First Quantized Pair Interactions
A. F. Bennett
The parametrized Dirac wave equation for multiple particles is shown here to yield the standard cross sections for creation and annihilation of fermion pairs, while avoiding the paradoxes arising from the standard single-particle Dirac equation and hole theory. These paradoxes were originally overcome with Quantum Field Theory. The creation and annihilation operators of Quantum Field Theory are subject to causality conditions which preclude entanglement across distances greater than the order of the Compton wavelength. The parametrized first-quantized formalism requires no causality condition for fermions, and it admits fermion entanglement across all of space and time, while ensuring covariant Dyson series and ${\mathcal TPC$ invariance.
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\begin{document}



\title{First Quantized Pair Interactions}

 


\author{A. F. Bennett}

\email{afbennett97333@post.harvard.edu}



\affiliation{College of Earth, Ocean and Atmospheric Sciences\\
Oregon State University\\104 CEOAS Administration Building\\ Corvallis, OR 97331-5503, USA}

\date{\today}


\begin{abstract}


The parametrized Dirac wave equation for multiple particles is shown here to yield the standard cross sections for  creation and annihilation of fermion pairs, while avoiding the paradoxes arising from the standard single-particle Dirac equation and hole theory. These paradoxes were originally overcome with Quantum Field Theory.  The creation and annihilation operators of Quantum Field Theory are subject to causality conditions which preclude entanglement across distances greater than the order of the Compton wavelength. The parametrized first-quantized formalism requires no causality condition for fermions, and it admits fermion entanglement across all of space and time, while ensuring  covariant Dyson series and  ${\mathcal TPC}$ invariance.


\end{abstract}

\maketitle
%
%\begin{keyword}
%Relativistic quantum mechanics; pair interactions; entanglement; discrete symmetries
%\end{keyword}
%
%
%
%\end{frontmatter}


\section{Introduction}\label{S:intro}
The parametrized Dirac wave equation is a first-quantized formalism for quantum electrodynamics \cite{Alv98,John69,Fan93,Bennett:2014mz}. Interactions are semiclassical, that is, electromagnetic interaction potentials are sustained by M\o ller currents \cite{Bennett:2014mz,BjDr64,Tay72}. Perturbation theory yields  the one--loop corrections and the axial anomaly  obtained originally with Quantum Field Theory (QFT) \cite{Bennett:2014mz}. The Bethe-Salpeter equation for bound states \cite{Salp51, Grei03} is obtained from the parametrized formalism without further conjecture \cite{Bennett:2014mz}. Unlike QFT  \cite{IZ, Wei95, Zee03, Sr07}, a causality condition for fermions is not essential to the parametrized formalism. The causality conditions for fermions and bosons render QFT almost incapable of representing quantum entanglement: in the massive vacuum state, entanglement decays exponentially with spacelike separation measured in Compton wavelengths \cite{Summ11}. The parametrized formalism provides a simple and unrestricted representation of fermionic entanglement in space-time. The formalism also yields the spin-statistics connection \cite{Benn2015}, again without presupposing a fermionic causality condition, by extension of the nonrelativistic first-quantized proof of Jabs \cite{Jabs10}. The fundamental criticism of the standard Dirac formalism \cite{Dir1930,Opp30}, and by implication the parametrized formalism, must however be answered. If the scattering of electrons into states of unboundedly negative energy must be excluded, then the negative energy states must be identified as different particles \cite{Dir1931}, namely, positrons \cite{And33}. How, then, in a particle--conserving formalism can pair annihilation such as 
\be\label{annih}
e^{-} + e^{+} \xrightarrow{t} \gamma+ \gamma
\ee
be represented? The symbol $t$ above the arrow indicates that coordinate time $t$ increases from left to right. It is shown here that the ostensibly single-particle scattering amplitude obtained \cite{BjDr64} from  the Dirac equation is in fact an amplitude for the two-particle parametrized Dirac equation. Creation of a muon pair $\mu^-\mu^+$ subsequent to $e^-e^+$ annihilation is similarly shown to be a scattering event for the four-particle equation. The argument is supported with a minimum of detail. The solution structure for the  parametrized Dirac equation is very close to that of the standard equation \cite{BjDr64,Grei03}, and comprehensive detail is available elsewhere \cite{Bennett:2014mz}.

The parametrized Dirac equation and its plane wave solutions are outlined in Section \ref{S:PDWE}. Scattering amplitudes for pair annihilation and pair creation are recovered in Sections \ref{S:pairann} and \ref{S:paircrea}, using the multiple-particle parametrized Dirac equation. Derivation of the standard cross sections from the parametrized scattering amplitudes is described in Section \ref{S:cross}, listing the differences in derivation between the parametrized and standard formalisms. Entanglement  is expressed in Section \ref{S:entg} in terms of nonadditivity of covariant currents. The lack of need of a causality condition for Dirac particles is discussed in Section \ref{S:cau}. The results are summarized in Section \ref{S:summ}.


\section{The parametrized Dirac wave equation}\label{S:PDWE}

For a single spin--1/2 particle  the parametrized Dirac \wf is a four--spinor  $\psi(x,\tau)$\,. The event $x$ is in ${\mathbb R}^4$, while the parameter  $\tau$ is an independent variable in ${\mathbb R}$\,.  The event $x$ is also denoted by $x^\mu$ having indices $\mu=0,1,2,3$, with $x^0=ct$ where $c$ is the speed of light and $t$ is coordinate time. The Lorentz metric $g^{\mu\nu}$ on  ${\mathbb R}^4$ has  signature $(- + + +)$\,. The position $\bf{x}$  is denoted by $x^j$ having indices $j=1,2,3$. Thus $x=(ct,\bf{x})$\,.

The parametrized Dirac wave equation for $\psi$ is  
\be\label{PDIR}
\frac{\hbar}{\rmi c}\frac{\partial}{\partial \tau}\psi+\gamma^\mu\Big(\frac{\hbar}{\rmi}\frac{\partial}{\partial x^\mu}-\frac
{e}{c}A_\mu\Big)\psi=0
\ee
where $e$ is the charge of the particle, $c$ is the speed of light and $\hbar$ is the reduced Planck's constant. The $\gamma^\mu$ are the four Dirac matrices,  while the  Maxwell electromagnetic potential $A^\mu(x)$ is  independent of the parameter $\tau$\,.  The summation convention is assumed with respect to repetitions of Greek indices such as $\mu =0,1,2,3$\,. The covariant and contravariant indices $\mu,\nu,\dots$ will be omitted wherever convenient, as in $x= x^\mu$\,, $p = p^\mu$\, and  $p\cdot x = p^\mu x_\mu$ \,. Henceforth the units are chosen such that  $c=\hbar=1$\,.  The covariance of the theory with respect to the homogeneous Lorentz transformation $(x^\mu)' =\Lambda^{\mu}_{\;\;\nu}x^\nu$ and $ \psi'(x,\tau)=S(\Lambda)\psi(\Lambda^{-1}x,\tau)$ follows for $S(\Lambda)$  generated in the standard way \cite{BjDr64}. No mass constant appears  in (\ref{PDIR}), but masses are introduced through boundary conditions as $\tau\to \pm \infty$\,. Feynman's development of quantum electrodynamics using (\ref{PDIR}) has been reviewed by Garcia Alvarez  and Gaioli \cite{Alv98}. The indefiniteness of the invariant bilinear form $\overline{\psi}\psi=\psi^\dag \gamma^0 \psi$ has impeded \cite{Barut85, Evans98} the development of the parametrized Dirac formalism as a relativistic extension of quantum mechanics. 

The plane \wf solutions of (\ref{PDIR}) in the absence of an electromagnetic field are of the form
\begin{subequations}\label{free}
\begin{gather}
{\sf f}^{(+)}_p(x,\tau)=\frac{{\sf u}_p}{(2\pi)^2}\exp[\rmi\chi^{(+)}]\,, \label{free1}\\ {\sf f}^{(-)}_p(x,\tau)=\frac{{\sf v}_p}{(2\pi)^2}\exp[\rmi\chi^{(-)}] \,.\label{free2}
\end{gather}
\end{subequations}
In (\ref{free}), $\chi^{(\pm)}=\chi^{(\pm)}(p,x,\tau)=p \cdot x \pm \varphi_p m_p \tau$,  $\varphi_p=\mathrm{sgn}(p^0)= p^0/E_p$ where $E_p=|p^0|$ is the energy of the particle, and $m_p=m(p)$ is the positive square root of $-p \cdot p$ for subluminal energy-momentum $p$\,. At constant phase $\chi^{(\pm)}$, $dx^0/d\tau=\pm m_p/E_p$ regardless of the value of $\varphi_p$\,, so ${\sf f}^{(+)}_p$ and ${\sf f}^{(-)}_p$ propagate respectively forward  and backward  in   coordinate time $t=x^0$ as $\tau$ increases. The $4 \times 4$ block  of amplitudes $({\sf u}_p, {\sf v}_p)$ is a  basis of four  Dirac spinors constructed in the standard way \cite{Bennett:2014mz} from the elementary basis (the unit matrix) for the rest frame where ${\bf p}= {\bf 0}$\,. The basis is an even function of $p$\,. The orthonormality of the basis is expressed as $\overline{{\sf u}}_p{\sf u}_p = {\sf I}_2\,, \overline{{\sf v}}_p {\sf v}_p =-{\sf I}_2, \overline{{\sf u}}_p {\sf v}_p  =\overline{{\sf v}}_p {\sf u}_p={\sf 0}$\,, where ${\sf I}_2$ is the $2 \times 2$ unit matrix. Antiparticle \wfs ${\sf h}^{(-)}_p$ are formed from particle \wfs ${\sf f}^{(+)}_p$ by combining the discrete symmetries \cite{Bennett:2014mz} of (\ref{PDIR}), yielding\footnote{The last equation in this string corrects an isolated error in Eq. (23) of Bennett (2014) \cite{Bennett:2014mz}.} ${\sf h}^{(-)}_p(x,\tau)=(\mathcal {TPC}\,{\sf f}^{(+)}_p)(x,\tau)=-\rmi\gamma^5 {\sf f}^{(+)}_p(-x,\tau)=-\rmi {\sf f}^{(-)}_{-p}(x,\tau)$\,. Physically real particles and antiparticles are associated here with free on-mass-shell \wfs  having positive energy and positive mass, that is, they have energy $p^0>0$ and their \wfs are proportional to $\exp[+\rmi m_p \tau]$\,, with $m_p$ equal for example to the electron mass $m_e$\,. Thus a backward-propagating,  positive-energy  positron has the \wf that might otherwise be attributed to a `backward-propagating, negative-energy electron'.   While the coordinate time $t$ for a free positron \wf ${\sf h}^{(-)}_p(x,\tau)$ at constant phase does decrease as the parameter $\tau$  increases, its spacelike coordinates $\bf x$ change in the same sense and at the same rates with respect to coordinate time $t$  as those of its $\mathcal {CPT}$ conjugate \wf ${\sf f}^{(+)}_p(x,\tau)$\,. 

It is readily verified that $\mathcal{TCP}$ conjugation commutes with not only the vector  interaction in (\ref{PDIR}), but with all the  invariant linear interactions (scalar, pseudoscalar, vector, axial vector and tensor) \cite{BjDr64}. In particular, the M\o ller currents and hence the semiclassical potentials transform as vectors.

The particles and antiparticles here are, again, positive-energy \wfs propagating respectively forward and backward in time as $\tau$ increases. Their Dirac spinor amplitudes,  respectively ${\sf u}_p$ and ${\sf v}_p$\,, are exactly the same as those for standard particles and antiparticles \cite{BjDr64}\,. It follows that the spinor-dependent factors or `invariant amplitudes' in the scattering amplitudes derived in Section \ref{S:cross}  are identical to the standard invariant amplitudes \cite{BjDr64} .


\section{Pair annihilation}\label{S:pairann}
An incident  free \wf $\phi_i$ scatters off an electromagnetic field as a final  free \wf $\phi_f$\,, with scattering amplitude \cite{Bennett:2014mz,Tay72} 
\be\label {scamp}
S_{fi}=(\pm)\lim_{\tau \to + \infty}\int d^4x \;\overline{\phi_f(x,\tau)}(\omega_+ \phi_i)(x,\tau)
\ee
where $\omega_+$ is the forward M\o ller operator \cite{Bennett:2014mz} for  (\ref{PDIR}). Consistent with the normalization of the spinor basis, the leading sign in (\ref{scamp})  is $(+)$ for a final \wf that is forward-propagating  and $(-)$ for one that is backward-propagating. There is a nonvanishing amplitude for the mathematical contingency of a forward-propagating, positive-energy \wf scattering off two free photons and into a backward-propagating, negative-energy wavefunction. To leading order, the nontrivial and nonvanishing amplitude is 
%\be\label{oppie}
\be\label{oppie}
%for multiline insert \\ \times  after w'
S_{fi}^{(2)}=-\rmi e^2\int d^5w\int d^5w'\,  \overline{{\sf h}^{(-)}_f(w)}\fy{A}(x)\Gamma^{0}_{+}(w-w')\fy{B}(x'){\sf f}^{(+)}_i(w)\,,
\ee
%\ee
where ${\sf h}^{(-)}_f={\sf h}^{(-)}_p$ for $p=p_f$ and  ${\sf f}^{(+)}_i={\sf f}^{(+)}_p$ for $p=p_i$\,, while $w=(x,\tau)$\,. The Feynman slashed notation is $\fy{A}=\gamma^\mu A_\mu$ and $\fy{B}=\gamma^\mu B_\mu$\,, where the classical potentials $A^\mu$ and $B^\mu$ represent the two free photons. Finally, $\Gamma^{0}_+$ is the free forward \infl  for positive-energy fermions \cite{Bennett:2014mz}\,. For clarity, the spins of the incident and final \wfs have not been stipulated and the additive amplitude with photons exchanged \cite{BjDr64} is suppressed. The integrals with respect to $\tau$ reduce to a delta function that enforce equality of the incident and final masses $m_i=m(p_i)$ and $m_f=m(p_f)$ respectively. Otherwise the parameterized scattering amplitude (\ref{oppie}) is exactly the standard Dirac amplitude \cite{BjDr64}. As discussed in Section \ref{S:cross}, the cross section corresponding to (\ref{oppie}) is exactly the standard cross section. 

Consider further the leading-order amplitude (\ref{oppie}). Identifying the plane \wf ${\sf h}^{(-)}_f$\,, which again is the $\mathcal{TCP}$ conjugate of the final  `negative-energy electron' ${\sf f}^{(-)}_f$  as the parameter $\tau \to +\infty$\,, with a positive-energy positron injected as the coordinate time $t \to -\infty$ yields the standard and routinely observed cross section for pair annihilation into gamma rays. The interaction is now expressed as 
\be\label{annihtau}
e^{-}  +\gamma \xrightarrow{\tau} e^{+} + \gamma\,.
\ee 
In (\ref{annihtau}) it is the parameter $\tau$ which increases from left to right.  The standard first-quantized Dirac formalism does not support more than one particle, much less the annihilation of one particle and the creation of another as expressed by (\ref{annihtau}). The issue is resolved by analyzing  the \infl or internal line in (\ref{oppie}) as
\begin{multline}\label{pq}
\Gamma^{0}_+(w-w')=\rmi \int d^4p\int d^4q\, \delta^4(p-q)\\ \times  \Bigg\{\theta (\tau-\tau') \theta(p^0)\theta(q^0)-\theta(\tau'-\tau)\theta(-p^0)\theta(-q^0)\Bigg\}  \\ \times  \Bigg[{\sf f}^{(+)}_p(w)\overline{{\sf f}^{(+)}_q(w')}-{\sf h}^{(-)}_p(w)\overline{{\sf h}^{(-)}_q(w')} \Bigg]\,,
\end{multline}
where $\theta$ is the Heaviside unit step function. The second-order amplitude (\ref{oppie}) is thereby analyzed as a uniformly weighted sum of products of first-order amplitudes, for 
\begin{itemize}
\item [(i)] a physical electron  ${\sf f}^{(+)}_i$ scattering into  a virtual (off mass shell) electron ${\sf f}^{(+)}_q$ of positive mass $m_q$\,, and a virtual electron ${\sf f}^{(+)}_p$ of positive mass $m_p$ scattering into a physical positron ${\sf h}^{(-)}_f$\,, or 
\item[(ii)] a physical electron ${\sf f}^{(+)}_i$ scattering into a virtual positron ${\sf h}^{(-)}_q$ of positive mass $m_q$, and a virtual positron ${\sf h}^{(-)}_p$ of positive mass $m_p$ scattering into a physical positron  ${\sf h}^{(-)}_f$.  
\end{itemize}
In case (i) the virtual energies $p^0$ and $q^0$ are positive, while in case (ii) they are negative. The minus sign preceding the second tensor product in (\ref{pq}) is consistent with the normalization of the spinor basis. The other two scattering contingencies $-{\sf f}^{(+)}_p\overline{{\sf h}^{(-)}_q}$ and $+{\sf h}^{(-)}_p\overline{{\sf f}^{(+)}_q}$ are kinematically excluded. Indicating  the virtual particles with braces $\{\dots\}$,  in case (i) above the interaction is now 
\be\label{annvirt1}
e^{-} + \{e^{-}\} +\gamma \xrightarrow{\tau}  \{e^{-}\}+e^{+} +\gamma\,,
\ee
while in case (ii) it is 
\be\label{annvirt2}
e^{-} + \{e^{+}\} +\gamma \xrightarrow{\tau}\{e^{+}\}+e^{+} +\gamma\,.
\ee
Since the double integral with respect to $p$ and $q$ is uniformly weighted, the abovementioned single-particle amplitudes vanish\footnote{This is most clearly seen by expanding  the single-particle amplitudes (\ref{scamp}) in the Born series.  Integrals over $\tau$ arise, and these may be postponed.}.   It is a textbook exercize \cite{BjDr64}, taking into account the kinematical constraints,  to verify that the amplitude products which are the  integrands on the right hand side of  (\ref{oppie}) modulo (\ref{pq}) are precisely those arising from the two-particle parametrized Dirac wave equation \cite{Bennett:2014mz}
 \be\label{TWO}
\frac{1}{\rmi}\frac{\partial}{\partial \tau}\Psi+\fy{\pi}(x)\otimes {\sf I}_4\,\Psi+{\sf I}_4\otimes \fy{\pi}(y)\,\Psi=0\,,
\ee
where $\Psi(x,y,\tau)$ is the two-particle wavefunction, while $\pi^\mu(x)=(1/\rmi)\partial /\partial x_\mu-e_1A^\mu(x)$ and $\pi^\mu(y)=(1/\rmi)\partial /\partial y_\mu-e_2A^\mu(y)$\,. The field $A^\mu$ in (\ref{TWO}) denotes the total field, or the sum of the two fields denoted $A^\mu$ and $B^\mu$ in (\ref{oppie}). Tensor products of the single-particle free \wfs  (\ref{free}) form a basis for the two-particle free wavefunctions.  The number of particles is conserved by the \textit{two}-particle scattering event (\ref{oppie}) analyzed with  (\ref{pq})\,: there are two incident particles, one physical and one virtual,  and there are two final particles, one virtual and one physical. 


\section{Pair creation}\label{S:paircrea}
It is observed that an electron and a positron can annihilate, followed by the creation of a muon and an antimuon. In the sense of $t$ increasing the interaction is
\be\label{teemupr}
e^{-} + e^{+} \xrightarrow{t}  \mu^{-} + \mu^{+}\,.
\ee    
The interaction is mediated by a virtual photon. In terms of $\tau$ increasing the interaction is
\be\label{taumupr}
e^{-} + \mu^{+} \xrightarrow{\tau} e^{+} + \mu^{-}\,.
\ee
The amplitude for the interaction may be constructed semi-classically using (\ref{PDIR}). Consider the scattering of the antimuon into a muon.  The leading-order nontrivial amplitude is 
\be\label{am2m}
S_{fi}^{(1)}=\rmi e\int d\tau \int d^4 x \,\overline{f^{(+)}_{f'}(x,\tau)}\fy{A}(x)h^{(-)}_{i'}(x,\tau)\,.
\ee
The primed subscripts $f'$ and $i'$ indicate that the \wfs have the energy-momenta of the final antimuon and incident muon respectively.
The scattering potential $A^\lambda$ is the (virtual) photon owing to the electron-positron current. That is, 
\be\label{Max}
A^{\lambda}(x)=e\int d^4y\, D^{\lambda\nu}(x-y) J_{\nu}(y)
\ee
where $D^{\lambda\nu}$ is the standard \infl for a massless vector boson \cite{Zee03}, while $J^{\nu}$ is the M\o ller current owing to the electron and positron. To leading order,
\be\label{e2p} 
J^{\lambda}(y)=\int d\sigma\,  \overline{h^{(-)}_{f}(y,\sigma)}\gamma^{\lambda}f^{(+)}_{i}(y,\sigma)\,.
\ee
Note that the current has been concatenated \cite{Bennett:2014mz,ArHoLa83} with respect to the parameter, that is, integrated for all $\sigma$\,.  

The amplitude (\ref{am2m})--(\ref{e2p}) again raises the issue for first-quantized formalism: a positive-energy physical antimuon is being scattered into a different particle, namely, a positive-energy physical muon. There is of course an identical amplitude for the physical electron being scattered into a physical positron by the potential  sustained by the antimuon-muon current. The issue is resolved in two stages. First, the boson \infl is analyzed as a sum of products. Second, the number  of vertices is doubled from two to four by introducing two trivial vertices. At each vertex there is an on-shell \wf for a physical particle and an off-shell \wf for a virtual particle. There is also a virtual photon at each of the two original vertices, but none at either of the two trivial vertices. The amplitude is then precisely of the form arising from the four-particle parametrized Dirac equation. Some details now follow. 

First, the concatenation in (\ref{e2p}) is postponed, and the massless vector boson \infl  is replaced with the \infl for a massive vector boson \cite{Zee03}. The spectral representation of the \infl becomes
\be\label{mbos}
D^{\lambda\nu}(k,\varpi)=\frac{G^{\lambda\nu}(k,\varpi)}{k\cdot k+\varpi^2}\,,
\ee
where $k$ is the four-vector wavenumber, while $\varpi$ is the mass (the frequency with respect to the parameter $\tau$ as in $\exp[+\rmi \varpi \tau]$\,). The numerator is variously expressed as
\be\label{numer}
G^{\lambda\nu}(k,\varpi)=g^{\lambda\nu}+\frac{k^\lambda k^\nu}{\varpi^2}=\sum_{j=1}^3 \varepsilon_j^\lambda \varepsilon_j^\nu\,,
\ee
where $\varepsilon_j^{\lambda}(k)$ is a polarization amplitude such that $k_\mu\varepsilon_j^{\mu}=0$ for $j=1,2,3$\,. It is evident that the boson \infl is an integral of products of virtual, $\tau$-dependent `preMaxwell' photons \cite{AhHo06}, or plane waves that are off the `shell' where $k\cdot k =-\varpi^2$\,.  
Second, the incident antimuon in (\ref{am2m}) is expressed as 
\begin{multline}\label{virtvex}
h^{(-)}_{i'}(x,\tau)=\frac{1}{\rmi}\int d^4z \{\theta(\tau-\rho)\Gamma^0_+(x-z,\tau-\rho)-\\ \theta(\rho-\tau)\Gamma^0_-(x-z,\tau-\rho)\}h^{(-)}_{i'}(z,\rho)\,.
\end{multline}
The fermion influence function $\Gamma^0_{-}$ has \cite{Bennett:2014mz} an  analysis  similar  to (\ref{pq}). The expression (\ref{virtvex}) can accordingly be interpreted as the trivial scattering of the incident physical antimuon $h^{(-)}_{i'}$ into a virtual particle. The other virtual particle (in the tensor product within either \infl) is incident to the physical vertex at $(x,\tau)$\,. The incident virtual particle scatters off the virtual photon and into the final physical muon $f^{(+)}_{f'}$\,. 

The analysis of the boson influence function,  the introduction of trivial vertices and the analysis of the fermion \infl convert both the $e^-e^+$ side and the $\mu^-\mu^+$ side of the single-particle {\it ansatz } (\ref{am2m})--(\ref{e2p}) to instances of (\ref{scamp}). Then (\ref{am2m})--(\ref{e2p}) is  expressed as an amplitude for a multiple-particle parametrized Dirac equation, without any physical particle being converted by scattering into a different physical particle. The expression is verbose, but the {\it ansatz} is exactly equivalent and provides a  convenient short-hand.
\section{Cross sections}\label{S:cross}
Series expansions of the scattering amplitude (\ref{scamp}) lead identically to the standard cross sections \cite{BjDr64,IZ, Wei95, Sr07}. First-order Coulomb scattering of an electron was considered in \cite{Bennett:2014mz}.  The second-order amplitude (\ref{oppie}) for pair annihilation into two gamma rays yields the standard cross section for Compton scattering (except that the final particle is a positron rather than an electron), while the first-order amplitude (\ref{am2m})-(\ref{e2p}) yields the standard cross section for Bahba scattering (except that a muon-antimuon pair is created rather than an electron-positron pair). The derivations here are almost standard and so only points of difference will be listed. As mentioned in Section \ref{S:PDWE}, the spinor-dependent factors or `invariant amplitudes' here are identical to the standard invariant amplitudes \cite{BjDr64} .

\begin{enumerate}
\item[1.] The free \wfs are normalized in a 4-cube of space-time with edge $L$, rather than in spatial 3-cube. The element $d^4x$ is invariant and so it suffices to replace the factor $1/(2\pi)^2$ in (\ref{free}) with $1/L^2$\,.
\item[2.] In addition to having the one factor $(2\pi)\delta^4(p_f - p_i -\dots)$ expressing conservation of total energy-momentum, the scattering amplitude has for each particle a factor $2\pi\delta(m_f-m_i)$ expressing conservation of each mass.
\item[3.] Division of  the squared magnitude of the amplitude by  $(2\pi)^4\delta^4(0)2\pi\delta(0)$ yields a rate with respect to the parameter $\tau$, per unit 4-volume of space-time.
\item[4.] The number of states in an element of phase space is a product of invariant factors all of the form $L^4d^4p/(2\pi)^4$\,, with one such factor for each particle. 
\item[5.] Division by an invariant incident flux with respect to $\tau$, such as
\be
J_{{\rm inc}}=m_im_{i'}/L^4\sqrt{{\bf p}_i\cdot{\bf p}_{i'}-m_im_{i'}}
\ee
in the case of two--particle scattering \cite{Wei95}, yields an invariant cross section.
\item[6.] As a consequence of the relation 
\be
\delta(m_f-m_i)dp^0_f=\delta(m_f-m_i)\frac{m_f}{E_f}dm_f
\ee
for fixed ${\bf  p}_f$ where $E_f^2=m_f^2+|{\bf p}_f|^2$\,,   the cross section in for example the case of two-particle scattering is exactly the  standard cross section multiplied by $2\pi\delta(0)$\,.
\item[7.] The cross section thus far is a 3-volume in space-time (per unit 4-volume of space-time), rather than an area. However  the interaction takes place over an  interval of proper time which may  \cite{HoLa82} be taken to be $2\pi\delta(0)$, and a final division by this factor completes exact agreement with the standard cross section.
\end{enumerate}
 
\section{Entanglement}\label{S:entg}

A nonseparable two-particle \wf provides an elementary and unrestricted representation of entanglement. Consider the nonseparable \wfs \be\label{ent}
\Psi_{\mp}(x,y,\tau)=\psi(x,\tau)\otimes\xi(y,\tau) \mp \xi(x,\tau)\otimes\psi(y,\tau)
\ee
 for two identical Dirac particles  with \wfs $\psi(x,\tau)$ and $\xi(x,\tau)$, each satisfying (\ref{PDIR}). It is assumed that the field $A^\mu$ is external, and so $\Psi_{\mp}(x,y,\tau)$ satisfy (\ref{TWO}). The \wf  $\Psi_-$ is antisymmetric in accordance with the spin--statistics connection after Jabs  \cite{Jabs10}, while $\Psi_+$ is symmetric.  The states represented by $\Psi_{\mp}$ are entangled for each value of $\tau$, and for all spacelike and timelike separations $x-y$\,. The entanglement is maximal in the following sense.  If instead the two-particle \wf has the  separable form $\Psi(x,y,\tau)=\psi(x,\tau) \otimes \xi(y,\tau)$\,, then \cite{Bennett:2014mz} the currents owing to the two particles are additive. That is, the total current is the sum of the individual currents. If the two-particle \wf is nonseparable, being a linear combination of $\psi \otimes \xi$ and $\xi \otimes \psi$, then the two linear combinations in (\ref{ent}) yield the maximal departure from additivity.  As discussed in \cite{Bennett:2014mz}, the symmetric or bosonic form $\Psi_+$ must be used for the calculations of M\o ller currents in order that the scattering matrix for a pair interaction be symmetric with respect to the two Dirac particles. The antisymmetric or fermionic form $\Psi_-$ must of course be used for the incident and final \wfs that define the scattering matrix. 

The standard quantum-mechanical discussion of entanglement involves low-energy approximations to invariantly normalized timelike components of the covariant total two-particle currents discussed above. The components are positive-definite and so, following concatentation, define correlations in the standard way. The principle of entanglement is common to both discussions. 


\section{Causality}\label{S:cau}
The commutation and anticommutation rules for QFT operators at two events in space-time are in general inhomogeneous. They are homogeneous for spacelike separations, as a causality condition \cite{Wei95}. The rules for parametrized quantum mechanics involve no causality  condition. The commutation rules for the event $x^\mu$ and the energy-momentum $p^\nu=(1/\rmi)\,\partial/\partial x_\nu$ at the same event are  
\be\label{commxp}
[x^\mu,p^\nu]=x^\mu p^\nu -p^\nu x^\mu = \rmi g^{\mu\nu}\,,
\ee
while the event $x^\mu$ commutes with the energy-momentum $q^\nu=(1/\rmi)\,\partial/\partial y_\nu$ at the event $y^\nu$  for all nonvanishing spacelike and timelike separations $x-y$\,. The Dirac matrices in (\ref{PDIR}) and (\ref{TWO}) obey the inhomogeneous anticommutation rules
\be\label{gamma}
\{\gamma^\mu,\gamma^\nu\}=-2g^{\mu\nu}\,,
\ee
without reference to events in space-time.

The \infls $D^{\nu\lambda}(x)$ and $D^{\nu\lambda}(x,\tau)$ in Section \ref{S:paircrea} are both constructed as a contour integral. The semiclassical path leads to retarded influence for nonnegative mass $\varpi$\, (influence upon or within the forward light cone only), and to advanced influence for negative $\varpi$ (upon or within the backward light cone only) \cite{BjDr64,IZ}.  That is, the path choice is a causality condition and so there is a causal electromagnetic interaction here between charged particles. There is no causality condition here upon the quantum mechanics of noninteracting particles, such as neutrinos or those entangled as in (\ref{ent})\,.  

The fermion influence function $\Gamma^0_+(x-x',\tau-\tau')$ in (\ref{oppie})
is also constructed as a contour integral. As may be seen in (\ref{pq}), the contour is chosen so that free states of positive mass (\wfs proportional to $\exp[+\rmi m_p \tau]$) only evolve for $\tau > \tau'$\,, while  negative mass states only evolve for $\tau > \tau'$\,. The choice is independent of light cone considerations and so is not a causality condition.

The proof of $\mathcal{TPC}$ invariance in Quantum Field Theory requires fermionic and bosonic causality conditions \cite{Sozzi08}, but no fermionic condition is required here. Perturbation solutions for QFT may be expressed as Dyson series, which may in turn be expressed as sums of coordinate-time-ordered ($t$-ordered) products. A causality condition ensures that the sums are covariant \cite{Wei95}. The Dyson series for the  parametrized quantum-mechanical formalism (\ref{PDIR}) are sums of parameter-ordered  ($\tau$-ordered) products, and (\ref{PDIR}) is manifestly covariant, hence  a fermionic causality condition is not necessary.


\section{Summary}\label{S:summ} Particles and their antiparticles may be represented in  relativistic quantum mechanics as the positive-energy free solutions of the parametrized Dirac wave equation, respectively propagating forward in time and backward in time with increasing parameter \cite{Bennett:2014mz}. It has been shown here that the annihilation and creation of physical (on-shell) Dirac particles are represented by either real or virtual electromagnetic waves scattering incident physical particles off mass shell, and by scattering  incident virtual particles  onto mass shell. The parametrized scattering amplitudes lead to exactly the standard cross sections.

Unrestricted space-time entanglement,  the covariance of Dyson series and  $\mathcal{TPC}$ invariance are simple, acausal consequences of the parametrized Dirac equation. 





\section*{References}
\bibliography{paraDirac_Bib}


\end{document}
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