# On the relativistic quantum mechanics of a photon between two electrons in
1+1 dimensions

Lawrence Frolov, Samuel E. Leigh, and A. Shadi Tahvildar-Zadeh

Department of Mathematics
Rutgers, The State University of New Jersey
110 Frelinghuysen Rd., Piscataway, NJ 08854, USA

###### Abstract

A Lorentz-covariant system of wave equations is formulated for a quantum-
mechanical three-body system in one space dimension, comprised of one photon
and two identical massive spin one-half Dirac particles, which can be thought
of as two electrons (or alternatively, two positrons). Manifest covariance is
achieved using Dirac’s formalism of multi-time wave functions, i.e, wave
functions
$\Psi(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})$
where
$\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}}$
are generic spacetime events of the photon and two electrons respectively.
Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths
boundary condition at the coincidence submanifolds
$\\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}_{1}}\\}$ and
$\\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}_{2}}\\}$ compatible with
conservation of probability current. The corresponding initial-boundary value
problem is shown to be well-posed under the additional assumption of anti-
symmetry given by the Pauli exclusion principle, and a closed-form solution to
the ensuing coupled system of Klein-Gordon and transport equations is given.

##  1 Introduction

> The Compton effect, at its discovery, was regarded as a simple collision of
> two bodies, and yet the detailed discussion at the present time involves the
> idea of the annihilation of one photon and the simultaneous creation of one
> among an infinity of other possible ones. We would like to be able to treat
> the effect as a two-body problem, with the scattered photon regarded as the
> same individual as the incident, in just the way we treat the collisions of
> electrons.
>
> C G. Darwin, Notes on the Theory of Radiation” (1932)

Despite it being over 90 years since the words above appeared in print, it is
still true that ”the detailed discussion [of the Compton effect] at the
present time involves the idea of annihilation of one photon and the
simultaneous creation of one among an infinity of others;” see [5, 4, 3, 2,
1]. Not only has Darwin’s goal, of treating electrons and photons on an equal
footing within a quantum-mechanical framework of a fixed number of particles,
remained elusive, but modern quantum-field theorists have even come to the
conclusion that “a relativistic quantum theory of a fixed number of particles,
is an impossibility;” [17].

The impossible was however done in [7] in the $1+1$ dimensional setting, where
the authors were able to successfully couple the relativistic quantum-
mechanical photon wave equation of [6] with Dirac’s relativistic quantum-
mechanical electron wave equation in a Lorentz-covariant manner to accomplish
Darwin’s goal: to “treat the [Compton] effect as a two-body problem, with the
scattered photon regarded as the same individual as the incident, in just the
way we treat the collisions of electrons”. They did not do this by studying a
quantization of the Maxwell-Dirac system, since Maxwell’s equations in $1+1$
dimensions do not describe any electromagnetic radiation that can be
quantized. Instead, they worked in Dirac’s manifestly Lorentz covariant
formalism of multi-time wave functions, with the Compton interaction induced
on the two body wave function
$\Psi^{(2)}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})$ by imposing a
suitable boundary condition on it at the subset of co-incident events
$\\{\textbf{x}_{\text{e}}=\textbf{x}_{\text{ph}}\\}$.

In this paper we see how, after reformulating their results in terms of
Feynman-like diagrams111As we will see, the main difference between these and
the traditional Feynman diagrams is that the vertices in our diagrams have
degree 4, not 3., that boundary condition induces a local pair interaction
between the electron and the photon which is worthy of being called ”Compton
Scattering in $1+1$ dimensions”. Of course, a $1+1$ dimensional treatment of
the Compton effect is too simplistic to allow a comparison with empirical
data, so our analysis of this toy model should only be seen as a proof of
concept. We will see that after enough time has passed for the photon to
interact with the electron, the two-body wave function becomes a sum of two
types of diagrams: diagrams where the electron and photon freely propagate,
and diagrams where the pair become incident before freely propagating:

$\Psi^{(2)}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})$ $=$ ${+}\quad$

The main purpose of this paper is to extend the results of [7] to the case of
two electrons interacting solely through their individual contact interactions
with a mediating photon. We expect that placing a photon between the two
freely propagating electrons will produce an effective interaction along the
electron’s light cones as the photon bounces back and forth between them.
However, allowing the photon to bounce arbitrarily between the two massive
electrons has a serious consequence. So long as the electrons remain
sufficiently close together throughout their contact interactions, any number
of bounces can occur in a finite time. Since our wave function must be a
super-position over all possible interactions, it becomes an infinite sum of
diagrams of the following form222Note that in this and future diagram sums
like it, one diagram represents also the ones symmetric to it under the
exchange of label for the two electrons.

$\Psi^{(3)}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})$
$=$ ${+}\quad$ ${+}\quad$ ${+}$

${+}\quad$ ${+}\quad$ ${+}\quad$ ${+}...$

Figure 1: The photon could have bounced arbitrarily many times between the
electrons before the three propagated to their final destinations.

This makes it effectively impossible to write down a closed-form solution for
the three-body wave function evolution, in contrast to what was done for the
electron-photon wave function in [7]. Worse, it is not clear that this
infinite sum converges in some appropriate function space.

Our goal here is to take one more step in the direction of a fully rigorous
and manifestly Lorentz-covariant quantum theory that incorporates creation,
annihilation, and scattering of any number of photons and electrons in one
space dimension. The following is an informal statement of the main result of
this paper (for the precise formulation, see Theorem 5.3.)

######  Theorem 1.1.

Let $\Psi_{\epsilon}$ denote the truncated sum where one discards all diagrams
in which the photon bounces off one electron and then the other in less than
$\epsilon$ time. Then $\lim_{\epsilon\to 0}\Psi_{\epsilon}$ exists in $L^{2}$,
and this limit returns the dynamics we set out to solve for.

######  Remark 1.1.

Although there is no $1+1$ dimensional toy QED for us to compare our $1+1$
dimensional relativistic toy electron-photon quantum mechanics with, our
approach is deeply related to a quantum field-theoretical formulation.
Historically there is a direct line from Dirac’s multi-time formalism to the
QFT formalism of Schwinger and Tomonaga, see [11]. To see this connection, let
$\hat{\Psi}$ stand for the Heisenberg field operators, $\ket{0}$ denote the
zero-particle state and $\ket{\Psi_{0}}$ the Heisenberg state vector. Then,
for all spacelike configurations $(\textbf{x}_{1},...,\textbf{x}_{N})$ of any
number $N$ of particles, the $N$-time $N$-body wave function is given by
(neglecting spin)

$\Psi^{(N)}(\textbf{x}_{1},...,\textbf{x}_{N})=\frac{1}{\sqrt{N!}}\braket{0}{\hat{\Psi}(\textbf{x}_{1})...\hat{\Psi}(\textbf{x}_{N})}{\Psi_{0}}.$
(1.1)

So, the multi-time picture should be regarded as a covariant particle-position
representation of the quantum state. Furthermore, for the quantum field-
theoretical dynamics, the particle creation/annihilation formalism requires
working with wave functions that belong to the multi-time analog of a Fock
space, represented by the sequence

$\Psi=(\Psi^{(0)},\Psi^{(1)},\Psi^{(2)},...)$ (1.2)

where each $\Psi^{(N)}$ is an $N$-particle multi-time wave function as above.
The dynamics for the multi-time wave functions then typically mix the
different particle numbers in the interaction terms, i.e, the right hand side
of the abstract Schrödinger equation

$i\partial_{t_{k}}\Psi^{(N)}=(H_{k}\Psi)^{(N)}$ (1.3)

generally contains $\Psi^{(M)}$ for $M\neq N$. This quantum-field formalism is
usually applied with all times synchronized, cf. [15]; but recently a multi-
time formulation has been achieved for certain quantum field theories
(essentially those with local interactions and finite propagation speed), see
[10].

Unique solvability of the system of evolution equations (1.3) requires
imposing suitable boundary conditions at the subset of co-incident events
$\\{\textbf{x}_{j}=\textbf{x}_{k}\\}$. We refer to the set of coincidence
points as an intrinsic boundary, because it is system-intrinsic and not
determined by some extraneous constraint (such as a container wall). In
particular, the quantum field-theoretical creation/annihilation formalism can
be implemented by formulating suitable intrinsic boundary conditions on the
$N$-body multi-time wave functions which involve, in addition to such boundary
points, interior points of the $N-1$ body wave functions, see [14, 13] for
single time wave functions. This has led to a notion of interior-boundary
conditions. For an extension to multi-time wave functions, see [9]. In this
framework, our quantum-mechanical approach can then be viewed as a special
case of the quantum field-theoretical approach, since we restrict our
attention to boundary conditions in which the $3$-body dynamics are disjoint
from all other $M$-body wave functions, so that we only have scattering.

The rest of this paper is structured as follows:
Section $2$ collects the basic mathematical ingredients needed to formulate
our model.
In section $3$ we define the three-time three-body wave function and derive
the interaction-inducing intrinsic boundary condition from probability
conservation.
Section $4$ reviews the results of [7] regarding the multi-time dynamics of a
photon-electron system, and reformulates their results in terms of Feynman
diagrams.
In Section $5$ we use said Feynman diagram formulation of photon-electron
interactions to study the multi-time dynamics for a photon between two
electrons.
In Section $6$ we close with an outlook on open questions left for future
work.

##  2 Preliminaries

For any $d\in\mathbb{N}$, we let $\boldsymbol{\eta}=(\eta_{\mu\nu})$ denote
the Minkowski metric on Minkowski spacetime $\mathbb{R}^{1,d}$:

$\boldsymbol{\eta}=\boldsymbol{\eta}^{-1}=\text{diag(1,-1,...,-1)}.$ (2.1)

For $d=1$, the complexified spacetime algebra $\mathcal{A}$ is isomorphic to
the algebra of $2\times 2$ complex matrices
$\mathcal{A}:=Cl_{1,1}(\mathbb{R})\cong M_{2}(\mathbb{C})$. We take our basis
to be $\\{\mathbb{1},\gamma^{0},\gamma^{1},\gamma^{0}\gamma^{1}\\}$ where

$\mathbb{1}\coloneqq\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\hskip
5.69046pt\gamma^{0}\coloneqq\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\hskip
5.69046pt\gamma^{1}\coloneqq\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$ (2.2)

satisfy the Clifford Algebra relations

$\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2\eta^{\mu\nu}\mathbb{1}.$
(2.3)

###  2.1 Electron wave function and Dirac Equation

For a single electron in 1+1 dimensions, the wave function for an electron is
given by a rank-one spinor field on the electron’s configuration space-time
$\mathbb{R}^{1,1}_{\text{e}}$, equipped with coordinates
$(t_{\text{e}},s_{\text{e}})$:

$\Psi_{\text{e}}=\begin{pmatrix}\psi_{-}\\\ \psi_{+}\end{pmatrix}.$ (2.4)

As a rank-one spinor field, the components of $\Psi_{e}$ transform under the
action of $O(1,1)$ as

$\psi_{\pm}\xrightarrow{\Lambda}e^{\mp
a/2}\psi_{\pm},\quad\psi_{\pm}\xrightarrow{\mathbf{P}}\psi_{\mp},\quad\psi_{\pm}\xrightarrow{\mathbf{T}}\psi_{\mp}^{*}$
(2.5)

where $\Lambda$ refers to proper Lorentz transformations, $P$ space
reflections, $T$ time reflections, and ∗ refers to the complex conjugate
operator.

The electron wave function satisfies the massive Dirac equation

$-i\hbar\gamma^{\mu}\frac{\partial}{\partial
x^{\mu}_{\text{e}}}\Psi_{\text{e}}+m_{\text{e}}\Psi_{\text{e}}=0$ (2.6)

where $m_{e}$ is the mass of the electron. In this paper, we will primarily
concern ourselves with solving initial-value problems for such equations.
Given initial data
$\mathring{\Psi}:\mathbb{R}^{1,1}_{\text{e}}\to\mathbb{C}^{2}$, the solution
to the Dirac equation IVP

$\left\\{\begin{array}[]{rcl}-i\hbar\gamma^{\mu}\frac{\partial}{\partial
x^{\mu}_{\text{e}}}\Psi_{\text{e}}+m_{\text{e}}\Psi_{\text{e}}&=&0\\\
\Psi_{\text{e}}\big{|}_{t_{\text{e}}=0}&=&\mathring{\Psi}_{\text{e}}\end{array}\right.$
(2.7)

can be found by recalling that each component of $\Psi$ satisfies a Klein-
Gordon equation. Solving the Dirac IVP becomes equivalent to solving the
Klein-Gordon IVP

$\left\\{\begin{array}[]{rcl}\Box_{\text{e}}\psi_{\pm}+\omega^{2}\psi_{\pm}&=&0\\\
\psi_{\pm}\big{|}_{t_{\text{e}}=0}&=&A_{\pm}(s_{\text{e}})\\\
\partial_{t_{\text{e}}}\psi_{\pm}\big{|}_{t_{\text{e}}=0}&=&B_{\pm}(s_{\text{e}})\end{array}\right.$
(2.8)

where the initial data $B_{\pm}$ is derived from the Dirac equation and will
depend on $A_{\mp}$ and $\partial_{1}A_{\pm}$. We use this to derive the time
evolution operator for the Dirac IVP, and write the solution as

$\Psi_{\text{e}}(t_{\text{e}},s_{\text{e}})=\mathcal{E}(t_{\text{e}},s_{\text{e}})\mathring{\Psi}_{\text{e}}$
(2.9)

The electronic time evolution operator
$\mathcal{E}(t_{\text{e}},s_{\text{e}})$ is defined explicitly by equation
(A.5) in the appendix.

Also relevant is a discussion of the Goursat problem for the Dirac equation.
The Goursat problem arises when one seeks to solve a Klein-Gordon equation
with data specified on two future null rays emanating from some point. So long
as this data is consistent with the Dirac equation, it will produce a solution
for the Dirac equation as well. Without loss of generality, we take the point
to be $(0,0)$, and write the Goursat problem as

$\begin{cases}\Box_{\text{e}}\Psi+\omega^{2}\Psi&=0\quad\text{for
}|s_{\text{e}}|<t_{\text{e}}\\\ \Psi(b,b)&=F(b)\\\
\Psi(c,-c)&=G(c)\end{cases}$ (2.10)

Written explicitly in the Appendix, the solution has us multiply the functions
$F$ and $G$ by certain propagators and integrate over the intersection of the
null-rays with the backwards light-cone of $(t_{\text{e}},s_{\text{e}})$:

Figure 2: Data along lines $\mathcal{B}$ and $\mathcal{C}$ are sufficient to
solve the Goursat problem in $\Omega$

$\Psi\big{|}_{\Omega}=G^{R}(t_{\text{e}},s_{\text{e}})F+G^{L}(t_{\text{e}},s_{\text{e}})G,$
(2.11)

where $G^{R}$ and $G^{L}$ are the right and left time evolution operators of
the Goursat problem, defined explicitly by equations (A.10), (A.11) in the
Appendix.

###  2.2 The Photon-KTZ Equation

For a single photon in 1+1 dimensions, the wave function for a photon is shown
in [6] to be given by a rank two bi-spinor field defined on the photon’s
configuration space-time $\mathbb{R}^{1,1}_{\text{ph}}$, equipped with
coordinates $(t_{\text{ph}},s_{\text{ph}})$:

$\Psi_{\text{ph}}=\begin{pmatrix}0&\chi_{-}\\\ \chi_{+}&0\end{pmatrix}.$
(2.12)

As a rank-two bi-spinor field, the components of $\Psi_{\text{ph}}$ transform
under the action of $O(1,1)$ as

$\chi_{\pm}\xrightarrow{\Lambda}e^{\mp
a}\chi_{\pm},\quad\chi_{\pm}\xrightarrow{\mathbf{P}}\chi_{\mp},\quad\chi_{\pm}\xrightarrow{\mathbf{T}}\chi_{\mp}^{*}.$
(2.13)

In 1+1 dimensions the KTZ photon wave equation reduces to the massless Dirac
equation

$-i\hbar\gamma^{\mu}\frac{\partial}{\partial
x^{\mu}_{\text{ph}}}\Psi_{\text{ph}}=0.$ (2.14)

Given initial data $\mathring{\Psi}_{\text{ph}}$, the solution to the
corresponding IVP is given by transports in each component

$\Psi_{\text{ph}}(t_{\text{ph}},s_{\text{ph}})=\mathcal{P}(t_{\text{ph}},s_{\text{ph}})\mathring{\Psi}_{\text{ph}}:=\begin{pmatrix}0&\mathring{\chi}_{-}(s_{\text{ph}}-t_{\text{ph}})\\\
\mathring{\chi}_{+}(s_{\text{ph}}+t_{\text{ph}})&0\end{pmatrix}.$ (2.15)

$\chi_{+}$ is transported along left moving null rays; while $\chi_{-}$ is
transported along right moving null rays.

####  2.2.1 Photon probability current

Similarly to the electron, the photonic wave equation admits a conserved
probability current. Given any Killing field $X$ of Minkowski space-time, the
manifestly covariant current

$j^{\mu}_{X}:=\text{tr}(\overline{\Psi}_{\text{ph}}\gamma^{\mu}\Psi_{\text{ph}}\gamma(X))$
(2.16)

is conserved, i.e

$\partial_{\mu}j^{\mu}_{X}=0.$ (2.17)

Here $\gamma(X)=\gamma^{\mu}X_{\mu}$.
In [6] it is proven that when $X$ is causal and future-directed, then so is
$j_{X}$, i.e $\eta(j_{X},j_{X})>0$ and $j^{0}_{X}\geq 0$. Thus, this satisfies
the appropriate version of the Born rule. However, there is some debate about
which time-like Killing field $X$ should be used to define the probability
current for a photon. Setting $X=\partial_{t}$, the time-like unit vector of
the ”lab frame”, returns a probability current formula that looks similar to
the well-known conserved current for a spin-half Dirac particle. However,
there are compelling reasons for why a quantum probability current for a
particle should only depend on the wave function of the particle, not on some
external “observer.” A proposal for an $X$ that only depends on the initial
wave function of the system was given in [6]. For a critique of theories with
observer-dependent probability currents, see Struyve et al. [12] and Tumulka
[16] (and references therein).

Since in this paper we are only interested in solving initial value problems
where the wave function is specified on some Cauchy hyperplane given as the
zero level-set of a time function $t$ defined on the Minkowski spacetime, we
will side-step these arguments by setting up and solving the initial value
problem in the fixed frame where $X=\partial_{t}$.

##  3 The Three Body El-Ph-El System

###  3.1 The multi-time wave function

Let
$\mathcal{M}=\mathbb{R}_{\text{ph}}^{1,1}\times\mathbb{R}_{\text{e}_{1}}^{1,1}\times\mathbb{R}_{\text{e}_{2}}^{1,1}$
be the configuration spacetime, where, as will be the case in the rest of the
paper, the subscripts ${}_{\text{ph}}$, ${}_{\text{e}_{1}}$,
${}_{\text{e}_{2}}$ refer to the photon, the first electron, and the second
electron, respectively. We denote by
$(x_{\text{ph}}^{\mu},x_{\text{e}_{1}}^{\nu},x_{\text{e}_{2}}^{\kappa})$ where
$\mu,\nu,\kappa\in\\{0,1\\}$ a global system of rectangular coordinates on
$\mathcal{M}$. The multi-time wave function of the electron-photon-electron
system in one space dimension is a mapping
$\Psi:\mathcal{S}\subset\mathbb{R}^{1,1}_{\text{ph}}\times\mathbb{R}^{1,1}_{\text{e}_{1}}\times\mathbb{R}^{1,1}_{\text{e}_{2}}\to\mathbb{B}$,
where $\mathcal{S}$ is the subset of pairwise spacelike configurations

$\mathcal{S}:=\left\\{(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})\in\mathbb{R}^{1,1}_{\text{ph}}\times\mathbb{R}^{1,1}_{\text{e}_{1}}\times\mathbb{R}^{1,1}_{\text{e}_{2}}:\eta(\textbf{x}_{\text{ph}}-\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{ph}}-\textbf{x}_{\text{e}_{1}})<0,\quad\eta(\textbf{x}_{\text{ph}}-\textbf{x}_{\text{e}_{2}},\textbf{x}_{\text{ph}}-\textbf{x}_{\text{e}_{2}})<0\right\\}.$
(3.1)

Here $\eta=\mbox{diag}\,(1,-1)$ is the Minkowski metric on $\mathbb{R}^{1,1}$
and $\mathbb{B}$ is the vector bundle over $\mathcal{S}$ whose fibers are
isomorphic to the set of anti-diagonal matrices in $M_{2}(\mathbb{C})$
tensored with the vector space $\mathbb{C}^{2}$ twice. We take a basis for
this bundle by taking tensor products of the basis elements for the photon
wave function $E_{\varsigma}$ with the basis elements for the Dirac wave
function $e_{\varsigma}$. In this basis,$\Psi$ can be expanded as

$\Psi(\mathbf{x}_{\text{ph}},\mathbf{x}_{\text{e}_{1}},\mathbf{x}_{\text{e}_{2}})=\sum_{\varsigma_{0},\varsigma_{1},\varsigma_{2}\in\\{-,+\\}}\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}(\mathbf{x}_{\text{ph}},\mathbf{x}_{\text{e}_{1}},\mathbf{x}_{\text{e}_{2}})E_{\varsigma_{0}}\otimes
e_{\varsigma_{1}}\otimes e_{\varsigma_{2}}$ (3.2)

where the eight components given by
$\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}$ are complex valued functions
on $\mathcal{M}$. These transform as

$\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}\xrightarrow{\Lambda}e^{-a(\varsigma_{0}1+\varsigma_{1}\frac{1}{2}+\varsigma_{2}\frac{1}{2})}\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}},\hskip
5.69046pt\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}\xrightarrow{P}\psi_{\overline{\varsigma}_{0}\overline{\varsigma}_{1}\overline{\varsigma}_{2}},\hskip
5.69046pt\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}\xrightarrow{T}\psi_{\overline{\varsigma}_{0}\overline{\varsigma}_{1}\overline{\varsigma}_{2}}^{*}$
(3.3)

where $\Lambda$ denotes the proper Lorentz transformations, $P$ space
reflections and $T$ time reflections. $\overline{\varsigma}$ denotes the sign
opposite of $\varsigma$, and ∗ denotes the complex conjugation operator
extended in the natural way for tensor space.

###  3.2 The multi-time wave equations

Let

$\gamma_{\text{ph}}^{\mu}\coloneqq\gamma^{\mu}\otimes\mathbbm{1}\otimes\mathbbm{1},\hskip
28.45274pt\gamma_{\text{e}_{1}}^{\nu}\coloneqq\mathbbm{1}\otimes\gamma^{\nu}\otimes\mathbbm{1}\hskip
28.45274pt\gamma_{\text{e}_{2}}^{\kappa}\coloneqq\mathbbm{1}\otimes\mathbbm{1}\otimes\gamma^{\kappa}$
(3.4)

and

$D_{\text{ph}}\coloneqq\gamma_{\text{ph}}^{\mu}\partial_{x_{\text{ph}}^{\mu}}\hskip
36.98866ptD_{\text{e}_{1}}\coloneqq\gamma_{\text{e}_{1}}^{\nu}\partial_{x_{\text{e}_{1}}^{\nu}}\hskip
36.98866ptD_{\text{e}_{2}}\coloneqq\gamma_{\text{e}_{2}}^{\kappa}\partial_{x_{\text{e}_{2}}^{\kappa}},$
(3.5)

where repeated indices imply Einstein summation.

The multi-time dynamics of the electron-photon-electron wave function are then
defined by the following:

  1. 1.

The free multi-time equations on $\mathcal{S}$

$\displaystyle\left\\{\begin{array}[]{rcl}-i\hbar D_{\text{ph}}\Psi&=&0\\\
-i\hbar D_{\text{e}_{1}}\Psi+m_{\text{e}}\Psi&=&0\\\ -i\hbar
D_{\text{e}_{2}}\Psi+m_{\text{e}}\Psi&=&0\end{array}\right..$ (3.9)

  2. 2.

Initial data specified on the surface

$\mathcal{I}\coloneqq\left\\{(t_{\text{ph}},s_{\text{ph}},t_{\text{e}_{1}},s_{\text{e}_{1}},t_{\text{e}_{2}},s_{\text{e}_{2}})\in\overline{\mathcal{S}}:t_{\text{ph}}=t_{\text{e}_{1}}=t_{\text{e}_{2}}=0\right\\},$
(3.10)

namely

$\Psi(0,s_{\text{ph}},0,s_{\text{e}_{1}},0,s_{\text{e}_{2}})=\mathring{\Psi}(s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}}).$
(3.11)

  3. 3.

Boundary conditions on the coincidence sets $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$. These implement contact interactions, and their exact form
will be determined in the next section by analyzing the conditions for
probability conservation. Since we are interested in studying the dynamics of
a photon between two electrons, we will restrict our attention to two disjoint
subsets of $\mathcal{S}$

$\displaystyle\mathcal{S}_{1}\coloneqq\left\\{(t_{\text{ph}},s_{\text{ph}},t_{\text{e}_{1}},s_{\text{e}_{1}},t_{\text{e}_{2}},s_{\text{e}_{2}})\in\mathcal{S}:s_{\text{e}_{1}}<s_{\text{ph}}<s_{\text{e}_{2}}\right\\}$
(3.12)
$\displaystyle\mathcal{S}_{2}\coloneqq\left\\{(t_{\text{ph}},s_{\text{ph}},t_{\text{e}_{1}},s_{\text{e}_{1}},t_{\text{e}_{2}},s_{\text{e}_{2}})\in\mathcal{S}:s_{\text{e}_{2}}<s_{\text{ph}}<s_{\text{e}_{1}}\right\\},$
(3.13)

and the boundary conditions will be linear relations between limits along
events in $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ towards the sets
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ where

$\begin{split}\mathcal{C}_{1}\coloneqq\left\\{(t_{\text{ph}},s_{\text{ph}},t_{\text{e}_{1}},s_{\text{e}_{1}},t_{\text{e}_{2}},s_{\text{e}_{2}})\in\partial\mathcal{S}:s_{\text{e}_{1}}=s_{\text{ph}},\hskip
5.69046ptt_{\text{e}_{1}}=t_{\text{ph}}\right\\}\\\
\mathcal{C}_{2}\coloneqq\left\\{(t_{\text{ph}},s_{\text{ph}},t_{\text{e}_{1}},s_{\text{e}_{1}},t_{\text{e}_{2}},s_{\text{e}_{2}})\in\partial\mathcal{S}:s_{\text{e}_{2}}=s_{\text{ph}},\hskip
5.69046ptt_{\text{e}_{2}}=t_{\text{ph}}\right\\}\end{split}$ (3.14)

###  3.3 Conserved currents and the boundary condition

For a pure product state
$\Psi=\psi_{\text{ph}}\otimes\psi_{\text{e}_{1}}\otimes\psi_{\text{e}_{2}}$,
we define its Dirac adjoint $\overline{\Psi}$ as

$\overline{\Psi}\coloneqq\overline{\psi_{\text{ph}}}\otimes\overline{\psi_{\text{e}_{1}}}\otimes\overline{\psi_{\text{e}_{2}}}=\gamma^{0}\psi_{\text{ph}}^{\dagger}\gamma^{0}\otimes\psi_{\text{e}_{1}}^{\dagger}\gamma^{0}\otimes\psi_{\text{e}_{2}}^{\dagger}\gamma^{0}.$
(3.15)

For a general three-time, three-body wave function we extend this definition
by linearity. We take the joint probability current of our three-time wave
function to be the tensor product of the three particle’s individual
probability currents. Let $X$ be a time-like Killing field on
$\mathbb{R}^{1,1}$, we define the joint probability tensor
$j^{\mu\nu\kappa}_{X}$ as

$j^{\mu\nu\kappa}_{X}[\Psi]\coloneqq\frac{1}{4}\text{tr}_{\text{ph}}\left\\{\overline{\Psi}\gamma_{\text{ph}}^{\mu}\gamma_{\text{e}_{1}}^{\nu}\gamma_{\text{e}_{2}}^{\kappa}\Psi\gamma_{\text{ph}}(X)\right\\},$
(3.16)

where $\text{tr}_{ph}=\text{tr}\otimes\mathbbm{1}\otimes\mathbbm{1}$ is the
operation of taking the trace of the photonic component. It follows (see [7])
that

######  Proposition 3.1.

Let $\Psi$ be a $C^{1}$ solution of (3.9) and let $X$ be any constant vector
field on $\mathbb{R}^{1,1}$. Then the current $j_{X}$ is jointly conserved,
that is:

$\begin{split}\begin{cases}\partial_{x_{\text{ph}}^{\mu}}j^{\mu\nu\kappa}_{X}&=0\hskip
5.69046pt\text{for}\hskip 5.69046pt\nu,\kappa\in\\{0,1\\}\\\
\partial_{x_{\text{e}_{1}}^{\nu}}j^{\mu\nu\kappa}_{X}&=0\hskip
5.69046pt\text{for}\hskip 5.69046pt\mu,\kappa\in\\{0,1\\}\\\
\partial_{x_{\text{e}_{2}}^{\kappa}}j^{\mu\nu\kappa}_{X}&=0\hskip
5.69046pt\text{for}\hskip 5.69046pt\mu,\nu\in\\{0,1\\}.\end{cases}\end{split}$
(3.17)

######  Theorem 3.1.

Let the vector field $X$ be time-like. Then the only Poincaré invariant class
of boundary conditions for the components of $\Psi$ that conserves the total
probability

$\int_{\Sigma^{3}\cap\mathcal{S}_{i}}\omega_{j}=\int_{\Sigma^{3}\cap\mathcal{S}_{i}}\sum_{\mu,\nu,\kappa\in\\{0,1\\}}(-1)^{\mu+\nu+\kappa}j_{X}^{\mu\nu\kappa}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})dx_{\text{ph}}^{1-\mu}\wedge
dx_{\text{e}_{1}}^{1-\nu}\wedge dx_{\text{e}_{2}}^{1-\kappa}$ (3.18)

independently of Cauchy surface $\Sigma$ is:

$\begin{split}&\lim_{\varepsilon\to
0}\psi_{-+\varsigma_{2}}(t,s\pm\varepsilon,t,s\mp\varepsilon,\textbf{x}_{\text{e}_{2}})=\lim_{\varepsilon\to
0}e^{\pm
i\theta_{1}}\sqrt{\frac{X^{0}+X^{1}}{X^{0}-X^{1}}}\psi_{+-\varsigma_{2}}(t,s\pm\varepsilon,t,s\mp\varepsilon,\textbf{x}_{\text{e}_{2}})\\\
&\lim_{\varepsilon\to
0}\psi_{+\varsigma_{1}-}(t,s\mp\varepsilon,\textbf{x}_{\text{e}_{1}},t,s\pm\varepsilon)=\lim_{\varepsilon\to
0}e^{\pm
i\theta_{2}}\sqrt{\frac{X^{0}-X^{1}}{X^{0}+X^{1}}}\psi_{+\varsigma_{1}-}(t,s\mp\varepsilon,\textbf{x}_{\text{e}_{1}},t,s\pm\varepsilon),\end{split}$
(3.19)

for constant phases $\theta_{1},\theta_{2}\in[0,2\pi)$. Taking the additional
anti-symmetry condition imposed by the Pauli exclusion, i.e.

$\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})=-\psi_{\varsigma_{0}\varsigma_{2}\varsigma_{1}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{2}},\textbf{x}_{\text{e}_{1}})$
(3.20)

implies that $\theta_{1}=\theta_{2}$.

We now have all the tools needed to set up an initial-boundary value problem
for the dynamics of our three body wave function. But before we do so, let us
review the results of [7] concerning the dynamics of a two-body electron-
photon wave function.

##  4 IBVP for two body wave function

In this section we review the results of [7] for the initial-boundary value
problem (IBVP) for the two-time two-body wave function. This IBVP consists of
the free two-time equations on a subset of the two-particle configuration
space-time

$\displaystyle\mathcal{S}_{1}\coloneqq\left\\{(t_{\text{ph}},s_{\text{ph}},t_{\text{e}},s_{\text{e}})\in\mathbb{R}^{(1,1)}_{\text{ph}}\times\mathbb{R}^{(1,1)}_{\text{e}}:\eta(\textbf{x}_{\text{ph}}-\textbf{x}_{\text{e}},\textbf{x}_{\text{ph}}-\textbf{x}_{\text{e}})<0,\quad
s_{\text{ph}}<s_{\text{e}}\right\\}$ (4.1)

combined with a probability conserving boundary condition.

######  Theorem 4.1.

[7, 8] Let the vector field $X$ be time-like. Then the only Poincaré invariant
class of boundary conditions for the components of $\Psi$ that preserves the
total probability in $\mathcal{S}_{1}$ is:

$\lim_{\varepsilon\to
0}\psi_{+-}(t,s\mp\varepsilon,t,s\pm\varepsilon)=\lim_{\varepsilon\to 0}e^{\pm
i\theta}\sqrt{\frac{X^{0}-X^{1}}{X^{0}+X^{1}}}\psi_{-+}(t,s\mp\varepsilon,t,s\pm\varepsilon).$
(4.2)

for a constant $\theta\in[0,2\pi)$.

For simplicity, we study the IBVP in the fixed Lorentz frame where
$X=\partial_{t}$.

######  Theorem 4.2.

[7, 9] Let $\theta\in[0,2\pi)$. Let $\mathcal{S}_{1}$ denote the set of space-
like configurations in $\mathcal{M}$, as was defined in (4.1), let
$\mathcal{C}\subset\partial\mathcal{S}_{1}$ be the coincidence set defined by
$\\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}}\\}$, and let $\mathcal{I}$ be
the initial surface. Let the function
$\mathring{\Psi}:\mathcal{I}\rightarrow\mathbb{C}^{4}$ be $C^{1}$ and
compactly supported in the half-space
$\mathcal{I}\cap\overline{\mathcal{S}_{1}}$, and suppose that the initial-
value function also satisfies the boundary conditions given by (4.2). Then the
following initial-boundary value problem for the wave function
$\Psi:\mathcal{M}\rightarrow\mathbb{C}^{4}$,

$\left\\{\begin{array}[]{rclr}-i\hbar D_{\text{ph}}\Psi&=&0&\\\ -i\hbar
D_{\text{e}}\Psi+m_{\text{e}}\Psi&=&0&\text{in }\mathcal{S}_{1}\\\
\Psi&=&\mathring{\Psi}&\text{on }\mathcal{I}\\\
\psi_{+-}&=&e^{i\theta}\psi_{-+}&\text{on }\mathcal{C}\end{array}\right.$
(4.3)

has a unique global-in-time solution that is supported in
$\overline{\mathcal{S}_{1}}$, and depends continuously on the initial data
$\mathring{\Psi}$.

###### Proof.

For a fixed-point argument of existence given by [9] we direct the reader to
the Appendix. The fixed-point argument is simpler than the proof given in [7],
but it is not constructive. The explicit solution constructed in [7], as we
shall see later, can be elegantly represented using Feynman-like diagrams. On
the other hand the proof given in [9] applies to a much larger class of
initial-boundary value problems, which will be useful when we generalize to
the case of a photon between two electrons. ∎

###  4.1 Construction of Solution

Now following [7], we will prove that $\Psi$ has a unique global-in-time
solution by writing it down. Firstly, we note that by finite speed of
propagation any data evolving sufficiently far from the boundary $\mathcal{C}$
is not influenced by the boundary condition, and is only governed by the free
multi-time equations. So, we begin by decomposing $\mathcal{S}_{1}$ into two
disjoint parts

$\mathcal{S}_{1}=\mathcal{F}\cup\mathcal{N}$ (4.4)

via

$\mathcal{F}:=\left\\{(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})\in\mathcal{S}_{1}:s_{\text{ph}}+t_{\text{ph}}\leq
s_{\text{e}}-t_{\text{e}}\right\\},$ (4.5)
$\mathcal{N}:=\left\\{(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})\in\mathcal{S}_{1}:s_{\text{ph}}+t_{\text{ph}}>s_{\text{e}}-t_{\text{e}}\right\\},$
(4.6)

and give distinct solution formulas in each region. Data evolving on
$\mathcal{F}$ is governed only by the free equations, and its solution formula
will be identical to the solution formula for a freely evolving two-body
system. This formula can easily be written in terms of the time evolution
operators of photonic transport and electronic Dirac equation. Let
$\mathcal{P}(t_{\text{ph}},s_{\text{ph}})$ denote the photonic evolution
operator defined by equation (2.15), and
$\mathcal{E}(t_{\text{e}},s_{\text{e}})$ denote the electronic evolution
operator defined by equation (A.5) in the Appendix.

######  Proposition 4.1.

([7]) The multi-time evolution operator for the freely evolving two body
system is the tensor product of the two single body time evolution operators.

So, for $(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})\in\mathcal{F}$, the
solution formula for $\Psi$ is given by

$\Psi(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})=\big{(}\mathcal{P}(\textbf{x}_{\text{ph}})\otimes\mathcal{E}(\textbf{x}_{\text{e}})\big{)}\mathring{\Psi}.$
(4.7)

Our goal now is to find a solution formula in $\mathcal{N}$. Fix
$(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})\in\mathcal{N}$. Looking
closely at the explicit free evolution solution formula given by equation
(4.7), we find that this formula is still valid in the near configuration
regime for the $\varsigma_{0}=-$ components, i.e for $\psi_{--}$ and
$\psi_{-+}$. This is because these components transport the photon along
right-moving null-rays, so in $\mathcal{S}_{1}$ this evolution cannot depend
on any data along $\mathcal{C}$. Letting $\Pi_{\varsigma_{0}\varsigma_{1}}$ be
the projection operator sending $\Psi$ to $\psi_{\varsigma_{0}\varsigma_{1}}$,
the solution formula for $\psi_{-\varsigma_{1}}$ on $\mathcal{N}$ is written
as

$\psi_{-\varsigma_{1}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})=\Pi_{-\varsigma_{1}}\big{(}\mathcal{P}(\textbf{x}_{\text{ph}})\otimes\mathcal{E}(\textbf{x}_{\text{e}})\big{)}\mathring{\Psi}.$
(4.8)

To solve for the remaining components, consider the forwards light-cone
emanating from the point $(0,s_{\text{ph}}+t_{\text{ph}})\in\mathbb{R}^{1,1}$.
For all $\textbf{x}_{\text{e}}^{\prime}$ which are on the right edge of this
cone, we have data for
$\psi_{+\varsigma_{1}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}}^{\prime})$
using equation (4.7). Since $\psi_{+\varsigma_{1}}$ each satisfy an electronic
Klein-Gordon equation, we are motivated to write down and solve a Goursat
problem in the electronic variables. For this we will need data along two
lines in configuration space-time

$\mathcal{B}=\left\\{(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}}^{\prime})\in\mathcal{S}_{1}:\textbf{x}_{\text{e}}^{\prime}=(b,s_{\text{ph}}+t_{\text{ph}}+b),\quad
b\in\mathbb{R}\right\\}$ (4.9)
$\mathcal{C}^{\prime}=\left\\{(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}}^{\prime})\in\mathcal{S}_{1}:\textbf{x}_{\text{e}}^{\prime}=(c,s_{\text{ph}}+t_{\text{ph}}-c),\quad
c\in\mathbb{R}\right\\}$ (4.10)

Figure 3: Keeping $\textbf{x}_{\text{ph}}$ fixed, data for
$\textbf{x}_{\text{e}}$ along the lines $\mathcal{B}$ and $\mathcal{C}$ is
sufficient to solve the Goursat problem for all
$\textbf{x}_{\text{e}}\in\Omega$.

To solve for the data along $\mathcal{C}^{\prime}$, we appeal to the fact that
$\varsigma_{0}=+$ components are constant as we transport the photonic
configuration along left moving null rays. Letting
$(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}}^{\prime})$ be an arbitrary
point in $\mathcal{C}^{\prime}$, the photonic transport equations imply

$\psi_{+-}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}}^{\prime})=\psi_{+-}(\textbf{x}_{\text{e}}^{\prime},\textbf{x}_{\text{e}}^{\prime})=e^{i\theta}\psi_{-+}(\textbf{x}_{\text{e}}^{\prime},\textbf{x}_{\text{e}}^{\prime}).$
(4.11)

The second equality of equation (4.11) will be our only application of the
contact boundary condition. Abusing notation slightly, we may say that
$\psi_{+-}\big{|}_{\mathcal{C^{\prime}}}=e^{i\theta}\psi_{-+}\big{|}_{\mathcal{C}}$.

Applying the Goursat problem solution formula (A.8) derived in the appendix,
the solution for $\psi_{+-}$ on near configurations is written as

$\psi_{+-}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})=G^{R}(t_{\text{e}},\tilde{s})\big{(}\psi_{+-}\big{|}_{\mathcal{B}}\big{)}+G^{L}(t_{\text{e}},\tilde{s})\big{(}e^{i\theta}\psi_{-+}\big{|}_{\mathcal{C}}\big{)}$
(4.12)

where $\tilde{s}=s_{\text{e}}-(s_{\text{ph}}+t_{\text{ph}})$. To write this
solution in terms of the initial data, we may plug equations (4.7) and (4.8)
into (4.12) for $\psi_{+-}\big{|}_{\mathcal{B}}$ and
$e^{i\theta}\psi_{-+}\big{|}_{\mathcal{C}}$ respectively.

To solve for the fourth and final component $\psi_{++}$, we plug equation
(4.12) into the sourced transport equation that this component satisfies and
receive

$\psi_{++}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})=\psi_{++}(\textbf{x}_{\text{ph}},0,s_{\text{e}}+t_{\text{e}})-i\omega\int_{0}^{t_{\text{e}}}\psi_{+-}(\textbf{x}_{\text{ph}},\tau,s_{\text{e}}+t_{\text{e}}-\tau)d\tau$
(4.13)

This completes our algorithm to solve for the two-body wave function
everywhere in $\mathcal{S}_{1}$. To save space, we will refer to this general
process as a ”Contact Evolution Operator”
$C_{\theta}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})$.

###  4.2 Feynman-like Diagrams for Photon-Electron Pairs

The evolution of our interacting two-body system can be more elegantly
represented using Feynman diagrams. We found that the evolution of the two-
body wave function for ”Far” configurations is identical to the free
evolution.

$\Psi\big{|}_{\mathcal{F}}$ ${=}$

Figure 4: We sum over all diagrams which depicts the electron and photon
freely propagating.

On configurations $(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})$ which are
”Near”, two of the wave function components no longer satisfy the free
evolution. The $\varsigma_{0}=+$ components become a sum of two terms. The
first is non-interacting data: $\varsigma_{0}=+$ data on $\mathcal{B}$ which
has freely propagated to $(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})$. The
other term is arose purely from the interaction: $\varsigma_{0}=-$ data which
has propagated through $\mathcal{C}$, but was reborn in the form of
$\varsigma_{0}=+$ and then freely propagated to
$(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})$.

$\Psi\big{|}_{\mathcal{N}}$ $=$ ${+}\quad e^{i\theta}$

Figure 5: There are now two diagrams we must sum over: freely propagated data,
and data resulting from a collision.

######  Remark 4.1.

Restricting the two-body wave function to $\mathcal{S}_{1}$ has produced a
contact interaction between the photon and the electron that is worthy of
being called ”Compton Scattering in $1+1$ dimensions”. The electron and the
photon propagate freely at first, until the photon inevitably reaches contact
with the electron at which point the photon is forced to bounce off the
electron and propagate away in the opposite direction. The number of photons
remains fixed during this interaction, with the photon before the incident
being the exact same as the one after, as originally desired by Darwin.

In the following sections, we will investigate how these contact boundary
interactions generalize to the case of one photon between two electrons. To
simplify our proofs we will write our solutions in terms of Feynman-like
diagrams. A diagram in which an electron-photon pair become incident and then
bounce off means that we are applying a contact evolution operator
$C_{\theta}$ in those variables. This means we are setting up and solving a
Goursat problem for the appropriate wave function component, with data on one
null line representing freely propagating data while data on the other is
given by the boundary condition, and thus represents interacting data. We then
plug the Goursat solution into a sourced transport to solve for the other
component.

##  5 IBVP for three-time three-body wave function

In this section we consider the initial-boundary value problem (IBVP) for the
three-time three-body wave function consisting of equations (3.9), (3.19). For
simplicity, we work in the fixed Lorentz frame where $X=\partial_{t}$.

######  Theorem 5.1.

Let $\theta_{1},\theta_{2}\in[0,2\pi)$ be two constant phases. Let
$\mathcal{S}_{1}$ denote the set of space-like configurations in
$\mathcal{M}$, as was defined in (3.12),
$\mathcal{C}_{1}\subset\partial\mathcal{S}_{1}$ and
$\mathcal{C}_{2}\subset\partial\mathcal{S}_{1}$ be the coincidence sets
defined by (3.14), and $\mathcal{I}$ be the initial surface as was defined in
(3.11). Let the function
$\mathring{\Psi}:\mathcal{I}\rightarrow\mathbb{C}^{8}$ be $C^{1}$ and
compactly supported in the half-space
$\mathcal{I}\cap\overline{\mathcal{S}_{1}}$, and suppose that the initial wave
function also satisfies the boundary conditions given by (3.19). Then the
following initial-boundary value problem for the wave function
$\Psi:\mathcal{M}\rightarrow\mathbb{C}^{8}$,

$\left\\{\begin{array}[]{rclr}-i\hbar D_{\text{ph}}\Psi&=&0&\\\ -i\hbar
D_{\text{e}_{1}}\Psi+m_{e}\Psi&=&0&\\\ -i\hbar
D_{\text{e}_{2}}\Psi+m_{\text{e}}\Psi&=&0&\text{in }\mathcal{S}_{1}\\\
\Psi&=&\mathring{\Psi}&\text{on }\mathcal{I}\\\
\psi_{-+\varsigma_{2}}&=&e^{i\theta_{1}}\psi_{+-\varsigma_{2}}&\text{on
}\mathcal{C}_{1}\\\
\psi_{+\varsigma_{1}-}&=&e^{i\theta_{2}}\psi_{-\varsigma_{1}+}&\text{on
}\mathcal{C}_{2}\end{array}\right.$ (5.1)

has a unique global-in-time solution that is supported in
$\overline{\mathcal{S}_{1}}$, is continuous with bounded derivatives, and
depends continuously on the initial data $\mathring{\Psi}$.

###### Proof.

For a fixed-point argument of existence modeled on the one in [9] we direct
the reader to the Appendix. ∎

###  5.1 Construction of Solution

Our attempt to construct an algorithm for $\Psi$ will repeat the exact same
steps as the ones taken for two-body interacting photon-electron wave
function. We begin by splitting up our configuration space-time into relevant
pieces, which are analogues to the ”Far” and ”Near” configurations from the
two-body construction.
Free Configurations: Configurations in $\mathcal{S}_{1}$ for which none of the
backwards light-cones of the particles cross.

$\mathcal{F}:=\left\\{(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})\in\mathcal{S}_{1}:s_{\text{e}_{1}}+t_{\text{e}_{1}}\leq
s_{\text{ph}}-t_{\text{ph}},\quad s_{\text{ph}}+t_{\text{ph}}\leq
s_{\text{e}_{2}}-t_{\text{e}_{2}}\right\\}$ (5.2)

Compton Configurations: Configurations for which at least one the backwards
light cones of the electrons cross the photon’s, but the electron’s cones do
not cross each other.

$\mathcal{X}:=\left\\{(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})\in\mathcal{S}_{1}:s_{\text{ph}}-t_{\text{ph}}<s_{\text{e}_{1}}+t_{\text{e}_{1}}\leq
s_{\text{e}_{2}}-t_{\text{e}_{2}}\quad\text{or}\quad
s_{\text{e}_{1}}+t_{\text{e}_{1}}\leq
s_{\text{e}_{2}}-t_{\text{e}_{2}}<s_{\text{ph}}+t_{\text{ph}}\right\\}$ (5.3)

Coulomb Configurations: Configurations for which the backwards light-cones of
the electrons cross before $t<0$

$\mathcal{S}_{1}-\big{(}\mathcal{F}\cup\mathcal{X}\big{)}=\left\\{(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})\in\mathcal{S}_{1}:s_{\text{e}_{1}}+t_{\text{e}_{1}}<s_{\text{e}_{2}}-t_{\text{e}_{2}}\right\\}$
(5.4)

####  5.1.1 Free Configurations

Similarly to the two-body case, the solution formula on ”Free” configurations
is given by the free three-time evolution operator

$\Psi(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})=\big{(}\mathcal{P}(\textbf{x}_{\text{ph}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{1}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{2}})\big{)}\mathring{\Psi}$
(5.5)

$\Psi\big{|}_{\mathcal{F}}$ ${=}$

####  5.1.2 Compton Configurations

For ”Compton” configurations, there are three relevant cases for the
configurations of our three particles.
Case 1: The left electron is ”Near” the photon, while the right electron is
not. The associated diagram for such configurations is given by

${+}\quad e^{i\theta_{1}}$

From this diagram we determine that the multi-time evolution is given by a
contact evolution operator in the photon and electron $1$ variables tensored
with the free evolution in the electron $2$ variables

$\Psi(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})=\big{(}C_{\theta_{1}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{2}})\big{)}\mathring{\Psi}.$
(5.6)

We note that the $\varsigma_{0}=+$ components of the wave function are freely
propagating in this case, and that the $\varsigma_{0}=-$ components are solved
using Goursat evolution operators and sourced transports in electron $1$
variables.
Case 2: The right electron is ”Near” the photon while the left electron is
not. In this case electron $1$ freely propagates while the photon undergoes a
contact interaction with electron $2$

$\Psi(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})=\big{(}\mathcal{E}(\textbf{x}_{\text{e}_{1}})\otimes
C_{\theta_{2}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{2}})\big{)}\mathring{\Psi}.$
(5.7)

Similarly, we note that the $\varsigma_{0}=-$ components of the wave function
are freely propagating in this case, and that the $\varsigma_{0}=+$ components
is solved by setting up and solving a Goursat problem in the electron $2$
variables.
Case 3: Configurations in which both backwards light-cones of the electrons
cross the backwards light-cone of the photon, but not each other’s. On such
configurations our wave function is given as a sum of three diagrams:

${+}\quad e^{i\theta_{1}}$ ${+}\quad e^{i\theta_{2}}$

In case 3 all components will be given by Contact evolution operators. Letting
$\Pi_{\varsigma_{1}\varsigma_{2}\varsigma_{3}}$ be the projection operator of
a three-body wave function onto its relevant component, we may write the
evolution of each component in terms of the initial data as

$\begin{split}\psi_{-\varsigma_{1}\varsigma_{2}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})&=\Pi_{-\varsigma_{1}\varsigma_{2}}\bigg{(}\big{(}C_{\theta_{1}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{2}})\big{)}\mathring{\Psi}\bigg{)}\\\
\psi_{+\varsigma_{1}\varsigma_{2}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})&=\Pi_{+\varsigma_{1}\varsigma_{2}}\bigg{(}\big{(}\mathcal{E}(\textbf{x}_{\text{e}_{1}})\otimes
C_{\theta_{1}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{2}})\big{)}\mathring{\Psi}\bigg{)}.\end{split}$
(5.8)

####  5.1.3 Coulomb Configurations

In this section we will discuss the wave function on configurations where the
electron light cones are allowed to intersect. These configurations are of
great interest to us because enough time has passed for the photon to bounce
from one electron to the other, which may produce an effective interaction
between the two electrons.

Unfortunately, allowing the photon to bounce off one electron and then the
other is not without consequence. When drawing the diagrams associated to
these configurations, we must remember the rules: the wave function at
$(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})$
is calculated by summing over all the possible diagrams which end in that
state. For ”Free” configurations there is only one type of diagram: free
evolution of all three particles. On ”Compton” configurations we were limited
to diagrams with only one bounce. However, when we open the door to two
bounces, we are in fact opening the door to any number of bounces. In the time
it takes for two bounces to occur, so too could have three bounces, or four,
etc, and so our diagram must be a super-position of all such interactions

${+}\quad e^{i\theta_{2}}$ ${+}\quad e^{i(\theta_{1}+\theta_{2})}$ ${+}\quad
e^{i(\theta_{1}+2\theta_{2})}$ ${+}$

$e^{i2(\theta_{1}+\theta_{2})}$ ${+}\quad e^{i(2\theta_{1}+3\theta_{2})}$
${+}\quad e^{i3(\theta_{1}+\theta_{2})}$ ${+}...$

Figure 6: The photon could have bounced arbitrarily many times between the
electrons before the three propagated to their final destinations.

This paints a bleak picture for our endeavor, as it is not clear whether the
infinite sum of diagrams converges in some appropriate function space. Even if
it does converge, we would still need to show that the limit is in some
appropriate weak sense a solution of the multi-time equations of our system.

###  5.2 Convergence of Infinite Diagram Sum

####  5.2.1 Motivation for altered evolution

In this section we prove convergence of the infinite sum of diagrams (6) which
represents the wave function on ”Coulomb” configurations. We consider a
slightly altered evolution for our three-body wave function which approaches
the original evolution under an appropriate limit. We will find it convenient
to set up the altered evolution in the equal time setting
$t_{\text{ph}}=t_{\text{e}_{1}}=t_{\text{e}_{2}}$, and later remark why an
equal time convergence is enough to construct the wave function for ”Coulomb”
configurations in the multi-time setting.

We rectify the infinite bouncing problem by setting the boundary conditions to
$0$ on configurations where $|s_{\text{e}_{1}}-s_{\text{e}_{2}}|<\epsilon$. By
finite speed of propagation in configuration space, this has the desired
effect of placing a lower bound of $\frac{\epsilon}{2}$ on the time it takes
for data to bounce off one boundary and propagate to the other.

In the language of Feynman diagrams, this altered evolution is equivalent to
deleting all terms from our sum in which a photon bounced off an electron
while the two electrons were too close. By linearity, taking $\epsilon$
smaller re-introduces the deleted diagrams into our sum. Our goal in this
section will be to prove the convergence of these ”leaky boundary” evolutions
in the limit as $\epsilon$ goes to $0$ and show that this limit returns the
correct dynamics.

####  5.2.2 Leaky Boundary IBVP

In this section we will prove convergence of the ”leaky boundary” evolution.
This will be done in the equal time setting, where
$t_{\text{ph}}=t_{\text{e}_{1}}=t_{\text{e}_{2}}$. The Hamiltonian of our
system is

$\hat{H}=\big{(}\hat{H}_{\text{ph}}\otimes\mathbb{1}\otimes\mathbb{1}\big{)}+\big{(}\mathbb{1}\otimes\hat{H}_{\text{e}_{1}}\otimes\mathbb{1}\big{)}+\big{(}\mathbb{1}\otimes\mathbb{1}\otimes\hat{H}_{\text{e}_{2}}\big{)}$
(5.9)

where

$\hat{H}_{\text{ph}}=-i\hbar\gamma^{0}\gamma^{1}\partial_{s_{\text{ph}}},\quad\hat{H}_{\text{e}_{1}}=\gamma^{0}(m_{\text{e}}-i\hbar\gamma^{1}\partial_{s_{\text{e}_{1}}})\quad\hat{H}_{\text{e}_{2}}=\gamma^{0}(m_{\text{e}}-i\hbar\gamma^{1}\partial_{s_{\text{e}_{2}}}).$
(5.10)

The equal time dynamics will take place on a subset of
$\mathbb{R}_{\text{ph}}\times\mathbb{R}_{\text{e}_{1}}\times\mathbb{R}_{\text{e}_{2}}$:

$\displaystyle\mathcal{S}_{1}\coloneqq\left\\{(s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}}):s_{\text{e}_{1}}<s_{\text{ph}}<s_{\text{e}_{2}}\right\\},$
(5.11)

with boundary conditions that are linear relations between limits along events
in $\mathcal{S}_{1}$ towards the coincidence sets $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$, defined by

$\begin{split}\mathcal{C}_{1}\coloneqq\left\\{(s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})\in\partial\mathcal{S}_{1}:s_{\text{e}_{1}}=s_{\text{ph}}\right\\}\\\
\mathcal{C}_{2}\coloneqq\left\\{(s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})\in\partial\mathcal{S}_{1}:s_{\text{e}_{2}}=s_{\text{ph}}\right\\}.\end{split}$
(5.12)

######  Theorem 5.2.

Let $\theta_{1},\theta_{2}\in[0,2\pi)$ be two constant phases, and let
$\mu_{\epsilon}(s):\mathbb{R^{+}}\rightarrow[0,1]$ be a smooth transition
function which is zero on $[0,\epsilon]$ and 1 on $[2\epsilon,\infty)$. Let
the function $\mathring{\Psi}:\mathcal{S}_{1}\rightarrow\mathbb{C}^{8}$ be
bounded and compactly supported in ${\mathcal{S}_{1}}$. Then the following
initial-boundary value problem for the wave function
$\Psi:\mathbb{R}^{+}\times\mathcal{S}_{1}\rightarrow\mathbb{C}^{8}$,

$\left\\{\begin{array}[]{rclr}i\hbar\partial_{t}\Psi&=&\hat{H}\Psi&\text{in
}\mathbb{R}^{+}\times\mathcal{S}_{1}\\\ \Psi\big{|}_{t=0}&=&\mathring{\Psi}\\\
\psi_{-+\varsigma_{2}}&=&\mu_{\epsilon}(|s_{\text{e}_{1}}-s_{\text{e}_{2}}|)e^{i\theta_{1}}\psi_{+-\varsigma_{2}}&\text{on
}\mathcal{C}_{1}\\\
\psi_{+\varsigma_{1}-}&=&\mu_{\epsilon}(|s_{\text{e}_{1}}-s_{\text{e}_{2}}|)e^{i\theta_{2}}\psi_{-\varsigma_{1}+}&\text{on
}\mathcal{C}_{2}\end{array}\right.$ (5.13)

has a unique global-in-time solution that is supported in
$\overline{\mathcal{S}_{1}}$, and depends continuously on the initial data
$\mathring{\Psi}$.

###### Proof.

See the explicit solution for $\Psi$ derived in the Appendix. ∎

We will now state the main result of this paper.

######  Theorem 5.3.

Let $\mathring{\Psi}:\mathcal{S}_{1}\to\mathbb{C}^{8}$ bounded and compactly
supported, and let $\Psi_{\epsilon}$ denote the one parameter family given by
solutions to the leaky boundary IBVP (5.13) with initial data
$\mathring{\Psi}$ and transition function $\mu_{\epsilon}$. Then taking a
limit of these solutions as $\epsilon\to 0$ returns the equal-time dynamics we
originally set out to solve for, i.e $\lim_{\epsilon\to
0}\Psi_{\epsilon}(t)=e^{\frac{-it}{\hbar}\hat{H}_{\theta}}\mathring{\Psi}$
where $\hat{H}_{\theta}$ is the self-adjoint extension of $\hat{H}$ with
domain

$D(\hat{H}_{\theta}):=\left\\{\Psi\text{ is abs. cont. on
}\overline{\mathcal{S}_{1}}:\psi_{-+\varsigma_{2}}=e^{i\theta_{1}}\psi_{+-\varsigma_{2}}\text{
on
}\mathcal{C}_{1},\quad\psi_{+\varsigma_{1}-}=e^{i\theta_{2}}\psi_{-\varsigma_{1}+}\text{
on }\mathcal{C}_{2}\right\\}.$ (5.14)

######  Corollary 5.1.

Let $\mathring{\Psi}:\mathcal{S}_{1}\to\mathbb{C}^{8}$ be $C^{1}$ and
compactly supported in $\mathcal{S}_{1}$. Then $\lim_{\epsilon\to
0}\Psi_{\epsilon}$ has a representative in $L^{2}(\mathcal{S}_{1})$ which is
continuous with bounded derivatives.

###### Proof.

The fixed point argument showed that there exists a solution to the multi-time
IBVP which is continuous and with bounded derivatives, and this solution must
be unique by probability conservation. Restricting this solution to the equal-
time setting must therefore be a representative of
$e^{\frac{-it}{\hbar}\hat{H}_{\theta}}\mathring{\Psi}$. ∎

Here is a brief outline for the proof of theorem (5.3).

  1. 1.

We first show that for each fixed time, $\Psi_{\epsilon}(t)$ converges in
$L^{2}(\overline{\mathcal{S}_{1}})$ to some $\Psi(t)$ with
$||\Psi(t)||_{2}=||\mathring{\Psi}||_{2}$. It follows that there exists a one
parameter family of unitary operators $U(t)$ on
$L^{2}(\overline{\mathcal{S}_{1}})$ such that $\Psi(t)=U(t)\mathring{\Psi}$.

  2. 2.

We also show that $\Psi(t)$ weakly satisfies the equal-time three body PDE
$i\hbar\partial_{t}\Psi=\hat{H}\Psi$.

  3. 3.

We then show $U(t)$ satisfies the requirements of Stone’s theorem, so there
exists a self-adjoint extension $\tilde{H}$ of $\hat{H}$ such that
$U(t)=e^{\frac{-it}{\hbar}\tilde{H}}$.

  4. 4.

We conclude our proof by classifying all self-adjoint extensions of $\hat{H}$,
and use process of elimination to show that $\tilde{H}$ must be the unique
extension with domain given by (5.14).

Steps $1$ and $2$ are done in proposition (5.1), step $3$ is done in
proposition (5.2), and step $4$ is done in propositions (5.3) and (5.4).
Throughout all of these steps, we will consistently make use of an energy
estimate for $\Psi_{\epsilon}$:

######  Lemma 5.1.

The equal-time evolution of the leaky boundary IBVP is not unitary, since it
does not preserve the total probability in $\mathcal{S}_{1}$. However, the
total probability in $\mathcal{S}_{1}$ is non-increasing, with the total
probability lost in any finite time interval proportional to $\epsilon$.

###### Proof.

Let $\Psi$ satisfy the leaky boundary IBVP (5.13). The evolution equation
admits a probability current $j^{\mu}$ defined by

$\displaystyle
j^{0}:=\rho:=\sum_{\varsigma_{0},\varsigma_{1},\varsigma_{2}}|\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}|^{2},\quad$
$\displaystyle
j^{1}:=\sum_{\varsigma_{1},\varsigma_{2}}|\psi_{-\varsigma_{1}\varsigma_{2}}|^{2}-|\psi_{+\varsigma_{1}\varsigma_{2}}|^{2},$
(5.15) $\displaystyle
j^{2}:=\sum_{\varsigma_{0},\varsigma_{2}}|\psi_{\varsigma_{0}-\varsigma_{2}}|^{2}-|\psi_{\varsigma_{0}+\varsigma_{2}}|^{2},\quad$
$\displaystyle
j^{3}:=\sum_{\varsigma_{0},\varsigma_{1}}|\psi_{\varsigma_{0}\varsigma_{1}-}|^{2}-|\psi_{\varsigma_{0}\varsigma_{1}+}|^{2}$
(5.16)

which satisfies a continuity equation

$\frac{\partial\rho}{\partial t}=-\vec{\nabla}\cdot\vec{j}\quad\text{on
}\mathcal{S}_{1}.$ (5.17)

Integrating this over $\mathcal{S}_{1}$ returns

$\frac{d}{dt}||\rho||_{1}=\int_{\mathcal{C}^{1}\cup\mathcal{C}^{2}}-\vec{j}\cdot\vec{n}dS,$
(5.18)

where $dS$ is the surface element of $\mathcal{C}^{1}\cup\mathcal{C}^{2}$ and
$\vec{n}$ its normals. We compute that

$-\vec{j}\cdot\vec{n}=\frac{1}{\sqrt{2}}\begin{cases}j^{1}-j^{2}\text{ on
}\mathcal{C}^{1}\\\ j^{3}-j^{1}\text{ on }\mathcal{C}^{2}.\end{cases}$ (5.19)

Plugging in our boundary conditions returns

$j^{1}-j^{2}=|\psi_{-+-}|^{2}+|\psi_{-++}|^{2}-|\psi_{+--}|^{2}-|\psi_{+-+}|^{2}=((\mu_{\epsilon})^{2}-1)\big{(}|\psi_{+--}|^{2}+|\psi_{+-+}|^{2}\big{)}\text{
on }\mathcal{C}_{1}$ (5.20)
$j^{3}-j^{1}=|\psi_{+--}|^{2}+|\psi_{++-}|^{2}-|\psi_{--+}|^{2}-|\psi_{-++}|^{2}=((\mu_{\epsilon})^{2}-1)\big{(}|\psi_{--+}|^{2}+|\psi_{-++}|^{2}\big{)}\text{
on }\mathcal{C}_{2}$ (5.21)

It follows that the flux of the probability current is zero through
$\mathcal{C}^{1}\cup\mathcal{C}^{2}$ except for configurations where
$|s_{\text{e}_{2}}-s_{\text{e}_{1}}|<2\epsilon$. We will call these regions
$\mathcal{C}_{1}^{\epsilon}$ and $\mathcal{C}_{2}^{\epsilon}$, so that

$\frac{d}{dt}||\rho||_{1}=\int_{\mathcal{C}_{1}^{\epsilon}\cup\mathcal{C}_{2}^{\epsilon}}-\vec{j}\cdot\vec{n}dS$
(5.22)

Since $-\vec{j}\cdot\vec{n}\leq 0$, we see that the total probability in
$\mathcal{S}_{1}$ is non-increasing. Moreover, we may bound the probability
lost in any finite time interval in terms of $\epsilon$ and the initial data
$\mathring{\Psi}$. We first note that

$\bigg{|}\frac{d}{dt}||\rho||_{L^{1}}\bigg{|}\leq\int_{C_{1}^{\epsilon}\cup
C_{2}^{\epsilon}}\rho dS.$ (5.23)

We bound the right hand side using a Grönwall type estimate on the growth of
$||\rho||_{\infty}$.

######  Lemma 5.2.

There exists an $A<\infty$ independent of $\epsilon$ such that for all
$t>t_{0}>0$

$||\rho||_{\infty}(t)\leq 8e^{2A(t-t_{0})}||\rho||_{\infty}(t_{0}).$ (5.24)

###### Proof.

This follows from the Grönwall type estimate for $\Psi(t)$ given by equation
(B.4) derived in the Appendix . ∎

Since our initial data was compactly supported, finite speed of propagation
implies that there exists a constant $B>0$ independent of $\epsilon$ such that
for all $t>0$

$\int_{\mathcal{C}^{\epsilon}_{1}\cup\mathcal{C}^{\epsilon}_{2}}\mathbb{1}_{\text{supp}(\rho(t))}dS\leq\epsilon
B(1+t).$ (5.25)

where here $\mathbb{1}_{\mathcal{A}}$ denotes the indicator function of a set
$\mathcal{A}$. Taking a sup norm of equation $(\ref{Brute})$, and integrating
in time from $t_{0}$ to $t$ returns

$0\leq||\rho||_{L^{1}}(t_{0})-||\rho||_{L^{1}}(t)\leq 8\epsilon
BF(t_{0},t)||\rho||_{\infty}(t_{0})\leq 64\epsilon
BF(t_{0},t)e^{2At_{0}}||\rho||_{\infty}(0).$ (5.26)

where $F(t_{0},t):=\int_{t_{0}}^{t}(1+t)e^{2A(t-t_{0})}dt<\infty$ for all
$t_{0},t$. This provides us with an upper bound for the total probability
which can be lost in any finite interval of time. We see that as $\epsilon$
goes to zero, the total probability lost in any finite interval of time does
as well.∎

######  Remark 5.1.

We’ve proven that although we lose probability by truncating our Feynman
diagram sum, the probability lost shrinks to zero in the limit as we add the
deleted terms back in.

######  Proposition 5.1.

Fix $\mathring{\Psi}:\mathcal{S}_{1}\to\mathbb{C}^{8}$ bounded and compactly
supported, and let $\Psi_{\epsilon}$ denote the one parameter family given by
solutions to the leaky boundary IBVP with initial data $\mathring{\Psi}$ with
transition function $\mu_{\epsilon}$. Then

  1. 1.

For each fixed time $t>0$, $\Psi_{\epsilon}(t)$ converges in
$L^{2}(\overline{\mathcal{S}_{1}})$ to some $\Psi(t)$ as $\epsilon\to 0$ with
$\Psi(0)=\mathring{\Psi}$.

  2. 2.

$\Psi(t)$ weakly satisfies the equal-time three body PDE

$i\hbar\partial_{t}\Psi=\hat{H}\Psi.$ (5.27)

  3. 3.

$||\Psi||_{2}(t)=||\mathring{\Psi}||_{2}$ for all time, and thus gives rise to
a one parameter family of unitary operators
$U(t):C_{c}^{0}(\mathcal{S}_{1})\to L^{2}(\overline{\mathcal{S}_{1}})$.

  4. 4.

By density of $C_{c}^{0}(\mathcal{S}_{1})$ in
$L^{2}(\overline{\mathcal{S}_{1}})$, we may extend $U(t)$ to a one parameter
family of unitary operators on $L^{2}(\overline{\mathcal{S}_{1}})$

###### Proof.

Fix $t>0$. We seek to prove that $\Psi_{\epsilon}(t)$ is cauchy in
$L^{2}(\overline{\mathcal{S}_{1}})$. Let $\delta<\epsilon$, and
$\Phi=\Psi_{\delta}-\Psi_{\epsilon}$. By linearity, $\Phi$ satisfies

$\left\\{\begin{array}[]{rclr}i\hbar\partial_{t}\Phi&=&\hat{H}\Phi&\text{in
}\mathcal{S}_{1}\\\ \Phi\big{|}_{t=0}&=&0\\\
\phi_{-+\varsigma_{2}}&=&e^{i\theta_{1}}\big{[}\mu_{\delta}(\psi_{+-\varsigma_{2}})_{\delta}-\mu_{\epsilon}(\psi_{+-\varsigma_{2}})_{\epsilon}\big{]}&\text{on
}\mathcal{C}_{1}\\\
\phi_{+\varsigma_{1}-}&=&e^{i\theta_{2}}[\mu_{\delta}(\psi_{-\varsigma_{1}+})_{\delta}-\mu_{\epsilon}(\psi_{-\varsigma_{1}+})_{\epsilon}]&\text{on
}\mathcal{C}_{2}\end{array}\right.$ (5.28)

So $\Phi$ also satisfies similar probability conservation equations. Applying
similar estimates to before and integrating from $0$ to $t$ returns

$||\Phi||^{2}_{2}(t)=||\Phi||^{2}_{2}(t)-||\Phi||^{2}_{2}(0)\leq 24\epsilon
B(1+t)te^{2At}||\mathring{\rho}||_{\infty}(0).$ (5.29)

So $||\Phi||_{2}$ goes to zero as $\epsilon$ goes to zero. It follows that for
each fixed time, $\Psi_{\epsilon}(t)$ is cauchy and thus converges in $L^{2}$.
Also, $\Psi_{\epsilon}(0)=\mathring{\Psi}$ for all $\epsilon$. This concludes
our proof of $(1)$.

Let $\Psi(t)=\lim_{\epsilon\to 0}\Psi_{\epsilon}(t)$. We wish to show that
$\Psi(t)$ satisfies the equal-time three-body evolution equation weakly.
Luckily, $\Psi_{\epsilon}$ is a family of functions which all satisfy the same
first order partial differential in $\mathcal{S}_{1}$, so it suffices to show
that $\Psi_{\epsilon}$ converges to $\Psi$ in
$L^{2}([t_{1},t_{2}]\times\mathcal{S}_{1})$ for any finite time interval.

######  Lemma 5.3.

For any finite time interval $[t_{1},t_{2}]$,
$||\Psi-\Psi_{\epsilon}||_{L^{2}([t_{1},t_{2}]\times\mathcal{S}_{1})}\to 0$ as
$\epsilon\to 0$.

###### Proof.

This follows by taking a limit as $\delta\rightarrow 0$ of equation (5.29) and
integrating over the time interval $[t_{1},t_{2}]$. ∎

This concludes our proof of $(2)$. To prove $(3)$, we recall from equation
(5.26) that

$0\leq||\mathring{\Psi}||_{2}^{2}-||\Psi_{\epsilon}||_{2}^{2}(t)\leq 8\epsilon
BF(0,t)||\mathring{\rho}||_{\infty}$ (5.30)

where $B>0$ is some constant, and $F(0,t)<\infty$ for all $t$. Taking a limit
of this equation as $\epsilon\rightarrow 0$ returns
$||\mathring{\Psi}||_{2}\leq||\Psi||_{2}(t)$. Since we also had
$||\Psi_{\epsilon}||_{2}(t)\leq||\mathring{\Psi}||_{2}$ for all $\epsilon$, it
follows that the $L^{2}$ norm of $\Psi(t)$ is preserved for all time. This
concludes our proof of $(3)$. ∎

We’ve proven that taking a limit of the leaky boundary evolutions gives rise
to a one parameter family of unitary operators on
$L^{2}(\overline{\mathcal{S}_{1}})$. We have yet to prove that this one
parameter family of unitary operators produces a unitary evolution on our
Hilbert space, and that this unitary evolution correctly corresponds to the
original initial-boundary value problem we set out to solve. Towards this
goal, we first prove that there exists a self-adjoint operator $\tilde{H}$
such that $U(t)=e^{\frac{-it}{\hbar}\tilde{H}}$.

######  Proposition 5.2.

The one parameter family of unitary operators $U(t)$ on
$L^{2}(\overline{\mathcal{S}_{1}})$ form a semi-group, i.e.

$U(t+s)=U(t)U(s)\quad\forall t,s\in\mathbb{R}_{+},$ (5.31)

and are strongly continuous, i.e

$\lim_{t\to t_{0}}U(t)\Psi=U(t_{0})\Psi\quad\forall
t_{0}\in\mathbb{R}_{+},\Psi\in L^{2}(\overline{\mathcal{S}_{1}}).$ (5.32)

###### Proof.

Let $t,s\in\mathbb{R}_{+}$, and $\Psi\in C_{c}^{0}(\mathcal{S}_{1})$. We wish
to prove that $U(t+s)\Psi=U(t)U(s)\Psi$. Let $U_{\epsilon}(t)$ denote the
evolution operator corresponding the leaky boundary problem with transition
function $\mu_{\epsilon}$. By the triangle inequality

$||\big{(}U(t+s)-U(t)U(s)\big{)}\Psi||_{2}\leq||\big{(}U(t+s)-U_{\epsilon}(t+s)\big{)}\Psi||_{2}+||\big{(}U_{\epsilon}(t+s)-U(t)U_{\epsilon}(s)\big{)}\Psi||_{2}+||\big{(}U(t)U_{\epsilon}(s)-U(t)U(s)\big{)}\Psi||_{2}.$
(5.33)

By the convergence of our leaky boundary evolution we have that the first term
vanishes in the limit as $\epsilon\to 0$. As does the last term, after
recalling that $U(t)$ is a unitary operator and thus

$||U(t)\big{(}U_{\epsilon}(s)-U(s)\big{)}\Psi||_{2}=||\big{(}U_{\epsilon}(s)-U(s)\big{)}\Psi||_{2}\xrightarrow{\epsilon\to
0}0.$ (5.34)

For the middle term, we adhere to the following lemma which states that for
each $\epsilon$, the one parameter family $U_{\epsilon}(t)$ form a semi-group:

######  Lemma 5.4.

For each $\epsilon>0$, $t,s\in\mathbb{R}_{+}$, we have

$U_{\epsilon}(t+s)\Psi=U_{\epsilon}(t)U_{\epsilon}(s)\Psi.$ (5.35)

###### Proof.

This follows from $U_{\epsilon}(t)$ being the time-evolution operator
associated to the leaky boundary IBVP (5.13). ∎

Applying this to the middle term returns

$||\big{(}U_{\epsilon}(t+s)-U(t)U_{\epsilon}(s)\big{)}\Psi||_{2}=||\big{(}U_{\epsilon}(t)-U(t)\big{)}U_{\epsilon}(s)\Psi||_{2}\xrightarrow{\epsilon\to
0}0.$ (5.36)

Since all terms on the right hand side of equation (5.33) vanish as
$\epsilon\to 0$, we have that these operators coincide on a dense subset of
$L^{2}(\overline{\mathcal{S}_{1}})$. Since these operators are also bounded,
they must coincide on the whole Hilbert space. This concludes our proof that
$U(t)$ forms a semigroup.
Let $\Psi\in L^{2}(\overline{\mathcal{S}_{1}})$, and $t_{0}\in\mathbb{R}_{+}$.
We wish to prove that $\lim_{t\to t_{0}}U(t)\Psi=U(t_{0})\Psi$. Let $\Psi_{n}$
be a sequence of functions in $C_{c}^{0}(\mathcal{S}_{1})$ which converges to
$U(t_{0})\Psi$. By triangle inequality

$\begin{split}||U(t)\Psi-U(t_{0})\Psi||_{2}&\leq||U(t-t_{0})\big{(}U(t_{0})\Psi-\Psi_{n}\big{)}||_{2}+||U(t-t_{0})\Psi_{n}-\Psi_{n}||_{2}+||\Psi_{n}-U(t_{0})\Psi||_{2}=\\\
&=||U(t-t_{0})\Psi_{n}-\Psi_{n}||_{2}+2||\Psi_{n}-U(t_{0})\Psi||_{2}.\end{split}$
(5.37)

holds for each $n$. To show that the R.H.S vanishes, we adhere to the
following Lemma

######  Lemma 5.5.

For each $\Psi^{\prime}\in C_{c}^{0}(\mathcal{S}_{1})$,

$\lim_{t\to 0}U(t)\Psi^{\prime}=\Psi^{\prime}.$ (5.38)

###### Proof.

Since $\Psi^{\prime}$ is compactly supported in $\mathcal{S}_{1}$, it is
supported away from the boundary $\mathcal{C}_{1}\cup\mathcal{C}_{2}$. By
finite speed of propagation, there exists some small $\delta$ which depends on
$\Psi^{\prime}$ such that for all $|t|<\delta$ we have

$U(t)\Psi^{\prime}=\big{(}\mathcal{P}\otimes\mathcal{E}\otimes\mathcal{E}\big{)}\Psi.$
(5.39)

The proof then follows from the strong continuity of the free propagator. ∎

Taking a limit as $t\to t_{0}$ of equation (5.37)

$\lim_{t\to t_{0}}||U(t)\Psi-U(t_{0})\Psi||_{2}\leq
2||\Psi_{n}-U(t_{0})\Psi||_{2}.$ (5.40)

Since this holds for all $n$, we may take a limit as $n\to\infty$ to show that
this limit must vanish. This concludes our proof for the strong continuity of
$U(t)$. ∎

######  Corollary 5.2.

By Stone’s theorem, there exists some self-adjoint operator $\tilde{H}$ such
that $U(t)=e^{\frac{-it}{\hbar}\tilde{H}}$.

Clearly, $\tilde{H}$ is a self-adjoint extension of the three particle
Hamiltonian operator $\hat{H}$. However, we have yet to show that $\tilde{H}$
is equal to $\hat{H_{\theta}}$, the self-adjoint extension with domain given
by (5.14). We will prove this by writing down all self-adjoint extensions of
$\hat{H}$ on $\mathcal{S}_{1}$, and use process of elimination to show that
$\tilde{H}$ is the particular extension we are interested in.

######  Proposition 5.3.

Each self-adjoint extension $\bar{H}$ of $\hat{H}$ in $\mathcal{S}_{1}$ can be
associated to a mapping $O:\mathcal{C}_{1}\cup\mathcal{C}_{2}\to
U(\mathbb{C}^{2})$ by

$D(\bar{H}):=\left\\{\Psi\text{ is abs. cont. on
}\overline{\mathcal{S}_{1}}:\begin{pmatrix}\psi_{-++}\\\
\psi_{-+-}\end{pmatrix}=O\begin{pmatrix}\psi_{+-+}\\\
\psi_{+--}\end{pmatrix}\text{ on
}\mathcal{C}_{1},\quad\begin{pmatrix}\psi_{++-}\\\
\psi_{+--}\end{pmatrix}=O\begin{pmatrix}\psi_{-++}\\\
\psi_{--+}\end{pmatrix}\text{ on }\mathcal{C}_{2}\right\\}.$ (5.41)

###### Proof.

Let $\Psi$, $\Phi$ be absolutely continuous on $\overline{\mathcal{S}_{1}}$.
Define the current $J^{\mu}(\Psi,\Phi)$ by

$\displaystyle
J^{0}:=\sum_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}\phi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}^{*},\quad$
$\displaystyle
J^{1}:=\sum_{\varsigma_{1}\varsigma_{2}}\psi_{-\varsigma_{1}\varsigma_{2}}\phi_{-\varsigma_{1}\varsigma_{2}}^{*}-\psi_{+\varsigma_{1}\varsigma_{2}}\phi_{+\varsigma_{1}\varsigma_{2}}^{*},$
(5.42) $\displaystyle
J^{2}:=\sum_{\varsigma_{0}\varsigma_{2}}\psi_{\varsigma_{0}-\varsigma_{2}}\phi_{\varsigma_{0}-\varsigma_{2}}^{*}-\psi_{\varsigma_{0}+\varsigma_{2}}\phi_{\varsigma_{0}+\varsigma_{2}},\quad$
$\displaystyle
J^{3}:=\sum_{\varsigma_{0}\varsigma_{1}}\psi_{\varsigma_{0}\varsigma_{1}-}\phi_{\varsigma_{0}\varsigma_{1}-}^{*}-\psi_{\varsigma_{0}\varsigma_{1}+}\phi_{\varsigma_{0}\varsigma_{1}+}^{*}$
(5.43)

For $\Psi$ and $\Phi$ to be in the domain of a self-adjoint extension
$\bar{H}$ of $\hat{H}$, we must have $\vec{J}(\Psi,\Phi)\cdot\vec{n}=0$ on
$\partial\mathcal{S}_{1}$, where $\vec{n}$ is the unit normal of the boundary.
This condition is equivalent to setting $J^{1}-J^{2}=0$ on $\mathcal{C}_{1}$,
and $J^{3}-J^{1}=0$ on $\mathcal{C}_{2}$, which returns the quadratic boundary
conditions

$\psi_{-+-}\phi_{-+-}^{*}+\psi_{-++}\phi_{-++}^{*}=\psi_{+--}\phi_{+--}^{*}+\psi_{+-+}\phi_{+-+}^{*}\text{
on }\mathcal{C}^{1},$ (5.44)
$\psi_{+--}\phi_{+--}^{*}+\psi_{++-}\phi_{+--}^{*}=\psi_{--+}\phi_{--+}^{*}+\psi_{-++}\phi_{-++}^{*}\text{
on }\mathcal{C}^{2}.$ (5.45)

These conditions are only satisfied for all $\Psi,\Phi\in D(\bar{H})$ if there
exists some complex-inner-product-preserving unitary-operator-valued mapping
$O$ on $\mathcal{C}_{1}\cup\mathcal{C}_{2}$ such that equation (5.41) holds. ∎

######  Proposition 5.4.

The unitary evolution operator $U(t)$ derived by taking limits of the leaky
boundary evolution problem is equal to
$e^{\frac{-it}{\hbar}\hat{H}_{\theta}}$.

###### Proof.

We have all the tools we need to easily prove this. By the previous two
theorems, there exists some unitary-operator-valued mapping $O$ on
$\mathcal{C}_{1}\cup\mathcal{C}_{2}$ such that
$U(t)=e^{\frac{-it}{\hbar}\tilde{H}}$ and

$D(\tilde{H}):=\left\\{\Psi\text{ is abs. cont. on
}\overline{\mathcal{S}_{1}}:\begin{pmatrix}\psi_{-++}\\\
\psi_{-+-}\end{pmatrix}=O\begin{pmatrix}\psi_{+-+}\\\
\psi_{+--}\end{pmatrix}\text{ on
}\mathcal{C}_{1},\quad\begin{pmatrix}\psi_{++-}\\\
\psi_{+--}\end{pmatrix}=O\begin{pmatrix}\psi_{-++}\\\
\psi_{--+}\end{pmatrix}\text{ on }\mathcal{C}_{2}\right\\}.$ (5.46)

To determine what $O$ is, we may take any smooth, compactly supported initial
data $\mathring{\Psi}\in C_{c}^{\infty}(\mathcal{S}_{1})$ and evolve it for a
very short period of time. Short enough that it bounces off one of the
boundaries, but has yet to hit the triple point line
$\mathcal{C}_{1}\cap\mathcal{C}_{2}$. Let $t_{0}$ be the time that this data
propagates to the triple point line, and $t<t_{0}$. By finite speed of
propagation, there exists some $\epsilon_{0}$ such that

$U(t)\mathring{\Psi}=U_{\epsilon}(t)\mathring{\Psi}\quad\forall\epsilon<\epsilon_{0}.$
(5.47)

Clearly, the boundary condition satisfied by this evolution matches the
desired boundary condition. Since $\mathring{\Psi}$ was arbitrary, it follows
that

$O(s_{\text{e}_{1}},s_{\text{e}_{2}})\big{|}_{\mathcal{C}_{k}}=\begin{pmatrix}e^{i\theta_{k}}&0\\\
0&e^{i\theta_{k}}\end{pmatrix}\quad\forall|s_{\text{e}_{1}}-s_{\text{e}_{2}}|\geq
2\epsilon_{0},$ (5.48)

and again, since $\Psi_{0}$ was arbitrary, this must hold for all
$\epsilon_{0}$. This uniquely determines the mapping $O$ everywhere except the
triple point line. Since $O$ is continuous on $\mathcal{C}_{k}$, it must be
the constant matrix as above, and this concludes our proof. ∎

####  5.2.3 Extension to Multi-Time Setting

In the previous section, we proved convergence of the infinite diagram sum
depicted in figure 6 in the equal-time setting. We did this by considering an
altered evolution of our wave function which was equivalent to discarding all
diagrams in which the photon bounced between the two electrons while they were
”too close” (i.e when $|s_{\text{e}_{2}}-s_{\text{e}_{1}}|<\epsilon$), and
then took a limit as our alteration went to zero.

A natural way to extend our method to the multi-time setting would be to
discard all diagrams in which the photon bounced between the two electrons
while
$-\eta(\textbf{x}_{\text{e}_{1}}-\textbf{x}_{\text{e}_{2}},\textbf{x}_{\text{e}_{1}}-\textbf{x}_{\text{e}_{2}})<\epsilon$.
This corresponds to the following multi-time IBVP

$\left\\{\begin{array}[]{rclr}-i\hbar D_{\text{ph}}\Psi&=&0&\\\ -i\hbar
D_{\text{e}_{1}}\Psi+m_{e}\Psi&=&0&\\\ -i\hbar
D_{\text{e}_{2}}\Psi+m_{\text{e}}\Psi&=&0&\text{in }\mathcal{S}_{1}\\\
\Psi&=&\mathring{\Psi}&\text{on }\mathcal{I}\\\
\psi_{-+\varsigma_{2}}&=&e^{i\theta_{1}}\mu_{\epsilon}\psi_{+-\varsigma_{2}}&\text{on
}\mathcal{C}_{1}\\\
\psi_{+\varsigma_{1}-}&=&e^{i\theta_{2}}\mu_{\epsilon}\psi_{-\varsigma_{1}+}&\text{on
}\mathcal{C}_{2}\end{array}\right.$ (5.49)

where here,
$\mu_{\epsilon}=\mu_{\epsilon}(-\eta(\textbf{x}_{\text{e}_{1}}-\textbf{x}_{\text{e}_{2}},\textbf{x}_{\text{e}_{1}}-\textbf{x}_{\text{e}_{2}}))$
is a smooth transition function. Unfortunately, the multi-time IBVP above is
not well-posed. To see this, consider the dynamics of the wave function on
$\mathcal{C}_{1}$. Both $\psi_{-+\varsigma_{2}}$ and $\psi_{+-\varsigma_{2}}$
satisfy the electron $2$ evolution equation on $\mathcal{C}_{1}$

$(\partial_{t_{\text{e}_{2}}}-\varsigma_{2}\partial_{s_{\text{e}_{2}}})\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}+i\omega\psi_{\varsigma_{0}\varsigma_{1}\overline{\varsigma}_{2}}=0.$
(5.50)

For this to be consistent with the boundary condition on $\mathcal{C}_{1}$, we
would need

$(\partial_{t_{\text{e}_{2}}}-\varsigma_{2}\partial_{s_{\text{e}_{2}}})\mu_{\epsilon}=0$
(5.51)

which cannot hold for non-constant functions of
$\eta(\textbf{x}_{\text{e}_{1}}-\textbf{x}_{\text{e}_{2}},\textbf{x}_{\text{e}_{1}}-\textbf{x}_{\text{e}_{2}})$.
This makes it impossible to generalize the equal-time ”leaky boundary” problem
to the multi-time setting.

Luckily, the equal-time evolution is enough for our purposes, so long as we
combine it with our previous multi-time results. Let
$(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})$
be a Coulomb configuration. We wish to write down the solution for $\Psi$
satisfying the original multi-time IBVP (5.1) at this configuration in terms
of the initial data $\mathring{\Psi}$. First, let
$t=\min(t_{\text{ph}},t_{\text{e}_{1}},t_{\text{e}_{2}})$. We may evolve the
initial data up to the equal-time hypersurface in $\mathcal{S}_{1}$ written as
$\\{t_{\text{ph}}=t_{\text{e}_{1}}=t_{\text{e}_{2}}=t\\}$ using the equal-time
evolution operator. Let
$\Psi\big{|}_{t}=e^{\frac{-it}{\hbar}\hat{H}_{\theta}}\mathring{\Psi}$. To
solve for $\Psi$ at the Coulomb configuration, we only need to evolve our new
initial data $\Psi\big{|}_{t}$ further in only two of the time variables. For
$\textbf{x}\in\mathbb{R}^{1,1}$, let $\textbf{x}^{t}:=(x^{0}-t,x^{1})$. There
are only three possible cases:
Free Evolution: If
$(\textbf{x}^{t}_{\text{ph}},\textbf{x}^{t}_{\text{e}_{1}},\textbf{x}^{t}_{\text{e}_{2}})\in\mathcal{F}$,
then

$\Psi(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})=\big{(}\mathcal{P}(\textbf{x}^{t}_{\text{ph}})\otimes\mathcal{E}(\textbf{x}^{t}_{\text{e}_{1}})\otimes\mathcal{E}(\textbf{x}^{t}_{\text{e}_{2}})\big{)}\Psi\big{|}_{t}.$
(5.52)

One more bounce: If
$(\textbf{x}^{t}_{\text{ph}},\textbf{x}^{t}_{\text{e}_{1}},\textbf{x}^{t}_{\text{e}_{2}})\in\mathcal{X}$,
then either
$(\textbf{x}^{t}_{\text{ph}},\textbf{x}^{t}_{\text{e}_{1}})\in\mathcal{N}$ or
$(\textbf{x}^{t}_{\text{ph}},\textbf{x}^{t}_{\text{e}_{2}})\in\mathcal{N}$.
Suppose the former. Then

$\Psi(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_{1}},\textbf{x}_{\text{e}_{2}})=\big{(}C_{\theta_{1}}(\textbf{x}^{t}_{\text{ph}},\textbf{x}^{t}_{\text{e}_{1}})\otimes\mathcal{E}(\textbf{x}^{t}_{\text{e}_{2}})\big{)}\Psi\big{|}_{t}$
(5.53)

while an analogous equation holds for the other case. This completes our
construction of the solution to the initial-boundary value problem (5.1).

##  6 Summary and Outlook

Using a relativistic quantum-mechanical treatment of the Compton effect in
$1+1$ dimensions, we have shown that placing a photon between two Dirac
electrons generates an effective interaction which can be decomposed into many
individual photon-electron Compton interactions. Our next goal is to further
study this effective interaction, and to see whether $1+1$ dimensional
electromagnetic interactions can be reproduced in terms of relativistic
quantum mechanics of a fixed finite number of electrons, photons, and their
anti-particles. Of course, this would only be one step in the direction of a
larger goal, which would be to find out whether such results can also be made
in $3+1$ dimensions in a way that accurately reproduces empirical
electromagnetism.

##  7 Acknowledgment

We thank Michael Kiessling and Matthias Lienert for helpful discussions.

## APPENDIX

##  Appendix A Solution formulas for the Klein-Gordon and Dirac equation

The initial value problem for the Dirac equation is given by

$\left\\{\begin{array}[]{rcl}-i\hbar
D_{\text{e}}\Psi_{\text{e}}+m_{\text{e}}\Psi_{\text{e}}&=&0\\\
\Psi_{\text{e}}\big{|}_{t_{\text{e}}=0}&=&\mathring{\Psi}_{\text{e}}\end{array}\right..$
(A.1)

where $\Psi_{\text{e}}=\begin{pmatrix}\psi_{-}\\\ \psi_{+}\end{pmatrix}$. To
solve the Dirac IVP, we decouple the components of $\Psi_{\text{e}}$ and write
down the second order initial value problems

$\left\\{\begin{array}[]{rcl}\Box_{\text{e}}\psi_{\varsigma}+\omega^{2}\psi_{\varsigma}&=&0\\\
\psi_{\varsigma}\big{|}_{t_{\text{e}}=0}&=&\mathring{\psi}_{\varsigma}(s_{\text{e}})\\\
\partial_{t_{\text{e}}}\psi_{\varsigma}\big{|}_{t_{\text{e}}=0}&=&\varsigma\partial_{s_{\text{e}}}\mathring{\psi}_{\varsigma}(s_{\text{e}})-i\omega\mathring{\psi}_{\overline{\varsigma}}(s_{\text{e}})\end{array}\right..$
(A.2)

Recall that the general solution to the Cauchy problem for the Klein-Gordon
Equation

$w_{tt}-w_{ss}+\omega^{2}w=0,\hskip 5.69046ptw(0,s)=A(s),\hskip
5.69046ptw_{t}(0,s)=B(s),$ (A.3)

is

$\begin{split}w(t,s)=\frac{1}{2}\\{A(s-t)+A(s+t)\\}-&\frac{\omega
t}{2}\int_{s-t}^{s+t}\frac{J_{1}(\omega\sqrt{t^{2}-(s-\sigma)^{2}})}{\sqrt{t^{2}-(s-\sigma)^{2}}}A(\sigma)d\sigma\\\
+&\frac{1}{2}\int_{s-t}^{s+t}J_{0}(\omega\sqrt{t^{2}-(s-\sigma)^{2}})B(\sigma)d\sigma\end{split}$
(A.4)

where $J_{\nu}$ is the Bessel function of index $\nu$. Applying this to the
Dirac equation and using integration by parts, our Dirac evolution can be
written as

$\begin{split}\psi_{\varsigma}(t_{\text{e}},s_{\text{e}})&=\mathring{\psi}_{\varsigma}(s_{\text{e}}+\varsigma
t_{\text{e}})\\\
&-\frac{\omega}{2}\int_{s_{\text{e}}-t_{\text{e}}}^{s_{\text{e}}+t_{\text{e}}}J_{1}(\omega\sqrt{t_{\text{e}}^{2}-(s_{\text{e}}-\sigma)^{2}})\frac{\sqrt{t_{\text{e}}-\varsigma(s_{\text{e}}-\sigma)}}{\sqrt{t_{\text{e}}+\varsigma(s_{\text{e}}-\sigma)}}\mathring{\psi}_{\varsigma}(\sigma)d\sigma\\\
&-\frac{i\omega}{2}\int_{s_{\text{e}}-t_{\text{e}}}^{s_{\text{e}}+t_{\text{e}}}J_{0}(\omega\sqrt{t_{\text{e}}^{2}-(s_{\text{e}}-\sigma)^{2}})\mathring{\psi}_{\overline{\varsigma}}(\sigma)d\sigma\end{split}$
(A.5)

We condense such equations by introducing the electronic time evolution
operator $\mathcal{E}(t_{\text{e}},s_{\text{e}})$ defined by
$\Psi_{\text{e}}(t_{\text{e}},s_{\text{e}})=\mathcal{E}(t_{\text{e}},s_{\text{e}})\mathring{\Psi}_{\text{e}}$.
Since $\frac{J_{1}(\omega x)}{x}\leq\frac{\omega}{2}$, and $J_{0}(x)\leq 1$,
we have the following small-time growth estimate

$||\mathcal{E}(t_{\text{e}},\cdot)\mathring{\Psi}||_{\infty}\leq(1+\omega^{2}t_{\text{e}}+\omega
t_{\text{e}})||\mathring{\Psi}||_{\infty}$ (A.6)

Also relevant is the Goursat problem for the 1-dimensional Klein-Gordon
equation,

$\left\\{\begin{array}[]{rcl}U_{tt}-U_{ss}+\omega^{2}U&=&0\quad\text{for
}|s|<t\\\ U(b,b)&=&F(b)\\\ U(c,-c)&=&G(c)\end{array}\right.,$ (A.7)

which has the solution

$\begin{split}U(t,s)=&F(\frac{t+s}{2})+G(\frac{t-s}{2})-\frac{1}{2}(F(0)+G(0))J_{0}(\omega\sqrt{t^{2}-s^{2}})\\\
&-\omega(t-s)\int_{0}^{\frac{t+s}{2}}F(b)\frac{J_{1}(\omega\sqrt{(t-s)(t+s-2b)})}{\sqrt{(t-s)(t+s-2b)}}db\\\
&-\omega(t+s)\int_{0}^{\frac{t-s}{2}}G(c)\frac{J_{1}(\omega\sqrt{(t+s)(t-s-2c)})}{\sqrt{(t+s)(t-s-2c)}}dc.\end{split}$
(A.8)

By linearity, we may introduce Goursat evolution operators $G^{R}(t,s)$ and
$G^{L}(t,s)$, defined so that

$U(t,s)=G^{R}(t,s)F+G^{L}(t,s)G.$ (A.9)

Explicitly, these operators are written as

$G^{R}(t,s)F=F(\frac{t+s}{2})-\frac{F(0)}{2}J_{0}(\omega\sqrt{t^{2}-s^{2}})-\omega(t-s)\int_{0}^{\frac{t+s}{2}}F(b)\frac{J_{1}(\omega\sqrt{(t-s)(t+s-2b)})}{\sqrt{(t-s)(t+s-2b)}}db$
(A.10)
$G^{L}(t,s)G=G(\frac{t-s}{2})-\frac{G(0)}{2}J_{0}(\omega\sqrt{t^{2}-s^{2}})-\omega(t+s)\int_{0}^{\frac{t-s}{2}}G(c)\frac{J_{1}(\omega\sqrt{(t+s)(t-s-2c)})}{\sqrt{(t+s)(t-s-2c)}}dc.$
(A.11)

##  Appendix B Solution and Convergence of Leaky Boundary Problem

In this section we construct and analyze the solution to the ”leaky boundary”
problem introduced in equation (5.13).

######  Theorem B.1.

Let $\theta_{1},\theta_{2}\in[0,2\pi)$ be two constant phases, and let
$\mu_{\epsilon}(s):\mathbb{R^{+}}\rightarrow[0,1]$ be a smooth transition
function which is zero on $[0,\epsilon]$ and 1 on $[2\epsilon,\infty)$. Let
the function $\mathring{\Psi}:\mathcal{S}_{1}\rightarrow\mathbb{C}^{8}$ be
bounded and compactly supported in ${\mathcal{S}_{1}}$. Then the following
initial-boundary value problem for the wave function
$\Psi:\mathbb{R}^{+}\times\mathcal{S}_{1}\rightarrow\mathbb{C}^{8}$,

$\left\\{\begin{array}[]{rclr}i\hbar\partial_{t}\Psi&=&\hat{H}\Psi&\text{in
}\mathbb{R}^{+}\times\mathcal{S}_{1}\\\ \Psi\big{|}_{t=0}&=&\mathring{\Psi}\\\
\psi_{-+\varsigma_{2}}&=&\mu_{\epsilon}(|s_{\text{e}_{1}}-s_{\text{e}_{2}}|)e^{i\theta_{1}}\psi_{+-\varsigma_{2}}&\text{on
}\mathcal{C}_{1}\\\
\psi_{+\varsigma_{1}-}&=&\mu_{\epsilon}(|s_{\text{e}_{1}}-s_{\text{e}_{2}}|)e^{i\theta_{2}}\psi_{-\varsigma_{1}+}&\text{on
}\mathcal{C}_{2}\end{array}\right.$ (B.1)

has a unique global-in-time solution that is supported in
$\overline{\mathcal{S}_{1}}$, and depends continuously on the initial data
$\mathring{\Psi}$.

###### Proof.

By construction, these boundary conditions ensure that data which bounces off
one boundary $\mathcal{C}_{1}$ takes at least $\frac{\epsilon}{2}$ time to
propagate to the other boundary $\mathcal{C}_{2}$ and vice versa. Because of
this, we will find it easier to only solve for the evolution to the leaky
boundary IBVP for all times up to $\frac{\epsilon}{2}$. Since $\epsilon$ is
fixed, we may repeatedly apply the evolution operator with time step size
$\frac{\epsilon}{2}$ until any desired time is reached. As we construct our
solution, we will find it convenient to prove the following estimate for the
rate of growth of $\Psi$.

######  Theorem B.2.

Let $\Psi(t)$ satisfy the leaky boundary IBVP given by (B.1) with
$\epsilon<2$. Let us define the following norm

$||\Psi(t)||_{\infty}:=\sup_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}\sup_{s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}}}|\psi_{\varsigma_{1}\varsigma_{2}\varsigma_{3}}(t)|.$
(B.2)

Then there exists an $A>0$ independent of $\epsilon$, such that for all
$t\leq\frac{\epsilon}{2}$

$||\Psi(t)||_{\infty}\leq(1+At)||\mathring{\Psi}||_{\infty}.$ (B.3)

By repeatedly evolving data with time step size $\frac{\epsilon}{2}$, we have
the global in time estimate

$||\Psi(t)||_{\infty}\leq e^{At}||\mathring{\Psi}||_{\infty}.$ (B.4)

Fix $t\leq\frac{\epsilon}{2}$. We wish to solve for $\Psi(t)$ on
$\mathcal{S}_{1}$. Similarly to previous interacting systems, we approach this
by splitting up our configuration space into regions for which the particle
configurations are ”Free” or ”Compton”.

$\mathcal{S}_{1}=\mathcal{F}\cup\mathcal{X},$ (B.5)

where

$\mathcal{F}:=\left\\{(s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})\in\mathcal{S}_{1}:s_{\text{e}_{1}}+t<s_{\text{ph}}-t,\quad
s_{\text{ph}}+t<s_{\text{e}_{2}}-t\right\\},$ (B.6)
$\mathcal{X}:=\mathcal{S}_{1}\setminus\mathcal{F}=\left\\{(s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})\in\mathcal{S}_{1}:s_{\text{e}_{1}}+t\geq
s_{\text{ph}}-t\ \mbox{ or }\quad s_{\text{ph}}+t\geq
s_{\text{e}_{2}}-t\right\\}.$ (B.7)

###  B.1 Free Configurations

Similarly to the interacting multi-time problems encountered earlier, the
solution for free configurations is simply given by the free three body
evolution operator

$\Psi(t,s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})=\big{(}\mathcal{P}(\textbf{x}_{\text{ph}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{1}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{2}})\big{)}\mathring{\Psi},$
(B.8)

where here $\textbf{x}_{\text{ph}}=(t,s_{\text{ph}})$,
$\textbf{x}_{\text{e}_{1}}=(t,s_{\text{e}_{1}})$ and analogously for
$\textbf{x}_{\text{e}_{2}}$. By (A.6), we have that for all Free
configurations

$|\psi_{\varsigma_{1}\varsigma_{2}\varsigma_{2}}(t,s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})|\leq(1+\omega^{2}t+\omega
t)^{2}||\mathring{\Psi}||_{\infty}\leq(1+A_{0}t)||\mathring{\Psi}||_{\infty},$
(B.9)

for some $A_{0}$, where here we’ve used that $t<1$ to absorb all higher order
terms.

###  B.2 Compton Configurations

We begin by splitting up the Compton Configurations into three cases:
Case 1: This case covers all configurations for which the first electron is
interacting with the photon, while the second electron is freely evolving. In
this regime, the $\varsigma_{0}=+$ components are freely evolving, and so they
are given by

$\psi_{+\varsigma_{1}\varsigma_{2}}=\Pi_{+\varsigma_{1}\varsigma_{2}}\bigg{(}\big{(}\mathcal{P}(\textbf{x}_{\text{ph}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{1}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{2}})\big{)}\mathring{\Psi}\bigg{)},$
(B.10)

where $\Pi$ is the projection operator onto the associated component. It
follows that the estimate (B.9) holds for these components as well. The
$\varsigma_{0}=-$ components require a different approach.

We begin by expanding the first equation of (B.1) in terms of the components
of $\Psi$. This returns sourced transports

$(\partial_{t}-\varsigma_{0}\partial_{s_{\text{ph}}}-\varsigma_{1}\partial_{s_{\text{e}_{1}}}-\varsigma_{2}\partial_{s_{\text{e}_{2}}})\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}+i\omega\psi_{\varsigma_{0}\overline{\varsigma}_{1}\varsigma_{2}}+i\omega\psi_{\varsigma_{0}{\varsigma}_{1}\overline{\varsigma}_{2}}=0.$
(B.11)

Applying the operator
$(\partial_{t}-\varsigma_{0}\partial_{s_{\text{ph}}}-\overline{\varsigma}_{1}\partial_{s_{\text{e}_{1}}}-\varsigma_{2}\partial_{s_{\text{e}_{2}}})$
to equation (B.11) returns

$((\partial_{\nu}-\varsigma_{2}\partial_{s_{\text{e}_{2}}})^{2}-\partial_{s_{\text{e}_{1}}}^{2})\psi_{-+\varsigma_{2}}+2i\omega\partial_{\nu}\psi_{-+\overline{\varsigma}_{2}}=0,$
(B.12)

where $\partial_{\nu}=\partial_{t}+\partial_{s_{\text{ph}}}$. It is then easy
to verify that the following solution formula for $\psi_{-+\varsigma_{2}}$
satisfies equation (B.12) in the space of Compton configurations.

$\psi_{-+\varsigma_{2}}(t,s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})=G^{R}(t,s_{\text{e}_{1}}-(s_{\text{ph}}-t))F_{\varsigma_{2}}+G^{L}(t,s_{\text{e}_{1}}-(s_{\text{ph}}-t))G_{\varsigma_{2}},$
(B.13)

where

$\begin{split}G_{\varsigma_{2}}(c)&=\Pi_{\varsigma_{2}}\bigg{(}\mathcal{E}(t-c,s_{\text{e}_{2}})\begin{pmatrix}\psi_{-++}\\\
\psi_{-+-}\end{pmatrix}(c,\textbf{c},s_{\text{e}_{2}})\bigg{)}=\psi_{-+\varsigma_{2}}(c,\textbf{c},s_{\text{e}_{2}}+\varsigma_{2}(t-c))\\\
&-\frac{\omega}{2}\int_{s_{\text{e}_{2}}-(t-c)}^{s_{\text{e}_{2}}+(t-c)}J_{1}(\omega\sqrt{(t-c)^{2}-(s_{\text{e}_{2}}-\sigma)^{2}})\frac{\sqrt{(t-c)-\varsigma(s_{\text{e}_{2}}-\sigma)}}{\sqrt{(t-c)+\varsigma(s_{\text{e}_{2}}-\sigma)}}\psi_{-+\varsigma_{2}}(c,\textbf{c},\sigma)d\sigma\\\
&-\frac{i\omega}{2}\int_{s_{\text{e}_{2}}-(t-c)}^{s_{\text{e}_{2}}+(t-c)}J_{0}(\omega\sqrt{(t-c)^{2}-(s_{\text{e}_{2}}-\sigma)^{2}})\psi_{-+\overline{\varsigma}_{2}}(c,\textbf{c},\sigma)d\sigma,\end{split}$
(B.14)

and
$\textbf{c}=(s_{\text{ph}}-t+c,s_{\text{ph}}-t-c)\in\mathbb{R}_{\text{ph}}\times\mathbb{R}_{\text{e}_{1}}$.
We define $F_{\varsigma_{2}}$ similarly, but taking care to correctly apply
our boundary condition

$F_{\varsigma_{2}}(b)=\Pi_{\varsigma_{2}}\bigg{(}\mathcal{E}(t-b,s_{\text{e}_{2}})\begin{pmatrix}\mu_{\epsilon}(|s_{\text{e}_{1}}-s_{\text{e}_{2}}|)e^{i\theta_{1}}\psi_{+-+}\\\
\mu_{\epsilon}(|s_{\text{e}_{1}}-s_{\text{e}_{2}}|)e^{i\theta_{1}}\psi_{+--}\end{pmatrix}(b,\textbf{b},s_{\text{e}_{2}})\bigg{)},$
(B.15)

where
$\textbf{b}=(s_{\text{ph}}-t+b,s_{\text{ph}}-t+b)\in\mathbb{R}_{\text{ph}}\times\mathbb{R}_{\text{e}_{1}}$,
which implies we are evaluating along the boundary $\mathcal{C}_{1}$. To write
this solution in terms of the initial data, we note that for all
$\sigma\in(s_{\text{e}_{2}}-(t-c),s_{\text{e}_{2}}+(t-c))$, we have that
$(\textbf{c},\sigma)$ are ”Free configurations”. So we may plug (B.8) into
equation (B.14). Similarly, we may plug equation (B.10) into (B.15). The
boundary condition
$\psi_{-+\varsigma_{2}}=\mu_{\epsilon}(|s_{\text{e}_{1}}-s_{\text{e}_{2}}|)e^{i\theta}\psi_{+-\varsigma_{2}}$
on $\mathcal{C}_{1}$ is satisfied by construction. To estimate
$\psi_{-+\varsigma_{2}}$ for Compton configurations, we note that
$|t\pm(s_{\text{e}_{1}}-(s_{\text{ph}}-t))|\leq 2t$. Plugging equation (B.13)
into the definition of the Goursat evolution operators (A.10), (A.11) it
follows that the integral terms are
$O(t^{2})\cdot||\mathring{\Psi}||_{\infty}$. Hence,

$\psi_{-+\varsigma_{2}}=F_{\varsigma_{2}}(\frac{s_{\text{ph}}-s_{\text{e}_{1}}}{2})+G_{\varsigma_{2}}(\frac{s_{\text{e}_{1}}+2t-s_{\text{ph}}}{2})-\frac{F_{\varsigma_{2}}(0)+G_{\varsigma_{2}}(0)}{2}J_{0}(\omega\sqrt{t^{2}-(s_{\text{e}_{1}}-(s_{\text{ph}}-t))^{2}})+O(t^{2})||\mathring{\Psi}||_{\infty}.$
(B.16)

Plugging equation (A.5) into equations (B.14), (B.15), we find that

$\begin{split}G_{\varsigma_{2}}(\frac{s_{\text{e}_{1}}+2t-s_{\text{ph}}}{2})&=\mathring{\psi}_{-+\varsigma_{2}}(s_{\text{ph}}-t,s_{\text{ph}}-t,s_{\text{e}_{2}}+\varsigma_{2}t)+O(t)||\mathring{\Psi}||_{\infty},\\\
\frac{F_{\varsigma_{2}}(0)+G_{\varsigma_{2}}(0)}{2}&=\mathring{\psi}_{-+\varsigma_{2}}(s_{\text{ph}}-t,s_{\text{ph}}-t,s_{\text{e}_{2}}+\varsigma_{2}t)+O(t)||\mathring{\Psi}||_{\infty},\\\
|F_{\varsigma_{2}}(\frac{s_{\text{ph}}-s_{\text{e}_{1}}}{2})|&\leq|\mathring{\psi}_{+-\varsigma_{2}}(s_{\text{e}_{1}}+t,s_{\text{ph}}-t,s_{\text{e}_{2}}+\varsigma_{2}t)|+O(t)||\mathring{\Psi}||_{\infty}.\end{split}$
(B.17)

We conclude our estimate for $\psi_{-+\varsigma_{2}}$ using

$1-J_{0}(\omega\sqrt{t^{2}-(s_{\text{e}_{1}}-(s_{\text{ph}}-t))^{2}})\leq(\omega
t)^{2}=O(t^{2}).$ (B.18)

Combining these estimates returns

$|\psi_{-+\varsigma_{2}}|\leq|\mathring{\psi}_{+-\varsigma_{2}}(s_{\text{e}_{1}}+t,s_{\text{ph}}-t,s_{\text{e}_{2}}+\varsigma_{2}t)|+O(t)||\mathring{\Psi}||_{\infty},$
(B.19)

as desired.
To solve for the remaining two components, we apply the operator
$(\partial_{t}-\varsigma_{0}\partial_{s_{\text{ph}}}-{\varsigma}_{1}\partial_{s_{\text{e}_{1}}}-\overline{\varsigma}_{2}\partial_{s_{\text{e}_{2}}})$
to equation (B.11) and find that $\psi_{--\varsigma_{2}}$ satisfies

$(\partial_{u}^{2}-\partial_{s_{\text{e}_{2}}}^{2}+\omega^{2})\psi_{--\varsigma_{2}}+i\omega(\partial_{u}+\overline{\varsigma}_{2}\partial_{\text{e}_{2}})\psi_{-+\varsigma_{2}}+\omega^{2}\psi_{-+\overline{\varsigma}_{2}}=0,$
(B.20)

where
$\partial_{u}=\partial_{t}+\partial_{s_{\text{ph}}}+\partial_{s_{\text{e}_{1}}}$.
Equation (B.20) is an inhomogenous Klein-Gordon equation with sources that are
known to us in terms of the initial data from equation (B.13). We solve for
the two remaining components by setting up and solving the initial value
problem

$\begin{cases}(\partial_{u}^{2}-\partial_{s_{\text{e}_{2}}}^{2}+\omega^{2})\psi_{--\varsigma_{2}}+i\omega(\partial_{u}+\overline{\varsigma}_{2}\partial_{\text{e}_{2}})\psi_{-+\varsigma_{2}}+\omega^{2}\psi_{-+\overline{\varsigma}_{2}}=0\\\
\psi_{--\varsigma_{2}}\bigg{|}_{u=0}=\mathring{\psi}_{--\varsigma_{2}}(s_{\text{ph}}-t,s_{\text{e}_{1}}-t,s_{\text{e}_{2}})\\\
\partial_{u}\psi_{--\varsigma_{2}}\bigg{|}_{u=0}=\bigg{(}\varsigma_{2}\partial_{\text{e}_{2}}\mathring{\psi}_{--\varsigma_{2}}+i\omega\mathring{\psi}_{-+\varsigma_{2}}+i\omega\mathring{\psi}_{--\overline{\varsigma}_{2}}\bigg{)}(s_{\text{ph}}-t,s_{\text{e}_{1}}-t,s_{\text{e}_{2}}).\end{cases}$
(B.21)

By linearity, we can write the solution for $\psi_{--\varsigma_{2}}$ as
$U_{\varsigma_{2}}+S_{\varsigma_{2}}$, where $U_{\varsigma_{2}}$ represents
the freely evolving initial data and $S_{\varsigma_{2}}$ the sourced data.
These solutions are given as

$\displaystyle
U_{\varsigma_{2}}(t,s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})$
$\displaystyle=$
$\displaystyle\mathring{\psi}_{--\varsigma_{2}}(s_{\text{ph}}-t,s_{\text{e}_{1}}-t,s_{\text{e}_{2}}+\varsigma_{2}t)$
$\displaystyle\mbox{}-\frac{\omega}{2}\int_{s_{\text{e}_{2}}-t}^{s_{\text{e}_{2}}+t}J_{1}(\omega\sqrt{t^{2}-(s_{\text{e}_{2}}-\sigma)^{2}})\frac{\sqrt{t-\varsigma(s_{\text{e}_{2}}-\sigma)}}{\sqrt{t+\varsigma_{2}(s_{\text{e}_{2}}-\sigma)}}\mathring{\psi}_{--\varsigma}(s_{\text{ph}}-t,s_{\text{e}_{1}}-t,\sigma)d\sigma$
$\displaystyle\mbox{}-\frac{i\omega}{2}\int_{s_{\text{e}_{2}}-t}^{s_{\text{e}_{2}}+t}J_{0}(\omega\sqrt{t^{2}-(s_{\text{e}_{2}}-\sigma)^{2}})\big{(}\mathring{\psi}_{-+\varsigma_{2}}+\mathring{\psi}_{--\overline{\varsigma}_{2}}\big{)}(s_{\text{ph}}-t,s_{\text{e}_{1}}-t,\sigma)d\sigma$

and

$\displaystyle
S_{\varsigma_{2}}(t,s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})$
$\displaystyle=$
$\displaystyle-\frac{\omega^{2}}{2}\int_{0}^{t}\int_{s_{\text{e}_{2}}-(t-u)}^{s_{\text{e}_{2}}+(t-u)}J_{0}(\omega\sqrt{(t-u)^{2}-(s_{\text{e}_{2}}-\sigma)^{2}}){\psi}_{-+\overline{\varsigma}_{2}}(u,s_{\text{ph}}-t+u,s_{\text{e}_{1}}-t+u,\sigma)d\sigma
du$ $\displaystyle-$
$\displaystyle\frac{i\omega}{2}\int_{0}^{t}\int_{s_{\text{e}_{2}}-(t-u)}^{s_{\text{e}_{2}}+(t-u)}J_{0}(\omega\sqrt{(t-u)^{2}-(s_{\text{e}_{2}}-\sigma)^{2}})(\partial_{u}-\overline{\varsigma}_{2}\partial_{\text{e}_{2}}){\psi}_{-+{\varsigma}_{2}}(u,s_{\text{ph}}-t+u,s_{\text{e}_{1}}-t+u,\sigma)d\sigma
du,$

This concludes our construction of $\Psi$ for this case of Compton
configurations. Before moving on to the next case of configurations, we
briefly note that

$|U_{\varsigma_{2}}(t,s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})|\leq(1+\omega^{2}t+2\omega
t)||\mathring{\Psi}||_{\infty},$ (B.24)

while, after integrating by parts and using the estimate from equation (B.19),
we find
$S_{\varsigma_{2}}(t,s_{\text{ph}},s_{\text{e}_{1}},s_{\text{e}_{2}})\leq
O(t)||\mathring{\Psi}||_{\infty}$. This, in combination with our previous
estimates imply that all $8$ components of $\Psi$ satisfy
$|\psi_{\varsigma_{0}\varsigma_{1}\varsigma_{2}}|\leq(1+O(t))||\mathring{\Psi}||_{\infty}$
for this case of Compton configurations.
Case 2: This case covers all configurations for which the second electron is
interacting with the photon, while the first electron is freely evolving. In
this regime, the $\varsigma_{0}=-$ components are freely evolving, and their
solution formulas are given by

$\psi_{-\varsigma_{1}\varsigma_{2}}=\Pi_{-\varsigma_{1}\varsigma_{2}}\bigg{(}\big{(}\mathcal{P}(\textbf{x}_{\text{ph}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{1}})\otimes\mathcal{E}(\textbf{x}_{\text{e}_{2}})\big{)}\mathring{\Psi}\bigg{)}$
(B.25)

Meanwhile, the $\varsigma_{0}=+$ components will be given by Goursat type
evolutions for $\psi_{+\varsigma_{1}-}$, which is then plugged into an
inhomogenous Klein-Gordon equation for $\psi_{+\varsigma_{1}+}$. The
construction of these solutions are nearly identical to the construction for
Case 1, with the desired estimates from Case 1 holding in this regime as well.
Case 3: For the final set of configurations we will construct $\Psi$ on are
those for which both electrons are interacting with the photon. This is
similar to the third case discussed in the multi-time setting, except now we
are not excluding configurations where the two electrons are interacting,
previously called Coulomb configurations. Since we are only evolving for this
short time, we will find that the solution formulas constructed for this case
also extends to Coulomb configurations identically. In this regime, non of the
wave function components are freely evolving, their solutions will be given by
Goursat evolutions and inhomogenous Klein-Gordon equations which are identical
to those found in the previous two cases.

For example, the solution formula for $\psi_{-+\varsigma_{2}}$ in this regime
is still given by (B.13). To see that this solution formula makes sense, we
note that the data necessary to define $G_{\varsigma_{2}}$ as in (B.14) is
data for which the first electron is freely evolving, while the second
electron interacts with the photon. This data for this is given by equation
(B.25). Of greater interest is the data necessary to define
$F_{\varsigma_{2}}$ as in (B.15), which written explicitly is

$\begin{split}F_{\varsigma_{2}}(b)&=\Pi_{\varsigma_{2}}\bigg{(}\mathcal{E}(t-b,s_{\text{e}_{2}})\begin{pmatrix}\mu_{\epsilon}(|s_{\text{e}_{1}}-s_{\text{e}_{2}}|)e^{i\theta_{1}}\psi_{+-+}\\\
\mu_{\epsilon}(|s_{\text{e}_{1}}-s_{\text{e}_{2}}|)e^{i\theta_{1}}\psi_{+--}\end{pmatrix}(b,\textbf{b},s_{\text{e}_{2}})\bigg{)}=\big{(}\mu_{\epsilon}e^{i\theta_{1}}\psi_{+-\varsigma_{2}}\big{)}(b,\textbf{b},s_{\text{e}_{2}}+\varsigma_{2}(t-b))\\\
&-\frac{\omega}{2}\int_{L_{\epsilon}}J_{1}(\omega\sqrt{(t-b)^{2}-(s_{\text{e}_{2}}-\sigma)^{2}})\frac{\sqrt{(t-b)-\varsigma_{2}(s_{\text{e}_{2}}-\sigma)}}{\sqrt{(t-b)+\varsigma_{2}(s_{\text{e}_{2}}-\sigma)}}\big{(}\mu_{\epsilon}e^{i\theta_{1}}\psi_{+-\varsigma_{2}}\big{)}(b,\textbf{b},\sigma)d\sigma\\\
&-\frac{i\omega}{2}\int_{L_{\epsilon}}J_{0}(\omega\sqrt{(t-b)^{2}-(s_{\text{e}_{2}}-\sigma)^{2}})\big{(}\mu_{\epsilon}e^{i\theta_{1}}\psi_{+-\overline{\varsigma}_{2}}\big{)}(b,\textbf{b},\sigma)d\sigma,\end{split}$
(B.26)

where
$\textbf{b}=(s_{\text{ph}}-t+b,s_{\text{ph}}-t+b)\in\mathbb{R}_{\text{ph}}\times\mathbb{R}_{\text{e}_{1}}$,
and

$L_{\epsilon}:=\left\\{\sigma\in\mathbb{R}_{\text{e}_{2}}:|\sigma-
s_{\text{e}_{2}}|<t-b,\quad|\sigma-(s_{\text{ph}}-t+b)|\geq\epsilon\right\\}.$
(B.27)

This extra condition on the region of integration comes from the lack of
support of $\mu_{\epsilon}$ for configurations where the two electrons are
$\epsilon$ distance away. We find that only freely evolving data is necessary
to define $F_{\varsigma_{2}}$ even on configurations where the two electrons
are interacting, since we are only evolving up to $t\leq\frac{\epsilon}{2}$.
It follows that for all $t\leq\frac{\epsilon}{2}$, we have constructed a
global solution for $\psi_{-+\varsigma_{2}}$ in $\mathcal{S}_{1}$ using
solution formulas (B.8) for free configurations, (B.25) for Compton
configurations of case 2, and (B.13) for Compton configuration cases 1 and 3.
The remaining six components can be treated similarly, with the formula
$\psi_{+\varsigma_{1}-}$ also given in terms of Goursat evolution operators
while the other four components are solved by studying inhomogeneous Klein-
Gordon equations. Since all of these formulas are identical to the ones given
in case 1, it can be shown that all of the desired estimates hold in this
regime as well. This concludes our construction of the small time solution to
the leaky boundary IBVP. ∎

##  Appendix C Well-Posedness by Fixed-Point Argument

In this section we very briefly outline an alternative proof of theorem (4.2)
given in [9] which relies on a fixed-point argument. For the photon-electron
system, the fixed-point argument may not seem as valuable as the proof given
by [7] because it does not construct an explicit algorithm for the time
evolution of $\Psi$. But this proof is easily generalized to the many-particle
case, as was done in full generality in [9]. We will now once again state the
theorem.

######  Theorem C.1.

Let $\theta\in[0,2\pi)$ be a constant phase. Let $\mathcal{S}_{1}$ denote the
set of space-like configurations in $\mathcal{M}$, as was defined in (4.1),
and let $\mathcal{C}\subset\partial\mathcal{S}_{1}$ be the coincidence set
defined by $\\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}}\\}$, and let
$\mathcal{I}$ be the initial surface. Let the function
$\mathring{\Psi}:\mathcal{I}\rightarrow\mathbb{C}^{4}$ be $C^{1}$ and
compactly supported in the half-space
$\mathcal{I}\cap\overline{\mathcal{S}_{1}}$, and suppose that the initial-
value function also satisfies the boundary conditions given by (4.2). Then the
following initial-boundary value problem for the wave function
$\Psi:\mathcal{M}\rightarrow\mathbb{C}^{4}$,

$\left\\{\begin{array}[]{rclr}-i\hbar D_{\text{ph}}\Psi&=&0&\\\ -i\hbar
D_{\text{e}}\Psi+m_{\text{e}}\Psi&=&0&\text{in }\mathcal{S}_{1}\\\
\Psi&=&\mathring{\Psi}&\text{on }\mathcal{I}\\\
\psi_{+-}&=&e^{i\theta}\psi_{-+}&\text{on }\mathcal{C}\end{array}\right.$
(C.1)

has a unique global-in-time solution that is supported in
$\overline{\mathcal{S}_{1}}$, and depends continuously on the initial data
$\mathring{\Psi}$.

###### Proof.

We present a brief outline of the work of [9], who proved this for a more much
general class of IBVPs which also allowed data to transfer from the $N$
particle sector to other sectors. Solving the initial-boundary value problem
above is equivalent to solving the following system of integral equations

$\psi_{-\varsigma_{1}}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})=\mathring{\psi}_{-\varsigma_{1}}(s_{\text{ph}}-t_{\text{ph}},s_{\text{e}}+\varsigma_{2}t)-i\omega\int_{0}^{t_{\text{e}}}\psi_{-\overline{\varsigma}_{1}}(0,s_{\text{ph}}-t_{\text{ph}},\tau,s_{\text{e}}+\varsigma_{1}(t-\tau))d\tau$
(C.2)
$\psi_{++}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})=\mathring{\psi}_{++}(s_{\text{ph}}+t_{\text{ph}},s_{\text{e}}+t)-i\omega\int_{0}^{t_{\text{e}}}\psi_{+-}(0,s_{\text{ph}}+t_{\text{ph}},\tau,s_{\text{e}}+(t-\tau))d\tau$
(C.3)
$\psi_{+-}(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})=\left\\{\begin{array}[]{ll}\mathring{\psi}_{+-}(s_{\text{ph}}+t_{\text{ph}},s_{\text{e}}-t)-i\omega\int_{0}^{t_{\text{e}}}\psi_{++}(\textbf{x}_{\text{ph}},\tau,s_{\text{e}}-(t-\tau))d\tau&\mbox{
on }\mathcal{F},\\\
e^{i\theta}\left[\mathring{\psi}_{-+}(s_{\text{e}}-t_{\text{e}},s_{\text{ph}}+t_{\text{ph}})-i\omega\int_{0}^{T}\psi_{--}(T,S,\tau,s_{\text{ph}}+(t_{\text{ph}}-\tau))d\tau\right]\\\
\hskip
72.26999pt\mbox{}-i\omega\int_{T}^{t_{\text{e}}}\psi_{++}(\textbf{x}_{\text{ph}},\tau,s_{\text{e}}-(t-\tau))d\tau&\mbox{
on }\mathcal{N}.\end{array}\right.$ (C.4)

Here

$T:=\frac{1}{2}(s_{\text{ph}}+t_{\text{ph}}-s_{\text{e}}+t_{\text{e}})>0,\quad
S:=\frac{1}{2}(s_{\text{ph}}+t_{\text{ph}}+s_{\text{e}}-t_{\text{e}})$ (C.5)

is the space-time location where the backwards light-cones of the photon and
electron cross. To prove existence of a solution to the integral equations we
use the standard fixed-point method. Let

$\mathcal{S}^{T}:=\left\\{(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}})\in\mathcal{S}_{1}|t_{\text{ph}},t_{\text{e}}\in[0,T]\right\\}$
(C.6)

and

$\mathcal{D}:=\left\\{\Psi\in
C^{1}_{b}(\mathcal{S}^{T},\mathbb{C}^{4})|\Psi=\mathring{\Psi}\text{ on
}\mathcal{I}\right\\}.$ (C.7)

Let $\gamma>0$. We equip the function space $\mathcal{D}$ with the weighted
norm

$||\Psi||_{\gamma}:=||e^{\gamma
t_{\text{e}}}\Psi||_{L^{\infty}(\mathcal{S}^{T})}+\max_{\mu\in\\{t_{\text{ph}},s_{\text{ph}},t_{\text{e}},s_{\text{e}}\\}}||e^{\gamma
t_{\text{e}}}\partial_{\mu}\Psi||_{L^{\infty}(\mathcal{S}^{T})}.$ (C.8)

Clearly $\mathcal{D}$ equipped with this norm is a Banach space. Let $F$
denote the mapping defined by equations (C.2,C.3,C.4), so that these equations
reduce to $\Psi=F(\Psi)$. It suffices to show that $F$ is a well-defined map
from $\mathcal{D}\to\mathcal{D}$, and that

$||F(\Psi)-F(\Phi)||_{\gamma}\leq\frac{4\omega}{\gamma}||\Psi-\Phi||_{\gamma}.$
(C.9)

Setting $\gamma>4\omega$, it follows that $F$ contraction mapping on
$\mathcal{D}$, so that by Banach’s fixed point theorem there exists some wave
function fixed by the integral mapping. ∎

This same technique is applied in [9] to systems with many massive particles,
and so also holds for our IBVP of one photon between two electrons.

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