{\mu}j_{\mu}$
$\displaystyle\overset{!}{=}$ $\displaystyle 0.$

This fixes the adjoint as:

(3.19) $\displaystyle\psi^{\dagger}$ $\displaystyle:=$
$\displaystyle\overline{\psi}^{T}\gamma_{\phi^{2}}.$

###### Proof.

Build the $\mathbb{C}$-hermitian transpose of the Dirac equation,
$\sum_{\mu=0}^{3}i\gamma_{\mu}\partial_{\mu}\psi=\widetilde{m}\psi$.
Conjugation is done in $\mathbb{C}$ since the quantum mechanical probability
amplitude is modeled in this subalgebra. The differentials $\partial_{\mu}$
are understood as acting on the wave function $\overline{\psi}^{T}$, i.e., to
the left in this case:

(3.20)
$\displaystyle\overline{\psi}^{T}\sum_{\mu=0}^{3}\left(-i\overline{\gamma}_{\mu}^{T}\partial_{\mu}\right)$
$\displaystyle=$ $\displaystyle\overline{\psi}^{T}\widetilde{m},$ (3.21)
$\displaystyle\overline{\psi}^{T}\sum_{\mu=0}^{3}\left(-i\gamma_{\phi^{2}}\gamma_{\mu}\gamma_{\underline{\phi}^{2}}\partial_{\mu}\right)$
$\displaystyle=$ $\displaystyle\overline{\psi}^{T}\widetilde{m}.$

Pull out $\gamma_{\phi^{2}}$ to the left, identify
$\widetilde{m}=\gamma_{\phi^{2}}\widetilde{m}\gamma_{\underline{\phi}^{2}}$,
then multiply with $\gamma_{\phi^{2}}$ from the right and identify the adjoint
$\psi^{\dagger}$:

(3.22)
$\displaystyle\overline{\psi}^{T}\gamma_{\phi^{2}}\sum_{\mu=0}^{3}\left(-i\gamma_{\mu}\gamma_{\underline{\phi}^{2}}\partial_{\mu}\right)$
$\displaystyle=$
$\displaystyle\overline{\psi}^{T}\gamma_{\phi^{2}}\widetilde{m}\gamma_{\underline{\phi}^{2}},$
(3.23)
$\displaystyle\overline{\psi}^{T}\gamma_{\phi^{2}}\sum_{\mu=0}^{3}\left(-i\gamma_{\mu}\partial_{\mu}\right)$
$\displaystyle=$
$\displaystyle\overline{\psi}^{T}\gamma_{\phi^{2}}\widetilde{m},$ (3.24)
$\displaystyle\psi^{\dagger}\sum_{\mu=0}^{3}\left(-i\gamma_{\mu}\partial_{\mu}\right)$
$\displaystyle=$ $\displaystyle\psi^{\dagger}\widetilde{m},$ (3.25)
$\displaystyle\sum_{\mu=0}^{3}\left(\left(\partial_{\mu}\psi^{\dagger}\right)\gamma_{\mu}\right)$
$\displaystyle=$ $\displaystyle i\widetilde{m}\psi^{\dagger}.$

The last line only reordered the terms. With this:

(3.26)
$\sum_{\mu=0}^{3}\partial_{\mu}j_{\mu}=\sum_{\mu=0}^{3}\partial_{\mu}\left(\psi^{\dagger}\gamma_{\mu}\psi\right)=\sum_{\mu=0}^{3}\left(\left(\partial_{\mu}\psi^{\dagger}\gamma_{\mu}\right)\psi+\psi^{\dagger}\left(\gamma_{\mu}\partial_{\mu}\psi\right)\right)=i\widetilde{m}\psi^{\dagger}\psi-i\widetilde{m}\psi^{\dagger}\psi=0.$

Probability density is conserved. ∎

Note that this proof holds regardless of whether $\widetilde{m}$ is real-
valued or complex-valued in $\mathbb{C}_{0}$. The restriction on
$\widetilde{m}$ is that it must not be complex-valued in $\mathbb{C}$, since
complex conjugation in that space is used for the proof.

The classical adjoint $\left.\psi^{\dagger}\right|_{\mathrm{QED}}$ is
recovered as expected for $\phi^{2}=-1$ as

(3.29) $\displaystyle\left.\psi^{\dagger}\right|_{\mathrm{QED}}$
$\displaystyle=$
$\displaystyle\overline{\psi}^{T}\left.\gamma_{0}\right|_{\mathrm{QED}}=\overline{\psi}^{T}\left(\begin{array}[]{cc}\sigma_{0}&0\\\
0&-\sigma_{0}\end{array}\right).$

###  3.2. Solutions of the free equation of motion

The free Dirac equation,
$\sum_{\mu=0}^{3}i\gamma_{\mu}\partial_{\mu}\psi=\widetilde{m}\psi$ (2.2) is
explicitly:

(3.30) $\displaystyle
i\left[\sum_{\mu=0}^{3}\gamma_{\mu}\partial_{\mu}\right]\psi$ $\displaystyle=$
$\displaystyle\widetilde{m}\psi,$ (3.35) $\displaystyle
i\left[\left(\begin{array}[]{cc}\sigma_{0}&0\\\
0&-\sigma_{0}\end{array}\right)\partial_{0}+\sum_{j=1}^{3}\left(\begin{array}[]{cc}0&\phi^{2}\sigma_{j}\\\
\sigma_{j}&0\end{array}\right)\partial_{j}\right]\psi$ $\displaystyle=$
$\displaystyle\widetilde{m}\psi,$ (3.44) $\displaystyle
i\left[\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{array}\right)\partial_{0}+\left(\begin{array}[]{cccc}0&0&0&\phi^{2}\\\
0&0&\phi^{2}&0\\\ 0&1&0&0\\\ 1&0&0&0\end{array}\right)\partial_{1}+\right.$
(3.53) $\displaystyle\left.\left(\begin{array}[]{cccc}0&0&0&-i\phi^{2}\\\
0&0&i\phi^{2}&0\\\ 0&-i&0&0\\\
i&0&0&0\end{array}\right)\partial_{2}+\left(\begin{array}[]{cccc}0&0&\phi^{2}&0\\\
0&0&0&-\phi^{2}\\\ 1&0&0&0\\\
0&-1&0&0\end{array}\right)\partial_{3}\right]\psi$ $\displaystyle=$
$\displaystyle\widetilde{m}\psi,$ (3.58)
$\displaystyle\left(\begin{array}[]{cccc}i\partial_{0}&0&\phi^{2}\left(i\partial_{3}\right)&\phi^{2}\left(i\partial_{1}+\partial_{2}\right)\\\
0&i\partial_{0}&\phi^{2}\left(i\partial_{1}-\partial_{2}\right)&\phi^{2}\left(-i\partial_{3}\right)\\\
i\partial_{3}&i\partial_{1}+\partial_{2}&-i\partial_{0}&0\\\
i\partial_{1}-\partial_{2}&-i\partial_{3}&0&-i\partial_{0}\end{array}\right)\psi$
$\displaystyle=$ $\displaystyle\widetilde{m}\psi.$

Finding eingenfunctions and eigenvalues requires an energy-momentum-mass
relation, which is
$\left|E\right|^{2}-\left|\vec{p}\right|^{2}=m_{\mathrm{Mink}}^{2}$ in the
classical case, and
$\left|E\right|^{2}+\left|\vec{p}\right|^{2}=m_{\mathrm{Eucl}}^{2}$ in the 4D
Euclidean case ([14]). This obviously has to be generalized now due to the
phase $\alpha$ contained in $\phi^{2}$. Equation (2.11) has the consistent
generalization,

(3.59) $\displaystyle m_{0}^{2}$ $\displaystyle=$ $\displaystyle
E^{2}+\phi^{2}\left|\vec{p}\right|^{2}.$

This is not consistent any more with a real-valued
$\left.\widetilde{m}\right|_{\mathrm{QED}}\equiv m_{\mathbb{R}}\in\mathbb{R}$
as in the classical case. Eigenfunctions of this linear differential equation
- to be found - must contain an exponential function part
$\sim\exp\left(f\left(p\right)\right)$, as well as a vector part. The vector
part will contain terms of $\vec{p}$ and $E$ that multiply with additional
terms $\vec{p}$ and $E$ from the differential on the exponential function
part. In order for $\widetilde{m}$ to be real-valued, all factors $\phi^{2}$
would have to cancel out directly from these differentials. However, since all
momentum differentials in (3.58) appear both with and without factors
$\phi^{2}$, this is impossible in principle. Therefore, $\widetilde{m}$ must
be complex-valued in $\mathbb{C}_{0}$ and the placeholder $\widetilde{m}$ is
determined to be $m_{0}$ (and not $m_{\mathbb{R}}$) going forward:

(3.60) $\displaystyle\widetilde{m}$ $\displaystyle:=$ $\displaystyle m_{0},$
(3.61) $\displaystyle
i\left[\sum_{\mu=0}^{3}\gamma_{\mu}\partial_{\mu}\right]\psi$ $\displaystyle=$
$\displaystyle m_{0}\psi.$

There is an underlying choice that is made to arrive at this identification:
The generalized Dirac equation is to remain as similar as possible to the
classical case, with as little as needed modification in the formulation as
possible.

With this, eigenfunctions $\widetilde{\Psi}_{1/2}^{\pm}$ can be found:

(3.70) $\displaystyle\widetilde{\Psi}_{1}^{+}$ $\displaystyle:=\exp
i\left(\vec{p}\vec{x}-Et\right)\left(\begin{array}[]{c}1\\\ 0\\\
-p_{3}/\left(m_{0}+E\right)\\\
\left(-p_{1}-ip_{2}\right)/\left(m_{0}+E\right)\end{array}\right),\qquad\widetilde{\Psi}_{1}^{-}:=\exp
i\left(\vec{p}\vec{x}-Et\right)\left(\begin{array}[]{c}0\\\ 1\\\
\left(-p_{1}+ip_{2}\right)/\left(m_{0}+E\right)\\\
p_{3}/\left(m_{0}+E\right)\end{array}\right),$ (3.79)
$\displaystyle\widetilde{\Psi}_{2}^{+}$ $\displaystyle:=\exp
i\left(\vec{p}\vec{x}+Et\right)\left(\begin{array}[]{c}-\phi^{2}p_{3}/\left(m_{0}+E\right)\\\
\phi^{2}\left(-p_{1}-ip_{2}\right)/\left(m_{0}+E\right)\\\ 1\\\
0\end{array}\right),\qquad\widetilde{\Psi}_{2}^{-}:=\exp
i\left(\vec{p}\vec{x}+Et\right)\left(\begin{array}[]{c}\phi^{2}\left(-p_{1}+ip_{2}\right)/\left(m_{0}+E\right)\\\
\phi^{2}p_{3}/\left(m_{0}+E\right)\\\ 0\\\ 1\end{array}\right).$

In comparison to the classical solutions ($\phi^{2}=-1$) the
$\widetilde{\Psi}_{1/2}^{\pm}$ can be identified as (anti)particle ($1/2$)
plane waves with spin up ($+$) or down ($-$). Here in the general case,
$m_{0}$ varies in $\mathbb{C}_{0}$ for all solutions, and there is a phase
$\phi^{2}$ in the vector parts of the eigenfunctions between particles and
antiparticles.

Using a spin vector $\chi^{\pm}$ with

(3.80) $\chi^{+}:=\left(\begin{array}[]{c}1\\\
0\end{array}\right),\qquad\qquad\chi^{-}:=\left(\begin{array}[]{c}0\\\
1\end{array}\right),$

the eigenfunctions can be expressed using Pauli spinors $\sigma_{\mu}$ and
$\vec{\sigma}:=\left(\sigma_{1},\sigma_{2},\sigma_{3}\right)$ as:

(3.83) $\displaystyle\widetilde{\Psi}_{1}^{\pm}$ $\displaystyle:=$
$\displaystyle\exp
i\left(\vec{p}\vec{x}-Et\right)\left(\begin{array}[]{c}\sigma_{0}\\\
-\vec{\sigma}\vec{p}/\left(m_{0}+E\right)\end{array}\right)\chi^{\pm},$ (3.86)
$\displaystyle\widetilde{\Psi}_{2}^{\pm}$ $\displaystyle:=$ $\displaystyle\exp
i\left(\vec{p}\vec{x}+Et\right)\left(\begin{array}[]{c}-\phi^{2}\vec{\sigma}\vec{p}/\left(m_{0}+E\right)\\\
\sigma_{0}\end{array}\right)\chi^{\pm}.$

###  3.3. Normalizing the plane-wave eigenfunctions

Norms of numbers $a\in\mathbb{C}\times\mathbb{C}_{0}$ are defined in
$\mathbb{C}$, $\mathbb{C}_{0}$, and $\mathbb{C}\times\mathbb{C}_{0}$ as:

(3.87) $\displaystyle\left|a\right|^{2}$
$\displaystyle:=a\overline{a}\in\mathbb{C}_{0},\qquad\qquad\left|a\right|_{0}^{2}:=a\underline{a}\in\mathbb{C},\qquad\qquad\left\|a\right\|^{4}:=a\overline{a}\underline{a\overline{a}}\in\mathbb{R}.$

The term “norm” is used loosely, in the sense that the composition property is
conserved for any $a,b\in\mathbb{C}\times\mathbb{C}_{0}$:

(3.88)
$\left|ab\right|^{2}=\left|a\right|^{2}\left|b\right|^{2},\qquad\qquad\left|ab\right|_{0}^{2}=\left|a\right|_{0}^{2}\left|b\right|_{0}^{2},\qquad\qquad\left\|ab\right\|^{4}=\left\|a\right\|^{4}\left\|b\right\|^{4}.$

Norms are, however, not positive definite or point separating. Note that when
taking the square (or fourth) root of these expressions it has to be made
clear which space the result is to be in ($\mathbb{C}$, $\mathbb{C}_{0}$, or
$\mathbb{C}\times\mathbb{C}_{0}$).

Since the generalized Dirac equation uses an eigenvalue
$m_{0}\in\mathbb{C}_{0}$, the calculation of the scattering cross section will
be executed in the $\mathbb{C}$ subalgebra only, i.e., in the same manner as
in the classical case. This is possible since all dynamic variables act in
this subalgebra, and the only variable in $\mathbb{C}_{0}$ is $\alpha$ itself
which is constant in space and time (and therewith in $p$). Only at the very
end, when asking for probabilities, will the $\mathbb{C}_{0}$ norm be taken,
to obtain a real value.

The $\widetilde{\Psi}_{1/2}^{\pm}$ above are obtained by fixing components
$\left(1,0\right)$ and $\left(0,1\right)$ in their vector parts. They are not
yet normed to conserve probability in field-free space (and time). The normed
eigenfunctions $\hat{\Psi}_{1/2}^{\pm}$ satisfy

(3.89)
$\displaystyle\left(\hat{\Psi}_{1/2}^{\pm}\right)^{\dagger}\hat{\Psi}_{1/2}^{\pm}$
$\displaystyle\overset{!}{=}$ $\displaystyle 1.$

They differ from the $\widetilde{\Psi}_{1/2}^{\pm}$ only by a constant factor
$N_{1/2}\in\mathbb{C}_{0}$,

(3.90) $\displaystyle\hat{\Psi}_{1/2}^{\pm}$ $\displaystyle:=$ $\displaystyle
N_{1/2}\widetilde{\Psi}_{1/2}^{\pm}.$

For $\widetilde{\Psi}_{1}^{+}$there is:

(3.91)
$\displaystyle\left(\hat{\Psi}_{1}^{+}\right)^{\dagger}\hat{\Psi}_{1}^{+}$
$\displaystyle=$
$\displaystyle\overline{\left(\hat{\Psi}_{1}^{+}\right)}^{T}\gamma_{\phi^{2}}\hat{\Psi}_{1}^{+}=\overline{N_{1}}\overline{\left(\widetilde{\Psi}_{1}^{+}\right)}^{T}\gamma_{\phi^{2}}N_{1}\widetilde{\Psi}_{1}^{+}$
(3.100) $\displaystyle=$
$\displaystyle\left|N_{1}\right|^{2}\left(1,0,\frac{-p_{3}}{m_{0}+E},\frac{-p_{1}+ip_{2}}{m_{0}+E}\right)\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&\phi^{2}&0\\\
0&0&0&\phi^{2}\end{array}\right)\left(\begin{array}[]{c}1\\\ 0\\\
-p_{3}/\left(m_{0}+E\right)\\\
\left(-p_{1}-ip_{2}\right)/\left(m_{0}+E\right)\end{array}\right)$ (3.101)
$\displaystyle=$
$\displaystyle\left|N_{1}\right|^{2}\left(1+\frac{\phi^{2}p_{3}^{2}}{\left(m_{0}+E\right)^{2}}+\frac{\phi^{2}\left(p_{1}^{2}+p_{2}^{2}\right)}{\left(m_{0}+E\right)^{2}}\right)$
(3.102) $\displaystyle=$
$\displaystyle\left|N_{1}\right|^{2}\left(1+\frac{\phi^{2}\left|\vec{p}\right|^{2}}{\left(m_{0}+E\right)^{2}}\right).$

Using $m_{0}^{2}=E^{2}+\phi^{2}\left|\vec{p}\right|^{2}$ this becomes:

(3.103)
$\displaystyle\left(\hat{\Psi}_{1}^{+}\right)^{\dagger}\hat{\Psi}_{1}^{+}$
$\displaystyle=$
$\displaystyle\left|N_{1}\right|^{2}\frac{\left(m_{0}+E\right)^{2}+\phi^{2}\left|\vec{p}\right|^{2}}{\left(m_{0}+E\right)^{2}}\,=\,\left|N_{1}\right|^{2}\frac{m_{0}^{2}+2m_{0}E+E^{2}+\phi^{2}\left|\vec{p}\right|^{2}}{\left(m_{0}+E\right)^{2}}$
(3.104) $\displaystyle=$
$\displaystyle\left|N_{1}\right|^{2}\frac{m_{0}^{2}+2m_{0}E+m_{0}^{2}}{\left(m_{0}+E\right)^{2}}\,=\,\left|N_{1}\right|^{2}\frac{2m_{0}\left(m_{0}+E\right)}{\left(m_{0}+E\right)^{2}}$
(3.105) $\displaystyle=$
$\displaystyle\left|N_{1}\right|^{2}\frac{2m_{0}}{m_{0}+E}.$

The normalizing factor $N_{1}\in\mathbb{C}_{0}$ therefore differs from the
classical case only by generalizing real mass $m_{\mathbb{R}}$ to $m_{0}$:

(3.106)
$\left|N_{1}\right|^{2}=\frac{m_{0}+E}{2m_{0}}\in\mathbb{C}_{0},\qquad\qquad
N_{1}=\sqrt{\frac{m_{0}+E}{2m_{0}}}.$

The square root is to be taken in $\mathbb{C}_{0}$. The identical calculation
can be done for the $\widetilde{\Psi}_{1}^{-}$.

For the $\widetilde{\Psi}_{2}^{+}$ we have:

(3.115)
$\displaystyle\left(\hat{\Psi}_{2}^{+}\right)^{\dagger}\hat{\Psi}_{2}^{+}$
$\displaystyle=$
$\displaystyle\left|N_{2}\right|^{2}\left(\frac{-\phi^{2}p_{3}}{m_{0}+E},\frac{-\phi^{2}p_{1}+i\phi^{2}p_{2}}{m_{0}+E},1,0\right)\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&\phi^{2}&0\\\
0&0&0&\phi^{2}\end{array}\right)\left(\begin{array}[]{c}-\phi^{2}p_{3}/\left(m_{0}+E\right)\\\
\phi^{2}\left(-p_{1}-ip_{2}\right)/\left(m_{0}+E\right)\\\ 1\\\
0\end{array}\right)$ (3.116) $\displaystyle=$
$\displaystyle\phi^{2}\left|N_{2}\right|^{2}\left(\frac{\phi^{2}p_{3}^{2}}{\left(m_{0}+E\right)^{2}}+\frac{\phi^{2}\left(p_{1}^{2}+p_{2}^{2}\right)}{\left(m_{0}+E\right)^{2}}+1\right)$
(3.117) $\displaystyle=$
$\displaystyle\phi^{2}\left|N_{2}\right|^{2}\frac{\phi^{2}\left|\vec{p}\right|^{2}+\left(m_{0}+E\right)^{2}}{\left(m_{0}+E\right)^{2}}$
(3.118) $\displaystyle=$
$\displaystyle\phi^{2}\left|N_{2}\right|^{2}\frac{2m_{0}}{m_{0}+E}.$

Since $\phi\underline{\phi}=1$ this yields:

(3.119)
$\left|N_{2}\right|^{2}=\underline{\phi}^{2}\frac{m_{0}+E}{2m_{0}}\in\mathbb{C}_{0},\qquad\qquad
N_{2}=\underline{\phi}\sqrt{\frac{m_{0}+E}{2m_{0}}}.$

The constant $N_{2}$ is rotated by a phase $\underline{\phi}$ as compared to
$N_{1}$,r

(3.120) $\displaystyle N_{2}$ $\displaystyle=$
$\displaystyle\underline{\phi}N_{1}.$

This will be of no consequence in this paper since particles and antiparticles
won’t change into one another during elastic scattering. For future work that
investigates interactions and transitions involving both particles and
antiparticles, e.g. fermion pair production and annihilation, this phase may
affect the measurement prediction.

You could of course multiply $\underline{\phi}^{2}$ into the eigenfunctions
$\widetilde{\Psi}_{2}^{\pm}$ themselves:

(3.123) $\displaystyle\widetilde{\Psi}_{1}^{\pm}$ $\displaystyle=$
$\displaystyle\exp
i\left(\vec{p}\vec{x}-Et\right)\left(\begin{array}[]{c}\sigma_{0}\\\
-\vec{\sigma}\vec{p}/\left(m_{0}+E\right)\end{array}\right)\chi^{\pm},$
(3.126) $\displaystyle\widetilde{\Psi}_{2}^{\pm}\underline{\phi}^{2}$
$\displaystyle=$ $\displaystyle\exp
i\left(\vec{p}\vec{x}+Et\right)\left(\begin{array}[]{c}-\vec{\sigma}\vec{p}/\left(m_{0}+E\right)\\\
\underline{\phi}^{2}\sigma_{0}\end{array}\right)\chi^{\pm}.$

While this would make the momentum parts
$-\vec{\sigma}\vec{p}/\left(m_{0}+E\right)$ symmetric between the
$\widetilde{\Psi}_{1}^{\pm}$ and
$\widetilde{\Psi}_{2}^{\pm}\underline{\phi}^{2}$, the $\sigma_{0}$ parts would
become asymmetric. Since there’s no real gain for the purpose of this paper,
the definitions are left unchanged.

The normalized component of the eigenfunctions that depends on the particle’s
spin and type, $u_{1/2}^{\pm}$, is defined as:

(3.140) $\displaystyle u_{1}^{\pm}$ $\displaystyle:=$ $\displaystyle
N_{1}\left(\begin{array}[]{c}\sigma_{0}\\\
-\vec{\sigma}\vec{p}/\left(m_{0}+E\right)\end{array}\right)\chi^{\pm}=\sqrt{\frac{m_{0}+E}{2m_{0}}}\left(\begin{array}[]{c}\sigma_{0}\\\
-\vec{\sigma}\vec{p}/\left(m_{0}+E\right)\end{array}\right)\chi^{\pm},$
$\displaystyle
u_{1}^{+}=\sqrt{\frac{m_{0}+E}{2m_{0}}}\left(\begin{array}[]{c}1\\\ 0\\\
-p_{3}/\left(m_{0}+E\right)\\\
\left(-p_{1}-ip_{2}\right)/\left(m_{0}+E\right)\end{array}\right),\qquad
u_{1}^{-}=\sqrt{\frac{m_{0}+E}{2m_{0}}}\left(\begin{array}[]{c}0\\\ 1\\\
\left(-p_{1}+ip_{2}\right)/\left(m_{0}+E\right)\\\
p_{3}/\left(m_{0}+E\right)\end{array}\right),$ (3.154) $\displaystyle
u_{2}^{\pm}$ $\displaystyle:=$ $\displaystyle
N_{2}\left(\begin{array}[]{c}-\phi^{2}\vec{\sigma}\vec{p}/\left(m_{0}+E\right)\\\
\sigma_{0}\end{array}\right)\chi^{\pm}=\sqrt{\frac{m_{0}+E}{2m_{0}}}\left(\begin{array}[]{c}-\phi\vec{\sigma}\vec{p}/\left(m_{0}+E\right)\\\
\underline{\phi}\sigma_{0}\end{array}\right)\chi^{\pm},$ $\displaystyle
u_{2}^{+}=\sqrt{\frac{m_{0}+E}{2m_{0}}}\left(\begin{array}[]{c}-\phi
p_{3}/\left(m_{0}+E\right)\\\
\phi\left(-p_{1}-ip_{2}\right)/\left(m_{0}+E\right)\\\ \underline{\phi}\\\
0\end{array}\right),\qquad
u_{2}^{-}=\sqrt{\frac{m_{0}+E}{2m_{0}}}\left(\begin{array}[]{c}\phi\left(-p_{1}+ip_{2}\right)/\left(m_{0}+E\right)\\\
\phi p_{3}/\left(m_{0}+E\right)\\\ 0\\\ \underline{\phi}\end{array}\right).$

This simply leaves out the oscillating plane-wave component of the
eigenfunctions:

(3.155) $\displaystyle\hat{\Psi}_{k}^{\pm}$ $\displaystyle=$ $\displaystyle
u_{k}^{\pm}\exp
i\left(\vec{p}\vec{x}+\left(-1\right)^{k}Et\right)\qquad\textrm{with }k=1,2.$

###  3.4. Normalizing to invariant volume

Following textbook calculation of the elastic (Rutherford) scattering cross
section on a fixed target in Born approximation, the wave functions $\Psi$ of
the incoming ($\mathrm{ini}$) and outgoing ($\mathrm{fin}$) particle will be
normalized to a small invariant volume $V_{0}$ in the reference frame of the
target. Per equation (2.16) the invariant volume $V_{0}$ is

(3.156) $\displaystyle V_{0}$ $\displaystyle=$
$\displaystyle\frac{VE}{m_{0}}=\frac{V}{\sqrt{1+\phi^{2}\left|\vec{v}\right|^{2}}},$

and the such normalized $\Psi_{k}^{\pm}$ are:

(3.157) $\displaystyle\Psi_{k}^{\pm}$ $\displaystyle:=$
$\displaystyle\sqrt{\frac{m_{0}}{VE}}\hat{\Psi}_{k}^{\pm}$ (3.158)
$\displaystyle=$ $\displaystyle\sqrt{\frac{m_{0}}{VE}}u_{k}^{\pm}\exp
i\left(\vec{p}\vec{x}+\left(-1\right)^{k}Et\right).$

This satisfies

(3.159) $\displaystyle\left(\Psi_{k}^{\pm}\right)^{\dagger}\Psi_{k}^{\pm}$
$\displaystyle=$ $\displaystyle\frac{1}{V_{0}}.$

###  3.5. Coulomb-type field of a point charge

In the classical $\phi^{2}=-1$ case, the electromagnetic field is introduced
by requiring invariance of the Dirac equation with field under a simple
$\exp\left(iq\chi\right)$ phase. There, $q$ is the electric charge of the
particle under the influence of the field, and space-time derivatives of
$\chi$ are identified as the electromagnetic potentials $A_{\mu}$ acting on
the particle proportionally to $q$ with inertia $m_{\mathbb{R}}$.

This is now generalized analogously, writing placeholder symbols
$\widetilde{q}$ and $\widetilde{\chi}$ for now until it is clarified exactly
which space they’re in:

(3.160) $\displaystyle\widetilde{A}_{\mu}$
$\displaystyle:=\frac{\partial\widetilde{\chi}}{\partial
x_{\mu}},\qquad\psi^{\prime}:=e^{i\widetilde{q}\widetilde{\chi}}\psi,$ (3.161)
$\displaystyle\sum_{\mu=0}^{3}\gamma_{\mu}\left(i\partial_{\mu}\right)\psi^{\prime}=m_{0}\psi^{\prime}$
$\displaystyle\qquad\longrightarrow\qquad
e^{i\widetilde{q}\widetilde{\chi}}\sum_{\mu=0}^{3}\gamma_{\mu}\left(i\partial_{\mu}-\widetilde{q}\widetilde{A}_{\mu}\right)\psi=m_{0}\psi^{\prime}.$

In the general case there is

(3.162) $\displaystyle\widetilde{q}\widetilde{A}_{\mu}$
$\displaystyle\in\mathbb{C}_{0},$ (3.163)
$\displaystyle\sum_{\mu=0}^{3}\gamma_{\mu}\left(i\partial_{\mu}+\widetilde{q}\widetilde{A}_{\mu}\right)\psi$
$\displaystyle=m_{0}\psi.$

Equation (3.163) is the generalized Dirac equation, invariant under
$\mathrm{U}\left(1\right)$ gauge (in $\mathbb{C}$).

This raises the question where exactly the phase in $\mathbb{C}_{0}$
originates from: Is one of the
$\left\\{\widetilde{q},\widetilde{A}_{\mu}\right\\}$ real or are both complex
in $\mathbb{C}_{0}$? By analogy, the purely electromagnetic and gravitational
edge cases have
$\left\\{m_{\mathrm{Eucl}},m_{\mathrm{Mink}}\right\\}\in\mathbb{R}$, and we
would expect in the gravitational case for the charge to become its mass,
$\widetilde{q}_{\mathrm{Eucl}}\equiv m_{\mathrm{Eucl}}$. In the
electromagnetic case we expect real charges as well, making
$\left\\{\widetilde{q}_{\mathrm{Eucl}},\widetilde{q}_{\mathrm{Mink}}\right\\}\in\mathbb{R}$.
Both edge cases appear, however, unphysical: Purely electromagnetic
interaction would assume a particle that is charged, however, doesn’t interact
gravitationally; and purely gravitational interaction would assume a particle
or field that has no kinetic component bound to Minkowskian spacetime.
Addressing these concerns is left for later.

Without needing to make speculations on the dynamics behind the
$\widetilde{A}_{\mu}$, a static Coulomb potential of a charge $\widetilde{Q}$
is now modeled as:

(3.164)
$\displaystyle\widetilde{A}_{0}:=-\frac{\widetilde{Q}}{\left|\vec{x}\right|},\qquad\widetilde{A}_{j}=0\textrm{
otherwise},$ (3.165) $\displaystyle\Longrightarrow$
$\displaystyle\qquad\sum_{\mu=0}^{3}\gamma_{\mu}\widetilde{q}\widetilde{A}_{\mu}=-\gamma_{0}\frac{\widetilde{q}\widetilde{Q}}{\left|\vec{x}\right|}.$

The generalized Dirac equation for a spin-1/2 particle in a static Coulomb
potential then is:

(3.166)
$\left(\sum_{\mu=0}^{3}i\gamma_{\mu}\partial_{\mu}-\gamma_{0}\frac{\widetilde{q}\widetilde{Q}}{\left|\vec{x}\right|}\right)\psi=m_{0}\psi^{\prime}.$

###  3.6. Lowest-order transition matrix element in Born approximation

For scattering on the fixed point target, initial (incoming) and final
(outgoing) wave functions $\Psi_{k,\mathrm{ini}}^{\pm}$ and
$\Psi_{k,\mathrm{fin}}^{\pm}$ of the spin-1/2 particle are assumed plane
waves. Only the lowest-order transition matrix element $S_{\mathrm{fi}}$ is
calculated. Using this customary approximation, results can then be compared
qualitatively with conventional QED results. Some quantitative estimates will
also be possible.

(3.167) $\displaystyle\Psi_{k,\mathrm{ini}}^{\pm}$
$\displaystyle:=\sqrt{\frac{m_{0}}{VE}}u_{k,\mathrm{ini}}^{\pm}\exp
i\left(\vec{p}\vec{x}+\left(-1\right)^{k}Et\right),\qquad\Psi_{k,\mathrm{fin}}^{\pm}:=\sqrt{\frac{m_{0}}{VE}}u_{k,\mathrm{fin}}^{\pm}\exp
i\left(\vec{p}\vec{x}+\left(-1\right)^{k}Et\right),$

For now omitting the annotations $\pm$ and $k$ for readability:

(3.168) $\displaystyle S_{\mathrm{fi}}$
$\displaystyle=\left\langle\Psi_{\mathrm{fin}}\right|S\left|\Psi_{\mathrm{ini}}\right\rangle=i\int
d^{4}x\Psi_{\mathrm{fin}}^{\dagger}\left(\widetilde{q}\sum_{\mu=0}^{3}\gamma_{\mu}A_{\mu}\right)\Psi_{\mathrm{ini}}$
(3.169) $\displaystyle=i\int
d^{4}x\Psi_{\mathrm{fin}}^{\dagger}\left(-\widetilde{q}\gamma_{0}\frac{\widetilde{Q}}{\left|\vec{x}\right|}\right)\Psi_{\mathrm{ini}}$
(3.170) $\displaystyle=-i\widetilde{q}\widetilde{Q}\int
d^{4}x\left(\sqrt{\frac{m_{0}}{VE_{\mathrm{fin}}}}u_{\mathrm{fin}}\exp
i\left(\vec{p}_{\mathrm{fin}}\vec{x}+\left(-1\right)^{k}E_{\mathrm{fin}}t\right)\right)^{\dagger}\left(\gamma_{0}\frac{1}{\left|\vec{x}\right|}\right)\left(\sqrt{\frac{m_{0}}{VE_{\mathrm{ini}}}}u_{\mathrm{ini}}\exp
i\left(\vec{p}_{\mathrm{ini}}\vec{x}+\left(-1\right)^{k}E_{\mathrm{ini}}t\right)\right)$
(3.171)
$\displaystyle=-i\frac{\widetilde{q}\widetilde{Q}m_{0}}{V\sqrt{E_{\mathrm{fin}}E_{\mathrm{ini}}}}u_{\mathrm{fin}}^{\dagger}\gamma_{0}u_{\mathrm{ini}}\int
d^{4}x\frac{\exp\left[i\sum_{\nu=0}^{3}\left(p_{\nu,\mathrm{fin}}-p_{\nu,\mathrm{ini}}\right)\right]}{\left|\vec{x}\right|}.$

As compared to the classical case, the differences are in the constants
$\widetilde{q}$, $\widetilde{Q}$, and $m_{0}$ (all valued in
$\mathbb{C}_{0}$), as well as the $u_{k}^{\pm}$ and their adjoint
$\left(u_{k}^{\pm}\right)^{\dagger}:=\overline{u_{k}^{\pm}}^{T}\gamma_{\phi^{2}}$
which contain constant terms in $\mathbb{C}_{0}$ as well. Here, “constant”
means independent of dynamic variables $\left(\vec{x},t\right)$ and properties
$\left(\vec{p},E\right)$.

When calculating the transition probability $dW$ for a single particle into a
particular state $dN$ (here, into a volume $V$ and momentum interval
$d^{3}p_{\mathrm{fin}}$),

(3.172) $dW=\left\|S_{\mathrm{fi}}\right\|^{2}dN,$

the norm in $\mathbb{C}\times\mathbb{C}_{0}$ with
$\left\|a\right\|^{4}:=a\overline{a}\underline{a}\overline{\underline{a}}$
will be used since it is guaranteed real-valued. Other than this intuitive
generalization, standard methods can be followed for calculating the spin-
independent part $d\widetilde{\sigma}_{\mathrm{class}}/d\Omega$ of the cross
section $d\widetilde{\sigma}$ into an angle element $d\Omega$:

(3.173) $\displaystyle\frac{d\widetilde{\sigma}}{d\Omega}$
$\displaystyle=\frac{d\widetilde{\sigma}_{\mathrm{class}}}{d\Omega}\frac{d\widetilde{\sigma}_{\mathrm{spin}}}{d\Omega},$
(3.174) $\displaystyle\frac{d\widetilde{\sigma}_{\mathrm{class}}}{d\Omega}$
$\displaystyle=\frac{\left|\widetilde{q}\widetilde{Q}m_{0}\right|^{2}}{4\left|\vec{p}\right|^{4}\sin^{4}\frac{\theta}{2}}.$

In order to obtain the real-valued $d\sigma$ at the end, we’ll simply take its
absolute again:

(3.175) $\displaystyle\frac{d\sigma}{d\Omega}$
$\displaystyle=\left\|\frac{d\widetilde{\sigma}}{d\Omega}\right\|\equiv\left|\frac{d\widetilde{\sigma}}{d\Omega}\right|_{0}.$

###  3.7. Spin contribution

The spin contribution to the cross section (3.173) is

(3.176)
$\frac{d\widetilde{\sigma}_{\mathrm{spin}}}{d\Omega}=\sum_{k,\pm}\left|u_{\mathrm{fin}}^{\dagger}\gamma_{0}u_{\mathrm{ini}}\right|^{2}.$

The symbols $k$ and $\pm$ indicate summation over all possible transitions
from the incoming to the outgoing wave.

####  3.7.1. Particle / antiparticle transition, spin flip

In the classical case the transition matrix elements for changing from a
particle into an antiparticle, and vice versa, are trivially zero. Here we
have additional factors $\phi$ and $\underline{\phi}$ to be tested. Using
definitions for the $u_{k}^{\pm}$ we have:

(3.181) $\displaystyle\left(u_{1}^{+}\right)^{\dagger}\gamma_{0}u_{2}^{+}$
$\displaystyle=\frac{m_{0}+E}{2m_{0}}\left(1,0,\frac{-p_{3}}{m_{0}+E},\frac{-p_{1}+ip_{2}}{m_{0}+E}\right)\gamma_{\phi^{2}}\gamma_{0}\left(\begin{array}[]{c}-\phi
p_{3}/\left(m_{0}+E\right)\\\
\phi\left(-p_{1}-ip_{2}\right)/\left(m_{0}+E\right)\\\ \underline{\phi}\\\
0\end{array}\right)=\phi\frac{-p_{3}+p_{3}}{2m_{0}}=0,$ (3.186)
$\displaystyle\left(u_{1}^{+}\right)^{\dagger}\gamma_{0}u_{2}^{-}$
$\displaystyle=\frac{m_{0}+E}{2m_{0}}\left(1,0,\frac{-p_{3}}{m_{0}+E},\frac{-p_{1}+ip_{2}}{m_{0}+E}\right)\gamma_{\phi^{2}}\gamma_{0}\left(\begin{array}[]{c}\phi\left(-p_{1}+ip_{2}\right)/\left(m_{0}+E\right)\\\
\phi p_{3}/\left(m_{0}+E\right)\\\ 0\\\
\underline{\phi}\end{array}\right)=\phi\frac{-p_{1}+ip_{2}+p_{1}-ip_{2}}{2m_{0}}=0.$

The other six cases of the
$u_{\mathrm{fin}}^{\dagger}\gamma_{0}u_{\mathrm{ini}}$ where
$k_{\mathrm{fin}}\neq k_{\mathrm{ini}}$ are symmetric through reordering of
the vector components, as well as swapping factors in the commutative product.
A particle cannot change into an antiparticle through elastic scattering, and
vice versa.

Spin flip is also excluded:

(3.191) $\displaystyle\left(u_{1}^{+}\right)^{\dagger}\gamma_{0}u_{1}^{+}$
$\displaystyle=\frac{m_{0}+E}{2m_{0}}\left(1,0,\frac{-p_{3}}{m_{0}+E},\frac{-p_{1}+ip_{2}}{m_{0}+E}\right)\gamma_{\phi^{2}}\gamma_{0}\left(\begin{array}[]{c}0\\\
1\\\ \left(-p_{1}+ip_{2}\right)/\left(m_{0}+E\right)\\\
p_{3}/\left(m_{0}+E\right)\end{array}\right)=0,$ (3.196)
$\displaystyle\left(u_{2}^{+}\right)^{\dagger}\gamma_{0}u_{2}^{+}$
$\displaystyle=\frac{m_{0}+E}{2m_{0}}\left(\frac{-\phi
p_{3}}{m_{0}+E},\frac{\phi\left(-p_{1}+ip_{2}\right)}{m_{0}+E},\underline{\phi},0\right)\gamma_{\phi^{2}}\gamma_{0}\left(\begin{array}[]{c}\phi\left(-p_{1}+ip_{2}\right)/\left(m_{0}+E\right)\\\
\phi p_{3}/\left(m_{0}+E\right)\\\ 0\\\ \underline{\phi}\end{array}\right)=0.$

####  3.7.2. Spin and particle type remains unchanged

Interaction of the particle with the central
$\widetilde{A}=\left(-\widetilde{Q}/\left|\vec{x}\right|,0,0,0\right)$
potential may only change the particle’s momentum distribution across
different angles, but not its spin or type. The only spin contribution
therefore is:

(3.197)
$\frac{d\widetilde{\sigma}_{\mathrm{spin}}}{d\Omega}=\left|\left(u_{k,\mathrm{fin}}^{\pm}\right)^{\dagger}\gamma_{0}u_{k,\mathrm{ini}}^{\pm}\right|^{2}\textrm{
for any }k,\pm\textrm{ unchanged}.$

The calculation is symmetric for particles and antiparticles, for spin up and
spin down, as well as rotational in space around the axis of incoming momentum
respective to the target. It is therefore calculated for a particle ($k=1$)
with spin up ($+$) in the $\left(x_{1},x_{2}\right)$ plane
($p_{3,\mathrm{ini}}=p_{3,\mathrm{fin}}=0$), assuming energy conservation
($E_{\mathrm{fin}}=E_{\mathrm{ini}}\equiv E$):

(3.202)
$\displaystyle\left(u_{1,\mathrm{fin}}^{+}\right)^{\dagger}\gamma_{0}u_{1,\mathrm{ini}}^{+}$
$\displaystyle=\frac{m_{0}+E}{2m_{0}}\left(1,0,0,\frac{-p_{1,\mathrm{fin}}+ip_{2,\mathrm{fin}}}{m_{0}+E}\right)\gamma_{\phi^{2}}\gamma_{0}\left(\begin{array}[]{c}1\\\
0\\\ 0\\\
\left(-p_{1,\mathrm{ini}}-ip_{2,\mathrm{ini}}\right)/\left(m_{0}+E\right)\end{array}\right)$
(3.203)
$\displaystyle=\frac{m_{0}+E}{2m_{0}}\left(1+\frac{\left(-p_{1,\mathrm{fin}}+ip_{2,\mathrm{fin}}\right)\phi^{2}\left(p_{1,\mathrm{ini}}+ip_{2,\mathrm{ini}}\right)}{\left(m_{0}+E\right)^{2}}\right)$
(3.204)
$\displaystyle=\frac{\left(m_{0}+E\right)^{2}-\phi^{2}\left|\vec{p}\right|^{2}\left(\cos\theta-i\sin\theta\right)}{2m_{0}\left(m_{0}+E\right)}.$

The last line expressed the difference between initial and final momentum in
terms of scattering angle $\theta$, while taking advantage of $p_{3}=0$:

(3.205) $\displaystyle
p_{1,\mathrm{fin}}p_{1,\mathrm{ini}}+p_{2,\mathrm{fin}}p_{2,\mathrm{ini}}$
$\displaystyle\equiv\vec{p}_{\mathrm{fin}}\vec{p}_{\mathrm{ini}}=\left|\vec{p}\right|^{2}\cos\theta,$
(3.206) $\displaystyle
p_{1,\mathrm{fin}}p_{2,\mathrm{ini}}-p_{2,\mathrm{fin}}p_{1,\mathrm{ini}}$
$\displaystyle\equiv\vec{p}_{\mathrm{fin}}\times\vec{p}_{\mathrm{ini}}=-\left|\vec{p}\right|^{2}\sin\theta.$

Using the identities

(3.207) $\displaystyle\cos^{2}\theta$ $\displaystyle=1-\sin^{2}\theta,$
(3.208) $\displaystyle\cos\theta$ $\displaystyle=1-2\sin^{2}\frac{\theta}{2},$

there is:

(3.209) $\displaystyle\frac{d\widetilde{\sigma}_{\mathrm{spin}}}{d\Omega}$
$\displaystyle=\left|\left(u_{1,\mathrm{fin}}^{+}\right)^{\dagger}\gamma_{0}u_{1,\mathrm{ini}}^{+}\right|^{2}$
(3.210)
$\displaystyle=\left|\frac{\left(m_{0}+E\right)^{2}-\phi^{2}\left|\vec{p}\right|^{2}\left(\cos\theta-i\sin\theta\right)}{2m_{0}\left(m_{0}+E\right)}\right|^{2}$
(3.211)
$\displaystyle=\frac{\left(\left(m_{0}+E\right)^{2}-\phi^{2}\left|\vec{p}\right|^{2}\cos\theta\right)^{2}+\left(\phi^{2}\left|\vec{p}\right|^{2}\sin\theta\right)^{2}}{\left(2m_{0}\left(m_{0}+E\right)\right)^{2}}$
(3.212)
$\displaystyle=\frac{\left(m_{0}+E\right)^{4}-2\left(m_{0}+E\right)^{2}\phi^{2}\left|\vec{p}\right|^{2}\cos\theta+\phi^{4}\left|\vec{p}\right|^{4}\cos^{2}\theta+\phi^{4}\left|\vec{p}\right|^{4}\sin^{2}\theta}{\left(2m_{0}\left(m_{0}+E\right)\right)^{2}}$
…using $\displaystyle\cos^{2}\theta=1-\sin^{2}\theta\textrm{ ...}$ (3.213)
$\displaystyle=\frac{\left(m_{0}+E\right)^{4}-2\left(m_{0}+E\right)^{2}\phi^{2}\left|\vec{p}\right|^{2}\cos\theta+\phi^{4}\left|\vec{p}\right|^{4}}{\left(2m_{0}\left(m_{0}+E\right)\right)^{2}}$
…using $\displaystyle\cos\theta=1-2\sin^{2}\frac{\theta}{2}\textrm{ ...}$
(3.214)
$\displaystyle=\frac{\left(m_{0}+E\right)^{4}-2\left(m_{0}+E\right)^{2}\phi^{2}\left|\vec{p}\right|^{2}+\phi^{4}\left|\vec{p}\right|^{4}+4\left(m_{0}+E\right)^{2}\phi^{2}\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}}{\left(2m_{0}\left(m_{0}+E\right)\right)^{2}}$
(3.215)
$\displaystyle=\frac{\left(\left(m_{0}+E\right)^{2}-\phi^{2}\left|\vec{p}\right|^{2}\right)^{2}}{\left(2m_{0}\left(m_{0}+E\right)\right)^{2}}+\frac{\phi^{2}\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}}{m_{0}^{2}}$
(3.216)
$\displaystyle=\frac{\left(m_{0}^{2}+2m_{0}E+E^{2}-\phi^{2}\left|\vec{p}\right|^{2}\right)^{2}}{\left(2m_{0}\left(m_{0}+E\right)\right)^{2}}+\frac{\phi^{2}\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}}{m_{0}^{2}}$
…using $\displaystyle m_{0}^{2}=E^{2}+\phi^{2}\left|\vec{p}\right|^{2}\textrm{
...}$ (3.217)
$\displaystyle=\frac{\left(2E\left(m_{0}+E\right)\right)^{2}}{\left(2m_{0}\left(m_{0}+E\right)\right)^{2}}+\frac{\phi^{2}\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}}{m_{0}^{2}}$
(3.218)
$\displaystyle=\frac{E^{2}}{m_{0}^{2}}+\phi^{2}\frac{\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}}{m_{0}^{2}}$
…using
$\displaystyle\frac{E^{2}}{m_{0}^{2}}=\frac{1}{1+\phi^{2}\left|\vec{v}\right|^{2}},\qquad\frac{\left|\vec{p}\right|^{2}}{E^{2}}=\left|\vec{v}\right|^{2}\textrm{
...}$ (3.219)
$\displaystyle=\frac{1+\phi^{2}\left|\vec{v}\right|^{2}\sin^{2}\frac{\theta}{2}}{1+\phi^{2}\left|\vec{v}\right|^{2}}.$

This correctly recovers the classical case for
$\left.\phi^{2}\right|_{\mathrm{QED}}=-1$.

###  3.8. Result and comparison with the classical case

Putting the (semi-)classical and spin contributions to the scattering cross
section together, the result is:

(3.220) $\displaystyle\frac{d\widetilde{\sigma}}{d\Omega}$
$\displaystyle=\frac{d\widetilde{\sigma}_{\mathrm{class}}}{d\Omega}\frac{d\widetilde{\sigma}_{\mathrm{spin}}}{d\Omega}$
(3.221) $\displaystyle\qquad\textrm{with
}\frac{d\widetilde{\sigma}_{\mathrm{class}}}{d\Omega}=\frac{\left|\widetilde{q}\widetilde{Q}m_{0}\right|^{2}}{4\left|\vec{p}\right|^{4}\sin^{4}\frac{\theta}{2}},\qquad\frac{d\widetilde{\sigma}_{\mathrm{spin}}}{d\Omega}=\frac{1}{m_{0}^{2}}\left(E^{2}+\phi^{2}\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}\right)=\frac{1+\phi^{2}\left|\vec{v}\right|^{2}\sin^{2}\frac{\theta}{2}}{1+\phi^{2}\left|\vec{v}\right|^{2}},$
(3.222)
$\displaystyle\Longrightarrow\qquad\frac{d\widetilde{\sigma}}{d\Omega}$
$\displaystyle=\frac{\left|\widetilde{q}\widetilde{Q}\right|^{2}}{4\left|\vec{p}\right|^{4}\sin^{4}\frac{\theta}{2}}\left(E^{2}+\phi^{2}\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}\right)=\frac{\left|\widetilde{q}\widetilde{Q}m_{0}\right|^{2}}{4\left|\vec{p}\right|^{4}\sin^{4}\frac{\theta}{2}}\cdot\frac{1+\phi^{2}\left|\vec{v}\right|^{2}\sin^{2}\frac{\theta}{2}}{1+\phi^{2}\left|\vec{v}\right|^{2}},$
(3.223) $\displaystyle\frac{d\sigma}{d\Omega}$
$\displaystyle=\left\|\frac{d\widetilde{\sigma}}{d\Omega}\right\|\equiv\left|\frac{d\widetilde{\sigma}}{d\Omega}\right|_{0}.$

As compared to the classical result,

(3.224) $\displaystyle\left.\frac{d\sigma}{d\Omega}\right|_{\mathrm{QED}}$
$\displaystyle=\frac{\left|qQ\right|^{2}}{4\left|\vec{p}\right|^{4}\sin^{4}\frac{\theta}{2}}\left(E^{2}-\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}\right)\qquad\left(\textrm{for
general }E,\left|\vec{p}\right|\right)$ (3.225)
$\displaystyle=\frac{\left|qQm\right|^{2}}{4\left|\vec{p}\right|^{4}\sin^{4}\frac{\theta}{2}}\cdot\frac{1-\left|\vec{v}\right|^{2}\sin^{2}\frac{\theta}{2}}{1-\left|\vec{v}\right|^{2}}\qquad\left(\textrm{only
for }m>0,\left|\vec{v}\right|<1\right),$

the following differences exist:

  * •

Particle charge $\widetilde{q}$, target charge $\widetilde{Q}$, and invariant
mass $m_{0}$ may be rotated against one another in $\mathbb{C}_{0}$ by
arbitrary real angles. In the classical case, charges may only have an
opposite sign. For the purpose of this paper this generalization to
$\mathbb{C}_{0}$ has no predictive value, since only the real absolute values
of products $\widetilde{q}\widetilde{Q}$ or $\widetilde{q}\widetilde{Q}m_{0}$
appear in the $d\sigma/d\Omega$ result, making it impossible to separate the
individual factors into their respective magnitudes and phases in
$\mathbb{C}_{0}$.

  * •

The phase $\phi^{2}$ appears in the spin-$\frac{1}{2}$ contribution part
$\phi^{2}\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}$. This makes for a
different value in the predicted measurement outcome.

The ratio $r_{\alpha/\mathrm{QED}}$ between general (any $\alpha$) and
classical ($\alpha=\pi/2$) cross section for fixed charges $q$ and $Q$ is:

(3.226) $\displaystyle
r_{\alpha/\mathrm{QED}}:=\frac{d\sigma}{d\Omega}/\left.\frac{d\sigma}{d\Omega}\right|_{\mathrm{QED}}$
$\displaystyle=\frac{\left\|E^{2}+\phi^{2}\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}\right\|}{E^{2}-\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}}=\frac{\sqrt{E^{4}+2\cos\left(2\alpha\right)E^{2}\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}+\left|\vec{p}\right|^{4}\sin^{4}\frac{\theta}{2}}}{E^{2}-\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}}.$

On first look, $r_{\alpha/\mathrm{QED}}$ is larger the higher the particle’s
incident speed is ($E^{2}\approx\left|\vec{p}\right|^{2}$,
$\left|\vec{v}\right|^{2}\rightarrow 1$) and the closer the scattering angle
is backwards ($\theta\rightarrow\pi$).

###  3.9. Approximation for slow moving incident particle

For slow moving particles, $E\gg\left|\vec{p}\right|$, there is
$\left|\vec{v}\right|=\left|\vec{p}\right|/E\ll 1$ and therefore

(3.227) $\left.r_{\alpha/\mathrm{QED}}\right|_{\left|\vec{v}\right|\ll
1}\approx
1+\left[2\left|\vec{v}\right|^{2}\cos^{2}\alpha\right]\sin^{2}\frac{\theta}{2}+\ldots$

At nonclassical $\alpha$ there is $\cos^{2}\alpha>0$ and therefore
backscattering ($\theta\approx\pi$) is enhanced by a term proportional to
$\left|\vec{v}\right|^{2}$ and $\cos^{2}\alpha$. This behavior can be
compared, at least qualitatively, with scattering of a classical body in
Schwarzschild geometry. This has been investigated [6] and confirms the
qualitative agreement (section 4 figure 2 shows enhanced nonvanishing
backscattering).

###  3.10. Approximation for fast incident particle

The faster the incoming particle gets, $\left|\vec{v}\right|^{2}\rightarrow
1$, the smaller the difference between $\left|\vec{p}\right|$ and $E$ becomes.
The ratio $r_{\alpha/\mathrm{QED}}$ grows approximately as

(3.228) $\left.r_{\alpha/\mathrm{QED}}\right|_{\left|\vec{v}\right|\rightarrow
1}\approx\left[\frac{\cos^{2}\alpha}{1-\left|\vec{v}\right|}\right]\sin^{2}\frac{\theta}{2}.$

As $\left|\vec{v}\right|$ approaches the speed of light, scattering becomes
infinitely much stronger for general $\alpha$ relative to the conventional
case where $\cos^{2}\alpha=0$, growing asymptotically like $1/x$ (with
$x=1-\left|\vec{v}\right|$). This might become a window to observability from
quantum gravitational effects, as all current assumptions for particle
scattering assume a negligible contribution from quantum gravity. The effect
can be compared qualitatively with the spacial distribution of hot inter- and
intragalactic gas, and estimated for lateral momentum transfer of a high-
energy neutrino through matter.

###  3.11. Approximation for purely gravitational interaction

Purely gravitational Rutherford scattering of a spin-$\frac{1}{2}$ particle
may be estimated with all phases in $\mathbb{C}_{0}$ zero (including
$\alpha=0$), target charge to be its mass at rest $\widetilde{Q}:=M$, and
incident particle’s charge its total energy $\widetilde{q}:=E$, so that
equation (3.223) becomes:

(3.229) $\displaystyle\left.\frac{d\sigma}{d\Omega}\right|_{\mathrm{grav}}$
$\displaystyle=\frac{M^{2}E^{2}}{4\left|\vec{p}\right|^{4}\sin^{4}\frac{\theta}{2}}\left(E^{2}+\left|\vec{p}\right|^{2}\sin^{2}\frac{\theta}{2}\right)=\frac{M^{2}}{4\left|\vec{v}\right|^{4}\sin^{4}\frac{\theta}{2}}\cdot\left(1+\left|\vec{v}\right|^{2}\sin^{2}\frac{\theta}{2}\right).$

As expected from the classical case, the purely gravitational elastic
scattering trajectory of the particle only depends on target charge $M$ and
particle speed $\vec{v}$, but not the particle’s gravitational charge222When
particle and target are of comparable energy, e.g. $1\,\mathrm{GeV}$ lepton on
proton, target recoil would have to be taken into account.. For fast moving
particles $\left|\vec{v}\right|\approx 1$ this becomes:

(3.230)
$\displaystyle\left.\frac{d\sigma}{d\Omega}\right|_{\mathrm{grav},\left|\vec{v}\right|\approx
1}$
$\displaystyle\approx\frac{M^{2}}{4\sin^{4}\frac{\theta}{2}}\cdot\left(1+\sin^{2}\frac{\theta}{2}\right)=\frac{M^{2}}{4}\left(\frac{1}{\sin^{4}\frac{\theta}{2}}+\frac{1}{\sin^{2}\frac{\theta}{2}}\right).$

In order to compare this effect with the strength of other forces, equation
(3.230) has to be multiplied with $G^{2}/c^{4}\approx
5.5*10^{-55}\,\mathrm{m}^{2}/\mathrm{kg}^{2}$ to obtain a magnitude in SI
units. For (near) pointlike targets such as electrons ($M_{e^{-}}\approx
9.1*10^{-31}\,\mathrm{kg}$) or neutrons ($M_{n}\approx
1.7*10^{-27}\,\mathrm{kg}$) this effect has a characteristic length scale
$GM/c^{2}$ in the order of $\sim 10^{-53\ldots-56}\,\mathrm{m}$. It will
therefore be overshadowed by electromagnetic interaction and inelastic
scattering, where possible. Observable effects can only be expected in the
absence of these interactions (e.g. elastic scattering of neutrinos on a point
mass) or at very large scales (e.g. elastic scattering of particles with gas
in and around galaxies).

###  3.12. Estimate for momentum transfer of a fast fermion

This section estimates the momentum transfer during elastic scattering of a
fast incoming point-like fermion on a stationary fermion target. Target recoil
is neglected for simplification, as it will not change the order of magnitude
of the momentum transfer.

The radially symmetric cross section $\sigma_{m}$ is the area within which an
incident particle will be scattered at a minimum outgoing angle $\theta_{m}$.
Writing $d\Omega=2\pi\sin\theta\,d\theta$, equation (3.230) can be integrated:

(3.231) $\displaystyle\sigma_{m}=\intop_{0}^{\sigma_{m}}d\sigma$
$\displaystyle=$
$\displaystyle\intop_{\theta_{m}}^{\pi}\frac{M^{2}}{4\sin^{4}\frac{\theta}{2}}\cdot\left(1+\sin^{2}\frac{\theta}{2}\right)2\pi\sin\theta\,d\theta.$

Using

(3.232) $\displaystyle\int\frac{\sin\theta}{\sin^{4}\frac{\theta}{2}}d\theta$
$\displaystyle=$
$\displaystyle-\frac{2}{\sin^{2}\frac{\theta}{2}}+\mathrm{const,}$ (3.233)
$\displaystyle\int\frac{\sin\theta}{\sin^{2}\frac{\theta}{2}}d\theta=2\int\cot\frac{\theta}{2}\,d\theta$
$\displaystyle=$ $\displaystyle
4\ln\left(\sin\frac{\theta}{2}\right)+\mathrm{const,}$

this becomes:

(3.234) $\displaystyle\sigma_{m}$ $\displaystyle=$ $\displaystyle\frac{\pi
M^{2}}{2}\left[-\frac{2}{\sin^{2}\frac{\theta}{2}}+4\ln\left(\sin\frac{\theta}{2}\right)\right]_{\theta_{m}}^{\pi}$
(3.235) $\displaystyle=$ $\displaystyle\frac{\pi
M^{2}}{2}\left[\left(-2+0\right)-\left(-\frac{2}{\sin^{2}\frac{\theta_{m}}{2}}+4\ln\left(\sin\frac{\theta_{m}}{2}\right)\right)\right]$
(3.236) $\displaystyle=$ $\displaystyle\pi
M^{2}\left[\frac{1}{\sin^{2}\frac{\theta_{m}}{2}}-\left(1+\ln\left(\sin^{2}\frac{\theta_{m}}{2}\right)\right)\right].$

Writing $\Delta p_{\theta}$ for momentum transfer at a given angle, and
relative momentum transfer $\Delta\hat{p}_{\theta}$ as

(3.237) $\displaystyle\Delta p_{\theta}$ $\displaystyle=$ $\displaystyle
2\left|\vec{p}\right|\sin\frac{\theta}{2},$ (3.238)
$\displaystyle\Delta\hat{p}_{\theta}$ $\displaystyle:=$
$\displaystyle\frac{\Delta
p_{\theta}}{2\left|\vec{p}\right|}=\sin\frac{\theta}{2},$

there is:

(3.239) $\displaystyle\sigma_{m}$ $\displaystyle=$ $\displaystyle\pi
M^{2}\left[\frac{1}{\left(\Delta\hat{p}_{\theta}\right)^{2}}-\left(1+\ln\left(\Delta\hat{p}_{\theta}\right)^{2}\right)\right],$
(3.240) $\displaystyle\hat{\sigma}_{m}:=\frac{\sigma_{m}}{\pi M^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{\left(\Delta\hat{p}_{\theta}\right)^{2}}-\left(1+\ln\left(\Delta\hat{p}_{\theta}\right)^{2}\right).$

Here, $\hat{\sigma}_{m}$ is the cross section relative to $\pi M^{2}$. This
can be solved for $\Delta\hat{p}_{\theta}$ using the Lambert function (product
log function) $W$ to:

(3.241) $\displaystyle\left(\Delta\hat{p}_{\theta}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{W\left(e^{\hat{\sigma}_{m}+1}\right)}.$

The relative cross section $\hat{\sigma}_{m}$ is typically very large. For
example, assuming $\sigma_{m}$ the face area of a neutron with radius
$r_{n}\approx 0.8*10^{-15}\mathrm{m}$, mass $M_{n}\approx
1.7*10^{-27}\,\mathrm{kg}$, and $G^{2}/c^{4}\approx
5.5*10^{-55}\,\mathrm{m}^{2}/\mathrm{kg}^{2}$ there is approximately:

(3.242) $\displaystyle\left.\hat{\sigma}_{m}\right|_{n}$
$\displaystyle\approx$ $\displaystyle\frac{\pi
r_{n}^{2}}{\pi\frac{G^{2}}{c^{4}}M_{n}^{2}}\approx\frac{\left(0.8*10^{-15}\mathrm{m}\right)^{2}}{5.5*10^{-55}\,\mathrm{m}^{2}*\left(1.7*10^{-27}\right)^{2}}\approx\frac{0.64*10^{-30}}{5.5*10^{-55}*2.9*10^{-54}}\approx
4*10^{77}.$

For large argument, the Lambert function is roughly a logarithm, and the
relative momentum transfer $\Delta\hat{p}_{\theta}$ can be estimated:

(3.243) $\displaystyle\lim_{x\rightarrow\infty}\frac{W\left(x\right)}{\ln x}$
$\displaystyle=$ $\displaystyle
1,\qquad\Longrightarrow\Delta\hat{p}_{\theta}\approx\frac{1}{\sqrt{\ln\left(e^{\hat{\sigma}_{m}+1}\right)}}\approx\frac{1}{\sqrt{\hat{\sigma}_{m}}}.$

The impact parameter $b_{m}$ is defined to be at the radius of $\sigma_{m}$,
i.e. the maximum distance from the direct path through the target:

(3.244) $\displaystyle\sigma_{m}$ $\displaystyle=$ $\displaystyle\pi
b_{m}^{2},\qquad\hat{\sigma}_{m}=\frac{\sigma_{m}}{\pi
M^{2}}=\frac{b_{m}^{2}}{M^{2}}.$

This allows to express the approximate relative momentum transfer
$\Delta\hat{p}_{m}$ in terms of a given impact parameter $b_{m}$ and target
mass $A$ as:

(3.245) $\displaystyle\Delta\hat{p}_{m}$ $\displaystyle\approx$
$\displaystyle\frac{M}{b_{m}}.$

In SI units, this is:

(3.246) $\displaystyle\Delta\hat{p}_{m}$ $\displaystyle\approx$
$\displaystyle\frac{GM}{c^{2}b_{m}}\approx
7.4*10^{-27}\frac{\mathrm{m}}{\mathrm{kg}}*\frac{M}{b_{m}}.$

As expected, the value is still very small compared to known particle
interactions through the other forces. When the number of interaction partners
becomes in the order of $10^{20}$ or more, or distances are in the order of
the diameter of the Milky Way ($\sim 10^{20}\,\mathrm{m}$) or larger, an
effect may become observable. Given the number of uncertainties on the exact
workings of such scattering in the real world, the effect here is small enough
to not have been noticed yet.

###  3.13. Qualitative prediction for intergalactic medium distribution

Another opportunity for observing effects predicted in this work is over large
distance scales. While individual gravitational scattering between particles
is very weak, the difference as compared to current model assumptions may
become apparent over galactic distance scales. Assuming that the origin of
most highly energized particles and gas in the universe is from within
galaxies, increased backscattering at higher energies as predicted in the
calculations here should - qualitatively - lead to a distribution of
intergalactic gas that is hotter and denser near galaxies as compared to
theoretical model predictions. Direct measurement is difficult, as such gas
typically does not radiate by itself (keywords include e.g. “warm-hot
intergalactic medium”, “intracluster medium”, “circumgalactic enrichment”). In
recent years, observation of absorption lines in the spectrum of remote
quasars (the “Lyman-$\alpha$ forest”) has become a powerful tool to constrain
and tune theoretical models of galactic development. It is envisioned that
within the next decade or two it should be possible to make quantitative
comparisons between these observations and model predictions.

## Acknowledgments

With many thanks to Michael J. Duff, Nichol Furey, and John Huerta for helpful
discussion and advice, and Wolfram Alpha for help with series approximations.
A special thanks to the organizers of the 4th Mile High Conference on
Nonassociative Mathematics (2017), University of Denver, CO, during which the
seeds for this work were planted.

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