###### Abstract

We present a new framework in which the interactions among quantum fields are
constrained by Lorentz-invariant thermalization. Thermalization, which occurs
as a natural and inevitable consequence of the interactions among the modes of
quantum fields, renders the scattering amplitudes naturally divergence-free at
all orders. We present the following calculations as a preliminary test of the
new framework. The 1-loop correction to electron’s anomalous magnetic moment,
calculated using the new framework, is shown to differ from the experimentally
measured value by less than 0.06%. The calculation of the Lamb shift yields a
result that differs from the experimentally measured value by less than 0.33%.
At momentum transfer scales that are one to four orders of magnitude above the
electron mass scale111 That is, over the range
$10\,m_{{}_{e}}\leq\sqrt{|\rule{0.0pt}{10.0pt}k^{2}|}\leq 10^{4}\,m_{{}_{e}}$
where $k$ is the transferred 4-momentum and $m_{{}_{e}}$, the physical rest
mass of the electron.
, the running of the fine structure constant, calculated in the new framework,
is shown to deviate from the previously known results by less than 0.8%
(deviating by less than 0.5% at the upper end of the range). At the tree
level, the angular distribution of the differential cross section for Compton
scattering predicted by the new framework is shown to be in good agreement
with the experimental results of Friedrich and Goldhaber [24] and the well-
known Klein-Nishina formula. Finally, the free Maxwell field’s contribution to
vacuum energy density predicted by the new framework is shown to be
approximately three orders of magnitude smaller than the WMAP’s latest upper
bound for dark energy density222 WMAP: Wilkinson Microwave Anisotropy Probe. ,
in contrast to the divergent contribution predicted by previous results.

Towards Divergence-free Theory of Quantum Fields

Nagabhushana Prabhu

##  1 Introduction

The laws of classical and quantum mechanics provide a satisfactory description
of the behavior of a system if the system has a small number of interacting
particles (modes). However, when the number of interacting particles in a
system becomes large an effective description of the behavior of the system
necessitates postulation of new laws, namely the laws of statistical
mechanics. In the words of Landau and Lifshitz [60], these new statistical
laws “are of a different kind” and “cannot in any way be reduced to purely
mechanical laws”. Further, the statistical mechanical laws, which are based
largely on combinatorial arguments, are applicable regardless of the nature of
the microscopic interactions among the constituent particles in the system,
and must be incorporated into any theory that seeks to describe the behavior
of a system with a large number of interacting degrees of freedom.

A quantum field has infinitely many degrees of freedom. Hence the behavior of
quantum fields is constrained not only by the laws that govern the microscopic
local interactions among their modes but also by the laws of statistical
mechanics that emerge as a result of the collective interactions of the
infinitely many modes. However, the calculational procedure that is currently
used to compute the scattering amplitudes in quantum field theory relies only
on the details of microscopic interactions among fields, and is not
constrained by the laws of statistical mechanics. In this paper, we describe a
new framework in which the laws of statistical mechanics are incorporated into
the theory of quantum fields. An immediate consequence of constraining the
interactions among quantum fields with the laws of statistical mechanics is
that the scattering amplitudes are rendered naturally divergence-free.

We present an overview of the new framework below, deferring the details to
later sections. An interacting quantum field can be regarded as a system that
is in thermal and diffusive contact with the other quantum fields.
Specifically, let $\left\\{{\cal Q}_{{}_{1}},{\cal
Q}_{{}_{2}},\ldots\right\\}$ be the set of all quantum fields that coinhabit
spacetime. Then, we can regard an arbitrary quantum field, say ${\cal
S}=\left\\{{\cal Q}_{{}_{1}}\right\\}$, as our system of interest and the
complement ${\cal R}_{{}_{{\cal S}}}=\left\\{{\cal Q}_{{}_{2}},{\cal
Q}_{{}_{3}},\ldots\right\\}$ as the reservoir with which the system is in
thermal and diffusive contact. Being in thermal and diffusive contact, the
system can exchange both energy and particles with the reservoir.

For example, we can take ${\cal Q}_{{}_{1}}$ (the system) to be the quantum
field corresponding to an electron. Then a process such as $W^{-}$ decay

$\displaystyle W^{-}\rightarrow e^{-}+\bar{\nu}_{e}$

represents the flow of a particle (electron) from the reservoir into the
system. When a particle flows from ${\cal R}_{{}_{{\cal S}}}$ into ${\cal S}$
we regard the event as creation of a particle.

Particle creation represents a transition between two microstates of the
system-reservoir complex ${\cal C}_{{}_{{\cal S}}}=\left\\{{\cal S},{\cal
R}_{{}_{{\cal S}}}\right\\}$. The field (system) ${\cal Q}_{{}_{1}}$
transitions between two microstates in its Fock space, with the final
microstate having one more electron than the initial microstate. In the
$W^{-}$ decay, the transition within the system’s Fock space is accompanied by
a complementary transition between two microstates of the reservoir ${\cal
R}_{{}_{{\cal S}}}$, with the final state of the reservoir having one fewer
$W^{-}$ boson and one more electron antineutrino333 The creation (or
annihilation) of a particle could result as a consequence of several different
complementary transitions in the reservoir. For example, in addition to
$W^{-}$ decay, mentioned above, an electron could also be created by a
$\mu^{-}$ decay  $\displaystyle\mu^{-}\rightarrow
e^{-}+\bar{\nu}_{e}+\nu_{\mu}$ .

Conversely, in electron capture

$\displaystyle e^{-}+p\rightarrow n+\nu_{e}$

a particle (electron) flows out of ${\cal S}$. The flow of a particle out of
${\cal S}$ is interpreted as annihilation of the particle in the system.

Particle annihilation also represents a transition between two microstates of
${\cal C}_{{}_{{\cal S}}}$. In the example of electron capture, the system’s
transition to a microstate that has one fewer particle (electron) is
accompanied by a complementary transition in the reservoir in which an initial
microstate comprising a quark triplet $uud$ (proton) transitions to a final
microstate that has the quark triplet $udd$ (neutron) and an electron
neutrino.

As the above examples show, when particles ‘flow’ between ${\cal S}$ and
${\cal R}_{{}_{{\cal S}}}$ they can change species, a possibility that was
first proposed in 1930 as the Ambarzumian-Ivanenko hypothesis [1]; the
hypothesis is one of the foundational principles of modern quantum field
theory. The electron that flows into ${\cal S}$ following a $W^{-}$ decay does
not pre-exist in the reservoir ${\cal R}_{{}_{{\cal S}}}$. Rather, the
reservoir contains particles of other species that have the necessary
conserved quantities (eg., energy, electric charge) to create an electron.

The vacuum of the system is taken to be the ensemble of all possible
microstates of the system-reservoir complex in which the system has no
particles. The system is said to be in a vacuum state whenever the system-
reservoir complex is in one of the microstates in the ensemble.

In addition to diffusive contact, the system and its reservoir are also
assumed to be in thermal contact. Thus they can exchange energy without
exchanging particles, as illustrated by Compton scattering444Compton
scattering  $\displaystyle e^{-}+\gamma\rightarrow e^{-}+\gamma$ represents
the flow of energy from ${\cal S}$ into ${\cal R}_{{}_{{\cal S}}}$ if the
incoming electron transfers some of its energy to the outgoing photon, or from
${\cal R}_{{}_{{\cal S}}}$ into ${\cal S}$ if the incoming photon transfers
some of its energy to the electron. In Compton scattering the system and
reservoir exchange energy without particles (electrons) flowing into or out of
the system..

A system that is allowed to exchange energy and particles with a reservoir
reaches thermal and diffusive equilibrium with the reservoir over a time
scale, called relaxation time scale, that is a characteristic of the system-
reservoir complex. We assume that the relaxation time scale of a quantum
field— that is, the time scale over which a quantum field returns to thermal
and diffusive equilibrium with its reservoir when perturbed—is significantly
smaller than the time resolution of our measuring instruments. Stated
differently, we assume that what we observe in a scattering process is the
equilibrium or near-equilibrium behavior of the participating fields.

In statistical mechanics, the behavior of a system that is in thermal and
diffusive equilibrium with a reservoir is described by the Grand Canonical
Distribution (GCD). The GCD states that the probability of a fluctuation away
from equilibrium is exponentially suppressed by the so-called Gibbs factor;
see (138). The Gibbs factor that is used in classical statistical mechanics,
however, is frame-dependent. In Section 2, we construct a Lorentz-invariant
analog of the Gibbs factor and incorporate it into the description of the
interactions among quantum fields. The Lorentz-invariant Gibbs factor serves
to ensure that the fluctuations of a quantum field from equilibrium are
constrained by the laws of statistical mechanics and are exponentially
suppressed. The rest of the calculational apparatus of quantum field theory is
left essentially unchanged, except for the small modifications necessitated by
the inclusion of the Lorentz-invariant Gibbs factor.

The paper is organized as follows. In Section 2 we construct a Lorentz-
invariant version of the Gibbs factor. The Grand Canonical Distribution, which
provides the background for Section 2, is described in Appendix A. In Section
3 we present the details of the procedure for calculating the scattering
amplitudes in the new framework. Appendix B supplements the discussion in
Section 3 and contains a fairly rigorous derivation of the LSZ reduction
formula for the new framework. The Lorentz invariance of the scattering
amplitudes described in Section 3 is established in Appendix C. In Section 3
we also discuss some ramifications of incorporating the Gibbs factor into the
description of the interactions among quantum fields. In Sections 4—8 we
compare the predictions of the new framework with known experimental and
observational data. Specifically, in Section 4 we compute the anomalous
magnetic moment of the electron, at 1-loop approximation. Selected details of
the computation are relegated to Appendix D. In Section 5 we compute the
(${}^{2}S_{{}_{1/2}}-^{2}P_{{}_{1/2}}$) Lamb shift in Hydrogen. The running of
the $QED$ coupling constant is calculated at 1-loop approximation in Section
6. In Section 7.2 we calculate the energy density of the zero-point
fluctuations of the free electromagnetic field. Whereas Sections 4, 5, 6 and
7.2 focus on calculations that give rise to divergences within the current
quantum field theoretic framework, in Section 7.1 we compare a tree-level
prediction of the new framework with experimental data. Specifically, we
compute the angular distribution of the differential cross section of Compton
scattering at tree level in the new framework and compare the prediction with
the well-known Klein-Nishina formula and the experimental measurements of
Friedrich and Goldhaber [24]. Section 7.3 contains a brief discussion on bare
and physical parameters. Section 8 contains the concluding remarks.

##  2 Lorentz-invariant Gibbs Factors

In this section we formulate Lorentz-invariant Gibbs factors using a heuristic
argument. A reader interested only in the forms of the Gibbs factors may skip
the heuristic argument and refer to (17)–(LABEL:eqn081). A background
discussion on Grand Canonical Distribution and Gibbs factors can be found in
Appendix A.

Let $\varphi$ be a free scalar quantum field of mass $m$. Let ${\cal F}$ be
some fixed Lorentz frame. Consider a vacuum-to-vacuum fluctuation of $\varphi$
in which a particle is created out of vacuum and then annihilated, as seen by
an observer at rest in ${\cal F}$. If we regard the quantum field to be a
system ${\cal S}=\left\\{\varphi\right\\}$ in thermal and diffusive
equilibrium with a reservoir555The complement of the quantum field, which
supplies the energy for the creation of a particle and absorbs the energy when
the particle is annihilated, can be regarded as a reservoir with which the
quantum field is in thermal and diffusive equilibrium. , then the fluctuation
can be interpreted as a twin-event comprising the flow of a particle into the
system (creation) followed by its flow out of the system (annihilation).
Denoting a microstate of the system ${\cal S}$ with energy $E_{{}_{{\cal
S}}}$, particle number $N_{{}_{{\cal S}}}$ and volume $V_{{}_{{\cal S}}}$ as
${\cal M}^{({\cal S})}(E_{{}_{{\cal S}}},N_{{}_{{\cal S}}},V_{{}_{{\cal
S}}})$, the vacuum fluctuation can be represented as the following transitions
among the microstates of the system:

$\displaystyle{\cal M}^{({\cal S})}(0,0,V_{{}_{{\cal S}}})\rightarrow{\cal
M}^{({\cal S})}_{k}(E_{{}_{{\cal S}}},1,V_{{}_{{\cal S}}})\rightarrow{\cal
M}^{({\cal S})}(0,0,V_{{}_{{\cal S}}}),\qquad E_{{}_{{\cal S}}}\neq 0$

The field starts in its vacuum microstate ${\cal M}^{({\cal
S})}(0,0,V_{{}_{S}})$, transitions to a single-particle microstate ${\cal
M}^{({\cal S})}_{k}(E_{{}_{S}},1,V_{{}_{S}})$, labeled by the subscript666The
subscript allows for degeneracy of single-particle states with energy
$E_{{}_{S}}$. $k$, before returning to the vacuum microstate.

Using (138), the joint probability, $P$, of finding the system in ${\cal
M}^{({\cal S})}(0,0,V_{{}_{S}})$,
${\cal M}^{({\cal S})}_{k}(E_{{}_{S}},1,V_{{}_{S}})$ and ${\cal M}^{({\cal
S})}(0,0,V_{{}_{S}})$ in three consecutive thought experiments, is given by

$\displaystyle P$ $\displaystyle=$ $\displaystyle\left({\cal
Z}\right)\cdot\left({\cal
Z}e^{\displaystyle-(E_{{}_{S}}-\mu)/\tau}\right)\cdot\left({\cal
Z}\right)={\cal
Z}^{\prime}e^{\displaystyle-\left(E_{{}_{S}}-\mu\right)/\tau}\propto
e^{\displaystyle-\left(E_{{}_{S}}-\mu\right)/\tau}$ (1)

where ${\cal Z}^{\prime}=({\cal Z})^{3}$, and ${\cal Z}$ is the probability of
finding the system in its vacuum state. In (1) we have assumed that the
postulate of equal a priori probabilities applies throughout the fluctuation;
see Appendix A.

The above vacuum fluctuation can be described using the current quantum field
theoretic framework as follows. The field operator
$\varphi(t_{{}_{1}},\vec{x})$, at ($t_{{}_{1}},\vec{x}$) acts on the vacuum
state and stochastically induces a transition to a single-particle state in
the Fock space. At a later time, $t_{{}_{2}}>t_{{}_{1}}$, the field operator
$\varphi(t_{{}_{2}},\vec{y})$ annihilates the single-particle excitation
inducing a transition back to the vacuum state.

In the current quantum field theoretic framework, the mode-expansion of the
free field $\varphi$ is taken to be

$\displaystyle\varphi(x)$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{d\vec{k}}{(2\pi)^{3}(2\omega(\vec{k}))}\
\left\\{{\bf\hat{a}}(\uline{\vec{k}})e^{\displaystyle-i\uline{\vec{k}}x}+{\bf\hat{a}}^{\dagger}(\uline{\vec{k}})e^{\displaystyle
i\uline{\vec{k}}x}\right\\}$ (2)

where
$\omega(\vec{k}\,):=\left[\,|\vec{k}|^{2}+m^{2}\,\right]^{1/2},\quad\uline{\vec{k}}:=(\omega(\vec{k}\,),\vec{k})$
and the creation and annihilation operators satisfy the commutation relations

$\displaystyle[{\hat{\bf a}(\vec{k}\,)},{\hat{\bf
a}^{\dagger}(\vec{k}\,^{\prime})}]=(2\pi)^{3}2\omega(\vec{k}\,)\delta^{3}(\vec{k}-\vec{k}\
^{\prime}),\quad[{\hat{\bf a}(\vec{k}\,)},{\hat{\bf
a}(\vec{k}\,^{\prime})}]=[{\hat{\bf a}^{\dagger}(\vec{k}\,)},{\hat{\bf
a}^{\dagger}(\vec{k}\,^{\prime})}]=0.$ (3)

We follow the convention of underlining a momentum 4-vector to indicate that
it is on mass shell 777We do not explicitly specify the mass parameter of the
mass-shell when it is evident from the context. .  Specifically, let
$k=(k^{0},\vec{k})$ be a momentum 4-vector. With some abuse of notation we
define

$\displaystyle\underline{\vec{k}}$ $\displaystyle:=$
$\displaystyle(\omega(\vec{k}),\vec{k}),\qquad\uline{k}:=(\omega(\vec{k}),\vec{k}),\qquad\mbox{where}\qquad\omega(\vec{k}):=\left[\,|\vec{k}|^{2}+m^{2}\,\right]^{1/2}$
(4)

Thus, $\uline{k}$ does not depend on $k^{0}$. And if $k$ is on mass shell then
$\uline{k}=k$.

The creation operator ${\hat{\bf a}^{\dagger}(\vec{k}\,)}$ acting on vacuum
creates a 1-particle momentum eigenstate888Definitions are denoted using the
symbol $:=$.  $\mid\vec{k}\ \rangle:={\hat{\bf
a}^{\dagger}(\vec{k}\,)}\left.\left|\,0\,\right\rangle\right.$. Up to a
normalizing factor, which is independent of $\vec{k}$, the complex number

$\displaystyle{\cal
A}(x,y;\vec{k}):=\left.\left\langle\,0\,\right.\right|\varphi(y)\mid\vec{k}\
\rangle\langle\ \vec{k}\
\mid\varphi(x)\left.\left|\,0\,\right\rangle\right.;\qquad
x:=(t_{{}_{1}},\vec{x}),\quad y:=(t_{{}_{2}},\vec{y})$

is the amplitude of the process that involves, first, the creation of a field
excitation corresponding to momentum eigenstate $\mid\vec{k}\ \rangle$ by the
operator $\varphi(x)$ followed by the annihilation of that excitation by the
operator $\varphi(y)$. Using (2) and (3) we have ${\cal A}(x,y;\vec{k}\
)=e^{\displaystyle i\uline{\vec{k}}(x-y)}$. Thus, the probability of the
creation and subsequent annihilation of an on-mass-shell particle of
3-momentum $\vec{k}$ is $P(x,y;\vec{k}\ )\propto|{\cal A}(x,y;\vec{k}\
)|\,^{2}.$ We note that $P(x,y;\vec{k}\ )$ is independent999We note that
$P(x,y;\vec{k}\ )$ is also independent of $x$ and $y$. of $\vec{k}$. In other
words, the current framework predicts that a vacuum-to-vacuum fluctuation of
arbitrarily high energy, as viewed in some fixed Lorentz frame ${\cal F}$, is
just as likely to occur as a fluctuation of very low energy—a prediction that
contradicts the conclusion (1) derived within the framework of statistical
mechanics.

The above discrepancy suggests that we scale the creation and annihilation
operators in the mode-expansion (2) with a real Lorentz-invariant factor to
ensure that quantum field theory is constrained by the laws of statistical
mechanics. We scale the creation and annihilation operators first in the
center-of-momentum frame of the scattering process of interest and
subsequently generalize to arbitrary frames.

Consider a scattering process ${\cal P}$. Let ${\cal C}_{{}_{\cal P}}$ be the
center-of-momentum frame of the process ${\cal P}$. Let $\varphi$ be the field
corresponding to one of the (real or virtual) particles participating in
${\cal P}$; that is, $\varphi$ is one of the fields in the Lagrangian of the
theory that is used to compute the scattering amplitude of ${\cal P}$. In the
interaction picture, $\varphi$ evolves as a free field. We expand the free
field $\varphi$ (in the interaction picture) in frame ${\cal C}_{{}_{\cal P}}$
as

$\displaystyle\varphi(x;{\cal C}_{{}_{\cal P}},{\cal P})$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{d\,\vec{k}}{(2\pi)^{3}2\omega(\vec{k}\,)}\cdot\left[\,\
g{{}_{b}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})\,\right]\cdot\left\\{{\bf\hat{a}}(\uline{\vec{k}})e^{\displaystyle-i\uline{\vec{k}}x}\
+{\bf\hat{a}}^{\dagger}(\uline{\vec{k}})e^{\displaystyle i\uline{\vec{k}}x}\
\right\\}.\qquad$ (5)

where $\ g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal P})$ is a
Lorentz-invariant scale factor. $\ g{{}_{b}}(\,\uline{\vec{k}}\,;{\cal
C}_{{}_{\cal P}},{\cal P})$ depends on the on-shell 4-momentum
$\uline{\vec{k}}$, the scattering process ${\cal P}$ of interest, and the
frame in which we are observing the scattering process. We observe that,
unlike in the current framework of quantum field theory in which field
representations are independent of the scattering process of interest, in the
new framework the field representation, such as $\varphi(x;{\cal P},{\cal
C}_{{}_{\cal P}})$, depends on the scattering process whose amplitude we seek
to compute. We elaborate on the dependence of the field representation on the
scattering process later. The subscript $b$ in $g_{{}_{b}}$ indicates that the
factor corresponds to the boson field $\varphi$. The commutation relations (3)
among the creation and annihilation operators of the field are left unchanged.

With the above modification, (3) and (5) imply that the amplitude ${\cal
A}(x,y;\vec{k})$ is

$\displaystyle{\cal A}(x,y;\vec{k})$ $\displaystyle=$ $\displaystyle
e^{\displaystyle i\uline{\vec{k}}(x-y)}\ \left[\,\
g{{}_{b}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal P})\,\right]^{2}$

The probability $P(\rule{0.0pt}{14.0pt}x,y;\vec{k}\ )$ then becomes

$\displaystyle P(\rule{0.0pt}{14.0pt}x,y;\vec{k}\ )\propto|\,{\cal
A}(x,y;\vec{k}\ )|\ ^{2}=\left[\,\ g{{}_{b}}(\,\uline{\vec{k}}\,;{\cal
C}_{{}_{\cal P}},{\cal P})\,\right]^{4}$ (6)

Guided by (1) and (6) we formulate the Lorentz-invariant scale factors for the
scattering process ${\cal P}$ in the center-of-momentum frame ${\cal
C}_{{}_{\cal P}}$ as follows. Let $k$ denote a momentum 4-vector in frame
${\cal C}_{{}_{\cal P}}$. We do not assume that $k$ is on mass shell. Let
$u({\cal F}_{{}_{1}};{\cal F}_{{}_{2}})$ denote the 4-velocity of the frame
${\cal F}_{{}_{1}}$ as seen by an observer who is at rest in frame ${\cal
F}_{{}_{2}}$. Then, using the abbreviation $\uline{k}\cdot u_{{}_{{\cal
C}_{{}_{\cal P}}}}:=\uline{k}\,_{\alpha}\,u^{\alpha}({\cal C}_{{}_{\cal
P}};{\cal C}_{{}_{\cal P}})$, we set

$\displaystyle\begin{array}[]{lcll}\ g{{}_{f}}\,(\,{k}\,;{\cal C}_{{}_{\cal
P}},{\cal
P})&:=&\left[\,\displaystyle\frac{d_{{}_{f}}}{\displaystyle{\mbox{\Large\it
e}}^{\rule{0.0pt}{14.0pt}\ ^{\displaystyle\left(\uline{k}\cdot u_{{}_{{\cal
C}_{{}_{\cal P}}}}-\mu_{{}_{f}}\right)/\tau_{{}_{{\cal P}}}\
}}+1}\,\right]^{1/4}&\mbox{(massive fermion)}\end{array}$ (8)
$\displaystyle\begin{array}[]{lcll}\ g{{}_{b}}\,(\,{k}\,;{\cal C}_{{}_{\cal
P}},{\cal
P})&:=&\left[\,\displaystyle\frac{d_{{}_{b}}}{\displaystyle{\mbox{\Large\it
e}}^{\rule{0.0pt}{14.0pt}\ ^{\displaystyle\left(\uline{k}\cdot u_{{}_{{\cal
C}_{{}_{\cal P}}}}-\mu_{{}_{b}}\right)/\tau_{{}_{{\cal P}}}\
}}-1}\,\right]^{1/4}&\mbox{(massive boson)}\end{array}$ (10)
$\displaystyle\begin{array}[]{lcll}\ g{{}_{p}}\,(\,{k}\,;{\cal C}_{{}_{\cal
P}},{\cal
P})&:=&\left[\,\displaystyle\frac{d_{{}_{p}}}{\displaystyle{\mbox{\Large\it
e}}^{\rule{0.0pt}{14.0pt}\ ^{\displaystyle\uline{k}\cdot u_{{}_{{\cal
C}_{{}_{\cal P}}}}/\tau_{{}_{{\cal P}}}\ +\tau_{{}_{{\cal P}}}/\uline{k}\cdot
u_{{}_{{\cal C}_{{}_{\cal P}}}}}}-1}\,\right]^{1/4}&\mbox{(massless
boson)}\end{array}$ (12)

where $g_{{}_{f}},g_{{}_{b}}$ and $g_{{}_{p}}$ represent the Gibbs factors for
a fermion, massive boson and massless boson fields respectively.

The subscripts $f$ and $b$ denote fermion and boson respectively. When the
fermion of interest is an electron we use the subscript $e$ instead of $f$.
The subscript for massless boson field is chosen to be $p$, since photon is
the only massless boson of interest to us in this paper.

$\tau_{{}_{{\cal P}}}$ is taken to be Lorentz-invariant ‘temperature’ that
depends on the process ${\cal P}$. $\mu_{{}_{f}}$ and $\mu_{{}_{b}}$ are taken
to be particle-dependent Lorentz-invariant chemical potentials. The chemical
potential of the photon is assumed to vanish. $d_{{}_{f}}$, $d_{{}_{b}}$ and
$d_{{}_{p}}$ are the particle-dependent degeneracy factors for massive
fermion, massive boson and massless boson respectively. The above terms are
described in greater detail in the following paragraphs.

Let $K_{{}_{{\cal P}}}=\left\\{k_{{}_{1}},\ldots,k_{{}_{n}}\right\\}$ denote
the collection of 4-vectors comprising the 4-momenta of incoming and outgoing
particles, as well as the 4-momentum transfers (among the on-shell external
particles) that occur in the scattering process ${\cal P}$. Then we take

$\displaystyle\tau_{{}_{{\cal P}}}$ $\displaystyle=$ $\displaystyle\max_{1\leq
i\leq n}\sqrt{|k_{{}_{i}}^{2}|}$

where $\sqrt{|k_{{}_{i}}^{2}|}$ is the Lorentz-invariant norm of the 4-vector
$k_{{}_{i}}$.

For example, in Section 4 we consider a scattering process ${\cal P}$ in which
an electron, initially at rest, acquires a vanishingly small 4-momentum that
is transferred to it from a background Maxwell field. Since the norm of the
momentum transfer is arbitrarily small, while the norm of the 4-momentum of
the incoming electron is $m_{{}_{e}}$ (in natural units), we take
$\tau_{{}_{{\cal P}}}$ for the process to be $\tau_{{}_{{\cal
P}}}=m_{{}_{e}}$, the physical mass of the electron. However, in the
calculation of the running coupling constant in Section 6, we consider
momentum transfers whose Lorentz-invariant norms are arbitrarily large
compared to the norms of the on-mass-shell 4-momenta of incoming electrons.
Therefore, in the running coupling constant calculation we take
$\tau_{{}_{{\cal P}}}=\sqrt{|k^{2}|}$ where $k$ represents the momentum
transfer between the scattering electrons.

The $\mu_{{}_{f}}$ for an electron, hereafter denoted $\mu_{{}_{e}}$, is
chosen to be a Lorentz-invariant quantity based on the following heuristic
argument. In finite temperature field theory it is known that the ‘chemical
potential’ of an electron plays a role analogous to the $A^{0}$ component of
the dynamical gauge (photon) field coupled to the electron [56, 79, 90].
Therefore, we take $\mu_{{}_{e}}\ =\ \langle A^{0}\rangle_{\lambda_{e}}$,
where the average101010 Using a classical heuristic argument, and regarding an
electron as a point particle, the magnitude of the average potential of an
electron over a sphere of radius $R$ centered at the electron is
$\displaystyle\langle
A^{0}\rangle_{R}=\left\\{\left[\,\frac{1}{(4\pi/3)R^{3}}\,\right]\int_{|x|\leq
R}\,d^{3}x\
\displaystyle\frac{|e|}{4\pi\epsilon_{{}_{0}}|x|}\right\\}=\left\\{\left[\,\frac{4\pi|e|}{(4\pi/3)(R^{3})\cdot
4\pi\epsilon_{{}_{0}}}\,\right]\int_{0}^{R}\,dr\ r\right\\}\ =\
\left(\frac{3}{2}\right)\left(\frac{|e|}{4\pi R}\right)$ is taken over the
characteristic Lorentz-invariant length scale of the electron, namely its
Compton wavelength $\lambda_{{}_{e}}\ =\ {\hbar}/{m_{{}_{e}}c}$. Thus we
set111111 We use units in which $\hbar=m_{{}_{e}}=c=\epsilon_{{}_{0}}=1$. From
footnote 10, we have $\langle A^{0}\rangle_{{}_{R=\lambda_{e}}}\ =\
\left(\frac{3}{2}\right)\left(\frac{|e|}{4\pi\epsilon_{{}_{0}}(\hbar/m_{{}_{e}}c)}\right)\
=\ \left(\frac{3}{2}\right)\sqrt{\displaystyle\frac{\alpha}{4\pi}}$. In our
units electric charge is dimensionless, and $A^{0}$ has the same dimension as
energy.

$\displaystyle\mu_{{}_{e}}\ =\
\left(\frac{3}{2}\right)\sqrt{\displaystyle\frac{\alpha}{4\pi}}.$ (13)

The degeneracy factors $d_{{}_{f}},\ d_{{}_{b}}$ and $d_{{}_{p}}$ for the
fermion fields , the massive boson fields and the photon field specify the
number of different modes that the field operator (and its conjugate) can
create at a fixed 3-momentum. In other words, the degeneracy factor is given
by the number of different creation operators, at a fixed 3-momentum, in the
free-field expansion of the field and its conjugate. The free field expansion
of the Dirac field has four creation operators—two creation operators for the
two spin states of the positive energy modes and two creation operators for
the two spin states of the negative energy modes. Therefore, for an electron
$d_{{}_{f}}=4$. A free Maxwell field on the other hand has only two creation
operators—corresponding to the two transverse polarization modes at a fixed
$3$-momentum. Therefore for a photon field $d_{{}_{p}}=2$.

In the frame ${\cal C}_{{}_{\cal P}}$, the velocity 4-vector of ${\cal
C}_{{}_{\cal P}}$ is $u({\cal C}_{{}_{\cal P}};{\cal C}_{{}_{\cal
P}})=(1,0,0,0)$. Therefore, in the frame ${\cal C}_{{}_{\cal P}}$, we have
$\uline{k}\cdot u_{{}_{{\cal C}_{{}_{\cal P}}}}\ =\
\uline{k}\,_{\alpha}u^{\alpha}({\cal C}_{{}_{\cal P}};{\cal C}_{{}_{\cal P}})\
=\ \sqrt{|\vec{k}|^{2}+m^{2}}:=E$, the energy of the excitation. The
probabilities (6) derived from the Gibbs factors defined in (8)–(12) resemble
the Fermi-Dirac and Bose-Einstein distributions, and have the desired
$e^{-(E-\mu)/\tau}$ dependence for large $E$, in the Lorentz frame ${\cal
C}_{{}_{\cal P}}$. In the small $E$ limit, although the Fermi-Dirac and Bose-
Einstein distributions deviate from the functional form shown in (1), the
Gibbs factors yield probabilities that remain bounded as required in (1).
Specifically, in the limit $E\rightarrow 0$ the Gibbs factors for fermions and
massive bosons do not have singularities on the real axis and remain bounded.
For massless bosons, however, we assume the chemical potential vanishes.
Therefore, as $E\rightarrow 0$ the Gibbs factor (10) diverges in the massless
limit, in contrast to (1). In order to ensure that (6) remains bounded for
massless bosons in the $E\rightarrow 0$ limit we have added the
$\tau_{{}_{{\cal P}}}/\uline{k}\,_{\alpha}u^{\alpha}({\cal C}_{{}_{\cal
P}};{\cal C}_{{}_{\cal P}})$ term to the exponent in (12). The justification
for the Gibbs factors formulated in (8)–(12) ultimately hinges on the
agreement of their predictions with experimental observations. In Sections 4,
5, 6 and 7.1 we compare the predictions of the Gibbs factors with experimental
data.

The Gibbs factors in (8)—(12) are defined in the center-of-momentum frame
${\cal C}_{{}_{\cal P}}$ of a scattering process ${\cal P}$. We chose to
present the definitions in ${\cal C}_{{}_{\cal P}}$ first, for the forms of
the Gibbs factors in ${\cal C}_{{}_{\cal P}}$ showcase the similarity to (1)
in the large $E$ limit. In what follows, we extend the definitions (8)—(12) to
an arbitrary frame ${\cal F}$.

Consider two observers ${\cal O}_{{}_{{\cal C}_{{}_{\cal P}}}}$ and ${\cal
O}_{{}_{{\cal F}}}$ who are at rest in frames ${\cal C}_{{}_{\cal P}}$ and
${\cal F}$, respectively. Let ${\cal O}_{{}_{{\cal C}_{{}_{\cal P}}}}$ and
${\cal O}_{{}_{{\cal F}}}$ assign coordinates $x$ and $x^{\prime}$ to an event
${\cal E}$. Let $\Lambda({\cal F},{\cal C}_{{}_{\cal P}})$ be the Lorentz
transformation that maps ${\cal C}_{{}_{\cal P}}$ to ${\cal F}$. In matrix
notation, $x^{\prime}=\Lambda({\cal F},{\cal C}_{{}_{\cal P}})\,x$. Note that
$\Lambda({\cal C}_{{}_{\cal P}},{\cal F})=\Lambda^{-1}({\cal F},{\cal
C}_{{}_{\cal P}})$. We abbreviate $\Lambda({\cal F},{\cal C}_{{}_{\cal P}})$
to $\Lambda$ and define a function $\xi$ as follows. Let $q^{\prime}$ be a
4-momentum vector in ${\cal F}$. Then,

$\displaystyle\xi(q^{\prime}\,;\,{\cal F},{\cal C}_{{}_{\cal P}})$
$\displaystyle:=$
$\displaystyle\Lambda\left[\,\uline{\Lambda^{-1}\,q^{\prime}}\,\right]\ =\
\Lambda({\cal F},{\cal C}_{{}_{\cal P}})\left[\,\uline{\Lambda({\cal
C}_{{}_{\cal P}},{\cal F})\,q^{\prime}}\,\right]$ (14)

where, as described in (4), we underline a 4-vector to indicate that it is on
mass shell. We note that if ${\cal F}\ =\ {\cal C}_{{}_{\cal P}}$, then
$\Lambda\ =\ I$ and we have for all $k$, $\xi(k;{\cal C}_{{}_{\cal P}},{\cal
C}_{{}_{\cal P}})=\uline{k}$.

Noting that the pre-image of $q^{\prime}$ under $\Lambda$ is $q\ =\
\Lambda^{-1}q^{\prime}\ =\ \Lambda({\cal C}_{{}_{\cal P}},{\cal F})q^{\prime}$
we have

$\displaystyle\xi_{\alpha}(q^{\prime}\,;\,{\cal F},\,{\cal C}_{{}_{\cal
P}})\,u^{\alpha}({\cal C}_{{}_{\cal P}};{\cal F})$ $\displaystyle=$
$\displaystyle\left[\,\Lambda\left[\,\uline{\Lambda^{-1}q^{\prime}}\,\right]\,\right]_{\alpha}\left[\,\Lambda
u({\cal C}_{{}_{\cal P}};{\cal C}_{{}_{\cal P}})\,\right]^{\alpha}\ =\
\left[\,\uline{\Lambda^{-1}q^{\prime}}\,\right]_{\alpha}u^{\alpha}({\cal
C}_{{}_{\cal P}};{\cal C}_{{}_{\cal P}})$ (15) $\displaystyle=$
$\displaystyle\uline{q}\,_{\alpha}u^{\alpha}({\cal C}_{{}_{\cal P}};{\cal
C}_{{}_{\cal P}})$

In other words,

$\displaystyle\xi_{\alpha}(q^{\prime}\,\,;\,{\cal F},{\cal C}_{{}_{\cal
P}})\,u^{\alpha}({\cal C}_{{}_{\cal P}};{\cal F})$ $\displaystyle=$
$\displaystyle\uline{q}\,_{\alpha}u^{\alpha}({\cal C}_{{}_{\cal P}};{\cal
C}_{{}_{\cal P}})\ =\ \xi_{\alpha}(\Lambda({\cal C}_{{}_{\cal P}},{\cal
F})q^{\prime};{\cal C}_{{}_{\cal P}},{\cal C}_{{}_{\cal P}})u^{\alpha}({\cal
C}_{{}_{\cal P}};{\cal C}_{{}_{\cal P}})\qquad$ (16)

Hence, using the abbreviation $u_{{}_{{\cal F}}}:=u({\cal C}_{{}_{\cal
P}};{\cal F})$, we define the Gibbs factors for a scattering process ${\cal
P}$ in an arbitrary frame ${\cal F}$ as

$\displaystyle g_{f}(k;{\cal F},{\cal P})$ $\displaystyle:=$
$\displaystyle\left[\,\displaystyle\frac{d_{{}_{f}}}{\displaystyle{\mbox{\Large\it
e}}^{\rule{0.0pt}{14.0pt}\ ^{\displaystyle\left[\,\xi_{\alpha}(k\,;\,{\cal
F},{\cal C}_{{}_{\cal P}})\,(u_{{}_{{\cal
F}}})^{\alpha}-\mu_{{}_{f}}\,\right]/\tau_{{}_{{\cal P}}}\
}}+1}\,\right]^{1/4}$ (17) $\displaystyle g_{b}(k;{\cal F},{\cal P})$
$\displaystyle:=$
$\displaystyle\left[\,\displaystyle\frac{d_{{}_{b}}}{\displaystyle{\mbox{\Large\it
e}}^{\rule{0.0pt}{14.0pt}\ ^{\displaystyle\left[\,\xi_{\alpha}(k\,;\,{\cal
F},{\cal C}_{{}_{\cal P}})\,(u_{{}_{{\cal
F}}})^{\alpha}-\mu_{{}_{b}}\,\right]/\tau_{{}_{{\cal P}}}\
}}-1}\,\right]^{1/4}$ (18) $\displaystyle g_{p}(k;{\cal F},{\cal P})$
$\displaystyle:=$
$\displaystyle\left[\,\displaystyle\frac{d_{{}_{p}}}{\displaystyle{\mbox{\Large\it
e}}^{\rule{0.0pt}{14.0pt}\ ^{\displaystyle\left[\,\xi_{\alpha}(k\,;\,{\cal
F},{\cal C}_{{}_{\cal P}})\,(u_{{}_{{\cal
F}}})^{\alpha}\,\right]/\tau_{{}_{{\cal P}}}\ +\tau_{{}_{{\cal
P}}}/\left[\,\xi_{\alpha}(k\,;\,{\cal F},{\cal C}_{{}_{\cal
P}})\,(u_{{}_{{\cal F}}})^{\alpha}\,\right]}}-1}\,\right]^{1/4}$

where $g_{{}_{f}},g_{{}_{b}}$ and $g_{{}_{p}}$ represent the Gibbs factors for
a fermion, massive boson and massless boson fields respectively.

Although the function $\xi$ depends on ${\cal C}_{{}_{\cal P}}$, as we show
below, $\xi_{\alpha}(k;{\cal F},{\cal C}_{{}_{\cal P}})u^{\alpha}({\cal
F};{\cal C}_{{}_{\cal P}})$ is invariant under spatial rotation121212Frame
${\cal C}_{{}_{\cal P}}$ is determined only up to spatial rotation, since
spatial rotation of ${\cal C}_{{}_{\cal P}}$ yields another center-of-momentum
frame of the process ${\cal P}$.  of the frame ${\cal C}_{{}_{\cal P}}$.

Let ${\cal C}_{{}_{\cal P}}$ and ${\cal C}_{{}_{\cal P}}^{\prime}$ be two
center-of-momentum frames of a scattering process ${\cal P}$. Let $R:{\cal
C}_{{}_{\cal P}}\rightarrow{\cal C}_{{}_{\cal P}}^{\prime}$ be the rotation
that maps 4-vectors in ${\cal C}_{{}_{\cal P}}$ to corresponding 4-vectors in
${\cal C}_{{}_{\cal P}}^{\prime}$. We note that $\Lambda({\cal C}_{{}_{\cal
P}}^{\prime},{\cal F})\ =\ R\,\Lambda({\cal C}_{{}_{\cal P}},{\cal F})$. For a
4-vector $q\in{\cal F}$, we have

$\displaystyle\underline{\Lambda({\cal C}_{{}_{\cal P}}^{\prime},{\cal F})q}$
$\displaystyle=$ $\displaystyle\underline{R\,\Lambda({\cal C}_{{}_{\cal
P}},{\cal F})q}\ =\ \underline{\Lambda({\cal C}_{{}_{\cal P}},{\cal F})q}.$
(20)

Therefore, from (15) and (20) we have

$\displaystyle\xi_{\alpha}(q;{\cal F},{\cal C}_{{}_{\cal
P}}^{\prime})u^{\alpha}({\cal C}_{{}_{\cal P}}^{\prime},{\cal F})$
$\displaystyle=$ $\displaystyle\left[\,\Lambda({\cal F},{\cal C}_{{}_{\cal
P}}^{\prime})\left[\,\underline{\Lambda({\cal C}_{{}_{\cal P}}^{\prime},{\cal
F})q}\,\right]\,\right]_{\alpha}\left[\,\Lambda({\cal F},{\cal C}_{{}_{\cal
P}}^{\prime})u({\cal C}_{{}_{\cal P}}^{\prime},{\cal C}_{{}_{\cal
P}}^{\prime})\,\right]^{\alpha}$ (21) $\displaystyle=$
$\displaystyle\underline{\Lambda({\cal C}_{{}_{\cal P}}^{\prime},{\cal
F})q}\cdot u({\cal C}_{{}_{\cal P}}^{\prime},{\cal C}_{{}_{\cal P}}^{\prime})\
=\ \underline{\Lambda({\cal C}_{{}_{\cal P}},{\cal F})q}\cdot u({\cal
C}_{{}_{\cal P}};{\cal C}_{{}_{\cal P}})$ $\displaystyle=$
$\displaystyle\xi_{\alpha}(q;{\cal F},{\cal C}_{{}_{\cal P}})u^{\alpha}({\cal
C}_{{}_{\cal P}};{\cal F})$

In the penultimate step we have used (20) and $u({\cal C}_{{}_{\cal
P}}^{\prime};{\cal C}_{{}_{\cal P}}^{\prime})\ =\ u({\cal C}_{{}_{\cal
P}};{\cal C}_{{}_{\cal P}})=(1,0,0,0)$. (21) shows that
$g_{{}_{f}},g_{{}_{b}}$ and $g_{{}_{p}}$ in (17)—(LABEL:eqn081) depend only on
${\cal P}$ and not on the choice of ${\cal C}_{{}_{\cal P}}$ corresponding to
${\cal P}$.

If ${\cal F}={\cal C}_{{}_{\cal P}}$, then $\Lambda=I$, and the definitions in
(17)—(LABEL:eqn081) reduce to the definitions in (8)–(12). Hereafter, we take
(17)—(LABEL:eqn081) to be the definitions of the Gibbs factors in an arbitrary
frame ${\cal F}$ with (8)—(12) being the special case corresponding to ${\cal
F}={\cal C}_{{}_{\cal P}}$.

Let ${\cal F}$ and ${\cal F}^{\prime}$ be two Lorentz frames related by a
Lorentz transformation. Then, from (16) we have

$\displaystyle\xi_{\alpha}(\Lambda({\cal F}^{\prime},{\cal F})q;{\cal
F}^{\prime},{\cal C}_{{}_{\cal P}})u^{\alpha}({\cal C}_{{}_{\cal P}};{\cal
F}^{\prime})$ $\displaystyle=$ $\displaystyle\xi_{\alpha}(\Lambda({\cal
C}_{{}_{\cal P}},{\cal F}^{\prime})\Lambda({\cal F}^{\prime},{\cal F})q;{\cal
C}_{{}_{\cal P}},{\cal C}_{{}_{\cal P}})u^{\alpha}({\cal C}_{{}_{\cal
P}};{\cal C}_{{}_{\cal P}})$ (22) $\displaystyle=$
$\displaystyle\xi_{\alpha}(\Lambda({\cal C}_{{}_{\cal P}},{\cal F})q;{\cal
C}_{{}_{\cal P}},{\cal C}_{{}_{\cal P}})u^{\alpha}({\cal C}_{{}_{\cal
P}};{\cal C}_{{}_{\cal P}})$ $\displaystyle=$
$\displaystyle\xi_{\alpha}(\Lambda({\cal F},{\cal C}_{{}_{\cal
P}})\Lambda({\cal C}_{{}_{\cal P}},{\cal F})q;{\cal F},{\cal C}_{{}_{\cal
P}})u^{\alpha}({\cal C}_{{}_{\cal P}};{\cal F})$ $\displaystyle=$
$\displaystyle\xi_{\alpha}(q;{\cal F},{\cal C}_{{}_{\cal P}})u^{\alpha}({\cal
C}_{{}_{\cal P}};{\cal F})$

Since we chose $d_{f},d_{b},d_{{}_{p}},\mu_{{}_{f}},\mu_{{}_{b}}$ and
$\tau_{{}_{{\cal P}}}$ to be Lorentz invariants, from (17)—(LABEL:eqn081) and
(22) we have for any two Lorentz frames ${\cal F}$ and ${\cal F}^{\prime}$
that are related by a Lorentz transformation,

$\displaystyle\ g{{}_{f}}\,(\,{k}\,;{\cal F},{\cal P})$ $\displaystyle=$
$\displaystyle g_{{}_{f}}(\Lambda({\cal F}^{\prime},{\cal F})\,k\,;{\cal
F}^{\prime};{\cal P});\qquad\ g{{}_{b}}\,(\,{k}\,;{\cal F},{\cal P})\ =\
g_{{}_{b}}(\Lambda({\cal F}^{\prime},{\cal F})\,k\,;{\cal F}^{\prime};{\cal
P});$ $\displaystyle\ g{{}_{p}}\,(\,{k}\,;{\cal F},{\cal P})$ $\displaystyle=$
$\displaystyle g_{{}_{p}}(\Lambda({\cal F}^{\prime},{\cal F})\,k\,;{\cal
F}^{\prime};{\cal P});$ (23)

establishing the Lorentz invariance of the Gibbs factors.

Given the significance of the center-of-momentum frame ${\cal C}_{{}_{\cal
P}}$ of a process ${\cal P}$ in the preceding discussion, we take a closer
look at the definition of ${\cal C}_{{}_{\cal P}}$. We restrict attention to
scattering processes, such as131313 For simplicity, we assume the incoming and
outgoing particles correspond to scalar fields. ,

$\displaystyle{\cal P}:\quad
A_{1}(p_{{}_{1}})+A_{2}(p_{{}_{2}})+\ldots+A_{m}(p_{{}_{m}})\longrightarrow
B_{1}(q_{{}_{1}})+B_{2}(q_{{}_{2}})+\ldots+B_{n}(q_{{}_{n}})$ (24)

in which a collection of incoming particles $A_{{}_{1}},\ldots,A_{{}_{m}}$
created in the asymptotic past with 4-momenta $p_{{}_{1}},\ldots,p_{{}_{m}}$
scatter to outgoing particles $B_{{}_{1}},\ldots,B_{{}_{n}}$ with momenta
$q_{{}_{1}},\ldots,q_{{}_{n}}$ that propagate into the asymptotic future.

We make the following assumptions about the scattering process ${\cal P}$.

  1. 1.

In the perturbative expansion of the scattering amplitude, there is at least
one connected Feynman diagram in which $A_{{}_{1}},\ldots,A_{{}_{m}}$ are the
incoming particles and $B_{{}_{1}},\ldots,B_{{}_{n}}$ the outgoing particles.
In other words, ${\cal P}$ does not represent two or more separate non-
interacting scattering processes. This assumption also precludes spectator
particles.

  2. 2.

Denote the set of incoming and outgoing particles of a process ${\cal S}$ as
${\cal I}({\cal S})$ and ${\cal O}({\cal S})$ respectively. We assume that
there is no other scattering process ${\cal P}^{\prime}$, besides ${\cal P}$,
for which

$\displaystyle{\cal I}({\cal P})\subseteq{\cal I}({\cal
P}^{\prime})\qquad\mbox{and}\qquad{\cal O}({\cal P})\subseteq{\cal O}({\cal
P}^{\prime}).$

That is, the ${\cal P}$ shown in (24) is not subsumed in some larger
scattering process. The ${\cal I}$(${\cal P}$) and ${\cal O}$(${\cal P}$)
represent the totality of incoming and outgoing particles in the ongoing
scattering process.

  3. 3.

The cluster decomposition property holds. Thus, the scattering amplitude of
${\cal P}$, is independent of other scattering processes that may be occurring
concurrently, and in calculating ${\cal P}$’s scattering amplitude we may
assume that it is the only scattering process occurring in spacetime.

  4. 4.

We assume that the scattering occurs in a background Lorentzian spacetime.

With the above assumptions, and taking
$p=p_{{}_{1}}+\ldots+p_{{}_{m}}=(p^{0},\vec{p})$, ${\cal C}_{{}_{\cal P}}$ is
a Lorentz frame in which the total 3-momentum $\vec{p}$ of the incoming
particles vanishes. ${\cal C}_{{}_{\cal P}}$ is determined only up to spatial
rotation.

Unlike in the current framework of quantum field theory, the representations
of fields in the new framework depend on the scattering process. We note that
the representation of a field is not physically relevant. The only physically
relevant and observable quantities in quantum field theory are the $S$-matrix
elements—the probability amplitudes—of scattering processes, as Dirac observed
[12, page 456]. Therefore, we do have—and we exploit—the latitude to choose
the representations of fields and customize the calculational framework to the
scattering process, subject to the constraint that the computed probability
amplitudes must be Lorentz-invariant.

Finally, we note that invoking a specific Lorentz frame—such as ${\cal
C}_{{}_{\cal P}}$—is neither unusual nor an impediment to preserving Lorentz
invariance of scattering amplitudes. Even in the conventional field
quantization procedure, for example, one chooses a specific Lorentz frame to
impose the equal-time commutation relations on the field and its conjugate
momentum [38]. The arguments in Appendix C show that the scattering amplitudes
are Lorentz-invariant in the new framework.

Hereafter, we will assume that even when the scattering process is not
described explicitly, the Gibbs factors and the field representations pertain
to an underlying scattering process ${\cal P}$, whose center-of-momentum frame
is ${\cal C}_{{}_{\cal P}}$. And though inelegant, we retain reference to
${\cal P}$ in the Gibbs factors and the field representations as reminder of
the departure from the process-independent field representations in the
current quantum field theoretic framework.

##  3 Field Quantization

The insertion of the Gibbs factor in the mode-expansion of free fields, as
shown in (5), has a significant consequence. It breaks the equivalence between
(a) the commutation (anticommutation) relations imposed on the
creation/annihilation operators and (b) the commutation (anticommutation)
relations imposed directly on the field and its conjugate momentum. The
history of the above equivalence dates back to the very beginnings of quantum
field theory. Before discussing the broken equivalence we recall the details
of its origin, its consequences and Dirac’s rejection of it immediately after
it was proposed.

###  3.1 The Heisenberg-Pauli Framework

The theory of quantum fields had its beginnings in the early work of Born,
Heisenberg and Jordan [7], and Ehrenfest [17] on quantization of the
electromagnetic field; in Dirac’s calculations of the spontaneous emission of
radiation [10, 19], scattering of radiation and lifetimes of excited states of
atoms [11]; and in Jordan and Pauli’s demonstration of Lorentz invariance of
field commutators of the electromagnetic field [48]. The quantization
procedure, which was initially developed primarily for the electromagnetic
field, was subsequently generalized to other fields by Jordan and Wigner [47,
49] and a general theory of quantum fields was first proposed by Heisenberg
and Pauli [39, 40] in 1929-30.

A hallmark of the Heisenberg-Pauli formalism [39, 40] was that it postulated,
for the first time, canonical commutation conditions141414Commutation
relations for boson fields and anticommutation relations for fermion fields.,
such as, for example,

$\displaystyle[\varphi(t,\vec{x}),\dot{}\varphi(t,\vec{y})]$ $\displaystyle=$
$\displaystyle
i\,\delta^{3}(\vec{x}-\vec{y});\qquad[\varphi(t,\vec{x}),\varphi(t,\vec{y})]\
=\ [\dot{}\varphi(t,\vec{x}),\dot{}\varphi(t,\vec{y})]\ =\ 0\hskip 36.135pt$
(25)

for the Klein-Gordon field, directly on the fields and their conjugate momenta
in contrast to the earlier approaches, which postulated commutation14.
relations for the coefficients of momentum eigenmodes, as shown in (3). At the
very outset, Dirac151515Dirac [12, Page 454]: “Thus the role of the field is
to provide a means for making observations. The very nature of an observation
requires an interplay between the field and the particles. We cannot therefore
suppose the field to be a dynamical system on the same footing as the
particles and thus something to be observed in the same way as particles. The
field should appear in the theory as something more elementary and
fundamental.” expressed ‘serious objections’ to the Heisenberg-Pauli approach
and recognized ‘the mathematical difficulties to which they lead’ [12, Page
454]. Notwithstanding Dirac’s objections the Heisenberg-Pauli approach has
persisted to this day and forms the basis of the current theory of quantum
fields. We call the field quantization approach that imposes the commutation
(anticommutation) relations directly on the fields and their conjugate momenta
the Heisenberg-Pauli framework.

The ‘mathematical difficulties’ that Dirac alluded to—the divergences that
arise in higher order corrections to scattering amplitudes—persist to this
day, although partially circumvented by the renormalization procedure.
Renormalization, which involves subtraction of divergent terms arising in loop
integrals from bare parameters that are also assumed to be infinite a priori,
is, in Feynman’s words161616Feyman [21, Page 128]: “The shell game that we
play … is technically called ‘renormalization’. But no matter how clever the
word, it is still what I would call a dippy process! …. I suspect that
renormalization is not mathematically legitimate”., ‘not mathematically
legitimate’, notwithstanding its successes. Even setting aside reservations
about its mathematical legitimacy, the renormalization procedure does not
fully overcome the ‘mathematical difficulties’ since it fails when confronted
with the divergences in perturbatively nonrenormalizable theories [50, 99].

Even in renormalizable theories several of the predictions of the Heisenberg-
Pauli framework disagree with experimental data. For example, the theoretical
predictions of triplet positronium decay rate and the high energy electron-
positron scattering processes ($e^{-}+e^{+}\rightarrow
e^{-}+e^{+}+e^{-}+e^{+}$ and $e^{-}+e^{+}\rightarrow
e^{-}+e^{+}+\mu^{-}+\mu^{+}$) in quantum electrodynamics disagree
significantly with the experimentally measured values [74, Page 198]. The
zero-point energy density of a quantum field, even if it is cut off at Planck
scale, yields a value that is several dozens of orders of magnitude larger
than the observational bounds on the vacuum energy density [93]. Even the
theoretical prediction of Lamb shift, regarded as one of the seminal
achievements of quantum electrodynamics, ‘is not in good agreement’ with
recent experimental data [62].

The persistent ‘mathematical difficulties’ and the lingering discrepancies
between experiments and theory, after nearly nine decades, appear to reinforce
Dirac’s insight about the Heisenberg-Pauli field quantization formalism and
his studied insistence that in its final form the theory should be divergence-
free171717 Dirac [54, Page 184]: “Most physicists are very satisfied with the
situation. They say: ’Quantum electrodynamics is a good theory and we do not
have to worry about it any more.’ I must say that I am very dissatisfied with
the situation, because this so-called ‘good theory’ does involve neglecting
infinities which appear in its equations, neglecting them in an arbitrary way.
This is just not sensible mathematics. Sensible mathematics involves
neglecting a quantity when it is small - not neglecting it just because it is
infinitely great and you do not want it!” , notwithstanding, what
Salam181818Salam and Strathdee [44, Page 1805]:“Field theoretic infinities …
have persisted … in quantum electrodynamics for some thirty-five years. These
long years of frustration have left in the subject a curious affection for the
infinities and a passionate belief that they are an inevitable part of nature;
so much so that even the suggestion of a hope that they may after all be
circumvented—and finite values for the renormalization constants computed—is
considered irrational”. and Strathdee describe as, ‘a curious affection for
the infinities and the passionate belief that they are an inevitable part of
nature’. For a more elaborate account of the history of quantum field theory
the reader is referred to [8, 13, 15, dyso, 20, 23, 30, 32, 34, 41, 66, 67,
71, 75, 81, 82, 88, 92, 94, 96]. A more recent account of quantum field theory
can be found in [5, 38, 43, 45, 63, 65, 74, 77, 99, 100].

###  3.2 A Broken Equivalence

As mentioned earlier, the insertion of the Gibbs factor into the mode-
expansion of the free field, shown in (5), breaks the equivalence between (3)
and (25). Using (3) and (5) we see that

$\displaystyle[\varphi(\vec{x},t;{\cal C}_{{}_{\cal P}},{\cal
P}),\dot{}\varphi(\vec{y},t;{\cal C}_{{}_{\cal P}},{\cal P})]$
$\displaystyle=$ $\displaystyle
i\int\displaystyle\frac{d\,\vec{k}}{(2\pi)^{3}}e^{\displaystyle-i\vec{k}\cdot(\vec{x}-\vec{y})}\cdot\
g{{}_{b}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal P})^{2}\neq i\
\delta^{3}(\vec{x}-\vec{y})$

Thus, the mode-expansion (5) allows us to retain either (3) or (25), but not
both. We adhere to the approach followed by Dirac [10], and impose the
commutation relations (3) on the creation and annihilation operators, and do
not impose the commutation conditions such as (25) on the fields.

Thus, in an arbitrary frame ${\cal F}$ the mode-expansion of the free Dirac
field is taken to be

$\displaystyle\psi(x;{\cal F},{\cal P})$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{d\vec{k}\,\
g{{}_{f}}(\,\uline{\vec{k}}\,;{\cal F},{\cal P})}{(2\pi)^{3}\
(2\omega(\vec{k}\,))}\left\\{\sum_{s=1}^{2}\left[\,u(\uline{\vec{k}},s)\
{\bf\hat{a}}(\uline{\vec{k}},s)\
e^{\displaystyle-i\uline{\vec{k}}x}+v(\uline{\vec{k}},s)\
{\bf\hat{b}}^{\dagger}(\uline{\vec{k}},s)\ e^{\displaystyle
i\uline{\vec{k}}x}\,\right]\right\\}$ $\displaystyle\bar{\psi}(x;{\cal
F},{\cal P})$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{d\vec{k}\,\
g{{}_{f}}(\,\uline{\vec{k}}\,;{\cal F},{\cal P})}{(2\pi)^{3}\
(2\omega(\vec{k}\,))}\left\\{\sum_{s=1}^{2}\left[\,\bar{u}(\uline{\vec{k}},s)\
{\bf\hat{a}}^{\dagger}(\uline{\vec{k}},s)\ e^{\displaystyle
i\uline{\vec{k}}x}+\bar{v}(\uline{\vec{k}},s)\
{\bf\hat{b}}(\uline{\vec{k}},s)\
e^{\displaystyle-i\uline{\vec{k}}x}\,\right]\right\\}$

where $\ g{{}_{f}}(\,\uline{\vec{k}}\,;{\cal F},{\cal P})$ is defined in (17).
Hereafter we write $\psi(x)$ and $\bar{\psi}(x)$, when the ${\cal F}$ and
${\cal P}$ are evident from the context. The creation and annihilation
operators of the free Dirac field are postulated to satisfy the following
anti-commutation relations:

$\displaystyle\\{{\bf\hat{a}}(\uline{\vec{k}},s),{\bf\hat{a}}^{\dagger}(\uline{\vec{q}},s^{\prime})\\}$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\cdot(2\omega(\vec{k}\,))\cdot\delta^{3}(\vec{k}-\vec{q})\cdot\delta_{s,s^{\prime}};$
(27)
$\displaystyle\\{{\bf\hat{b}}(\uline{\vec{k}},s),{\bf\hat{b}}^{\dagger}(\uline{\vec{q}},s^{\prime})\\}$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\cdot(2\omega(\vec{k}\,))\cdot\delta^{3}(\vec{k}-\vec{q})\cdot\delta_{s,s^{\prime}}$

All of the other possible anticommutators among the ${\bf\hat{a}},\
{\bf\hat{a}}^{\dagger},\ {\bf\hat{b}}$ and ${\bf\hat{b}}^{\dagger}$ operators,
besides those listed in (27), are postulated to vanish. We do not impose any
anticommutation conditions on the Dirac field and its conjugate.

Similarly, in a frame ${\cal F}$ the mode-expansion of the free Maxwell field,
in Lorenz gauge191919 In Coulomb gauge, the free-field is expanded as
$\displaystyle A^{\mu}(x;{\cal F},{\cal P})$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{d\vec{k}\cdot\
g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal F},{\cal P})}{(2\pi)^{3}(2\,|\,\vec{k}\
|)}\cdot\left\\{\sum_{s=1}^{2}\epsilon^{\mu}(\uline{\vec{k}},s)\left[\,{\hat{\bf
c}(\uline{\vec{k}},s)}\ e^{\displaystyle-i\uline{\vec{k}}x}+{\hat{\bf
c}^{\dagger}(\uline{\vec{k}},s)}\ e^{\displaystyle
i\uline{\vec{k}}x}\,\right]\right\\}$ and the commutation relations between
the creation and annihilation operators are postulated to be
$\displaystyle\begin{array}[]{l}\ [{\hat{\bf c}(\uline{\vec{k}},s)},{\hat{\bf
c}^{\dagger}(\uline{\vec{k}}\,^{\prime},s^{\prime})}]\ =\
(2\pi)^{3}\cdot(2\,|\,\vec{k}\,|)\cdot\delta(\vec{k}-\vec{k}\
^{\prime})\cdot\delta_{s,s^{\prime}},\\\ \ [{\hat{\bf
c}(\uline{\vec{k}},s)},{\hat{\bf c}(\uline{\vec{k}}\,^{\prime},s^{\prime})}]\
=\ [{\hat{\bf c}^{\dagger}(\uline{\vec{k}},s)},{\hat{\bf
c}^{\dagger}(\uline{\vec{k}}\,^{\prime},s^{\prime})}]\ =\ 0,\qquad\qquad\quad\
s,\,s^{\prime}=1,2\end{array}$ (30) , taken to be

$\displaystyle A^{\mu}(x;{\cal F},{\cal
P})=\int\displaystyle\frac{d\vec{k}\cdot\ g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal
F},{\cal P})}{(2\pi)^{3}(2\,|\,\vec{k}\
|)}\cdot\left\\{\sum_{s=0}^{3}\epsilon^{\mu}(\uline{\vec{k}},s)\left[\,{\hat{\bf
c}(\uline{\vec{k}},s)}\ e^{\displaystyle-i\uline{\vec{k}}x}+{\hat{\bf
c}^{\dagger}(\uline{\vec{k}},s)}\ e^{\displaystyle
i\uline{\vec{k}}x}\,\right]\right\\}$ (31)

Of the four polarization modes, only two—the transverse modes—are physical.
The nonphysical scalar and longitudinal modes are rendered irrelevant by
postulating the following well-known commutation relations on the creation and
annihilation operators of the free Maxwell field

$\displaystyle\begin{array}[]{l}\ [{\hat{\bf c}(\vec{k},s)},{\hat{\bf
c}^{\dagger}(\vec{k}^{\prime},s^{\prime})}]\ =\
-(2\pi)^{3}\cdot(2\,|\,\vec{k}\,|)\cdot\delta(\vec{k}-\vec{k}\
^{\prime})\cdot\eta_{s,s^{\prime}},\quad\ \ s,\,s^{\prime}=0,1,2,3\\\ \
[{\hat{\bf c}(\vec{k},s)},{\hat{\bf c}(\vec{k}^{\prime},s^{\prime})}]\ =\
[{\hat{\bf c}^{\dagger}(\vec{k},s)},{\hat{\bf
c}^{\dagger}(\vec{k}^{\prime},s^{\prime})}]\ =\ 0,\end{array}\hskip 36.135pt$
(34)

and defining physical states as those states in the Fock space that contain
equal numbers of scalar and longitudinal photons (see Section 7.2). No
commutation conditions are imposed on the Maxwell field and its conjugate
momentum.

For later reference we note that the creation and annihilation operators of
the Dirac field can be expressed in terms of the field, as listed below.

$\displaystyle{\hat{\bf a}(\vec{k},s)}\ =\ \displaystyle\frac{1}{\
g{{}_{f}}\,(\,{\uline{\vec{k}}}\,;{\cal F},{\cal P})}\int d\,\vec{x}\
e^{\displaystyle i\uline{\vec{k}}x}\ \bar{u}(\uline{\vec{k}},s)\ \gamma^{0}\
\psi(x);$ $\displaystyle{\hat{\bf a}^{\dagger}(\vec{k},s)}\ =\
\displaystyle\frac{1}{\ g{{}_{f}}\,(\,{\uline{\vec{k}}}\,;{\cal F},{\cal
P})}\int d\,\vec{x}\ e^{\displaystyle-i\uline{\vec{k}}x}\ \bar{\psi}(x)\
\gamma^{0}\ u(\uline{\vec{k}},s)$ $\displaystyle{\hat{\bf b}(\vec{k},s)}\ =\
\displaystyle\frac{1}{\ g{{}_{f}}\,(\,{\uline{\vec{k}}}\,;{\cal F},{\cal
P})}\int d\,\vec{x}\ e^{\displaystyle i\uline{\vec{k}}x}\ \bar{\psi}(x)\
\gamma^{0}\ v(\uline{\vec{k}},s);$ $\displaystyle{\hat{\bf
b}^{\dagger}(\vec{k},s)}\ =\ \displaystyle\frac{1}{\
g{{}_{f}}\,(\,{\uline{\vec{k}}}\,;{\cal F},{\cal P})}\int d\,\vec{x}\
e^{\displaystyle-i\uline{\vec{k}}x}\ \bar{v}(\uline{\vec{k}},s)\ \gamma^{0}\
\psi(x)$

###  3.3 LSZ Reduction and Lorentz Invariance

The appearance of the Gibbs factors in (LABEL:eqn601) leads to a
straightforward modification of the LSZ reduction procedure. In Appendix B we
have presented a rigorous derivation of the LSZ reduction formula taking Gibbs
factors into account. As an example, consider a simple scattering process
${\cal P}$ in QED in which $m$ incoming electrons and $\tilde{m}$ positrons
scatter to $n$ outgoing electrons and $\tilde{n}$ positrons. The momenta and
$z$-components of the spins—hereafter called the $z$-spins—of the incoming
electrons (respectively positrons) are denoted as pairs
($p_{{}_{i}},r_{{}_{i}}$), $1\leq i\leq m$ (respectively
($\tilde{p}_{{}_{j}},\tilde{r}_{{}_{j}}$), $1\leq j\leq\tilde{m}$). Similarly
the (momentum, $z$-spin) pairs of outgoing electrons (respectively positrons)
are denoted ($q_{{}_{i}},s_{{}_{i}}$), $1\leq i\leq n$ (respectively
($\tilde{q}_{{}_{j}},\tilde{s}_{{}_{j}}$), $1\leq j\leq\tilde{n}$). The
ingoing and outgoing particles are on mass-shell. That is,
$\left(\uline{p_{{}_{i}}}\right)^{2}\ =\
\left(\tilde{\uline{p_{{}_{j}}}}\right)^{2}=\left(\uline{q_{k}}\right)^{2}=\left(\tilde{\uline{q_{l}}}\right)^{2}=m_{{}_{e}}^{2}$,
where $m_{{}_{e}}$ is the physical mass of the electron and $1\leq i\leq
m,1\leq j\leq\tilde{m}$, $1\leq k\leq n$, $1\leq l\leq\tilde{n}$. The
scattering amplitude of the process ${\cal P}$, in an arbitrary frame ${\cal
F}$ is given by

$\displaystyle{\cal A}({\cal P},{\cal F})$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{(-i)^{n+\tilde{n}}(i)^{m+\tilde{m}}\
\prod_{i=1}^{m}d^{4}x_{{}_{i}}\prod_{j=1}^{\tilde{m}}d^{4}\tilde{x}_{j}\prod_{k=1}^{n}d^{4}y_{k}\prod_{l=1}^{\tilde{n}}d^{4}\tilde{y}_{l}}{\left(\sqrt{\rule{0.0pt}{10.0pt}Z}\right)^{m+n+\tilde{m}+\tilde{n}}\displaystyle\prod_{i=1}^{m}\
g{{}_{e}}\,(\,{p_{i}}\,;{\cal F},{\cal P})\prod_{j=1}^{\tilde{m}}\
g{{}_{e}}\,(\,{\tilde{p}_{j}}\,;{\cal F},{\cal P})\prod_{k=1}^{n}\
g{{}_{e}}\,(\,{q_{k}}\,;{\cal F},{\cal P})\prod_{l=1}^{\tilde{n}}\
g{{}_{e}}\,(\,{\tilde{q}_{l}}\,;{\cal F},{\cal P})}$ (36) $\displaystyle
e^{\footnotesize{\left[\,-i\sum_{i=1}^{m}p_{i}x_{i}-i\sum_{j=1}^{\tilde{m}}{\tilde{p}}_{j}\tilde{x}_{j}+i\sum_{k=1}^{n}q_{k}y_{k}+i\sum_{l=1}^{\tilde{n}}{\tilde{q}}_{l}\tilde{y}_{l}\,\right]}}$
$\displaystyle\prod_{k=1}^{n}\bar{u}({q_{k}},s_{k})(i\partial\\!\\!\\!/_{y_{k}}-m_{{}_{e}})\prod_{j=1}^{\tilde{m}}\bar{v}({{\tilde{p}}_{j}},\tilde{r}_{j})(i\partial\\!\\!\\!/_{\tilde{x}_{j}}-m_{{}_{e}})\left\langle\,0\left|\,T\left\\{\psi(y_{k})\
\bar{\psi}(\tilde{y}_{l})\bar{\psi}(x_{i})\psi(\tilde{x}_{j})\right\\}\,\right|0\,\right\rangle$
$\displaystyle\prod_{l=1}^{\tilde{n}}(i\overleftarrow{\partial\\!\\!\\!/}_{\tilde{y}_{l}}+m_{{}_{e}})\
v({{\tilde{q}}_{l}},\tilde{s}_{l})\prod_{i=1}^{m}(i\overleftarrow{\partial\\!\\!\\!/}_{x_{i}}+m_{{}_{e}})\
u({p_{i}},r_{i})$

The LSZ reduction formula for other fields is derived similarly. $\sqrt{Z}$,
the wavefunction renormalization constant , defined and shown to be Lorentz-
invariant in (159), gives the amplitude for the interacting field
$\psi(\vec{x},t)$ to create an on-mass-shell 1-particle state. The spinors
$u(\uline{k},s)$ and $v(\underline{q},s)$, in (36), satisfy the equations
$(\uline{k}\\!\\!\\!/-m_{{}_{e}})u(\uline{k},s)=0$,
$(\underline{q}\\!\\!\\!/+m_{{}_{e}})v(\underline{q},s)=0$, where
$\uline{k}\in\left\\{\uline{p}_{{}_{1}},\ldots,\uline{p}_{{}_{m}},\underline{q}_{{}_{1}},\ldots,\underline{q}_{{}_{n}}\right\\}$
and
$\underline{q}\in\left\\{\uline{\tilde{p}}_{{}_{1}},\ldots,\uline{\tilde{p}}_{{}_{\tilde{m}}},\uline{\tilde{q}}_{{}_{1}},\ldots,\uline{\tilde{q}}_{{}_{\tilde{n}}}\right\\}$
and $s=1,2$. Lorentz invariance of (36) is established in Appendix C.

###  3.4 Propagators

We compute the propagators for a fermion of mass $m$ and for a photon, first
in the center-of-momentum frame ${\cal C}_{{}_{\cal P}}$ of the scattering
process ${\cal P}$ that is of interest, and subsequently in an arbitrary frame
${\cal F}$.

####  3.4.1 Dirac Propagator

Using (LABEL:eqn11), (27) and the properties of $u(\uline{\vec{k}},s)$ and
$v(\uline{\vec{k}},s)$, we have, in the center-of-momentum frame ${\cal
C}_{{}_{\cal P}}$,

$\displaystyle D(x,y;{\cal C}_{{}_{\cal P}})$ $\displaystyle:=$
$\displaystyle\left\langle\,0\left|\,T\left\\{\psi(x)\bar{\psi}(y)\right\\}\,\right|0\,\right\rangle\
=\
\left\\{\begin{array}[]{ll}\left\langle\,0\left|\,\psi(x)\bar{\psi}(y)\,\right|0\,\right\rangle&x^{0}>y^{0}\\\\[7.22743pt]
-\left\langle\,0\left|\,\bar{\psi}(y)\psi(x)\,\right|0\,\right\rangle&y^{0}>x^{0}\end{array}\right.$
(38) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\int\displaystyle\frac{d\,\vec{k}\,\left\\{\
g{{}_{f}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})\right\\}^{2}}{(2\pi)^{3}\,2\omega(\vec{k}\,)}\cdot(i\gamma^{0}\,\partial_{{}_{x^{0}}}-\vec{k}\cdot\vec{}\gamma+m)e^{\displaystyle-i\uline{\vec{k}}(x-y)},&x^{0}-y^{0}>0\\\
\\\ \displaystyle\int\displaystyle\frac{d\,\vec{k}\,\left\\{\
g{{}_{f}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})\right\\}^{2}}{(2\pi)^{3}\,2\omega(\vec{k}\,)}\cdot(i\gamma^{0}\,\partial_{{}_{x^{0}}}+\vec{k}\cdot\vec{}\gamma+m)e^{\displaystyle
i\uline{\vec{k}}(x-y)},&x^{0}-y^{0}<0\end{array}\right.$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{d\,\vec{k}\,\left\\{\
g{{}_{f}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})\right\\}^{2}}{(2\pi)^{3}\,2\omega(\vec{k}\,)}\left[\,\left(i\gamma^{0}\partial_{{}_{x^{0}}}-\vec{k}\cdot\vec{}\gamma+m\right)e^{\displaystyle
i\vec{k}\cdot(\vec{x}-\vec{y})}\,\right]e^{\displaystyle-i\omega(\vec{k}\,)|(x-y)^{0}|}$

In the last step, we have changed the variables $\vec{k}\rightarrow-\vec{k}$
for the case where $x^{0}-y^{0}<0$. Since $u({\cal C}_{{}_{\cal P}};{\cal
C}_{{}_{\cal P}})=(1,0,0,0)$, we have $\uline{\vec{k}}\
_{\alpha}u^{\alpha}({\cal C}_{{}_{\cal P}};{\cal C}_{{}_{\cal P}})\ =\
\left[\,\uline{-\vec{k}}\,\right]_{\alpha}u^{\alpha}({\cal C}_{{}_{\cal
P}};{\cal C}_{{}_{\cal P}})$ and therefore $\
g{{}_{f}}\,(\,{\uline{-\vec{k}}}\,;{\cal C}_{{}_{\cal P}},{\cal P})\ =\ \
g{{}_{f}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal P}).$

Noting that

$\displaystyle e^{\displaystyle-i\,\omega(\vec{k}\,)|(x-y)^{0}|}$
$\displaystyle=$ $\displaystyle\lim_{\epsilon\rightarrow
0^{+}}\left[\,\frac{i\cdot\left(2\,\omega(\vec{k}\,)\right)}{2\pi}\,\right]\int
dk^{0}\cdot\displaystyle\frac{e^{\displaystyle-
ik^{0}(x-y)^{0}}}{(k^{0})^{2}-\left[\,\ \omega(\vec{k}\,)-i\epsilon\
\,\right]^{2}},$ (42)

where $\omega(\vec{k}\,)=\sqrt{\rule{0.0pt}{12.0pt}|\vec{k}|^{2}+m^{2}}$, we
have

$\displaystyle D(x,y;{\cal C}_{{}_{\cal P}})$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\rightarrow
0^{+}}\left\\{i\int\displaystyle\frac{d^{4}k}{(2\pi)^{4}}\cdot\left\\{\
g{{}_{f}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal
P})\right\\}^{2}\cdot\displaystyle\frac{k\\!\\!\\!/+m}{k^{2}-m^{2}+i\epsilon}\cdot
e^{\displaystyle-ik(x-y)}\right\\}$

In (LABEL:eqn502) we have rewritten $\ g{{}_{f}}(\,\uline{\vec{k}}\,;{\cal
C}_{{}_{\cal P}},{\cal P})$ as $\ g{{}_{f}}\,(\,{k}\,;{\cal C}_{{}_{\cal
P}},{\cal P})$ using (4) and (8). We will omit writing
$\lim_{\epsilon\rightarrow 0^{+}}$ hereafter with the understanding that in
the end the limit is taken wherever $i\epsilon$ occurs in a propagator.

Next, we compute the Dirac propagator in an arbitrary frame ${\cal F}$. Let
$\Lambda({\cal F},{\cal C}_{{}_{\cal P}}):{\cal C}_{{}_{\cal
P}}\rightarrow{\cal F}$ denote the proper orthochronous Lorentz transformation
that maps ${\cal C}_{{}_{\cal P}}$ to ${\cal F}$. We use the abbreviation
$\Lambda=\Lambda({\cal F},{\cal C}_{{}_{\cal P}})$. Let $x$ and $x^{\prime}$
be the coordinates assigned to a spacetime point by observers at rest in
${\cal C}_{{}_{\cal P}}$ and ${\cal F}$ respectively. Further let $\psi(x)$
and $\psi^{\prime}(x^{\prime})$ be the Dirac fields in ${\cal C}_{{}_{\cal
P}}$ and ${\cal F}$. Then we have

$\displaystyle\psi^{\prime}(x^{\prime})$ $\displaystyle=$ $\displaystyle
S[\Lambda]\,\psi(x),\qquad S[\Lambda]\ =\
\displaystyle\exp\left\\{(i/2)\omega_{\mu\nu}\Sigma^{\mu\nu}\right\\},\quad\Sigma^{\mu\nu}:=\left(\frac{i}{4}\right)[\gamma^{\mu},\gamma^{\nu}]$

As shown in (219), $S^{\dagger}[\Lambda]\,\gamma^{0}\ =\
\gamma^{0}\,S^{-1}[\Lambda]$. Therefore, the propagator in frame ${\cal F}$ is

$\displaystyle D(x^{\prime},y^{\prime};{\cal F})$ $\displaystyle=$
$\displaystyle\left\langle\,0\left|\,T\left\\{\psi^{\prime}(x^{\prime})\bar{\psi}^{\prime}(y^{\prime})\right\\}\,\right|0\,\right\rangle\
=\ S[\Lambda]\,D(x,y;{\cal C}_{{}_{\cal P}})\,S^{-1}[\Lambda]$

where $x\ =\ \Lambda^{-1}x^{\prime},\ y\ =\ \Lambda^{-1}y^{\prime}$. Noting
that

$\displaystyle k_{\alpha}S[\Lambda]\gamma^{\alpha}S^{-1}[\Lambda]$
$\displaystyle=$ $\displaystyle
k_{\alpha}\Lambda_{\beta}\,^{\alpha}\gamma^{\beta}\ =\ (\Lambda
k)_{\beta}\gamma^{\beta}$

we have

$\displaystyle D(x^{\prime},y^{\prime};{\cal F})$ $\displaystyle=$
$\displaystyle i\int\displaystyle\frac{d^{4}k}{(2\pi)^{4}}\ \
g{{}_{f}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal P})^{2}\left[\,\frac{(\Lambda
k)_{\alpha}\gamma^{\alpha}+m}{k^{2}-m^{2}+i\epsilon}\,\right]e^{\displaystyle-
ik(x-y)}$

Changing variables from $k$ to $q=\Lambda k$ and noting
$k_{\alpha}(x-y)^{\alpha}\ =\ q_{\alpha}(x^{\prime}-y^{\prime})^{\alpha}$, we
have

$\displaystyle D(x^{\prime},y^{\prime};{\cal F})$ $\displaystyle=$
$\displaystyle i\int\displaystyle\frac{d^{4}q}{(2\pi)^{4}}\ \
g{{}_{f}}\,(\,{\Lambda^{-1}q}\,;{\cal C}_{{}_{\cal P}},{\cal
P})^{2}\left[\,\frac{q\\!\\!\\!/+m}{q^{2}-m^{2}+i\epsilon}\,\right]e^{\displaystyle-
iq(x^{\prime}-y^{\prime})}$

Using (23) we get the final form of the Dirac propagator in an arbitrary frame
${\cal F}$,

$\displaystyle D(x^{\prime},y^{\prime};{\cal F})$ $\displaystyle=$
$\displaystyle i\int\displaystyle\frac{d^{4}q}{(2\pi)^{4}}\ \left\\{\
g{{}_{f}}\,(\,{q}\,;{\cal F},{\cal
P})\right\\}^{2}\left[\,\frac{q\\!\\!\\!/+m}{q^{2}-m^{2}+i\epsilon}\,\right]e^{\displaystyle-
iq(x^{\prime}-y^{\prime})}$ (44)

where $\ g{{}_{f}}\,(\,{q}\,;{\cal F},{\cal P})$ is given by (17).

####  3.4.2 Photon Propagator

We calculate the photon propagator in frame ${\cal C}_{{}_{\cal P}}$, using
the Lorenz gauge. The 4-potential $A^{\mu}$ is expanded in the Lorentz frame
${\cal C}_{{}_{\cal P}}$, as

$\displaystyle A^{\mu}(x;{\cal C}_{{}_{\cal P}},{\cal P})$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{d\,\vec{k}\ \
g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})}{(2\pi)^{3}2\omega(\vec{k}\,)}\displaystyle\sum_{\lambda=0}^{3}\epsilon^{\mu}(\uline{\vec{k}},\lambda)\left\\{{\bf\hat{a}}(\uline{\vec{k}},\lambda)e^{\displaystyle-i\uline{\vec{k}}x}\
+{\bf\hat{a}}^{\dagger}(\uline{\vec{k}},\lambda)e^{\displaystyle
i\uline{\vec{k}}x}\ \right\\}\qquad$ (45)

In the frame ${\cal C}_{{}_{\cal P}}$, the polarization vectors are chosen as

$\displaystyle\epsilon^{\mu}(\uline{\vec{k}},0)\ =\ u({\cal C}_{{}_{\cal
P}};{\cal C}_{{}_{\cal P}});\quad\epsilon^{\mu}(\uline{\vec{k}},1)\ =\
(0,\hat{q}_{{}_{1}});\quad\epsilon^{\mu}(\uline{\vec{k}},2)\ =\
(0,\hat{q}_{{}_{2}});\quad\epsilon^{\mu}(\uline{\vec{k}},3)\ =\ (0,\hat{k})$
(46)

where $(\hat{q}_{{}_{1}},\hat{q}_{{}_{2}},\hat{k})$ form an orthonormal set
with $\hat{q}_{{}_{1}}\times\hat{q}_{{}_{2}}\ =\ \hat{k}$, and
$\hat{k}=\displaystyle\frac{\vec{k}}{|\,\vec{k}\,|}$. The polarization vectors
satisfy the relationship

$\displaystyle\sum_{\lambda,\lambda^{\prime}=0}^{3}\eta_{{}_{\lambda\,\lambda^{\prime}}}\,\epsilon^{\mu}(\uline{\vec{k}},\lambda)\,\epsilon^{\nu}(\uline{\vec{k}},\lambda^{\prime})$
$\displaystyle=$ $\displaystyle\eta^{\mu\nu}$ (47)

where $\eta$ is the Lorentzian metric tensor.

We impose the following commutation relations on the operators ${\bf\hat{a}}$
and ${\bf\hat{a}}^{\dagger}$:

$\displaystyle[{\bf\hat{a}}(\uline{\vec{k}},\lambda),{\bf\hat{a}}^{\dagger}(\uline{\vec{q}},\lambda^{\prime})]$
$\displaystyle=$
$\displaystyle-\eta_{{}_{\lambda\,\lambda^{\prime}}}\cdot(2\pi)^{3}\cdot
2\omega(\vec{k}\,)\cdot\delta^{3}(\vec{k}-\vec{q})$ (48)

Then

$\displaystyle\left\langle\,0\left|\,A^{\mu}(x;{\cal C}_{{}_{\cal P}},{\cal
P})A^{\nu}(y;{\cal C}_{{}_{\cal P}},{\cal P})\,\right|0\,\right\rangle$
$\displaystyle=$ $\displaystyle-\eta^{\mu\nu}\
\int\displaystyle\frac{d\vec{k}}{(2\pi)^{3}(2\omega(\vec{k}))}\ \cdot\left\\{\
g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})\right\\}^{2}e^{\displaystyle-i\uline{\vec{k}}(x-y)}$

Using arguments similar to those in (LABEL:eqn1101) and (42), we obtain the
Feynman propagator for the Maxwell field:

$\displaystyle M^{\mu\nu}(x,y;{\cal C}_{{}_{\cal P}})$ $\displaystyle:=$
$\displaystyle\left\langle\,0\left|\,T\left\\{A^{\mu}(x;{\cal C}_{{}_{\cal
P}})\,A^{\nu}(y;{\cal C}_{{}_{\cal P}})\right\\}\,\right|0\,\right\rangle$
(49) $\displaystyle=$ $\displaystyle
i\int\displaystyle\frac{d^{4}k}{(2\pi)^{4}}\cdot\ g{{}_{p}}\,(\,{k}\,;{\cal
C}_{{}_{\cal P}},{\cal P})^{2}\cdot\displaystyle\frac{e^{\displaystyle-
ik(x-y)}}{k^{2}+i\epsilon}\cdot\left(-\eta^{\mu\nu}\right)$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{d^{4}k}{(2\pi)^{4}}\
\tilde{M}^{\mu\nu}(k;{\cal C}_{{}_{\cal P}},{\cal P})e^{\displaystyle-
ik(x-y)},$

where

$\displaystyle\tilde{M}^{\mu\nu}(k;{\cal C}_{{}_{\cal P}},{\cal P})$
$\displaystyle:=$ $\displaystyle i\ \left[\,\frac{(-\eta^{\mu\nu})\left[\,\
g{{}_{p}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal
P})\,\right]^{2}}{k^{2}+i\epsilon}\,\right]$

Using an argument similar to that used for Dirac propagators, it is easily
verified that the Maxwell propagator in an arbitrary frame ${\cal F}$ is given
by

$\displaystyle M^{\mu\nu}(x^{\prime},y^{\prime};{\cal F})$ $\displaystyle=$
$\displaystyle i\int\displaystyle\frac{d^{4}k^{\prime}}{(2\pi)^{4}}\
\tilde{M}^{\mu\nu}(k^{\prime};{\cal F},{\cal P})e^{\displaystyle-
ik^{\prime}(x^{\prime}-y^{\prime})},$ (50)

where

$\displaystyle\tilde{M}^{\mu\nu}(k^{\prime};{\cal F},{\cal P})$
$\displaystyle:=$ $\displaystyle i\ \left[\,\frac{(-\eta^{\mu\nu})\left[\,\
g{{}_{p}}\,(\,{k^{\prime}}\,;{\cal F},{\cal
P})\,\right]^{2}}{(k^{\prime})^{2}+i\epsilon}\,\right]$

##  4 Anomalous Magnetic Moment of the Electron

In this section we calculate the 1-loop correction to electron’s anomalous
magnetic moment, using the new framework. The calculation is divergence-free
and yields a correction that deviates from the experimentally measured value
of 0.001159652180 by less than 0.06% at 1-loop approximation.

We recall that the spin operator for the Dirac field of is
$\vec{S}=[\sigma^{23},\sigma^{31},\sigma^{12}]$, where
$\sigma^{\mu\nu}:=\displaystyle\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}],\
\mu,\nu=0,\ldots,3$ are the generators of the Lorentz algebra. The magnetic
moment of the electron is related to its spin by the spin gyromagnetic ratio
as $\vec{\mu}=g\left(\frac{e}{2m_{e}}\right)\vec{S}$. We define $\kappa$, the
correction to the gyromagnetic ratio, as $g=2(1+\kappa)$. At the tree level,
$g_{tree}=2$, and $\kappa_{tree}=0$. Our objective is to compute
$g_{{}_{1-loop}}=g_{{}_{tree}}\cdot\kappa_{{}_{1-loop}}$, at 1-loop level.

Consider the interaction of an electron with a weak background classical
electromagnetic field $\delta A_{{}_{c}}(x)$. The strength of the classical
electromagnetic field is determined by the dimensionless parameter $\delta\ll
1$, which has been included explicitly to facilitate book-keeping in
perturbative calculations at the weak-field limit.

The magnetic moment of the electron $\vec{\mu}$ interacts with an external
classical magnetic field $\vec{B}_{\,c}$ through the interaction term
$-\vec{\mu}\cdot\vec{B}_{\,c}$, which is contained in the term of the form
$\sigma^{\mu\nu}\partial_{\mu}(A_{\,c})_{\nu}(x)$ as is readily seen by
setting
$\vec{d}=g\left(\frac{e}{2m_{{}_{e}}}\right)[\sigma^{01},\sigma^{02},\sigma^{03}]$
and observing that, at the classical (tree) level,

$\displaystyle-\vec{\mu}\cdot\vec{B}_{\,c}(x)+\vec{d}\cdot\vec{E}_{\,c}(x)$
$\displaystyle=$
$\displaystyle\delta\left[\,2\left(\frac{e}{2m_{{}_{e}}}\right)\sigma^{\mu\nu}\,\right]\partial_{\mu}(A_{c}(x))_{\nu}$
$\displaystyle=$ $\displaystyle
i\,\delta\left[\,g\left(\frac{e}{2m_{{}_{e}}}\right)\,\right]\int\left[\,\,\sigma^{\mu\nu}\
p_{\mu}\,(A_{c}(p))_{\nu}\,\right]\,e^{\displaystyle
i\,p\,x}\,\displaystyle\frac{d^{4}p}{(2\pi)^{4}}$

Hence, our focus will be on terms of the form
$\sigma^{\mu\nu}p_{\mu}(A_{c}(p))_{\nu}$ in momentum space.

In order to compute $\kappa_{{}_{1-loop}}$, following Schwinger [81], we
consider the scattering of an electron, which is initially at rest in frame
${\cal C}_{{}_{\cal P}}$, off a weak classical background electromagnetic
field $\delta A^{\mu}_{c}(x)$:

$\displaystyle{\cal P}:\qquad e^{-}(p,s)\quad\stackrel{{\scriptstyle\delta
A_{\,c}\ }}{{\longrightarrow}}\quad e^{-}(q,s^{\prime})$ (51)

$p$ and $q$ represent the 4-momenta of the incoming and outgoing electron,
while $s$ and $s^{\prime}$ represent their polarizations or
$z$-spins202020$z$-components of their spin.. We will consider the limit
$|\vec{q}\,|/m_{{}_{e}}\ll 1$.

In the presence of a classical background field, the total electromagnetic
field is

$\displaystyle A^{\mu}_{tot}(x)=\delta
A_{c}^{\mu}(x)+A^{\mu}(x),\qquad\mu=0,\ldots,3,\quad\delta\ll 1$

where $A^{\mu}(x)$ is the dynamical gauge field. The QED interaction
Hamiltonian density is

$\displaystyle{\cal H}_{QED}$ $\displaystyle=$ $\displaystyle-
e_{{}_{0}}\bar{\psi}\left(A\\!\\!\\!/+\delta A\\!\\!\\!/_{c}\right)\psi+\Delta
m\ \bar{\psi}\psi,\qquad\Delta m:=m_{{}_{e}}-m_{{}_{0}},$ (52)

We use the natural units setting $c=\hbar=\epsilon_{{}_{0}}=m_{{}_{e}}=1$,
where $c,h,\epsilon_{{}_{0}}$ and $m_{{}_{e}}$ represent the usual
quantities212121Namely the speed of light in vacuum, Planck’s constant,
electric permittivity of vacuum and physical rest mass of the electron..
$m_{{}_{0}}$ and $e_{{}_{0}}<0$ represent the bare mass and charge parameters
of an electron. In the natural units, the physical charge of the electron,
denoted $e$, is given by $e^{2}=4\pi\alpha$, where $\alpha$ is the fine
structure constant222222In the natural units, $e$ and $e_{{}_{0}}$ are also
dimensionless and we use them as the parameters for the perturbative
expansions.. We also note that $\Delta m$ vanishes at $O(e_{{}_{0}})$ and is
nonzero at $O(e_{{}_{0}}^{2})$. The tree-level and 1-loop 1-particle
irreducible Feynman diagrams describing the above scattering process are shown
in Figure 1.

Figure 1: Tree-level and 1-loop 1-particle irreducible vertex diagrams for an
electron scattering off a classical background electromagnetic field. The
solid circles represent the classical source.

The scattering amplitude ${\cal A}(p,s,q,s^{\prime})$ of the process ${\cal
P}$ can be expanded as a power series in $\delta$ as

$\displaystyle{\cal A}(p,s,q,s^{\prime})$ $\displaystyle=$ $\displaystyle{\cal
A}^{(0)}(p,s,q,s^{\prime})+{\cal A}^{(1)}(p,s,q,s^{\prime})\ \delta+{\cal
A}^{(2)}(p,s,q,s^{\prime})\ \delta^{2}+\ldots$ (53)

At $O(\delta^{0})$ the electron does not interact with the background field.
We are interested in the limit in which the classical background field $\delta
A_{\,c}$ is weak and hence restrict attention only to the scattering amplitude
at $O(\delta)$, or ${\cal A}^{(1)}(p,s,q,s^{\prime})$.

At $O(\delta)$ the scattering amplitude can be expanded in powers of the
dimensionless parameter $e_{{}_{0}}$ to yield

$\displaystyle{\cal A}^{(1)}(p,s,q,s^{\prime})$ $\displaystyle=$
$\displaystyle{\cal A}^{(1)}_{{}_{1}}(p,s,q,s^{\prime})\ e_{{}_{0}}+\ {\cal
A}^{(1)}_{{}_{2}}(p,s,q,s^{\prime})\ e_{{}_{0}}^{2}+{\cal
A}^{(1)}_{{}_{3}}(p,s,q,s^{\prime})\ e_{{}_{0}}^{3}+\ldots\quad$ (54)

where ${\cal A}^{(i)}_{j}\,e_{{}_{0}}^{j},\ j=1,2,\ldots$ represents the sum
of the amplitudes of all the Feynman diagrams with $j$ interaction vertices,
at $O(\delta\,^{i})$. We note that ${\cal A}^{(1)}_{{}_{2}}=0$.

Grouping the terms that contain the factor
$\sigma^{\mu\nu}({p-q})_{\mu}(A_{c})_{\nu}({p-q})$ one can write for
$j=1,3,\ldots$,

$\displaystyle{\cal A}^{(1)}_{{}_{j}}(p,s,q,s^{\prime})$ $\displaystyle=$
$\displaystyle\
g_{tree}\cdot\xi_{{}_{j}}(e,m_{{}_{e}})\cdot\bar{u}{(q,s^{\prime})}\left\\{\sigma^{\mu\nu}(p-q)_{\mu}(A_{c})_{\nu}(p-q)\right\\}u(p,s)$
(55) $\displaystyle+{\cal B}_{{}_{j}}(p,s,q,s^{\prime})$

where ${\cal B}_{{}_{j}}(p,s,q,s^{\prime}),\ j=1,3,\ldots$ represents the sum
of all the terms in ${\cal
A}^{(1)}_{{}_{j}}(p,s,q,s^{\prime})\,e_{{}_{0}}^{j}$ that do not contain the
factor $\sigma^{\mu\nu}(p-q)_{\mu}(A_{c})_{\nu}(p-q)$. $e$ and $m_{{}_{e}}$
represent the physical charge and mass of an electron.

Since ${\cal A}^{(1)}_{2}=0$, the amplitude ${\cal A}^{(1)}$ can be written as

$\displaystyle{\cal A}^{(1)}$ $\displaystyle=$
$\displaystyle\left[\,g_{tree}\left\\{1+e_{{}_{0}}^{2}\cdot\displaystyle\frac{\xi_{{}_{3}}(e,m_{{}_{e}})}{\xi_{{}_{1}}(e,m_{{}_{e}})}+\ldots\right\\}\,\right]\left[\,e_{{}_{0}}\
\xi_{{}_{1}}(e,m_{{}_{e}})\ \bar{u}{(q,s^{\prime})}\
\sigma^{\mu\nu}(p-q)_{\mu}(A_{c})_{\nu}(p-q)\ u(p,s)\,\right]$
$\displaystyle+\left[\,{\cal B}_{{}_{1}}+{\cal B}_{{}_{3}}+\ldots\,\right]$

Hence, $\kappa$, the correction to electron’s $g$-factor is

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
e_{{}_{0}}^{2}\cdot\displaystyle\frac{\xi_{{}_{3}}(e,m_{{}_{e}})}{\xi_{{}_{1}}(e,m_{{}_{e}})}+\ldots$

Since $e_{{}_{0}}=e+O(e^{3})$, and we are interested in the
$O(e_{{}_{0}}^{2})$ (that is, leading order) correction to $\kappa$, we can
replace $e_{{}_{0}}$ by $e$ in the above expansion232323 The physical mass and
charge of the electron can be written as a perturbartive series in
$e_{{}_{0}}$ as  $\displaystyle e$ $\displaystyle=$ $\displaystyle
e_{{}_{0}}+c_{{}_{3}}\,e_{{}_{0}}^{3}+\ldots\qquad\qquad m\ =\
m_{{}_{0}}+d_{{}_{2}}\,e_{{}_{0}}^{2}+\ldots$ The coefficients
$c_{{}_{3}},\ldots$ and $d_{{}_{2}}\ldots$ are computed using the Feynman
diagrams. Inverting the above power series we obtain  $\displaystyle
e_{{}_{0}}$ $\displaystyle=$ $\displaystyle
e-c_{{}_{3}}e^{3}+\ldots\qquad\qquad m_{{}_{0}}\ =\ m-d_{{}_{2}}e^{2}+\ldots$
The reader is referred to [38, 74, 86] for a more detailed discussion..
Expanding
$f_{31}(e,m_{{}_{e}}):=\displaystyle\frac{\xi_{{}_{3}}(e,m_{{}_{e}})}{\xi_{{}_{1}}(e,m_{{}_{e}})}$
in powers of $e$ as

$\displaystyle f_{31}(e,m_{{}_{e}})$ $\displaystyle=$ $\displaystyle
F_{31}^{(0)}+F_{31}^{(1)}\cdot e+\ldots$ (56)

we then obtain the 1-loop correction to $g$, denoted as
$g_{{}_{1-loop}}=g_{{}_{tree}}\cdot\kappa_{{}_{1-loop}}$ where

$\displaystyle\kappa_{{}_{1-loop}}$ $\displaystyle=$ $\displaystyle e^{2}\cdot
F_{31}^{(0)}\ =\ 4\pi\alpha\cdot F_{31}^{(0)}$ (57)

In summary, computation of the 1-loop correction to $g$ reduces to the
calculation of $F_{31}^{(0)}$, which involves calculations of
$\xi_{{}_{3}}(e,m)$ and $\xi_{{}_{1}}(e,m)$. We calculate $\xi_{{}_{1}}(e,m)$
and $\xi_{{}_{3}}(e,m)$ in the next section leading to the results shown in
(60) and (63) respectively.

###  4.1 $O(\alpha)$ Correction to Electron’s Magnetic Moment

Consider the scattering process ${\cal P}$ shown in (51). From (36),
(LABEL:eqnvcf3) and (194) we note that in the Lorentz frame ${\cal
C}_{{}_{\cal P}}$, the scattering amplitude ${\cal A}(p,s,q,s^{\prime})$ of
${\cal P}$ is given by

$\displaystyle{\cal A}(p,s,q,s^{\prime})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{(-i)(i)}{Z\cdot{\cal N}\cdot\
g{{}_{e}}\,(\,{p}\,;{\cal C}_{{}_{\cal P}},{\cal P})\cdot\
g{{}_{e}}\,(\,{q}\,;{\cal C}_{{}_{\cal P}},{\cal P})}\int d^{4}x\ d^{4}y\
e^{\displaystyle-i\uline{p}\cdot x+i\uline{q}\cdot y}$
$\displaystyle\bar{u}{(q,s^{\prime})}(i\cancel{\partial}_{y}-m_{{}_{e}})\
\left.\left\langle\,0_{{}_{F}}\,\right.\right|\ T{\psi_{{}_{I}}(y)\
\bar{\psi}_{{}_{I}}(x)\ e^{\displaystyle-i\int{\cal H}_{int}(z)\
d^{4}z}}\left.\left|\,0_{{}_{F}}\,\right\rangle\right.$
$\displaystyle(i\overleftarrow{\cancel{\partial}}_{x}+m_{{}_{e}})u(p,s)$

where $Z$ is the wave-function renormalization factor; ${\cal
N}=\left.\left\langle\,0_{{}_{F}}\,\right.\right|U(\infty,-\infty)\left.\left|\,0_{{}_{F}}\,\right\rangle\right.$;
$U$ is the time-evolution operator in the interaction picture; $\
g{{}_{e}}\,(\,{p}\,;{\cal C}_{{}_{\cal P}},{\cal P}),\
g{{}_{e}}\,(\,{q}\,;{\cal C}_{{}_{\cal P}},{\cal P})$ are the Gibbs factors
for the incoming and outgoing electron, and

$\displaystyle\left.\left|\,0_{{}_{F}}\,\right\rangle\right.$ $\displaystyle=$
$\displaystyle\left.\left|\,0_{{}_{F}}\,\right\rangle\right._{electron}\otimes\left.\left|\,0_{{}_{F}}\,\right\rangle\right._{photon}$

is the vacuum of the electron and photon fields corresponding to the free-
field Hamiltonian242424 $H_{{}_{0}}\ =\ \int
d^{3}x\left\\{\bar{\psi}\left(i\cancel{\partial}-m_{{}_{e}}\right)\psi-\displaystyle\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right\\}$
, $\psi_{{}_{I}}$ the Dirac field of the electron in the interaction picture,
and ${\cal H}_{int}$ the interaction Hamiltonian density of $QED$ in the
interaction picture [43]

$\displaystyle{\cal H}_{int}$ $\displaystyle=$ $\displaystyle-e_{{}_{0}}\
\bar{\psi}_{{}_{I}}(x)\gamma_{\mu}\psi_{{}_{I}}(x)\left\\{A_{{}_{I}}^{\mu}(x)+\delta
A_{\,c}^{\mu}(x)\right\\}+\Delta m\,\bar{\psi}_{{}_{I}}(x)\,\psi_{{}_{I}}(x)$

The term containing $\Delta m$ does not contribute to the $O(\alpha)$
correction $\kappa_{{}_{1-loop}}$; therefore, we ignore that term in the
following calculation252525 At $O(e_{{}_{0}}^{3})$, the $\Delta
m\,\bar{\psi}\,\psi$ term does not appear in the 1-loop diagram shown in
Figure 1. It can appear as an insertion in the external legs of the tree-level
diagram in Figure 1. The effect of such insertions, as well as 1-loop
corrections to the external legs, is to adjust the value of the bare mass
$m_{{}_{0}}$, so as to make the pole of the propagator on an external leg
occur at the physical mass $m_{{}_{e}}$ of the electron in the tree diagram.
The effect of the above adjustment of the bare mass will appear when the
$\Delta m\,\bar{\psi}\,\psi$ term is inserted in an internal leg in diagrams
at $O(e_{{}_{0}}^{4})$ and higher. Therefore, at $O(e_{{}_{0}}^{3})$ we can
ignore the $\Delta m\,\bar{\psi}\,\psi$ term in the interaction Hamiltonian. .
For notational simplicity, we also drop the subscript $I$ for the fields in
the following discussion with the understanding that the rest of the
calculation is done in the interaction picture.

At $O(\delta)$, the scattering amplitude can be expanded in powers of
$e_{{}_{0}}$, as shown in (54), to yield

$\displaystyle{\cal A}^{(1)}(p,s,q,s^{\prime})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{(-i)(i)}{Z\cdot{\cal N}\cdot\
g{{}_{e}}\,(\,{p}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{e}}\,(\,{q}\,;{\cal C}_{{}_{\cal P}},{\cal P})}$ (58) $\displaystyle\int
d^{4}x\ d^{4}y\ e^{\displaystyle-ip\cdot x+iq\cdot y}\ \bar{u}(p,s)\
(i\cancel{\partial}_{y}-m_{{}_{e}})\left.\left\langle\,0_{{}_{F}}\,\right.\right|$
$\displaystyle(-ie_{{}_{0}})\int d^{4}z\ T\left\\{\psi(y)\
\bar{\psi}(x)\left[\,\bar{\psi}(z)A\\!\\!\\!/_{\,c}(z)\psi(z)\,\right]\right\\}+$
$\displaystyle\displaystyle\frac{3\cdot(-ie_{{}_{0}})^{3}}{3!}\int d^{4}z\
d^{4}v\ d^{4}w\ T\left\\{\psi(y)\
\bar{\psi}(x)\left[\,\bar{\psi}(z)A\\!\\!\\!/_{\,c}(z)\psi(z)\,\right]\left[\,\bar{\psi}(v)A\\!\\!\\!/(v)\psi(v)\,\right]\right.$
$\displaystyle\left.\left[\,\bar{\psi}(w)A\\!\\!\\!/(w)\psi(w)\,\right]\right\\}\left.\left|\,0_{{}_{F}}\,\right\rangle\right.(i\overleftarrow{\cancel{\partial}}_{x}+m_{{}_{e}})\
u(q,s^{\prime})+O(e_{{}_{0}}^{4})$ $\displaystyle:=$ $\displaystyle{\cal
A}^{(1)}_{1}(p,s,q,s^{\prime})\cdot e_{{}_{0}}+{\cal
A}^{(1)}_{3}(p,s,q,s^{\prime})\cdot e_{{}_{0}}^{3}+O(e_{{}_{0}}^{4})$

The factor 3 in the $O(e_{{}_{0}}^{3})$ term accounts for the three ways in
which the $A_{\,c}$ can be chosen from the product $(-i\int{\cal
H}_{{}_{I}}(z)d^{4}z)(-i\int{\cal H}_{{}_{I}}(v)d^{4}v)(-i\int{\cal
H}_{{}_{I}}(w)d^{4}w)$. The last line in (58) follows from the definitions in
(54).

$\displaystyle{\cal A}_{{}_{1}}^{(1)}(p,s,q,s^{\prime})$ $\displaystyle=$
$\displaystyle\left[\,\frac{(-i)\ g{{}_{e}}\,(\,{p}\,;{\cal C}_{{}_{\cal
P}},{\cal P})\ g{{}_{e}}\,(\,{q}\,;{\cal C}_{{}_{\cal P}},{\cal P})}{Z\ {\cal
N}\ }\,\right]\left[\,\bar{u}(q,s^{\prime})\ \gamma^{\mu}\
u(p,s)\,\right](A_{c})_{\mu}(p-q)$

Since $p$ and $q$ are on mass shell, using Gordon’s identity

$\displaystyle\bar{u}(q,s^{\prime})\gamma^{\mu}u(p,s)$ $\displaystyle=$
$\displaystyle\bar{u}(q,s^{\prime})\left\\{\displaystyle\frac{(p+q)^{\mu}}{2m_{{}_{e}}}+\displaystyle\frac{2i\sigma^{\nu\mu}(p-q)_{\nu}}{2m_{{}_{e}}}\right\\}u(p,s)$

where $\sigma^{\nu\mu}:=\left(\frac{i}{4}\right)[\gamma^{\nu},\gamma^{\mu}]$
and noting that $g_{tree}=2$, ${\cal A}^{(1)}_{{}_{1}}$ can be rewritten as

$\displaystyle A^{(1)}_{{}_{1}}(p,s,q,s^{\prime})\hskip 433.62pt$

$\displaystyle=$ $\displaystyle\left[\,g_{tree}\cdot\left[\,\frac{(i)(-i)\
g{{}_{e}}\,(\,{p}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{e}}\,(\,{q}\,;{\cal C}_{{}_{\cal P}},{\cal P})}{Z{\cal
N}(2m_{{}_{e}})}\,\right]\bar{u}\,\sigma^{\nu\mu}(p-q)_{\nu}\,(A_{c})_{\mu}(p-q)\,u(p,s)\,\right]+{\cal
B}_{{}_{1}}$

where

$\displaystyle{\cal B}_{{}_{1}}(p,s,q,s^{\prime})$ $\displaystyle=$
$\displaystyle\left[\,\frac{(-i)\ g{{}_{e}}\,(\,{p}\,;{\cal C}_{{}_{\cal
P}},{\cal P})\ g{{}_{e}}\,(\,{q}\,;{\cal C}_{{}_{\cal P}},{\cal P})}{\ Z{\cal
N}(2m_{{}_{e}})}\,\right]\bar{u}\ (p+q)^{\mu}\ (A_{c})_{\mu}(p-q)\ u(p,s)$

Comparing (55) with (LABEL:eqn4) we have

$\displaystyle\xi_{{}_{1}}(e,m_{{}_{e}})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\ g{{}_{e}}\,(\,{p\ }\,;{\cal C}_{{}_{\cal
P}},{\cal P})\ g{{}_{e}}\,(\,{q\ }\,;{\cal C}_{{}_{\cal P}},{\cal P})}{Z{\cal
N}(2m_{{}_{e}})}$ (60)

From (58) we have

$\displaystyle{\cal A}^{(1)}_{3}(p,s,q,s^{\prime})$ $\displaystyle=$
$\displaystyle\lim_{\varsigma\rightarrow
0}\left\\{-\xi_{{}_{1}}(e,m_{{}_{e}})\cdot(2m_{{}_{e}})\int\ \
\displaystyle\frac{d^{4}k}{(2\pi)^{4}}\right.$ $\displaystyle\left[\,\
g{{}_{e}}\,(\,{(p+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{e}}\,(\,{(q+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{p}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal P})\,\right]^{2}$
$\displaystyle\left.\left[\,\frac{\bar{u}(q,s^{\prime})\left\\{\gamma^{\alpha}(q\\!\\!\\!/+k\\!\\!\\!/+m_{{}_{e}})\gamma^{\mu}(p\\!\\!\\!/+k\\!\\!\\!/+m_{{}_{e}})\gamma_{\alpha}\right\\}u(p,s)}{\left((p+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left((q+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left(k^{2}-\varsigma^{2}+i\epsilon\right)}\,\right]\
(A_{c})_{\mu}(p-q)\right\\}$ $\displaystyle=$
$\displaystyle-\xi_{{}_{1}}(e,m_{{}_{e}})\cdot(2m_{{}_{e}})\int\ \
\displaystyle\frac{d^{4}k}{(2\pi)^{4}}$ $\displaystyle\left[\,\
g{{}_{e}}\,(\,{(p+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{e}}\,(\,{(q+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{p}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal P})\,\right]^{2}$
$\displaystyle\left[\,\frac{\bar{u}(q,s^{\prime})\left\\{\gamma^{\alpha}(q\\!\\!\\!/+k\\!\\!\\!/+m_{{}_{e}})\gamma^{\mu}(p\\!\\!\\!/+k\\!\\!\\!/+m_{{}_{e}})\gamma_{\alpha}\right\\}u(p,s)}{\left((p+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left((q+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left(k^{2}+i\epsilon\right)}\,\right]\
(A_{c})_{\mu}(p-q)$

Setting $a=q+k$ and $b=p+k$ and recalling that

$\displaystyle\gamma^{\alpha}\gamma_{\alpha}=4;\
\gamma^{\alpha}\gamma^{\mu}\gamma_{\alpha}=-2\gamma^{\mu};\
\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\gamma_{\alpha}=4g^{\rho\mu};\
\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\gamma^{\sigma}\gamma_{\alpha}=-2\gamma^{\sigma}\gamma^{\mu}\gamma^{\rho}$

the numerator within the brackets in (LABEL:eqn103) can be rewritten as

$\displaystyle\bar{u}(q,s^{\prime})\left\\{\gamma^{\alpha}(a\\!\\!\\!/+m_{{}_{e}})\gamma^{\mu}(b\\!\\!\\!/+m_{{}_{e}})\gamma_{\alpha}\right\\}u(p,s)$
$\displaystyle=$
$\displaystyle(-2m_{{}_{e}}^{2})\left[\,\bar{u}(q,s^{\prime})\ \gamma^{\mu}\
u(p,s)\,\right]$ $\displaystyle+(4m_{{}_{e}})\left[\,\bar{u}(q,s^{\prime})\
\left\\{a^{\mu}+b^{\mu}\right\\}\ u(p,s)\,\right]$
$\displaystyle-2\left[\,\bar{u}(q,s^{\prime})\left(b\,\\!\\!\\!/\
\gamma^{\mu}\ a\\!\\!\\!/\right)u(p,s)\,\right]$

The middle term does not contribute to the anomalous magnetic moment because
it contracts $(A_{c})_{\mu}(p-q)$ with $(a+b)^{\mu}$. Consider the third term.

$\displaystyle-2\left[\,\bar{u}(q,s^{\prime})\left\\{\gamma^{\sigma}\gamma^{\mu}\gamma^{\rho}\right\\}u(p,s)\,b_{\sigma}\,a_{\rho}\,\right]\hskip
252.94499pt$

$\displaystyle=$ $\displaystyle
2\left[\,\bar{u}(q,s^{\prime})\left\\{\gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}-2\gamma^{\sigma}\
g^{\mu\rho}\right\\}u(p,s)\,b_{\sigma}\,a_{\rho}\,\right]$ $\displaystyle=$
$\displaystyle
2\left[\,\bar{u}(q,s^{\prime})\left\\{\left(g^{\sigma\rho}-2i\sigma^{\sigma\rho}\right)\gamma^{\mu}-2\gamma^{\sigma}\
g^{\mu\rho}\right\\}u(p,s)\,b_{\sigma}\,a_{\rho}\,\right]$

If $\delta=0$, that is the classical background field is turned off, then
$|\,\vec{q}-\vec{p}\ |=0$. Therefore, in the weak-field limit
$\delta\rightarrow 0$, $|\ \vec{q}-\vec{p}\ |\rightarrow 0$; that is, $|\
\vec{q}-\vec{p}\ |$ can be made as small as we please by taking $\delta$ to be
sufficiently small. Therefore, in the weak-field limit
$\sigma^{\sigma\rho}b_{\sigma}a_{\sigma}\ =\
\sigma^{\sigma\rho}(p+k)_{\sigma}(q+k)_{\sigma}\rightarrow 0$ and

$\displaystyle-2\left[\,\bar{u}(q,s^{\prime})\left(b\\!\\!\\!/\ \gamma^{\mu}\
a\\!\\!\\!/\right)u(p,s)\,\right]$ $\displaystyle=$ $\displaystyle
2(a^{\alpha}b_{\alpha})\left[\,\bar{u}(q,s^{\prime})\,\gamma^{\mu}\,u(p,s)\,\right]-4\,\bar{u}(q,s^{\prime})\
b\\!\\!\\!/\ u(p,s)\ a^{\mu}$

Putting it all together and using Gordon’s identity again we get

$\displaystyle\bar{u}(q,s^{\prime})\left\\{\gamma^{\alpha}(a\\!\\!\\!/+m_{{}_{e}})\gamma^{\mu}(b\\!\\!\\!/+m_{{}_{e}})\gamma_{\alpha}\right\\}u(p,s)(A_{c})_{\mu}(p-q)\hskip
216.81pt$

$\displaystyle=$ $\displaystyle
g_{tree}\cdot(a^{\alpha}b_{\alpha}-m_{{}_{e}}^{2})\left(\frac{2i}{2m_{{}_{e}}}\right)\left[\,\bar{u}(q,s^{\prime})\
\sigma^{\nu\mu}(p-q)_{\nu}\ (A_{c})_{\mu}(p-q)\ u(p,s)\,\right]$ (62)
$\displaystyle+\bar{u}(q,s^{\prime})\left[\,2(a^{\alpha}b_{\alpha}-m_{{}_{e}}^{2})\displaystyle\frac{(p+q)^{\mu}}{2m_{{}_{e}}}+4m_{{}_{e}}(a+b)^{\mu}-4b\\!\\!\\!/\
a^{\mu}\,\right]u(p,s)(A_{c})_{\mu}(p-q)$

The second term contracts $(p+q)^{\mu}$, $(a+b)^{\mu}$ and $a^{\mu}$ with
$(A_{c})_{\mu}(p-q)$ and hence does not contribute to $\xi_{{}_{3}}(e,m)$.

From (55), (60), (LABEL:eqn103) and (62) we have,

$\displaystyle\xi_{{}_{3}}(e,m_{{}_{e}})$ $\displaystyle=$
$\displaystyle-(2i)\cdot\xi_{{}_{1}}(e,m_{{}_{e}})\int\displaystyle\frac{d^{4}k}{(2\pi)^{4}}\left[\,\
g{{}_{e}}\,(\,{(p+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{e}}\,(\,{(q+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{p}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal P})\,\right]^{2}$ (63)
$\displaystyle\left[\,\frac{(p+k)^{\alpha}(q+k)_{\alpha}-m_{{}_{e}}^{2}}{\left((p+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left((q+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left(k^{2}+i\epsilon\right)}\,\right]$

From (60) and (63) we have

$\displaystyle F^{(0)}_{31}$ $\displaystyle=$
$\displaystyle-\left(2i\right)\int\ \displaystyle\frac{d^{4}k}{(2\pi)^{4}}\
\left[\,\ g{{}_{e}}\,(\,{(p+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{e}}\,(\,{(q+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{p}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal P})\,\right]^{2}$ (64)
$\displaystyle\left[\,\frac{(p+k)^{\alpha}(q+k)_{\alpha}-m_{{}_{e}}^{2}}{\left((p+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left((q+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left(k^{2}+i\epsilon\right)}\,\right]$

Using (56), (57) and (64), we have

$\displaystyle\kappa_{1-loop}$ $\displaystyle=$
$\displaystyle-8\pi\alpha\,i\int\ \ \displaystyle\frac{d^{4}k}{(2\pi)^{4}}\
\left[\,\ g{{}_{e}}\,(\,{(p+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{e}}\,(\,{(q+k)}\,;{\cal C}_{{}_{\cal P}},{\cal P})\
g{{}_{p}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal P})\,\right]^{2}$ (65)
$\displaystyle\hskip
36.135pt\left[\,\frac{(p+k)^{\alpha}(q+k)_{\alpha}-m_{{}_{e}}^{2}}{\left((p+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left((q+k)^{2}-m_{{}_{e}}^{2}+i\epsilon\right)\left(k^{2}+i\epsilon\right)}\,\right]$

The integral in (65) is evaluated numerically. The evaluation of the integral
is described in Appendix D and yields a numerical value of 0.001158990105,
which deviates from the experimentally measured value of 0.001159652180 [36,
37] by less than 0.06%.

##  5 Lamb Shift

Lamb and Retherford’s measurements [58] of the fine structure of the
${}^{2}S_{\frac{1}{2}}$ and the ${}^{2}P_{\frac{1}{2}}$ levels in the Hydrogen
atom showed that the energy of the ${}^{2}S_{\frac{1}{2}}$ level is higher
than that of the ${}^{2}P_{\frac{1}{2}}$ level, by about 1 GHz or about 4.14
$\mu\,eV$ [4, 42]; also see [2, 18, 51, 55, 57]. The measured difference
between the energies of the two levels, called the Lamb shift, was significant
because it conflicted with Dirac’s theory, which predicted equal energies for
the two levels [28, 42]. The subsequent theoretical explanation of the Lamb
shift, by Bethe [4], using renormalization argument262626 Bethe’s calculation
involves subtraction of one infinite quantity—the self energy of a free
electron due to its interaction with radiation—from another infinite quantity
—the self-energy of an electron in a quantum state due to its interaction with
radiation. has been hailed as one of the pivotal advances in the development
of quantum electrodynamics.

The shift calculated by Bethe depends on the value of the ultra-violet cutoff
$K$ chosen for photon frequencies [4, Eq. 11]:

$\displaystyle\delta\nu_{{}_{Lamb}}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\alpha^{5}\cdot m_{e}c^{2}}{6\pi
h}\cdot\log\left(\frac{K}{242.15\,eV}\right)$ (66)

The current theoretically estimated value of Lamb shift due to the self energy
of the electron [26] is $1011.45Hz$, which corresponds to $K\approx
0.82\,m_{e}\,c^{2}$. Varying the cutoff from this value can lead to
discrepancies between the expected value of the shift and that predicted by
(66). Specifically, as Kroll and Lamb [55, Footnote, page 388] observed, if
the retardation and recoil effects are included in Bethe’s non-relativistic
approximation then $K=2m_{e}c^{2}$, which leads (66) to predict a Lamb shift
of 1131.9 $MHz$. Bethe chose a value of $K=m_{e}c^{2}$ to obtain a prediction
of 1038 $MHz$ for the shift.

In the next subsection we compute the Lamb shift, treating the electron in
Hydrogen atom as a nonrelativistic particle, and the electromagnetic field
with which it interacts as a quantum field that is described by our new
framework. Unlike Bethe’s argument however, our calculation is divergence-free
and does not involve a cutoff for photon frequencies.

We treat the motion of the electron in Hydrogen atom nonrelativistically
because the relativistic correction to the energy levels in Hydrogen is known
to be small and much of the shift can be explained using a nonrelativistic
electron. In fact, Baranger, Bethe and Feynman [3] showed that the
relativistic correction, which is of $O(\alpha\cdot(Z\alpha)^{5})$,
contributes about 7.13 MHz, while the experimentally measured value of Lamb
shift is about 1057 MHz. $Z$ is the atomic number of the atom.

Before proceeding, we digress to briefly summarize the known theoretical and
experimental results on Lamb shift. The current theoretical prediction of the
Lamb shift is actually the sum of contributions from three main sources [26]:
(i ) a contribution of +1011.45 MHz from electron’s self-energy, (ii ) a
contribution of -27.13 MHz from vacuum polarization, and (iii ) a contribution
of +67.82 MHz from the anomalous magnetic moment, giving rise to a total of
about 1052.14 MHz. Our calculation, like Bethe’s, focuses only on the
contribution due to electron’s self energy. The theoretical predictions have
been further improved by including fourth order radiative corrections, and by
correcting for the finite size and mass of the proton[18] to 1057.814(4) $MHz$
[83] for proton rms272727root mean square radius of 0.805(11) fm [14, 35]; to
1057.833(4) $MHz$ [83] for proton rms radius of 0.862(12) fm [84]; to
1057.843(6) $MHz$ for proton rms radius of 0.891(18) fm. The experimentally
determined value of the total shift on the other hand has been improved to
about 1057 $MHz$ [18]282828 1057.77(6) $MHz$ [87, 89], 1057.90(6) $MHz$ [76,
87], , 1057.8576 (2.1) $MHz$ [72, 85] 1057.862(20) $MHz$, [69], 1057.845(9)
$MHz$ [62], 1057.852(15) $MHz$ [91] and 1057.842(12) $MHz$ [33].. Further
discussion of the theoretical and experimental results on Lamb shift may be
found in [18] and the references therein.

###  5.1 Line Shift in Hydrogen Atom

In Dirac’s theory of the Hydrogen atom the energy of an electron with quantum
numbers ($n,j$) is given by [28, page 244]

$\displaystyle E_{nj}$ $\displaystyle=$ $\displaystyle
m_{{}_{e}}c^{2}\left\\{\left[\,1+\left(\frac{\alpha}{n-(j+1/2)+\sqrt{(j+1/2)^{2}-\alpha^{2}}}\right)^{2}\,\right]^{-1/2}-1\right\\}$

where $\alpha$ is the fine structure constant. Thus Dirac’s theory predicts
that the ${}^{2}S_{1/2}$ and the ${}^{2}P_{1/2}$ electrons, both of which have
$n=2$ and $j=1/2$ have the same energy. The degeneracy is broken by the
different interactions that the ${}^{2}S_{1/2}$ and the ${}^{2}P_{1/2}$
electrons have with the background electromagnetic field.

Welton [95] showed that the difference in energy between the ${}^{2}S_{1/2}$
and the ${}^{2}P_{1/2}$ levels can be explained using a semiclassical
calculation of the perturbations of the average squared orbital distance of
the levels due to the interactions between electrons in those levels and the
vacuum fluctuations of the electromagnetic field. Specifically, the
perturbations are shown to contain a factor $|\psi_{nlm}(0)|\,^{2}$, where
$\psi_{nlm}$ is the wavefunction of an electron in the energy eigenstate with
quantum numbers $n,l$ and $m$. The factor is nonzero for the ${}^{2}S_{1/2}$
electron and vanishes for the ${}^{2}P_{1/2}$ electron, giving rise to the
energy difference between the levels.

In the following discussion, which is adapted from Dyson’s calculation [16],
we use the new framework to calculate the contribution to the Lamb shift due
to the correction to electron’s self-energy. We use the natural units
$c=\hbar=\epsilon_{{}_{0}}=1$.

The total Hamiltonian for an atomic electron interacting with the Maxwell
field is given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{A}+H_{M}+\lambda H_{int}$

where $H_{A}$ is the Hamiltonian of an orbital electron292929 In Hydrogen
atom., whose dynamics is described by nonrelativistic quantum mechanics,
$H_{M}$, the Hamiltonian of the free Maxwell field, which is regarded as a
quantum field, and
$H_{int}=\int\,d\,\vec{x}\,j^{\mu}(\vec{x},t)A_{\mu}(\vec{x},t)$, the
Hamiltonian describing the interaction between an electron and the Maxwell
field; $j^{\mu}(\vec{x},t)$ is the electron current that couples to the
Maxwell field $A^{\mu}(\vec{x},t)$. $\lambda$ is a dimensionless parameter
that is used for book-keeping and will be set to 1 eventually.

We denote the time-independent orthonormal family of energy eigenstates of
$H_{A}$ as
$\left.\left|\,1\,\right\rangle\right.,\left.\left|\,2\,\right\rangle\right.,\ldots,$
$\left.\left|\,k\,\right\rangle\right.,\ldots$ with eigenvalues
$E_{{}_{1}},E_{{}_{2}},\ldots,E_{k},\ldots$. We seek to compute the correction
to energy level $\left.\left|\,n\,\right\rangle\right.$; thus, the label $n$
is to be distinguished from all other labels $k\neq n$ in the following
discussion. Labels like $k$ and $n$ represent triplets of principal, azimuthal
and magnetic quantum numbers.

In the interaction picture, let
$\left.\left|\,\psi_{{}_{I}}(t)\,\right\rangle\right.$ denote the state of the
electron interacting with the electromagnetic field–that is with the $\lambda
H_{int}$ term present in the Hamiltonian– with the initial condition
$\left.\left|\,\psi_{{}_{I}}(0)\,\right\rangle\right.=\left.\left|\,n\,\right\rangle\right.$.
In the interaction picture303030 Evolution of operators is determined by the
unperturbed Hamiltonian $H_{{}_{0}}:=H_{{}_{A}}+H_{{}_{M}}$, and the evolution
of states is determined by the interaction Hamiltonian operator $\lambda
H^{I}_{{}_{int}}$ in the interaction picture.  we can expand
$\left.\left|\,\psi_{{}_{I}}(t)\,\right\rangle\right.$ as

$\displaystyle\left.\left|\,\psi_{{}_{I}}(t)\,\right\rangle\right.$
$\displaystyle=$ $\displaystyle
a_{{}_{1}}(t)\left.\left|\,1\,\right\rangle\right.+a_{{}_{2}}(t)\left.\left|\,2\,\right\rangle\right.+\ldots
a_{{}_{n}}(t)\left.\left|\,n\,\right\rangle\right.+\ldots$ (67)

Clearly, $a_{k}(t)=\left\langle\,k\,|\,\psi_{{}_{I}}(t)\,\right\rangle,\
k=1,2,\ldots$, $a_{{}_{n}}(0)=1$, $a_{k}(0)=0$ for $\ k\neq n$. Define

$\displaystyle\left.\left|\,\Psi_{{}_{I}}(t)\,\right\rangle\right.$
$\displaystyle:=$
$\displaystyle\left.\left|\,\psi_{{}_{I}}(t)\,\right\rangle\right.\otimes\left.\left|\,0_{{}_{M}}\,\right\rangle\right.,\qquad\left.\left|\,K\,\right\rangle\right.\
:=\
\left.\left|\,k\,\right\rangle\right.\otimes\left.\left|\,0_{{}_{M}}\,\right\rangle\right.,\qquad
k=1,\,2,\,\ldots$

where $\left.\left|\,0_{{}_{M}}\,\right\rangle\right.$ is the vacuum of the
free electromagnetic field. We note that

$\displaystyle\left\langle\,K\,|\,\Psi_{{}_{I}}(t)\,\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\,k\,|\,\psi_{{}_{I}}(t)\,\right\rangle\cdot\left\langle\,0_{M}\,|\,0_{M}\,\right\rangle\
=\ a_{k}(t),\qquad k=1,\,2,\,\ldots$

The evolution of $\left.\left|\,\Psi_{{}_{I}}(t)\,\right\rangle\right.$ in the
interaction picture is given by the equation

$\displaystyle
i\partial_{t}\left.\left|\,\Psi_{{}_{I}}(t)\,\right\rangle\right.$
$\displaystyle=$
$\displaystyle\lambda\,H_{int}^{I}(t)\left.\left|\,\Psi_{{}_{I}}(t)\,\right\rangle\right.$
(68)

At $O(\lambda^{0})$, that is in the absence of the interaction term,
$\left.\left|\,\Psi_{{}_{I}}(t)\,\right\rangle\right.$ and hence
$\left.\left|\,\psi_{{}_{I}}(t)\,\right\rangle\right.$ is time-independent,
$a_{k}(t)=0$ for $k\neq n$ and $a_{{}_{n}}(t)=1$.

Following Dyson [16], we will show that at $O(\lambda^{2})$ and for $t\gg 0$,
$a_{{}_{n}}(t)=e^{-i(\Delta E_{{}_{n}}-i\Gamma_{{}_{n}}/2)t}$. The above
asymptotic form of $a_{{}_{n}}(t)$ at $O(\lambda^{2})$ is interpreted as
follows.

A state $\left.\left|\,\psi_{{}_{S}}(t)\,\right\rangle\right.$ in Schrodinger
picture can be written as

$\displaystyle\left.\left|\,\psi_{{}_{S}}(t)\,\right\rangle\right.$
$\displaystyle=$ $\displaystyle e^{\displaystyle-
iH_{A}t}\left.\left|\,\psi_{{}_{I}}(t)\,\right\rangle\right.$ (69)
$\displaystyle=$ $\displaystyle a_{{}_{1}}(t)e^{\displaystyle-
iH_{A}t}\left.\left|\,1\,\right\rangle\right.+\ldots+e^{\displaystyle-(\Gamma_{{}_{n}}/2)t}\left[\,e^{\displaystyle-i\left(E_{{}_{n}}+\Delta
E_{{}_{n}}\right)t}\,\left.\left|\,n\,\right\rangle\right.\,\right]+\ldots$

If $\lambda=0$, then
$\left.\left|\,\psi_{{}_{S}}(t)\,\right\rangle\right.=e^{\displaystyle-
iE_{{}_{n}}t}\left.\left|\,n\,\right\rangle\right.$, and $a_{k}(t)=0$ for
$k\neq n$. If $\lambda$ is nonzero, that is if the interaction is turned on,
then the time-dependent phase of the $n^{th}$ term on the right hand side of
(69) changes from $e^{\displaystyle-iE_{{}_{n}}t}$ to
$e^{\displaystyle-i(E_{{}_{n}}+\Delta E_{{}_{n}})t}$. The change in phase is
interpreted as the shift in energy of the $n^{th}$ level by the amount $\Delta
E_{{}_{n}}$. The coefficient of the $n^{th}$ term, namely
$e^{\displaystyle-(\Gamma_{{}_{n}}/2)t}$ shows that the probability of finding
the electron in the state $\left.\left|\,n\,\right\rangle\right.$ decreases
exponentially. Asymptotically, the electron exits the initial state
$\left.\left|\,n\,\right\rangle\right.$ as a result of its repeated
interaction with the electromagnetic field.

At $O(\lambda)$, for $k=1,\,2,\,\ldots$, using (67) and (68) we have

$\displaystyle i\,\dot{a}_{k}(t)$ $\displaystyle=$
$\displaystyle\lambda\langle\,K\,|H_{int}^{I}(t)\left.\left|\,\Psi_{{}_{I}}(t)\,\right\rangle\right.\
=\ \lambda\cdot\left\langle k\,\left|\,j^{\mu}(\vec{x},t)\
\right|\psi_{{}_{I}}(t)\,\right\rangle\left\langle
0_{{}_{M}}\,\left|\,A_{\mu}(\vec{x},t)\ \right|0_{{}_{M}}\,\right\rangle$ (70)
$\displaystyle=$ $\displaystyle 0$

since the vacuum expectation value of $A_{\mu}(\vec{x},t)$ vanishes. Thus, at
$O(\lambda)$ we can consistently set $a_{{}_{n}}(t)=1$ and $a_{k}(t)=0$ for
$k\neq n$. And at $O(\lambda)$ the solution of (68) can be written as

$\displaystyle\left.\left|\,\Psi_{{}_{I}}(t)\,\right\rangle\right.$
$\displaystyle=$
$\displaystyle\left\\{1-i\int_{0}^{t}\lambda\,H_{int}^{I}(\tau)\,d\tau\right\\}\cdot
a_{{}_{n}}(t)\cdot\left.\left|\,N\,\right\rangle\right.;\qquad\left.\left|\,N\,\right\rangle\right.:=\left.\left|\,n\,\right\rangle\right.\otimes\left.\left|\,0_{{}_{M}}\,\right\rangle\right.$
(71)

The time-evolution of $a_{{}_{n}}(t)$ at $O(\lambda^{2})$ can be obtained by
inserting (71) into (70).

$\displaystyle i\,\dot{a}_{{}_{n}}(t)$ $\displaystyle=$
$\displaystyle-i\,\lambda^{2}\,\left\langle
N\,\left|\,\,H_{int}^{I}(t)\int_{0}^{t}\,H_{int}^{I}(\tau)\,d\tau\
\right|N\,\right\rangle\cdot a_{{}_{n}}(t)$ (72)

or

$\displaystyle i\cdot\displaystyle\frac{\dot{a}_{{}_{n}}(t)}{a_{{}_{n}}(t)}$
$\displaystyle=$
$\displaystyle-i\,\lambda^{2}\,\int\,d\,\vec{x}\,d\,\vec{y}\,\int_{0}^{t}\,d\tau\,\left\langle
n\,\left|\,j^{\mu}(\vec{x},t)j^{\nu}(\vec{y},\tau)\ \right|n\,\right\rangle\
\cdot\ \left\langle
0_{{}_{M}}\,\left|\,A_{\mu}(\vec{x},t)A_{\nu}(\vec{y},\tau)\
\right|0_{{}_{M}}\,\right\rangle$

In the interaction picture the electromagnetic field evolves as a free field.
Therefore, in Coulomb gauge, we can expand $A_{\mu}(x;{\cal C}_{{}_{\cal P}})$
as

$\displaystyle A_{\mu}(x;{\cal C}_{{}_{\cal P}})$ $\displaystyle=$
$\displaystyle\int\displaystyle\frac{d\vec{k}}{(2\pi)^{3}\ 2|\vec{k}|}\,\
g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})\sum_{s=1}^{2}\,\epsilon_{\mu}(\vec{k},s)\left\\{\hat{a}(\vec{k},s)e^{\displaystyle-i\,\uline{k}\,x}+\hat{a}^{\dagger}(\vec{k},s)e^{\displaystyle
i\,\uline{k}\,x}\right\\}\quad$ (74)

where $\uline{k}=(|\vec{k}\,|,\vec{k})$, the $\epsilon_{\mu}(\vec{k},1)$ and
$\epsilon_{\mu}(\vec{k},2)$ are chosen to be two polarization vectors such
that $\epsilon^{\mu}(\vec{k},s)=(0,\hat{}\epsilon(\vec{k},s))$, for $s=1,2$,
and ($\hat{}\epsilon(\vec{k},1),\hat{}\epsilon(\vec{k},2),\hat{k}$) form a
right-handed orthonormal triad, where $\hat{k}$ is the unit vector along
$\vec{k}$. We take ${\cal C}_{{}_{\cal P}}$ to be the Lorentz frame in which
the center of mass of the Hydrogen atom is at rest. Further, using the Coulomb
gauge commutation relations (30)—shown in Footnote 19—we obtain

$\displaystyle\left\langle 0_{{}_{M}}\,\left|\,A_{\mu}(\vec{x},t;{\cal
C}_{{}_{\cal P}})A_{\nu}(\vec{y},\tau;{\cal C}_{{}_{\cal P}})\
\right|0_{{}_{M}}\,\right\rangle\hskip 289.07999pt$

$\displaystyle=$ $\displaystyle\int\displaystyle\frac{d\,\vec{k}\cdot\
g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})^{2}}{(2\pi)^{3}\cdot|\vec{k}|}\cdot\sum_{s=1}^{2}\epsilon_{\mu}(\vec{k},s)\epsilon_{\nu}(\vec{k},s)e^{\displaystyle-i(|\vec{k}|\,(t-\tau)+\vec{k}\cdot(\vec{x}-\vec{y}))}$
(75)

Recalling that in the interaction picture $j^{\mu}(\vec{x},t)=e^{\displaystyle
iH_{A}t}j^{\mu}(\vec{x},0)e^{\displaystyle-iH_{A}t}$ and inserting
$1=\sum_{r}\left.\left|\,r\,\right\rangle\right.\,\left.\left\langle\,r\,\right.\right|$
between $j^{\mu}(\vec{x},t)$ and $j^{\nu}(\vec{y},\tau)$ we obtain

$\displaystyle\left\langle n\,\left|\,j^{\mu}(\vec{x},t)j^{\nu}(\vec{y},\tau)\
\right|n\,\right\rangle$ $\displaystyle=$
$\displaystyle\sum_{r}e^{\displaystyle-i(E_{r}-E_{{}_{n}})(t-\tau)}\langle\,j^{\mu}(\vec{x},0)\,\rangle_{nr}\,\langle\,j^{\nu}(\vec{y},0)\,\rangle_{rn}$
(76)

where $\langle\,j^{\mu}(\vec{x},0)\,\rangle_{nr}:=\left\langle
n\,\left|\,j^{\mu}(\vec{x},0)\ \right|r\,\right\rangle$. Noting that the
electron current is a Hermitian operator we have

$\displaystyle\langle\,j^{\mu}(\vec{x},0)\,\rangle_{nr}$ $\displaystyle=$
$\displaystyle\langle\,j^{\mu}(\vec{x},0)\,\rangle_{rn}^{*}$ (77)

With a slight abuse of notation we define
$\langle\,j^{\mu}(\vec{k})\,\rangle_{nr}:=\int
e^{\displaystyle-i\vec{k}\cdot\vec{x}}\langle\,j^{\mu}(\vec{x},0)\,\rangle_{nr}\
d\,\vec{x}$. Using (77) we have

$\displaystyle\langle\,j^{\mu}(-\vec{k})\,\rangle_{rn}$ $\displaystyle=$
$\displaystyle j^{\mu}(\vec{k})_{nr}^{*}$ (78)

Then, noting that $\epsilon_{{}_{0}}(\vec{k},s)=0$,
$\epsilon_{\mu}(\vec{k},s)$ is real, for $\mu=1,2,3$, and using (78) we have

$\displaystyle\int d\,\vec{x}d\,\vec{y}\
e^{\displaystyle-i\vec{k}\cdot(\vec{x}-\vec{y})}\left[\,\langle\,j^{\mu}(\vec{x},0)\,\rangle_{nr}\epsilon_{\mu}(\vec{k},s)\,\right]\left[\,\langle\,j^{\nu}(\vec{y},0)\,\rangle_{rn}\
\epsilon_{\nu}(\vec{k},s)\,\right]\hskip 216.81pt$

$\displaystyle=$
$\displaystyle\langle\,j^{\mu}(\vec{k})\,\rangle_{nr}\epsilon_{\mu}(\vec{k},s)\langle\,j^{\nu}(-\vec{k})\,\rangle_{rn}\epsilon_{\nu}(\vec{k},s)\
=\ |j^{\mu}(\vec{k})\ \epsilon_{\mu}(\vec{k},s)|\,^{2}$ (79)

Inserting (75), (76) and (79) into (LABEL:eqn209) we have

$\displaystyle
i\cdot\displaystyle\frac{\dot{a}_{{}_{n}}(t)}{a_{{}_{n}}(t)}=-i\,\lambda^{2}\int_{0}^{t}d\tau\int\displaystyle\frac{d\,\vec{k}\cdot\
g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})^{2}}{(2\pi)^{3}\
2|\vec{k}|}\sum_{r}\sum_{s=1}^{2}\left|\langle\,j^{\mu}(\vec{k})\,\rangle_{nr}\
\epsilon_{\mu}(\vec{k},s)\right|^{2}e^{\displaystyle-i(E_{r}-E_{{}_{n}}+|\vec{k}|)(t-\tau)}$

Using the identity [16, Equation 258],
$\displaystyle\int_{0}^{\infty}e^{\displaystyle-ia\tau}\
d\tau=\pi\delta(a)+\displaystyle\frac{1}{ia}$, we have for $t\gg 0$,

$\displaystyle\displaystyle\frac{\dot{a}_{{}_{n}}(t)}{a_{{}_{n}}(t)}$
$\displaystyle=$
$\displaystyle-\displaystyle\frac{\lambda^{2}}{16\pi^{3}}\int\displaystyle\frac{d\,\vec{k}}{|\,\vec{k}|}\
\ g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})^{2}\cdot\sum_{r}\sum_{s=1}^{2}\left|\langle\,j^{\mu}(\vec{k})\,\rangle_{nr}\
\epsilon_{\mu}(\vec{k},s)\right|^{2}$ (80) $\displaystyle\hskip
36.135pt\left[\,\pi\delta(E_{{}_{m}}-E_{{}_{n}}+|\vec{k}|)+\displaystyle\frac{1}{i(E_{{}_{m}}-E_{{}_{n}}+|\vec{k}|)}\,\right]$

Recalling the initial condition $a_{{}_{n}}(0)=1$, we conclude that the
solution of (80) is (setting $\lambda=1$),

$\displaystyle a_{{}_{n}}(t)$ $\displaystyle=$ $\displaystyle
e^{\displaystyle-i(\Delta E_{{}_{n}}-i\Gamma_{{}_{n}}/2)t},\qquad\mbox{ where
}$ $\displaystyle\Delta E_{{}_{n}}$ $\displaystyle=$
$\displaystyle-\displaystyle\frac{1}{16\pi^{3}}\int\displaystyle\frac{d\,\vec{k}}{|\,\vec{k}|}\
\ g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})^{2}\cdot\sum_{r}\displaystyle\frac{\sum_{s=1}^{2}\left|\langle\,j^{\mu}(\vec{k})\,\rangle_{nr}\
\epsilon_{\mu}(\vec{k},s)\right|^{2}}{E_{r}-E_{{}_{n}}+|\vec{k}|}$
$\displaystyle\Gamma_{{}_{n}}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\lambda^{2}}{8\pi^{2}}\int\displaystyle\frac{d\,\vec{k}}{|\vec{k}|}\
\ g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})^{2}\sum_{r}\sum_{s=1}^{2}\left|\langle\,j^{\mu}(\vec{k})\,\rangle_{nr}\
\epsilon_{\mu}(\vec{k},s)\right|^{2}\cdot\delta(E_{r}-E_{{}_{n}}+|\vec{k}|)$
(81)

Following Dyson we regard the electron to be a nonrelativistic particle and
use the dipole approximation, in which $\vec{k}\cdot\vec{r}\approx 0$, to set
$\vec{j}(\vec{k})=e\,\vec{v}=\left(\frac{e}{m_{{}_{e}}}\right)\vec{p}$ where
$\vec{v}$ and $\vec{p}$ are the velocity and momentum operators. Recalling
that $\epsilon_{{}_{0}}(\vec{k},s)=0$, we have

$\displaystyle\sum_{s=1}^{2}\left|\langle\,j^{\mu}(\vec{k})\,\rangle_{nr}\
\epsilon_{\mu}(\vec{k},s)\right|^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{e}{m_{{}_{e}}}\right)^{2}\sum_{s=1}^{2}\left|\langle\,\vec{p}\,\rangle_{nr}\cdot\hat{}\epsilon\,(\vec{k},s)\right|^{2}$
(82)

As noted earlier, the choice of $\hat{}\epsilon(\vec{k},1)$ determines
$\hat{}\epsilon(\vec{k},2)$. In the expansion (82) we chose a single
$\hat{}\epsilon(\vec{k},1)$ vector on the unit circle in the plane
perpendicular to $\vec{k}$ in the ($k^{1},k^{2},k^{3}$) space. The
$A_{\mu}(x)$ acting on $\left.\left|\,0_{{}_{M}}\,\right\rangle\right.$
produces for each $\vec{k}$ two photons that have a net polarization of
$\hat{}\epsilon(\vec{k},1)+\hat{}\epsilon(\vec{k},2)$ and a phase
$e^{\displaystyle i\uline{k}(x)}$. The net polarization vector could have been
chosen to be any vector on a circle of radius $\sqrt{2}$ in the plane
perpendicular to $\vec{k}$ in ($k^{1},k^{2},k^{3}$) space. Thus summing over
all possible net polarizations in (81)—that is considering the interaction of
the electron with photon-pairs of all possible net polarizations—involves
multiplying the right hand side of (81) with $2\pi$. Using (82) and summing
over all net polarizations we obtain

$\displaystyle\Delta E_{{}_{n}}$ $\displaystyle=$
$\displaystyle-\displaystyle\frac{e^{2}}{8\pi^{2}\,m^{2}}\int\displaystyle\frac{d\,\vec{k}}{|\,\vec{k}|}\
\ g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P})^{2}\cdot\sum_{r}\displaystyle\frac{\sum_{s=1}^{2}\left|\langle\,\vec{p}\,\rangle_{nr}\cdot\hat{}\epsilon\,(\vec{k},s)\right|^{2}}{E_{r}-E_{{}_{n}}+|\,\vec{k}|}$
(83)

We note that

$\displaystyle\sum_{s=1}^{2}\left|\langle\,\vec{p}\,\rangle_{nr}\cdot\hat{}\epsilon\,(\vec{k},s)\right|^{2}$
$\displaystyle=$
$\displaystyle\left|\langle\,\vec{p}\,\rangle_{nr}\right|^{2}(1-\cos^{2}\theta)$

where $\theta$ is the angle between $k^{3}$ and
$\langle\,\vec{p}\,\rangle_{nr}$.

Therefore setting $\kappa=|\vec{k}|$, and

$\displaystyle\tilde{g}(\kappa)$ $\displaystyle:=$
$\displaystyle\left[\,\displaystyle\frac{2}{e^{\displaystyle\kappa/\tau_{p}+\tau_{p}/\kappa}-1}\,\right]^{1/4}\
=\ \ g{{}_{p}}(\,\uline{\vec{k}}\,;{\cal C}_{{}_{\cal P}},{\cal
P});\qquad\tau_{{}_{{\cal P}}}\ =\ m_{{}_{e}}c^{2}$

(83) can be rewritten as

$\displaystyle\Delta E_{{}_{n}}$ $\displaystyle=$
$\displaystyle-\displaystyle\frac{e^{2}}{8\pi^{2}\,m_{{}_{e}}^{2}}\int_{0}^{\infty}d\kappa\cdot\kappa\cdot\tilde{g}(\kappa)^{2}\cdot\sum_{r}\displaystyle\frac{\left|\langle\,\vec{p}\,\rangle_{nr}\right|^{2}}{E_{r}-E_{{}_{n}}+\kappa}\int_{0}^{\pi}\sin\theta\
(1-\cos^{2}\theta)\ d\theta\int_{0}^{2\pi}d\varphi$ (84) $\displaystyle=$
$\displaystyle-\displaystyle\frac{e^{2}}{3\pi\,m_{{}_{e}}^{2}}\int_{0}^{\infty}d\kappa\cdot\kappa\cdot\tilde{g}(\kappa)^{2}\cdot\sum_{r}\displaystyle\frac{\left|\langle\,\vec{p}\,\rangle_{nr}\right|^{2}}{E_{r}-E_{{}_{n}}+\kappa}$

The interaction with the background electromagnetic field shifts the energy of
a Hydrogen orbital as well as the self-energy of a free electron. An electron
in a Hydrogen atom can inhabit several energy levels. In contrast, a free
electron inhabits a single energy level. We can obtain the shift in the self-
energy of a free electron from (84) by replacing the sum over several energy
levels—labeled $E_{r}$—with a sum over a single energy level $E_{{}_{n}}$.
Observing the single state $E_{{}_{n}}$ of a free electron is an eigenstate of
momentum operator, that is $\vec{p}\left.\left|\,n\,\right\rangle\right.\ =\
\vec{P}_{{}_{n}}\left.\left|\,n\,\right\rangle\right.$, where
$\vec{P}_{{}_{n}}$ is the eigenvalue of the momentum operator $\vec{p}$, we
obtain

$\displaystyle\Delta E^{(free)}$ $\displaystyle=$
$\displaystyle-\displaystyle\frac{e^{2}}{3\pi\,m_{{}_{e}}^{2}}\int_{0}^{\infty}d\kappa\cdot\tilde{g}(\kappa)^{2}\left|\langle\,\vec{p}\,\rangle_{nn}\right|^{2}=-\displaystyle\frac{1}{m_{{}_{e}}}\left[\,\displaystyle\frac{2e^{2}\
}{3\pi\,}\int_{0}^{\infty}d\kappa\cdot\tilde{g}(\kappa)^{2}\,\right]\displaystyle\frac{(\vec{P}_{{}_{n}})^{2}}{2m_{{}_{e}}}\qquad$
(85)

We note that $\Delta E^{(free)}$ is finite. In order to interpret (85) we look
at the shift in the kinetic energy of a free electron of momentum $\vec{P}$
when a mass correction $\Delta m$ is added to its bare mass $m$, to account
for the interactions between the electron and the background electromagnetic
field, to obtain the physical mass $m_{{}_{e}}\ =\ m+\Delta m$. At $O(\Delta
m)$, we have

$\displaystyle\Delta E^{(free)}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{(\vec{P})^{2}}{2m_{{}_{e}}}-\displaystyle\frac{(\vec{P})^{2}}{2(m_{{}_{e}}-\Delta
m)}\ =\ -\displaystyle\frac{1}{m_{{}_{e}}}\cdot\Delta
m\cdot\displaystyle\frac{(\vec{P})^{2}}{2m_{{}_{e}}}\qquad$ (86)

Comparing (85) and (86) we conclude that the energy shift given in (85) is the
change in the kinetic energy of the free electron resulting from the addition
of electromagnetic mass $\Delta m\ =\
\displaystyle\frac{2e^{2}}{3\pi}\int_{0}^{\infty}\tilde{g}(\kappa)^{2}\,d\kappa$
to the bare mass of the electron to obtain the physical mass $m_{{}_{e}}$. For
a bound electron, in state $\left.\left|\,n\,\right\rangle\right.$, we obtain
the shift in kinetic energy $\Delta E_{{}_{\Delta m}}^{(n)}$ due to the
addition of the electromagnetic mass by replacing $(\vec{P}_{{}_{n}})^{2}$
with $\left.\left\langle\,n\,\right.\right|\vec{p}\
^{2}\left.\left|\,n\,\right\rangle\right.=\langle\vec{p}\
^{2}\rangle_{{}_{nn}}$, the expected value of the $\vec{p}\ ^{2}$ operator,

$\displaystyle\Delta E_{{}_{\Delta m}}^{(n)}$ $\displaystyle=$
$\displaystyle-\displaystyle\frac{1}{m_{{}_{e}}}\left[\,\displaystyle\frac{2e^{2}\
}{3\pi\,}\int_{0}^{\infty}d\kappa\cdot\tilde{g}(\kappa)^{2}\,\right]\displaystyle\frac{\langle\vec{p}\
^{2}\rangle_{{}_{nn}}}{2m_{{}_{e}}}\ =\ -\left[\,\displaystyle\frac{e^{2}\
}{3\pi\,m_{{}_{e}}^{2}}\int_{0}^{\infty}d\kappa\cdot\tilde{g}(\kappa)^{2}\,\right]\sum_{r}|\langle\vec{p}\,\rangle_{nr}|^{2}$

The shift $\Delta E_{{}_{\Delta m}}^{(n)}$ is already included in the energy
$E_{{}_{n}}$ of $\left.\left|\,n\,\right\rangle\right.$, that we compute using
the Bohr model of the Hydrogen atom in which we use not the bare mass of the
electron but the physical mass $m_{{}_{e}}$. The Lamb shift we measure is the
change in energy relative to $E_{{}_{n}}$. Therefore, from the total shift
computed in (84) we need to subtract the shift in kinetic energy $\Delta
E_{{}_{\Delta m}}^{(n)}$ due to mass correction to obtain the theoretically
predicted Lamb shift.

Subtracting $\Delta E_{{}_{\Delta m}}^{(n)}$ from the $\Delta E_{{}_{n}}$ in
(84) and inserting the factors $\hbar,c$ and $\epsilon_{{}_{0}}$ (including in
$\tau_{{}_{{\cal P}}}=m_{{}_{e}}c^{2}$) to obtain an expression in SI units,
we get the prediction for the observed energy shift

$\displaystyle\Delta E_{{}_{n}}^{(predicted)}$ $\displaystyle=$
$\displaystyle\Delta E_{{}_{n}}-\Delta
E_{{}_{n}}^{(free)}=\displaystyle\frac{e^{2}}{3\pi\,\epsilon_{{}_{0}}\,m_{{}_{e}}^{2}\,c^{2}}\int_{0}^{\infty}d\kappa\cdot\tilde{g}(\hbar\,c\,\kappa)^{2}\cdot\sum_{r}\displaystyle\frac{\left|\langle\,\vec{p}\,\rangle_{nr}\right|^{2}\cdot(E_{r}-E_{{}_{n}})}{E_{r}-E_{{}_{n}}+\hbar\,c\,\kappa}$
(87) $\displaystyle=$
$\displaystyle\displaystyle\frac{e^{2}}{3\pi\,\epsilon_{{}_{0}}\,m_{{}_{e}}^{2}\,c^{3}\,\hbar}\sum_{r}\left|\langle\,\vec{p}\,\rangle_{nr}\right|^{2}\cdot(E_{r}-E_{{}_{n}})\left[\,\int_{0}^{\infty}dz\cdot\displaystyle\frac{\tilde{g}(z)^{2}}{E_{r}-E_{{}_{n}}+z}\,\right]$

where $z=\hbar\,c\,\kappa$.

For ${}^{2}S_{1/2}$ and ${}^{2}P_{1/2}$ levels the principal quantum number is
2. To avoid conflict with the symbol $n$ used above, we’ll use
$n_{{}_{principal}}$ to denote the principal quantum number of an energy level
in the Hydrogen atom. As mentioned before, the symbols $r$ and $n$, which are
used to denote the orbitals, actually stand for triples comprising the
principal, azimuthal and magnetic quantum numbers. Using the energy
eigenfunctions of the Hydrogen atom one can verify that

$\displaystyle\left\langle\psi_{2,\,0,\,0}\,\left|\,\nabla\
\right|\psi_{1,\,0,\,0}\,\right\rangle=\left\langle\psi_{2,\,1,\,0}\,\left|\,\nabla\
\right|\psi_{1,\,0,\,0}\,\right\rangle=\left\langle\psi_{2,\,1,\,1}\,\left|\,\nabla\
\right|\psi_{1,\,0,\,0}\,\right\rangle=\left\langle\psi_{2,\,1,\,-1}\,\left|\,\nabla\
\right|\psi_{1,\,0,\,0}\,\right\rangle=0$

Further at $n_{{}_{principal}}=2$,
$E_{(2,\,l^{\prime}\,,\,m^{\prime})}-E_{(2\,,\,l\,,\,m)}=0$. So the sum in
(87) starts at the principal quantum number $n_{{}_{principal}}=3$ and
$\left|\langle\,\vec{p}\,\rangle_{nr}\right|^{2}(E_{r}-E_{{}_{n}})>0$ for all
$r$ in the sum. Therefore, we conclude that $\Delta
E_{{}_{n}}^{(predicted)}\geq 0$. Further, the integral in (87) decreases
monotonically with increasing $E_{r}$. Therefore, we can bound the integral
using the two limits $n_{{}_{principal}}=3$ and
$n_{{}_{principal}}\rightarrow\infty$. Denoting $E_{n_{{}_{principal}}=j}$ as
$E_{[j]}$ and noting that $E_{[\infty]}=0$, we have for $j=3,4,5,\ldots$,

$\displaystyle
I_{min}:=\int_{0}^{\infty}dz\cdot\displaystyle\frac{\tilde{g}(z)^{2}}{0-E_{[2]}+z}<\int_{0}^{\infty}dz\cdot\displaystyle\frac{\tilde{g}(z)^{2}}{E_{[j]}-E_{[2]}+z}<\int_{0}^{\infty}dz\cdot\displaystyle\frac{\tilde{g}(z)^{2}}{E_{[3]}-E_{[2]}+z}:=I_{max}$

Hence, from (87) we have

$\displaystyle\displaystyle\frac{e^{2}\
I_{{}_{min}}}{3\pi\,\epsilon_{{}_{0}}\,m_{{}_{e}}^{2}\,c^{3}\,\hbar}\sum_{r}\left|\langle\,\vec{p}\,\rangle_{nr}\right|^{2}\cdot(E_{r}-E_{{}_{n}})<\Delta
E_{{}_{n}}^{(predicted)}<\displaystyle\frac{e^{2}\
I_{{}_{max}}}{3\pi\,\epsilon_{{}_{0}}\,m_{{}_{e}}^{2}\,c^{3}\,\hbar}\sum_{r}\left|\langle\,\vec{p}\,\rangle_{nr}\right|^{2}\cdot(E_{r}-E_{{}_{n}})$
(88)

Using (12) and $E_{[j]}=-\displaystyle\frac{E_{{}_{1}}}{j^{2}}$, where
$E_{{}_{1}}=\alpha^{2}m_{e}c^{2}/2$, we have

$\displaystyle I(j)$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\ dz\
\left[\,\frac{2}{e^{\displaystyle
z/(m_{e}c^{2})+(m_{e}c^{2})/z}-1}\,\right]^{1/2}\left[\,\frac{1}{E_{{}_{1}}\left(\displaystyle\frac{1}{4}-\displaystyle\frac{1}{j^{2}}\right)+z}\,\right]\qquad
j=3,4,5,\ldots$ (89) $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\ dq\
\left[\,\frac{2}{e^{\displaystyle
q+1/q}-1}\,\right]^{1/2}\left[\,\frac{1}{\left\\{\displaystyle\frac{\alpha^{2}}{2}\left(\displaystyle\frac{1}{4}-\displaystyle\frac{1}{j^{2}}\right)+q\right\\}}\,\right],\qquad
q:=\displaystyle\frac{z}{m_{{}_{e}}c^{2}}$

As in the calculation of electron’s anomalous magnetic moment, we have taken
the equilibrium temperature for the photon field in this calculation also to
be the only Lorentz-invariant energy scale occurring in the interaction of the
photon field with the electron, namely the mass of the electron.

Numerical evalution of (89) yields

$\displaystyle I_{min}=I(\infty)=1.244289,\qquad I_{max}=I(3)=1.244294$ (90)

Noting that the nonrelativistic Hamiltonian of the Hydrogen atom
$H_{A}=\displaystyle\frac{\vec{p}\cdot\vec{p}}{2m}-\displaystyle\frac{e^{2}}{4\pi\epsilon_{{}_{0}}|\,\vec{x}\,|}$
we have

$\displaystyle\sum_{r}|\langle\,\vec{p}\,\rangle_{nr}|\,^{2}\,(E_{r}-E_{{}_{n}})$
$\displaystyle=$ $\displaystyle\left\langle n\,\left|\,\vec{p}\cdot
H\vec{p}-H\vec{p}\cdot\vec{p}\ \right|n\,\right\rangle\ =\
\displaystyle\frac{e^{2}\hbar^{2}}{2\epsilon_{{}_{0}}}|\psi_{{}_{n}}(0)|^{2}$
(91)

Inserting (91) into (88) we have

$\displaystyle\displaystyle\frac{e^{2}I_{{}_{min}}}{3\pi\,\epsilon_{{}_{0}}\,m_{{}_{e}}^{2}\,c^{3}\,\hbar}\left[\,\frac{e^{2}\hbar^{2}}{2\epsilon_{{}_{0}}}\,\right]|\psi_{{}_{n}}(0)|^{2}\
<\ \Delta E_{{}_{n}}^{(predicted)}\ <\ \displaystyle\frac{e^{2}\
I_{{}_{max}}}{3\pi\,\epsilon_{{}_{0}}\,m_{{}_{e}}^{2}\,c^{3}\,\hbar}\left[\,\frac{e^{2}\hbar^{2}}{2\epsilon_{{}_{0}}}\,\right]|\psi_{{}_{n}}(0)|^{2}$
(92)

We note that $|\psi_{(2\,,\,1\,,\,m)}(0)|^{2}=0$, where $m=-1,0,1$. Therefore,
from (87) and (91) we conclude that there is no energy shift in the
${}^{2}P_{1/2}$ level.

On the other hand,
$|\psi_{2\,,\,0\,,\,0}(0)|^{2}=\displaystyle\frac{1}{8\pi\,a_{{}_{0}}^{3}}$,
where $a_{{}_{0}}=\displaystyle\frac{\hbar}{\alpha m_{e}c}$. From (90) and
(92) we have

$\displaystyle
1.244289\left[\,\frac{e^{4}\hbar}{48\pi^{2}\epsilon_{{}_{0}}^{2}m_{{}_{e}}^{2}c^{3}a_{{}_{0}}^{3}}\,\right]\
<\ E_{{}_{2}}^{(predicted)}\ <\
1.244294\left[\,\frac{e^{4}\hbar}{48\pi^{2}\epsilon_{{}_{0}}^{2}m_{{}_{e}}^{2}c^{3}a_{{}_{0}}^{3}}\,\right]$

Dividing throughout by $h$ we get the Lamb shift predicted by the above
calculation to be

$\displaystyle 1060.476\,MHz\ <\ \Delta\nu^{(predicted)}_{{}_{LS}}\ <\
1060.480\,MHz$ (93)

##  6 Running Coupling Constant in QED

The calculations in Sections 4 and 5 pertain to ‘soft’ scattering processes in
that they involve momentum transfers that are well below the electron mass
threshold. In this section we present a 1-loop calculation that involves
momentum transfers that are considerably above the electron mass threshold.
Specifically, we calculate, at 1-loop, the running of the fine structure
‘constant’ at scales that are from one to four orders of magnitude above the
electron mass threshold. The prediction of the new framework, denoted
$\alpha_{{}_{Gibbs}}(k)$, agrees with the previously known result, denoted
$\alpha_{{}_{HP}}(k)$, to within 0.8% as shown in Figure 4.

Consider the electron-electron scattering process ${\cal P}$ shown in Figure
2. As before, ${\cal C}_{{}_{\cal P}}$ represents the center-of-momentum frame
of the process. The 4-momentum transferred between the scattering electrons is
denoted $k$.

Figure 2: Electron-electron scattering with 1-loop correction to photon
propagator.

As is well known the full photon propagator can be written, in momentum space,
as

$\displaystyle\tilde{M}^{\mu\nu}_{{}_{full}}(k)$ $\displaystyle=$
$\displaystyle\tilde{M}^{\mu\nu}(k)+\tilde{M}^{\mu\alpha}(k)\Pi_{\alpha\beta}(k)\tilde{M}^{\beta\nu}(k)+\tilde{M}^{\mu\alpha}(k)\Pi_{\alpha\beta}(k)\tilde{M}^{\beta\rho}(k)\
\Pi_{\rho\sigma}(k)\,\tilde{M}^{\sigma\nu}(k)+\ldots$

Due to conservation of electron current $\Pi_{{}_{\alpha\beta}}(k)$ satisfies
$k^{\alpha}\Pi_{{}_{\alpha\beta}}(k)=k^{\beta}\Pi_{{}_{\alpha\beta}}(k)=0$.
Therefore, we can take $\Pi_{\alpha\beta}(k)$ to be

$\displaystyle\Pi_{{}_{\alpha\beta}}(k)$ $\displaystyle=$ $\displaystyle
i\,e_{{}_{0}}^{2}\,(k^{2}\eta_{{}_{\alpha\beta}}-k_{\alpha}k_{\beta})\,\chi(k),\qquad\chi(k):=\displaystyle\frac{\Pi^{\mu}\,_{\mu}(k)}{3\,i\,e_{{}_{0}}^{2}\,k^{2}}$
(95)

$e_{{}_{0}}$ is the bare charge. $\eta$ is the Lorentzian metric tensor. From
(49) the tree-level photon propagator $\tilde{M}^{\mu\nu}(k)$ is

$\displaystyle\tilde{M}_{{}_{\alpha\beta}}(k)=i\cdot\displaystyle\frac{(-\eta_{{}_{\alpha\beta}})\cdot\
g{{}_{p}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal P})^{2}}{k^{2}+i\epsilon}$

Before proceeding, we note a difference between (95) and the corresponding
definition313131In the notation used in [43], the function $\chi$ in (95) is
called $\Pi$. [43, Equation 13.47] in the Heisenberg-Pauli framework. In [43,
Equation 13.47], $\chi$ is taken to be a function of $k^{2}$ only. In our
framework, there are two Lorentz-invariant scalar-valued functions of $k$,
namely $k^{2}$ and $k_{\mu}u^{\mu}({\cal C}_{{}_{\cal P}};{\cal C}_{{}_{\cal
P}})$, where $u({\cal C}_{{}_{\cal P}};{\cal C}_{{}_{\cal P}})$ is the
velocity 4-vector of the center-of-momentum frame observed by an observer who
is at rest in ${\cal C}_{{}_{\cal P}}$. Therefore, in our framework $\chi$ is
not just a function of $k^{2}$ but also of $k_{\mu}u({\cal C}_{{}_{\cal
P}};{\cal C}_{{}_{\cal P}})^{\mu}$ at fixed $k^{2}$. Hence, we write $\chi(k)$
instead of $\chi(k^{2})$ in (95).

Defining

$\displaystyle P^{\mu}\,_{\nu}(k)$ $\displaystyle:=$
$\displaystyle\delta^{\mu}\,_{\nu}-\displaystyle\frac{k^{\mu}k_{\nu}}{k^{2}},$

and noting that
$P^{\mu}\,_{\alpha}(k)P^{\alpha}\,_{\nu}(k)=P^{\mu}\,_{\nu}(k)$, we have

$\displaystyle\prod_{j=1}^{n}\Pi^{\mu_{{}_{j}}}\,_{\alpha_{{}_{j}}}(k)\tilde{M}^{\alpha_{{}_{j}}}\,_{\nu_{{}_{j}}}(k)$
$\displaystyle=$ $\displaystyle\lambda^{n}(k)\
P^{\mu_{{}_{1}}}\,_{\nu_{{}_{n}}},\qquad\lambda(k):=e_{{}_{0}}^{2}\,\
g{{}_{p}}\,(\,{k}\,;{\cal C}_{{}_{\cal P}},{\cal P})^{2}\,\chi(k),\quad$ (96)

Inserting (96) into (LABEL:eqnfppr) we have

$\displaystyle\tilde{M}^{\mu\nu}_{{}_{full}}(k)$ $\displaystyle=$
$\displaystyle\tilde{M}^{\mu}\,_{\alpha}(k)\left\\{\displaystyle\frac{\eta^{\alpha\nu}}{1-\lambda}-\displaystyle\frac{\lambda}{1-\lambda}\