# Decoherence of a 2-Path System by Infrared Photons

Colby DeLisle<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of British Columbia, 6224 Agricultural Rd., Vancouver, B.C., Canada
V6T 1Z1  Pacific Institute of Theoretical Physics, University of British
Columbia, 6224 Agricultural Rd., Vancouver, B.C., Canada V6T 1Z1  P.C.E. Stamp
<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics and
Astronomy, University of British Columbia, 6224 Agricultural Rd., Vancouver,
B.C., Canada V6T 1Z1  Pacific Institute of Theoretical Physics, University of
British Columbia, 6224 Agricultural Rd., Vancouver, B.C., Canada V6T 1Z1
Theoretical Astrophysics, Cahill, California Institute of Technology, 1200 E.
California Boulevard, MC 350-17, Pasadena CA 91125, USA

###### Abstract

We calculate the decoherence caused by photon emission for a charged particle
travelling through an interferometer; the decoherence rate gives a
quantitative measure of how much “which-path” quantum information is gained by
the electromagnetic field. We isolate the quantum information content of both
leading and sub-leading soft photons, and show that it can be extracted
entirely from information about the endpoints of the particle’s paths. When
infrared dressing is properly incorporated, the leading order soft photons
give no contribution to decoherence, and carry no quantum information. The
sub-leading soft photons carry finite which-path information, and the sub-
leading contribution to decoherence takes an extremely simple, time-
independent form depending only on the size of the interferometer.

##  I Introduction

Our subject in this paper is the well-known “2-path” problem in quantum
mechanics [1], in the present case for a charged particle coupled to the
electromagnetic (EM) field. We thus assume a pointlike particle with electric
charge $e$, which is travelling through an interferometer, and is put into a
superposition of trajectories falling into two distinct classes [2], which we
label $L$ and $R$. We will sometimes refer to this particle as an electron,
although it could of course be some other charged particle of larger mass.

An external observer then has the choice of either (i) measuring which path
the particle has followed, or (ii) letting it simply pass through the
interferometer unmolested by any measuring system. As is well known, in the
former case no interference is seen between the 2 paths, whereas in the latter
case, a standard interference pattern is seen on any screen which intercepts
the 2 paths.

However, it is also well known that on its way through the interferometer, the
particle couples to the electromagnetic (EM) field, which may be either at
temperature $T=0$ (in its ground state) or at finite $T$ in the rest frame of
the interferometer. In either case, the EM field is able to, in effect,
passively “measure” which path the particle has followed; in quantum-
information-theoretic language, it can act as a “witness” as to which path is
followed. This mechanism is of course quite general, and is usually called
“environmental decoherence” [3, 4, 5, 6]; because of it one expects partial or
total destruction of the interference between the 2 paths.

The particular case of decoherence for 2-path systems in which electrons
coupled to photons has been repeatedly discussed in the literature [7, 8, 9].
In this paper we wish to revisit this topic, and clarify the role of low
frequency infrared (IR) photons in bringing about the decoherence. Our main
result will be explained in more detail below – we show that decoherence
actually comes not from the leading IR divergent terms in the effective
electron-photon coupling (which actually give zero contribution), but rather
the sub-leading terms.

We stress that in any real experiment there will be other processes also
causing decoherence – these include coupling of the electron to any gas
particles in the experiment, as well as the long-range interaction of the
electron with “charge fluctuators” in the apparatus. The former can be
understood using scattering theory [10], and the latter using a “spin bath”
theory [11]. We will ignore these contributions in what follows, since we are
primarily interested in the EM decoherence mechanism.

The topic of IR divergences, and their influence on the quantum electronic
dynamics, of course has a long history, dating all the way back to the work of
Bloch and Nordsieck [12]. Early work clarified the ways in which the leading
divergences coming from radiative corrections to scattering processes [12, 13,
14] were cancelled by divergences from virtual soft particles, leading to the
derivation of multiplicative “soft factors” (see, eg., refs. [16, 17, 15], and
also refs. [18, 19, 20, 21]) in scattering processes. These factors, when
correctly handled, give finite multiplicative corrections to scattering cross-
sections.

One can write soft factors for any electron-photon scattering process. Here we
are interested in the 3-point electron-photon scattering process (see Fig. 1),
in which the incoming state is an electron with 4-momentum $p^{a}$, and the
outgoing state is an electron and a photon, with momenta $p^{a}-q^{a}$ and
$q^{a}$ respectively. The “soft photon theorem” then says that one can write
the fully renormalized vertex for this process in the form

$\Lambda(p,q)\;=\;K_{2}(p)\,\left[{\cal S}_{(0)}(q)+{\cal
S}_{(1)}(q)\right]\;+\;\mathcal{O}(|{\bf q}|)$ (1)

where $K_{2}(p)$ is the fully renormalized 2-point propagator for the electron
alone, with no photon emission. Here the divergent leading soft factor ${\cal
S}_{(0)}(q)\sim\mathcal{O}(1/|{\bf q}|)$, where ${\bf q}$ is the 3-momentum,
and the sub-leading factor ${\cal S}_{(1)}(q))\sim\mathcal{O}(1)$.

Figure 1: The 3-point electron-photon vertex $\Lambda(p,q)$ written, for small
$|{\bf q}|$, as a product of the single electron propagator $K_{2}(p)$ and the
“soft factor” ${\cal S}(q)$; compare eq. (1) in the text.

Later work has shown that this result can also be understood by using either
an eikonal analysis [22, 23, 24], or a coherent state representation for the
coupled electron-photon system [25, 26, 27, 28, 29, 30]. The latter
representation, in which the system is represented by “dressed” electron
states, is quite illuminating, for two reasons.

First, it highlights that the naïve photon vacuum is non-unique; there are
infinitely many field configurations with zero energy. Thus, one needs to be
careful in trying to calculate decoherence: the decoherence rate one finds
will diverge if one evaluates scattering processes between the usual photon
Fock states, due to transitions between these superselected vacua, but will
will be convergent if one instead uses dressed states [31, 32]. This is
physically rather obvious – the electrons are at no point ever undressed, so
one should always work with dressed states.

Second, the coherent state representation affords a nice way of separating the
leading contributions to the scattering processes from all sub-leading
contributions, first discussed in perturbative calculations [33, 34, 35, 36]
(this separation is also done in a very clear way in the eikonal expansion
[22, 23]). The separation of terms will be crucial in what follows, since what
we wish to establish here is that the leading IR contributions to the soft
factor in QED make no contribution at all to decoherence. It is instead the
sub-leading contributions that are responsible for the lowest-energy
decoherence in QED.

To show this we will use the recent discovery that the leading and sub-leading
“soft photon factors” are encoded at the endpoints of charged particle
worldlines [37]. We shall see that we can use this result to isolate the
quantum information content of both leading and sub-leading soft photons in
our model; and we shall see that the former is zero, and the latter is finite.
What is more, it is not necessary to go to any asymptotic limit to show this –
we will see that even in some experiment in which all processes are confined
to a finite spacetime region, these results still hold – the leading term
still gives identically zero, and the sub-leading contribution is time-
independent, depending only on the size of the interferometer.

The results we obtain for the total amount of decoherence is in fact
consistent with a result derived two decades ago by Breuer and Petruccione
[9]. It is satisfying that their result – which was found without worrying
about the subtleties of the different contributions to soft factors – be
confirmed, since its derivation was clearly based on physical arguments.
However it is also satisfying to be able to unravel the different soft
contributions – from leading and sub-leading terms – to the decoherence
functional, entirely in terms of the boundary (endpoint) currents. This method
makes for a simple derivation, since we don’t have to worry too much about the
details of the paths between the endpoints. It also allows for a rather novel
and transparent interpretation of decoherence in the infrared.

###  I.1 Summary & Structure of the Paper

The rest of the paper proceeds as follows. First, in Section II we introduce
what is by now a standard semiclassical interferometry model. We describe the
assumed geometry of the interferometer, and discuss the way in which we
evaluate decoherence in the model using the decoherence functional $\Gamma$.

In Section III we discuss relevant features of the momentum space
electromagnetic current $j^{a}(q)$ which appears in the decoherence
functional. This will allow us to review results from our recent work [37],
which showed that boundary terms in the electromagnetic current encode the
“soft factors” appearing in both the leading and sub-leading soft photon
theorems.

We then show in Section IV that superpositions of the leading soft factor
generically cause the decoherence functional $\Gamma$ to diverge in the
infrared – even for finite-time experiments. We must deal with these infrared
divergences properly in order to obtain sensible results for the quantum
information content of the photon field, so Section IV also discusses how we
ought to “dress” the time evolution operator in our semiclassical model in
order to render the decoherence functional infrared-finite. After adding the
appropriate dressing, the leading soft photons cease to obtain any quantum
information about the matter particle.

Section V contains the evaluation of the corrected expression for $\Gamma$ in
our model. We show that sub-leading soft photons still contribute to $\Gamma$,
and isolate the contribution coming from the difference in the sub-leading
soft current on each branch of the superposition. The sub-leading contribution
to decoherence turns out to be time-independent, and to depend only upon the
spatial extent of the model interferometer.

The paper concludes in Section VII with a discussion of our results, and of
some open questions.

Here we use the mostly-negative metric signature,
$\eta_{ab}\equiv\textrm{diag}(1,-1,-1,-1)$, units in which
$\hbar=c=\epsilon_{0}=1$, and a bar over a quantity will indicate its complex
conjugate.

##  II Interferometer Model

Let us begin by introducing the simple model we will use to describe a charged
particle traversing an interferometer - this model is fairly standard in
quantum optics [38]. The spatial trajectories taken by the superposed particle
are illustrated in Figure 2. The particle enters the apparatus along the
$\hat{x}$ direction with four-velocity $\dot{X}^{a}_{2}$; upon reaching the
point $X^{a}_{1}$ at time $t=0$, the particle is put into a superposition, in
which it is either kicked into the $\hat{y}$ direction to follow the
trajectory $X_{L}^{a}(s)$, or allowed to proceed in the $\hat{x}$ direction
and follow the trajectory $X_{R}^{a}(s)$. Here $s$ is the proper time
experienced by the particle as it evolves along the trajectories.

$\dot{X}^{a}_{1}$$\dot{X}^{a}_{2}$$\dot{X}^{a}_{2}$$\dot{X}^{a}_{1}$$x$$y$$l$$X^{a}_{i}$$X^{a}_{L}$$X^{a}_{R}$$\mathcal{D}$
Figure 2: Top-down view of the two-path geometry traversed by the charged
particle.

As the system then evolves, that branch of the superposition following the
trajectory $X_{L}^{a}(s)$ (the solid line in Figure 2) proceeds with the four-
velocity $\dot{X}_{1}^{a}$ to the point $X_{L}^{a}$, and then with four-
velocity $\dot{X}_{2}^{a}$ until it reaches the detector $\mathcal{D}$. The
branch following the trajectory $X_{R}^{a}(s)$ (dashed line) proceeds with the
four-velocity $\dot{X}_{2}^{a}$ to the point $X_{R}^{a}$, and then with four-
velocity $\dot{X}_{1}^{a}$ to the detector. On both paths, the speed of the
particle is taken to be constant at $v\equiv l/\tau$, where $l$ is the length
of one side of the interferometer, so that the particle reaches either
$X^{a}_{L}$ or $X^{a}_{R}$ at time $t=\tau$ and reaches the detector at
$t=2\tau$.

As noted in the introduction, Fig. 2 is schematic in the sense that in
reality, a single path shown going along the left or right trajectory actually
represents the set of all paths passing through the left arm of an
interferometer (this shorthand way of representing entire classes of path
appears to have originated in Feynman’s celebrated description of two-path
experiments [1]).

In the computations that follow however, we will use a semiclassical model of
interferometry, consisting of a simple discrete superposition of two classical
electromagnetic currents, $j^{a}_{L}$ and $j^{a}_{R}$ in the interferometer.
The electromagnetic current of a pointlike particle with charge $e$, which
follows either of the trajectories $X_{L/R}^{a}(s)$, is

$j_{L/R}^{a}(x)=e\int_{s_{i}}^{s_{f}}ds\,\dot{X}_{L/R}^{a}(s)\delta^{(4)}(x-X_{L/R}(s)),$
(2)

where again $s$ is the particle’s proper time, and
$\dot{X}_{L/R}^{a}(s)\equiv\frac{d}{ds}X_{L/R}(s)$ its four-velocity. The
bounds on this integral are chosen such that the initial and final proper
times coincide with the beginning and end of the experiment,
$X_{L/R}^{0}(s_{i})=0$ and $X_{L/R}^{0}(s_{f})=2\tau$.

Assuming the photon field is in vacuum at the time the experiment starts,
under time evolution the electromagnetic environment will be placed into a
superposition of states $|L\rangle$ and $|R\rangle$ which will be coherent
photon states, sourced by either the $L$ or $R$ current:

$|L/R\rangle\equiv\mathcal{T}e^{-i\int
d^{4}x\;j_{L/R}^{a}(x)\hat{A}_{a}(x)}|0\rangle$ (3)

These coherent states are obtained by the action of the interaction-picture
time evolution operator acting on the photon vacuum $|0\rangle$, and here
$\mathcal{T}$ denotes time-ordering. In writing this expression we have
assumed as in (2) that the currents $j_{L/R}^{a}(x)$ have support only for the
duration of the experiment (between $s_{i}$ and $s_{f}$), so we can let the
integral $\int d^{4}x$ run over all values of $t\in(-\infty,+\infty)$.

The state of the matter+radiation system as the particle enters the detector
$\mathcal{D}$ is taken to be

$\frac{1}{\sqrt{2}}\left[|\psi_{L}\rangle|L\rangle+|\psi_{R}\rangle|R\rangle\right],$
(4)

with $|\psi_{L/R}\rangle$ the states of the matter particle after traversing
the $L/R$ arm of the interferometer. As the particle enters the detector, its
position space probability distribution is then

$\displaystyle P(x)$
$\displaystyle=\frac{1}{2}|\psi_{L}(x)|^{2}+\frac{1}{2}|\psi_{R}(x)|^{2}$ (5)
$\displaystyle+\frac{1}{2}\langle
R|L\rangle\,\psi_{L}(x)\bar{\psi}_{R}(x)+\frac{1}{2}\langle
L|R\rangle\,\bar{\psi}_{L}(x)\psi_{R}(x),$

where we have defined the position space wavefunctions
$\psi_{L/R}(x)\equiv\langle x|\psi_{L/R}\rangle$.

The quantity governing the quantum coherence of the particle in the two-path
experiment is then the inner product between the photon coherent states,

$\langle R|L\rangle\equiv e^{i\Theta-\Gamma};\hskip
10.00002pt\Theta,\Gamma\in\mathbb{R},\hskip 10.00002pt\Gamma\geq 0.$ (6)

Here, $\Gamma\equiv-\textrm{Re}\left(\ln\langle R|L\rangle\right)$, and
$\Theta\equiv\textrm{Im}\left(\ln\langle R|L\rangle\right)$. Because the
coherent states $|L/R\rangle$ are fully determined by the classical currents
$j^{a}_{L/R}$, we can think of the inner product (6) as a functional of these
currents,

$\langle R|L\rangle\equiv\langle R|L\rangle[j_{L},j_{R}].$ (7)

Consequently $\Gamma\equiv\Gamma[j_{L},j_{R}]$ and
$\Theta\equiv\Theta[j_{L},j_{R}]$ are functionals of the classical currents as
well.

The quantity $\Gamma$ will be of particular importance to us, and is called
the decoherence functional [7, 5, 39, 40]. It will be a functional of the
entire currents along the $L$ and $R$ paths, and is entirely responsible for
the suppression of interference terms when measuring the matter state, as it
alone governs the modulus of the electromagnetic inner product

$|\langle R|L\rangle|=e^{-\Gamma}.$ (8)

The electromagnetic decoherence functional was first derived long ago [40,
41], and has been discussed more recently e.g. by Ford [8], and by Breuer and
Petruccione [9]. At zero temperature it takes the form

$\Gamma[j_{L},j_{R}]={1\over
2}\int\frac{d^{3}q}{(2\pi)^{3}2\omega}\;P_{ab}\,\delta j^{a}(q)\delta
j^{b}(-q),$ (9)

where $\delta j^{a}\equiv j_{L}^{a}-j_{R}^{a}$ is the difference in
electromagnetic currents along the two superposed paths,
$q^{a}\equiv\omega(1,\hat{n})$, with $\hat{n}$ an angular unit vector, is a
null four-momentum, and
$\int\frac{d^{3}q}{(2\pi)^{3}}\equiv\oint{dS^{2}(\hat{n})\over(2\pi)^{2}}\int_{0}^{\infty}{d\omega\over
2\pi}\omega^{2}$. Finally, the transverse projector $P_{ab}$ can be written in
terms of the momentum $q^{a}$ as
$P_{jk}\equiv\delta_{jk}-\frac{q_{j}q_{k}}{\omega^{2}}$, with $P_{0a}=0$.

Note that $\Gamma$ as written is a functional of currents which may have
support for times $t\in(-\infty,+\infty)$. However we have restricted to
finite-time dynamics here by restricting the range of the proper time
integrals in (2).

##  III Form of Electromagnetic Currents & Decoherence Functional

Let us now further discuss the matter currents we will insert into the
decoherence functional (9); we will wish to understand in particular the
various contributions to the decoherence functional – which is central to any
analysis of decoherence, for any system – coming from the endpoints of the
particle’s paths through the interferometer.

What appears in the decoherence functional is actually the Fourier transform
of the current, $j^{a}(q)\equiv\int d^{4}x\,e^{iq\cdot x}j^{a}(x)$, which
using (2) is

$j^{a}(q)\;=\;e\int ds\,\dot{X}^{a}(s)\,e^{iq\cdot X(s)}.$ (10)

This current can be rewritten as

$j^{a}(q)\;=\;e\int
ds\,\dot{X}^{a}\left(\frac{1}{iq\cdot\dot{X}}\right)\frac{d}{ds}e^{iq\cdot
X},$ (11)

so that integrating by parts in $s$ gives

$\begin{split}j^{a}(q)=&-ie\int ds\,\frac{d}{ds}\left(e^{iq\cdot
X}\frac{\dot{X}^{a}}{q\cdot\dot{X}}\right)\\\ &+ie\int ds\,e^{iq\cdot
X}\frac{d}{ds}\left(\frac{\dot{X}^{a}}{q\cdot\dot{X}}\right).\end{split}$ (12)

Discarding the boundary term appearing here [42, 43, 37] ensures that the
current is conserved, $q_{a}j^{a}(q)=0$, leaving us with the following
corrected expression for the particle current in momentum space:

$j^{a}(q)=ie\int ds\,e^{iq\cdot
X}\frac{d}{ds}\left(\frac{\dot{X}^{a}}{q\cdot\dot{X}}\right)$ (13)

Notice that the form (13) of the matter current has the property that the
current vanishes when the particle is not accelerating ($\ddot{X}^{a}=0$).
Plugging the current into the decoherence functional (9), we note immediately
the important fact that there is no decoherence without acceleration.

###  III.1 Soft Electromagnetic Currents

Recently, we showed [37] that the electromagnetic current of a point particle
(13) possesses two pieces which are entirely localized to the boundary of the
worldline the particle traces out in space-time. These boundary terms turn out
to be the parts of the current which dominate at low frequencies. In
particular we showed that, expanding (13) in powers of $q$, we get

$ij^{a}(-q)\;=\;e\left[\Delta\mathcal{S}^{a}_{(0)}(q,m\dot{X})+\Delta\mathcal{S}^{a}_{(1)}(q,X,m\dot{X})\right]+\mathcal{O}(q).$
(14)

If the proper time integral in (13) runs from $s_{i}$ to $s_{f}$, then the
$\Delta$ in this expression has the meaning

$\Delta f(s)\equiv f(s_{f})-f(s_{i})=\int_{s_{i}}^{s_{f}}\partial_{s}f(s),$
(15)

showing that these contributions live on the endpoints of the particle
worldline, at $s_{i}$ and $s_{f}$, and we have defined the leading and sub-
leading electromagnetic “soft factors”

$\mathcal{S}_{(0)}^{a}(q,p)\equiv\frac{p^{a}}{q\cdot p},\hskip
10.00002pt\mathcal{S}_{(1)}^{a}(q,x,p)\equiv i\frac{q_{b}{J}^{ba}}{q\cdot p}.$
(16)

Here, ${J}^{ab}\equiv 2p^{[a}x^{b]}$ is the angular momentum tensor. Ref. [37]
further showed that these boundary terms explain the factorization of the
leading and sub-leading soft photon theorems at tree level.

We emphasize that these boundary terms on the worldline are not necessarily
located at asymptotic coordinate times $t\rightarrow\pm\infty$. They exist and
take the form (14) for any choice of bounds on the proper time integral in
(13).

From eq. (14) it is then clear that we can always split the point particle
current into three pieces,

$j^{a}(q)\equiv j_{div}^{a}(q)+j_{sub}^{a}(q)+j_{hard}^{a}(q),$ (17)

with

$\begin{split}j_{div}^{a}(q)&\equiv
ie\Delta\frac{\dot{X}^{a}}{q\cdot\dot{X}}\\\ j_{sub}^{a}(q)&\equiv
e\Delta\frac{q_{b}[X^{a}\dot{X}^{b}-\dot{X}^{a}X^{b}]}{q\cdot\dot{X}}\\\
j_{hard}^{a}(q)&\equiv j^{a}(q)-j_{div}^{a}(q)-j_{sub}^{a}(q).\end{split}$
(18)

We interpret the currents $j_{div}^{a}$, $j_{sub}^{a}$, and $j_{hard}^{a}$ as
the sources of leading soft radiation, sub-leading soft radiation, and hard
radiation, respectively. In this work, we then define “leading soft radiation”
as radiation which is sourced by $j_{div}^{a}$, “sub-leading soft radiation”
as radiation which is sourced by $j_{sub}^{a}$, and “hard radiation” as
radiation which is sourced by $j_{hard}^{a}$. This interpretation is
illustrated in Figure 3.

We note immediately that the leading soft current $j^{a}_{div}$ is infrared
divergent, as it goes like $1/\omega$ and blows up as $\omega\rightarrow 0$.
The sub-leading soft current $j_{sub}^{a}$ is however $\mathcal{O}(1)$ and
perfectly infrared-finite, and the hard current $j_{hard}^{a}$ is
$\mathcal{O}(\omega)$ and vanishes completely in the soft $\omega\rightarrow
0$ limit. Additionally we note that each of these currents is individually
conserved, i.e.,

$q_{a}j_{div}^{a}(q)=q_{a}j_{sub}^{a}(q)=q_{a}j_{hard}^{a}(q)=0.$ (19)

The identification of these soft currents will allow us to rather directly
assess the contribution to the decoherence functional (9) from leading and
sub-leading soft photon modes. In particular we will expand the decoherence
functional (9) in terms of the splitting (17) to get

$\begin{split}\Gamma={e^{2}\over
2}&\int\frac{d^{3}q}{(2\pi)^{3}2\omega}\;P_{ab}\times\\\ &\left[\delta
j_{div}^{a}(q)+\delta j_{sub}^{a}(q)+\delta j_{hard}^{a}(q)\right]\times\\\
&\left[\delta j_{div}^{b}(-q)+\delta j_{sub}^{b}(-q)+\delta
j_{hard}^{b}(-q)\right],\end{split}$ (20)

displaying clearly the contributions from the leading and sub-leading soft
currents to $\Gamma$.

We will see later on that this expression for $\Gamma$ is unphysical, since it
contains divergent terms that in the end make no contribution. The true
physical form for $\Gamma$ will be given in eq. (32) below.

$s_{i}$$s_{f}$

TimeSpace

$s_{i}$$s_{f}$

Figure 3: Left – two charged particle trajectories which source different hard
radiation, but the same soft radiation. Right – two trajectories which source
different hard and soft radiation, due to differences in the soft currents
$j_{div}^{a}$ and $j_{sub}^{a}$ at the worldline boundary. The hard radiation
is shown schematically in purple, the leading soft radiation in red, and the
sub-leading soft radiation in orange.

###  III.2 Decoherence Functional for Interferometer

The general form of a decoherence functional $\Gamma$ is independent of the
geometry of the system – but we wish to have a specific form for our 2-path
system. Later we will see what is specific to our interferometric setup, and
what is more generally true.

To do this, we first write the electromagnetic currents along the left and
right trajectories as

$\begin{split}j^{a}_{L}(q)=ie\int ds\,e^{iq\cdot
X_{L}(s)}\partial_{s}\left[\frac{\dot{X}_{L}^{a}(s)}{q\cdot\dot{X}_{L}(s)}\right],\\\
j^{a}_{R}(q)=ie\int ds\,e^{iq\cdot
X_{R}(s)}\partial_{s}\left[\frac{\dot{X}_{R}^{a}(s)}{q\cdot\dot{X}_{R}(s)}\right].\end{split}$
(21)

The difference between the two currents on the superposed paths,

$\delta j^{a}(q)\equiv j^{a}_{L}(q)-j^{a}_{R}(q)$ (22)

can then be approximated in the geometry illustrated in Figure 2 as

$\delta j^{a}(q)=ie\,\left[e^{iq\cdot X_{i}}-e^{iq\cdot X_{L}}-e^{iq\cdot
X_{R}}\right]\left[\frac{\dot{X}_{1}^{a}}{q\cdot\dot{X}_{1}}-\frac{\dot{X}_{2}^{a}}{q\cdot\dot{X}_{2}}\right].$
(23)

Notice that the current difference consists of terms associated with the
points $X_{i}^{a},X_{L}^{a},$ and $X_{R}^{a}$. These are the space-time events
at which the particle accelerates and would radiate photons.

If we assume that the particle travels non-relativistically – by which we mean
that

$v\equiv{l\over\tau}\ll 1$ (24)

in units with $c=1$ – then we can assume that the spatial parts of the phases
appearing in (23) can be neglected to leading order. This is just the well-
known dipole approximation [46]; we are assuming that the radiation emitted
during the experiment will have a wavelength that is long compared to the
spatial extent of the interferometer, and so cannot resolve the spatial
separation of the points at which the particle radiates.

In this approximation, the current difference can be further simplified to

$\delta j^{a}(q)\approx
ie\left(1-2e^{i\omega\tau}\right)\left[\frac{\dot{X}_{1}^{a}}{q\cdot\dot{X}_{1}}-\frac{\dot{X}_{2}^{a}}{q\cdot\dot{X}_{2}}\right].$
(25)

Plugging this current difference into the decoherence functional gives

$\displaystyle\Gamma$ $\displaystyle\approx{e^{2}\over
4(2\pi)^{3}}\int_{\lambda}^{\Omega}{d\omega\over\omega}\,\left(5-4\cos{\omega\tau}\right)\times$
(26) $\displaystyle\oint
dS^{2}(\hat{n})\,\omega^{2}\,\left[2\frac{\dot{X}_{1}\cdot\dot{X}_{2}}{(q\cdot\dot{X}_{1})(q\cdot\dot{X}_{2})}-\frac{1}{(q\cdot\dot{X}_{1})^{2}}-\frac{1}{(q\cdot\dot{X}_{2})^{2}}\right].$

Note that the sphere integral here is independent of $\omega$, since the term
in square brackets goes like ${1\over\omega^{2}}$ and this gets multiplied by
the factor $\omega^{2}$ from the integration measure.

Note here that although the decoherence functional formally includes a
frequency integration from $\omega=0$ to $\omega\rightarrow\infty$, we have
included a hard ultraviolet cutoff $\Omega$ in order to characterize the size
of the “scattering” regions $X^{a}_{\\{i,L,R\\}}$ inside of which the particle
experiences acceleration. An ultraviolet cutoff of this sort also emerges
naturally if one smears out the superposed trajectories we are considering,
allowing the currents to have support along worldtubes of finite width $\sim
1/\Omega$, rather than on perfectly-localized one-dimensional worldlines [9].
The length scale $l_{0}\equiv 1/\Omega$ is then the shortest length scale
relevant to our analysis.

The result (26) for $\Gamma$ incorporates all the soft contributions – what we
will now show is that the contribution from the leading divergent
contributions is actually zero.

##  IV Contribution from Leading Divergent Term

In order for the integral (26) to be well-defined, we have also been forced to
introduce an infrared cutoff $\lambda$. The integrand of (26) diverges like
$1/\omega$ as $\omega\rightarrow 0$, so the frequency integral goes to
infinity like $-\ln\lambda$ as we take the infrared cutoff $\lambda\rightarrow
0$. Taken naïvely, this leads one to predict an infinite amount of
decoherence, since $\Gamma\rightarrow\infty$ in the IR limit; this is clearly
unphysical.

In earlier discussion of IR divergences, noted in the introduction, it was
necessary to introduce complicated devices (eg., measuring systems designed to
observe IR photons) to deal with these divergences, and show how they were
either cancelled or otherwise eliminated. However in what follows we will show
how our understanding of soft currents allows these divergences to be handled
in a fairly straightforward way, which simply reduces their contribution to
$\Gamma$ to zero.

###  IV.1 Infrared Divergences

Begin by splitting the current (25) into its relevant soft
($\mathcal{O}(\omega^{-1})$ and $\mathcal{O}(1)$) and hard
($\mathcal{O}(\omega)$) pieces. In this model we have $\delta
j^{a}\equiv\delta j_{div}^{a}+\delta j_{sub}^{a}+\delta j_{hard}^{a}$, with

$\displaystyle\delta j_{div}^{a}(q)$ $\displaystyle\approx-
ie\left[\frac{\dot{X}_{1}^{a}}{q\cdot\dot{X}_{1}}-\frac{\dot{X}_{2}^{a}}{q\cdot\dot{X}_{2}}\right]$
$\displaystyle\delta j_{sub}^{a}(q)$ $\displaystyle\approx
2e\omega\tau\left[\frac{\dot{X}_{1}^{a}}{q\cdot\dot{X}_{1}}-\frac{\dot{X}_{2}^{a}}{q\cdot\dot{X}_{2}}\right]$
(27) $\displaystyle\delta j_{hard}^{a}(q)$ $\displaystyle\approx
2ie\left(1-e^{i\omega\tau}+i\omega\tau\right)\left[\frac{\dot{X}_{1}^{a}}{q\cdot\dot{X}_{1}}-\frac{\dot{X}_{2}^{a}}{q\cdot\dot{X}_{2}}\right]$

in the dipole approximation.

From the general form of the decoherence functional, eq. (20), we see that
$\Gamma$ contains a term involving only the leading soft current
$j_{div}^{a}$:

$\displaystyle\Gamma\supset{1\over
2}\int\frac{d^{3}q}{(2\pi)^{3}2\omega}\;P_{ab}\,\delta j_{div}^{a}(q)\delta
j_{div}^{b}(-q)$ $\displaystyle={e^{2}\over
4(2\pi)^{3}}\int_{\lambda}^{\Omega}{d\omega\over\omega}\times$ (28)
$\displaystyle\oint
dS^{2}(\hat{n})\,\omega^{2}\,\left[2\frac{\dot{X}_{1}\cdot\dot{X}_{2}}{(q\cdot\dot{X}_{1})(q\cdot\dot{X}_{2})}-\frac{1}{(q\cdot\dot{X}_{1})^{2}}-\frac{1}{(q\cdot\dot{X}_{2})^{2}}\right].$

It is this term that causes the infrared divergence when taking
$\lambda\rightarrow 0$, seen here as the logarithmically divergent integral
$\int_{\lambda}^{\Omega}{d\omega\over\omega}$. The infrared divergence is
caused by a superposition of the leading soft current, i.e.
$\Gamma\rightarrow\infty$ because $\delta j_{div}^{a}\neq 0$. The resulting
infinite amount of decoherence should thus be understood as coming from
information loss into leading soft photon modes.

###  IV.2 Infrared Dressing

If this result turned out to be the end of the story, then the leading-order
soft photons would always make a perfect which-path measurement of the charged
particle in any experiment of the sort shown in Figure 2. This is obviously
wrong: if it were the case, we would have $\Gamma\rightarrow\infty$ for any
choice of experimental parameters ($e,l,\tau$), and from eq. (5) we see that
there would never be any observable interference pattern once the particle
reaches the detector! Because quantum interference is observed in Nature, we
must reject this result.

This conclusion is not new, and indeed has informed all investigations of IR
divergences in QED since Bloch and Nordsieck [12]. That one needs to properly
handle these divergences is also obvious in the eikonal formulation of QED
[22, 23] and was a motivating factor in the coherent state formulation of QED
in terms of dressed states [25, 26, 27, 28, 29, 30, 31]. In the coherent state
formulation, the IR divergences are eliminated by dressing the electron, so
that soft photons appear already in the time evolution operator.

It turns out to be incredibly simple to effect IR dressing in our treatment.
We show in Appendix A that the net effect of applying a “minimal”
implementation of the dressed formalism, which we define in that appendix, is
that we can simply set the divergent soft current $j_{div}^{a}$ to zero
everywhere in our calculations – the physical result of IR dressing can then
be interpreted as the total decoupling of $j_{div}^{a}$ from the Maxwell
field. After incorporating the dressing, when studying the electromagnetic
radiation caused by the charged particle moving through our model
interferometer, what we do is define dressed coherent states in the same way
as we defined the coherent radiation states in eq. (3), but now explicitly
subtracting out the divergent current $j^{a}_{div}$, i.e., we write:

$||L/R\rangle\rangle\equiv\mathcal{T}e^{-i\int
d^{4}x\;[j_{L/R}^{a}-j_{L/R,div}^{a}](x)\hat{A}_{a}(x)}|0\rangle$ (29)

We can then define a dressed decoherence functional in the same way we defined
$\Gamma$ in the “undressed” case, but using the dressed coherent states
instead,

$|\langle\langle R|L\rangle\rangle|\equiv e^{-\Gamma_{dressed}}.$ (30)

The dressed decoherence functional $\Gamma_{dressed}$ can be obtained from the
undressed decoherence functional (20) by simply setting the divergent current
$j_{div}^{a}$ to zero, giving

$\begin{split}\Gamma_{dressed}={e^{2}\over
2}&\int\frac{d^{3}q}{(2\pi)^{3}2\omega}\;P_{ab}\times\\\ &\left[\delta
j_{sub}^{a}(q)+\delta j_{hard}^{a}(q)\right]\times\\\ &\left[\delta
j_{sub}^{b}(-q)+\delta j_{hard}^{b}(-q)\right].\end{split}$ (31)

The expression (31) makes it clear that in the dressed formalism, the leading
soft current no longer plays a role in decoherence of the matter state.
Because infrared divergences necessitate this dressing, we conclude that
leading soft photons do not cause decoherence, and therefore do not carry any
quantum information about the matter.

##  V Sub-Leading Soft Photons

After having “dressed away” the leading soft photon contributions to
decoherence, we can now begin our study of the remaining sub-leading
contributions. We will do this by once more evaluating the decoherence
functional, but this time using the dressed expression (31).

###  V.1 True Dressed Decoherence Functional

We begin by inserting the sub-leading and hard parts of the current difference
from (IV.1) into (31) gives

$\displaystyle\Gamma_{dressed}\approx{2e^{2}\over(2\pi)^{3}}\int_{0}^{\Omega}{d\omega\over\omega}\,\left(1-\cos{\omega\tau}\right)\times$
(32) $\displaystyle\oint
dS^{2}(\hat{n})\,\omega^{2}\,\left[2\frac{\dot{X}_{1}\cdot\dot{X}_{2}}{(q\cdot\dot{X}_{1})(q\cdot\dot{X}_{2})}-\frac{1}{(q\cdot\dot{X}_{1})^{2}}-\frac{1}{(q\cdot\dot{X}_{2})^{2}}\right].$

This expression incorporates both the sub-leading current difference $\delta
j_{sub}^{a}(q)$, and the hard current difference $\delta j_{hard}^{a}(q)$; and
cross-terms between them. We stress that since the contribution form the
divergent leading photon terms to $\Gamma$ is zero, eq. (32) should be
regarded as the correct expression for the decoherence functional.

We see that after incorporating the infrared dressing, the integrand of the
frequency integral now goes to zero as $\omega\rightarrow 0$, killing off the
divergence we found before, and rendering the integral finite so that we can
evaluate it properly, even with the infrared cutoff $\lambda$ now taken to
zero. The dressing is working as intended, and we note that since there are no
further infrared divergences, there seems to be no immediate need to add any
sort of “sub-leading dressing.”

To arrive at (32), we have again assumed that the particle travels through the
interferometer non-relativistically, $v\ll 1$, which, as discussed earlier,
allowed us to invoke the dipole approximation, in which we ignored spatial
parts of the phases appearing in the decoherence functional. In almost any
realistic interferometer, $v\ll 1$; thus, when evaluating $\Gamma_{dressed}$
we can now also assume that $\Omega\tau\gg 1$, i.e. that the the time of
completion of the experiment, $\tau$, is much greater than the time it takes
light to cross the “scattering” regions at the corners of the interferometer,
at which the particle accelerates, so that $\tau\gg 1/\Omega$.

We evaluate the expression (32) using these approximations in Appendix B.1,
with the result

$\Gamma_{dressed}\approx{e^{2}v^{2}\over 3\pi^{2}}\ln\Omega\tau.$ (33)

which agrees with previous calculations of electromagnetic decoherence [9,
45].

Note also that the decoherence is dipolar as expected from electromagnetic
radiation – we can write $\Gamma_{dressed}\propto(\partial_{t}e\delta x)^{2}$,
where $e\delta x$ is the difference in electric dipole moments between the $L$
and $R$ branches.

###  V.2 Purely Sub-Leading Contribution

In addition to computing the full amount of photon decoherence in our model,
we would now like to focus on the decoherence coming only from the sub-leading
terms in our dressed expression (32) for $\Gamma_{dressed}$.

In the general form (31) of the dressed decoherence functional, we see that
the purely sub-leading term $P_{ab}\delta j_{sub}^{a}\delta j_{sub}^{b}$, the
cross terms $P_{ab}\delta j_{sub}^{a}\delta j_{hard}^{b}$, and the purely hard
part $P_{ab}\delta j_{hard}^{a}\delta j_{hard}^{b}$ are all of different
orders in $\omega\tau$, and so can be classified according to this order. The
first purely sub-leading term gives decoherence caused purely by
superpositions of the sub-leading soft current. Writing this contribution as
$\Gamma_{sub}$, we then have

$\displaystyle\Gamma_{sub}\equiv{1\over
2}\int\frac{d^{3}q}{(2\pi)^{3}2\omega}\;P_{ab}\,\delta j_{sub}^{a}(q)\delta
j_{sub}^{b}(-q)$
$\displaystyle=\textdagger{e^{2}\over(2\pi)^{3}}\int_{0}^{\Omega}{d\omega}\;\omega\tau^{2}\times$
(34) $\displaystyle\oint
dS^{2}(\hat{n})\,\omega^{2}\,\left[2\frac{\dot{X}_{1}\cdot\dot{X}_{2}}{(q\cdot\dot{X}_{1})(q\cdot\dot{X}_{2})}-\frac{1}{(q\cdot\dot{X}_{1})^{2}}-\frac{1}{(q\cdot\dot{X}_{2})^{2}}\right],$

where we have used the expression for $\delta j_{sub}^{a}$ found in eq.
(IV.1), and where we see that this result is nothing but the lowest order term
in the expansion of the integrand of (32) in powers of $\omega\tau$.

The quantity $\Gamma_{sub}$ has an intuitive interpretation. Imagine allowing
the electromagnetic field to couple only to the sub-leading soft current
$j_{sub}^{a}$ during our hypothetical interferometry experiment. The resulting
states of the electromagnetic radiation sourced by the sub-leading current on
the $L/R$ branches would then be

$||l/r\rangle\rangle\equiv\mathcal{T}e^{-i\int
d^{4}x\;j_{L/R,sub}^{a}(x)\hat{A}_{a}(x)}|0\rangle,$ (35)

and $\Gamma_{sub}$ would encode the modulus of the inner product between these
two states as

$|\langle\langle r|l\rangle\rangle|\equiv e^{-\Gamma_{sub}}.$ (36)

$\Gamma_{sub}$ is then a decoherence functional in its own right, and is the
natural measure of the decoherence caused by the electromagnetic field’s
direct response to a superposition of sub-leading soft currents.

We evaluate the sub-leading contribution (V.2) in Appendix B.2, with the
result

$\Gamma_{sub}\approx{e^{2}\over 3\pi^{2}}\Omega^{2}v^{2}\tau^{2}.$ (37)

Because $v\equiv l/\tau$, this expression is equivalent to

$\Gamma_{sub}\approx{e^{2}\over 3\pi^{2}}\Omega^{2}l^{2}\equiv{e^{2}\over
3\pi^{2}}\left({l\over l_{0}}\right)^{2},$ (38)

where we have used the fundamental length scale $l_{0}\equiv 1/\Omega$ defined
by our ultraviolet cutoff.

The result (38) is then that the purely subleading contributions to
$\Gamma_{dressed}$ have a remarkable form: they only depend upon the strength
of the coupling $e$ of the particle to the electromagnetic field, and on the
size of the interferometer $l$, measured in units of $l_{0}$, the size of a
“pixel” in the implicit coarse-graining of space-time implied by our use of an
ultraviolet cutoff. Unlike the expression (33) for the total amount of
decoherence, this purely sub-leading soft contribution does not depend
directly upon the particle’s velocity $v$ or on the time $\tau$ taken to carry
out the experiment.

##  VI Discussion

Interpreting the result (38) is mostly straightforward. The full
electromagnetic field causes an amount of decoherence given by (33). The
amount of this decoherence attributable solely to the difference between the
sub-leading soft radiation emitted by each branch of the matter superposition
is given by (38), which shows that the sub-leading soft contribution grows
quadratically in $l$, the size of the interferometer. This makes sense,
because sub-leading soft photons are by definition excitations of the
electromagnetic field with a very long wavelength, and are capable of
resolving only large-scale details of the matter system. The interpretation of
(38) is then: the larger the matter system, the more capable sub-leading soft
photons are of making quantum measurements of it. For larger experiment
geometries, more of the quantum information broadcast by the charged particle
then ends up being stored in sub-leading soft modes, relative to the hard
modes. Keep in mind of course that we have obtained our results using the
dipole approximation, in which we assume that the wavelength of any emitted
radiation is still much larger than the size of the experiment. It would be
interesting to go beyond this approximation in future work.

It is also interesting to reflect upon the different roles played in our model
by the leading and sub-leading soft photon modes. Leading soft radiation was
completely removed from our discussion by the incorporation of infrared
dressing, meaning that leading soft photons do not carry quantum information
about the matter. Sub-leading soft photons in contrast do carry quantum
information away from the charged particle, causing an amount of decoherence
given by (38).

We could, if we wanted, also consider “dressing away” the sub-leading modes
[44]. In our semiclassical model we could try adding sub-leading dressing in
the same way as we dressed away the leading modes, by setting $j_{sub}^{a}$ to
zero everywhere. However, although we can do this, it isn’t clear why we would
need to; infrared divergences appear to mandate that we dress away
$j_{div}^{a}$, but the contribution of $j_{sub}^{a}$ is perfectly well-
behaved. Furthermore the soft currents $j_{div}^{a}$ and $j_{sub}^{a}$ have
been shown [37] to encode the leading and sub-leading soft photon theorems at
tree level. While the leading soft photon theorem is exact at tree level, the
sub-leading soft photon theorem in general receives loop corrections [47, 48,
49, 50]. It then seems prudent to explore quantum corrections to our simple
semiclassical model, if we hope to fully understand whether any sub-leading
dressing is required, and what form it might take.

##  VII Acknowledgements

We thank J. Wilson-Gerow for discussions of this work, which was supported by
NSERC in Canada.

##  Appendix A Dressing, in Detail

In this appendix, we will illustrate the usual “dressed” formalism [25, 30,
26, 27, 28, 29] using a scattering amplitude involving only a single charged
particle in QED. This simple example suffices for our needs, since our
interferometry model too involves only a single charged particle. We then show
how the dressing works in our finite-time model, proving that a minimal form
of infrared dressing can be accomplished simply by setting the divergent soft
current $j_{div}^{a}$ to zero everywhere.

###  A.1 Dressed Scattering

In textbook QED, the scattering amplitude for a charged particle to transition
from the momentum eigenstate $|p_{i}\rangle$ to the momentum eigenstate
$|p_{f}\rangle$, while the photon field transitions from $|\alpha\rangle$ to
$|\beta\rangle$ is

$\langle p_{f},\beta|\hat{S}|p_{i},\alpha\rangle,$ (39)

where $\hat{S}$ is the QED S-matrix. Applying infrared dressing to this
formalism amounts to making the replacement

$\langle p_{f},\beta|\hat{S}|p_{i},\alpha\rangle\rightarrow\langle
p_{f},\beta|e^{-\hat{R}_{f}}\,\hat{S}\,e^{\hat{R}_{i}}|p_{i},\alpha\rangle,$
(40)

where the “dressing operator,” which acts on the photon Hilbert space, is

$\hat{R}_{i/f}\equiv\sum_{h=+,-}\int\frac{d^{3}q}{(2\pi)^{3}\sqrt{2\omega}}\left[F^{h}_{i/f}(q)\hat{a}^{\dagger}_{h}(q)-\bar{F}^{h}_{i/f}(q)\hat{a}_{h}(q)\right],$
(41)

in which $\hat{a}^{\dagger}_{h}(q)$ and $\hat{a}_{h}(q)$ respectively create
and annihilate a photon with momentum $q^{a}\equiv\omega(1,\hat{n})$ and
helicity $h$, and obey

$\left[\hat{a}_{h}(q),\hat{a}^{\dagger}_{h^{\prime}}(q^{\prime})\right]=\delta_{hh^{\prime}}(2\pi)^{3}\delta^{(3)}(\vec{q}-\vec{q}^{\prime}).$
(42)

We have also defined

$F^{h}_{i/f}(q)\equiv e\frac{\epsilon_{h}(q)\cdot p_{i/f}}{q\cdot
p_{i/f}}\,\phi(q,p_{i/f}),$ (43)

with $\epsilon_{h}^{a}$ the polarization vector corresponding to the helicity
$h$, and $\phi(q,p)$ is a mostly arbitrary function, which must go smoothly to
$1$ as $|\vec{q}|\rightarrow 0$ in order for the dressing to properly remove
the infrared divergence we saw in the main text.

To remove the divergence, it suffices to choose $\phi(q,p)$ nonzero only in a
neighborhood of $|\vec{q}|=0$, where the divergence occurs. Such a choice
would however necessitate the introduction of an arbitrary new energy scale
“$\Lambda$” to our model, in order to characterize the size of the momentum
neighborhood involved in the dressing. There is no physical energy scale in
our interferometry model with which we could identify $\Lambda$, so instead we
make a more conservative choice involving no new parameters at all, simply
letting

$\phi(q,p)=1,$ (44)

a choice which has been referred to elsewhere as “minimal” dressing [51]. With
this choice, the frequency integral in (41) should be understood as extending
from the infrared cutoff $\lambda$ up to the ultraviolet cutoff $\Omega$, with
$\lambda$ eventually being taken to zero. We will use this choice of
$\phi(q,p)$ in what follows, and we will see that the minimal dressing has a
very clean interpretation in terms of the infrared divergent soft matter
current $j_{div}^{a}$, making it a natural choice in light of the deep
interplay between the soft currents and infrared radiation [37].

Evolving states using the dressed S-matrix
$e^{-\hat{R}_{f}}\,\hat{S}\,e^{\hat{R}_{i}}$ instead of the bare S-matrix
$\hat{S}$ allows one to avoid unphysical infrared divergent decoherence rates.
This has been shown already in scattering calculations [31], and we will now
show how it works in our finite-time interferometry model. In particular we
want to dress the interaction picture time evolution operator, which we used
in eq. (3), as

$\begin{split}\mathcal{T}e^{-i\int d^{4}x\;j^{a}(x)\hat{A}_{a}(x)}\\\
\rightarrow e^{-\hat{R}_{f}}\,&\left[\mathcal{T}e^{-i\int
d^{4}x\;j^{a}(x)\hat{A}_{a}(x)}\right]\,e^{\hat{R}_{i}},\end{split}$ (45)

with $\hat{R}_{i/f}$ given by (41), but now with the factors $F_{i/f}^{h}(q)$
of (43) evaluated at the beginning and end of the interferometry experiment
(at proper times $s_{i/f}$), rather than at the asymptotic times
$t\rightarrow\pm\infty$ relevant in scattering scenarios. We will now show
that in order to effect minimal dressing, using (44), we need only set the
divergent part $j_{div}^{a}$ of the electromagnetic current $j^{a}$ appearing
here to zero. That is, we will show that

$\begin{split}e^{-\hat{R}_{f}}\,&\left[\mathcal{T}e^{-i\int
d^{4}x\;j^{a}(x)\hat{A}_{a}(x)}\right]\,e^{\hat{R}_{i}}\\\
&=\mathcal{T}e^{-i\int
d^{4}x\;[j^{a}-j_{div}^{a}](x)\hat{A}_{a}(x)}.\end{split}$ (46)

The mechanism by which minimal infrared dressing removes infrared divergences
is thus the wholesale decoupling of the Maxwell field from the infrared
divergent soft current $j_{div}^{a}$.

###  A.2 Minimal Dressing Removes the Divergent Soft Current

To prove eq. (46), let us begin with the second line and work our way
backward. The second term in the exponent there is

$i\int d^{4}x\;j_{div}^{a}(x)\hat{A}_{a}(x),$ (47)

with $j_{div}^{a}$ defined by eq. (18). Using the usual mode expansion of
$\hat{A}_{a}(x)$,

$\hat{A}_{a}(x)=\sum_{h=+,-}\int\frac{d^{3}q}{(2\pi)^{3}\sqrt{2\omega}}[\epsilon^{h}_{a}\hat{a}^{\dagger}_{h}(q)e^{iq\cdot
x}+\bar{\epsilon}^{h}_{a}\hat{a}_{h}(q)e^{-iq\cdot x}],$ (48)

we can rewrite (47) as

$\displaystyle i\int
d^{4}x\;j_{div}^{a}(x)\sum_{h=+,-}\int\frac{d^{3}q}{(2\pi)^{3}\sqrt{2\omega}}[\epsilon^{h}_{a}\hat{a}^{\dagger}_{h}e^{iq\cdot
x}+\bar{\epsilon}^{h}_{a}\hat{a}_{h}e^{-iq\cdot x}]$
$\displaystyle=i\sum_{h=+,-}\int\frac{d^{3}q}{(2\pi)^{3}\sqrt{2\omega}}[\epsilon^{h}_{a}\hat{a}^{\dagger}_{h}j_{div}^{a}(q)+\bar{\epsilon}^{h}_{a}\hat{a}_{h}j_{div}^{a}(-q)]$
(49)
$\displaystyle=\sum_{h=+,-}\int\frac{d^{3}q}{(2\pi)^{3}\sqrt{2\omega}}[\bar{\epsilon}^{h}_{a}\hat{a}_{h}-\epsilon^{h}_{a}\hat{a}^{\dagger}_{h}]\frac{\dot{X}^{a}(s_{f})}{q\cdot\dot{X}(s_{f})}-(s_{f}\rightarrow
s_{i}).$

Comparing to eqs. (41) and (43), this is nothing but

$-\hat{R}_{f}+\hat{R}_{i},$ (50)

with $p^{a}_{i/f}=m\dot{X}^{a}(s_{i/f})$ and the choice of minimal dressing
$\phi(q,p)=1$.

This result allows us to conclude that

$\begin{split}&\mathcal{T}e^{-i\int
d^{4}x\;[j^{a}-j_{div}^{a}](x)\hat{A}_{a}(x)}\\\ =\;&\mathcal{T}e^{-i\int
d^{4}x\;j^{a}(x)\hat{A}_{a}(x)-\hat{R}_{f}+\hat{R}_{i}}.\end{split}$ (51)

We can rewrite this using the form (2) of the position space current
$j^{a}(x)$ as

$\mathcal{T}e^{-ie\int_{s_{i}}^{s_{f}}ds\;\dot{X}^{a}(s)\hat{A}_{a}(X(s))-\hat{R}_{f}+\hat{R}_{i}},$
(52)

in order to see explicitly that the time ordering of operators using the
coordinate time $t$ gives the same result as ordering operators using the
proper time $s$. Because the dressing operators here act at the beginning and
end of the experiment – i.e. $\hat{R}_{i/f}$ contains a contribution only when
the proper time is $s_{i/f}$ – we can thus pull $\hat{R}_{f}$ out from under
the time ordering operator $\mathcal{T}$ to the left, and we can pull
$\hat{R}_{i}$ out to the right, leaving

$e^{-\hat{R}_{f}}\,\left[\mathcal{T}e^{-ie\int_{s_{i}}^{s_{f}}ds\;\dot{X}^{a}(s)\hat{A}_{a}(X(s))}\right]\,e^{\hat{R}_{i}},$
(53)

or equivalently

$e^{-\hat{R}_{f}}\,\left[\mathcal{T}e^{-i\int
d^{4}x\;j^{a}(x)\hat{A}_{a}(x)}\right]\,e^{\hat{R}_{i}},$ (54)

proving eq. (46).

##  Appendix B Integrals

This appendix includes further detail on the evaluation of the dressed
decoherence functional $\Gamma_{dressed}$, and the sub-leading soft
contribution $\Gamma_{sub}$ to this, which are defined in the main text. Our
approach to these calculations is similar to that of Breuer & Petruccione [9].

###  B.1 The Dressed Decoherence Functional

First write $\Gamma_{dressed}$ (eq. (32)) as

$\Gamma_{dressed}\equiv{2e^{2}\over(2\pi)^{3}}\mathcal{I}_{\omega}\mathcal{I}_{\hat{n}}$
(55)

with

$\mathcal{I}_{\omega}\equiv\int_{0}^{\Omega}{d\omega}\,\frac{\left(1-\cos{\omega\tau}\right)}{\omega}$
(56)

and

$\displaystyle\mathcal{I}_{\hat{n}}\equiv$ (57) $\displaystyle\oint
dS^{2}(\hat{n})\,\omega^{2}\,\left[2\frac{\dot{X}_{1}\cdot\dot{X}_{2}}{(q\cdot\dot{X}_{1})(q\cdot\dot{X}_{2})}-\frac{1}{(q\cdot\dot{X}_{1})^{2}}-\frac{1}{(q\cdot\dot{X}_{2})^{2}}\right].$

We will evaluate each of these integrals one at a time.

The frequency integral $\mathcal{I}_{\omega}$ can be written as (using
$\varpi\equiv\omega\tau$)

$\int_{0}^{\Omega\tau}{d\varpi}\frac{(1-\cos{\varpi})}{\varpi}.$ (58)

We can then use the following identity [52] involving the cosine integral,

$\begin{split}\mathrm{Ci}(\Omega\tau)&\equiv-\int_{\Omega\tau}^{\infty}d\varpi{\cos\varpi\over\varpi}\\\
&=\gamma_{EM}+\ln{\Omega\tau}-\int_{0}^{\Omega\tau}d\varpi{(1-\cos\varpi)\over\varpi},\end{split}$
(59)

where $\gamma_{EM}\approx 0.577$ is the Euler-Mascheroni constant. from which
it follows that $\mathcal{I}_{\omega}$ is

$\int_{0}^{\Omega\tau}d\varpi{(1-\cos\varpi)\over\varpi}=\gamma+\ln{\Omega\tau}-\mathrm{Ci}(\Omega\tau).$
(60)

In the limit $\Omega\tau\gg 1$ discussed in the main text, the logarithm
dominates this expression, giving the approximate result

$\mathcal{I}_{\omega}\approx\ln{\Omega\tau}.$ (61)

The angular integral $\mathcal{I}_{\hat{n}}$ is a sum of three terms with
integrals of the form

$\begin{split}&\oint
dS^{2}(\hat{n})\,\omega^{2}\frac{\dot{X}_{A}\cdot\dot{X}_{B}}{(q\cdot\dot{X}_{A})(q\cdot\dot{X}_{B})}\\\
=&\oint
dS^{2}(\hat{n})\,\frac{(1-\vec{v}_{A}\cdot\vec{v}_{B})}{(1-\hat{n}\cdot\vec{v}_{A})(1-\hat{n}\cdot\vec{v}_{B})},\end{split}$
(62)

with $A,B\in\\{1,2\\}$. The Lorentz-invariance of this integral allows us to
evaluate it in any reference frame. For convenience we can choose the frame in
which $\vec{v}_{B}=0$, bringing $\mathcal{I}_{\hat{n}}$ into the simple form

$\oint dS^{2}(\hat{n})\,\frac{1}{(1-\hat{n}\cdot\vec{v}_{AB})}={4\pi\over
v_{AB}}\tanh^{-1}v_{AB},$ (63)

where $v_{AB}$ is the relative velocity,

$v_{AB}\equiv\sqrt{1-{1\over(\dot{X}_{A}\cdot\dot{X}_{B})^{2}}}.$ (64)

We can use the result (63) to see that the angular integral (57) is

$\mathcal{I}_{\hat{n}}=8\pi\left[{1\over v_{12}}\tanh^{-1}v_{12}-1\right].$
(65)

In the geometry illustrated in Figure 2, the relative velocity $v_{12}$ is
approximately

$v_{12}\approx\sqrt{2}v,$ (66)

in the case of non-relativistic speeds, $v\ll 1$,where again $v\equiv l/\tau$.
We can further approximate

${1\over v_{12}}\tanh^{-1}{v_{12}}=1+{1\over 3}v_{12}^{2}+\dots\approx
1+{2\over 3}v^{2}$ (67)

In this limit, the angular integral is

$\mathcal{I}_{\hat{n}}\approx{16\pi\over 3}v^{2}.$ (68)

Plugging eq. (61) and eq. (68) into eq. (55) yields eq. (33),

$\Gamma_{dressed}\approx{4e^{2}v^{2}\over 3\pi^{2}}\ln\Omega\tau.$ (69)

###  B.2 The Sub-Leading Soft Contribution to the Decoherence Functional

The sub-leading soft part of the dressed decoherence functional, given by
equation (V.2), can also be written in the form of a frequency integral
multiplied by an angular integral:

$\Gamma_{sub}\equiv{e^{2}\over(2\pi)^{3}}\mathcal{I}^{\prime}_{\omega}\mathcal{I}_{\hat{n}}$
(70)

Here, the angular integral $\mathcal{I}_{\hat{n}}$ is the same one we have
already evaluated, so we can simply use the result (68). The frequency
integral is different, however; we have

$\mathcal{I}^{\prime}_{\omega}\equiv\int_{0}^{\Omega}{d\omega}\;\omega\tau^{2}=\frac{1}{2}\Omega^{2}\tau^{2}.$
(71)

Plugging these results into (70) gives the result (37)

$\Gamma_{sub}\approx{e^{2}\over 3\pi^{2}}\Omega^{2}v^{2}\tau^{2}.$ (72)

## References

  * [1] R.P. Feynman, R.B. Leighgton, and M. Sands, The Feynman Lectures on Physics, vol. 3, Addison-Wesley, (1965).
  * [2] To rigorously divide the matter paths into discrete classes requires a little work – see A. Auerbach, S. Kivelson, and D. Nicole, Path Decomposition for Multidimensional Tunneling, PRL 53, 411 (1984); A. Auerbach, S. Kivelson, The path decomposition expansion and multidimensional tunneling, Nucl. Phys B 257, 799 (1985), for non-relativistic discussions appropriate to, e.g., a 2-path system.
  * [3] E. Joos and H.D. Zeh, The emergence of classical properties through interactions with the environment, Zeitschrift für Physik, B59, 223 (1985).
  * [4] E. Joos et al., Decoherence and the appearance of a classical world in quantum theory (2nd ed.), Springer (2003)
  * [5] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
  * [6] P.C.E. Stamp, The decoherence puzzle, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 37, 467 (2006).
  * [7] A. Stern, Y. Aharonov, and Y. Imry, Phase uncertainty and loss of interference: A general picture, Phys. Rev. A41, 3436 (1990).
  * [8] L.H. Ford, Electromagnetic vacuum fluctuations and electron coherence, Phys. Rev. D 47, 5571 (1993).
  * [9] H.-P. Breuer and F. Petruccione, Destruction of quantum coherence through emission of bremsstrahlung, Phys. Rev. A 63, 032102 (2001).
  * [10] see, eg., K. Hornberger, J. E. Sipe, and M. Arndt, Theory of decoherence in a matter wave Talbot-Lau interferometer, Phys. Rev. A 70, 053608 (2004).
  * [11] N.V. Prokof’ev and P.C.E. Stamp, Theory of the Spin Bath, Rep. Prog. Phys. 63, 669 (2000).
  * [12] F. Bloch and A. Nordsieck, Note on the radiation field of the electron, Phys. Rev. 52 (1937) 54
  * [13] V. V. Sudakov, Vertex Parts at Very High Energies in Quantum Electrodynamics, Sov. Phys. J.E.T.P. 3, 65 (1956).
  * [14] A.A. Abrikosov, The Infrared Catastrophe in Quantum Electrodynamics, Sov. Phys. J.E.T.P. 3, 71 (1956).
  * [15] T. Murota, On radiative corrections due to soft photons, Prog. Th. Phys. 24, 1109 (1960).
  * [16] T. Kinoshita, Note on the infrared catastrophe, Prog. Theor. Phys. 5, 1045 (1950)
  * [17] N. Nakanishi, General theory of infrared divergence, Prog. Theor. Phys. 19, 159 (1958)
  * [18] D.R. Yennie, S.C. Frautschi, and H. Suura, The infrared divergence phenomenon and high-energy processes, Ann. Phys. 13, 379 (1961).
  * [19] T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3, 650 (1962)
  * [20] G. Grammer and D.R. Yennie, improved treatment for the infrared divergence problem in Quantum Electrodynamics, Phys. Rev. D 8, 4332 (1973).
  * [21] S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140, B516 (1965).
  * [22] E.S. Fradkin, Application of functional methods in quantum field theory and quantum statistics. (I). Divergence-free field theory with local non-linear interaction, Nucl. Phys. 49, 624 (1963)
  * [23] E.S. Fradkin, Application of functional methods in quantum field theory and quantum statistics. (II), Nucl. Phys. 76, 588 (1966)
  * [24] J. Wilson-Gerow, C. DeLisle and P.C.E. Stamp, A functional approach to soft graviton scattering and BMS charges, Class. Quant. Grav. 35, 164001 (2018)
  * [25] V. Chung, Infrared Divergence in Quantum Electrodynamics, Phys. Rev. 140, B1110 (1965).
  * [26] T.W.B. Kibble, Coherent soft-photon states and infrared divergences. I. classical currents, Lect. Theor. Phys. D11, 387 (1969).
  * [27] T.W.B. Kibble, Coherent soft-photon states and infrared divergences. II. mass shell singularities of Green’s functions, Phys. Rev. 173, 1527 (1968).
  * [28] T.W.B. Kibble, Coherent soft-photon states and infrared divergences. III. asymptotic states and reduction formulas, Phys. Rev. 174, 1882 (1968).
  * [29] T.W.B. Kibble, Coherent soft-photon states and infrared divergences. IV. the scattering operator, Phys. Rev. 175, 1624 (1968).
  * [30] P.P. Kulish and L.D. Faddeev, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys. 4, 745 (1970)
  * [31] D. Carney, L. Chaurette, D. Neuenfeld, and G. Semenoff, On the need for soft dressing, JHEP 2018, 121, (2018).
  * [32] G.W. Semenoff, Entanglement and the Infrared, Dobrev, V. (eds) Lie Theory and Its Applications in Physics, LT 2019, Springer Proceedings in Mathematics & Statistics, vol 335, Springer, Singapore (2020).
  * [33] F.E. Low, Scattering of light of very low frequency by systems of spin 1/2, Phys. Rev. 96, 1428 (1954).
  * [34] F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110, 974 (1958).
  * [35] T.H. Burnett and N.M. Kroll, Extension of the Low soft photon theorem, Phys. Rev. Lett. 20, 86 (1968).
  * [36] M. Gell-Mann and M.L. Goldberger, Scattering of low-energy photons by particles of spin 1/2, Phys. Rev. 96, 1433 (1954).
  * [37] C. DeLisle, J. Wilson-Gerow, and P.C.E. Stamp, Soft theorems from boundary terms in the classical point particle currents, JHEP 2021, 290 (2021).
  * [38] G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics, Cambridge University Press (2010).
  * [39] Decoherence functionals are described in, eg., A. Schmid, J. Low Temp. Phys. 49, 609 (1982).
  * [40] R.P. Feynman and F.L. Vernon, Theory of a general quantum system interacting with a linear dissipative system, Ann. Phys. (NY) 24, 118 (1963).
  * [41] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, Dover (1965).
  * [42] V. Dinu, T. Heinzl, and A. Ilderton, Infrared divergences in plane wave backgrounds, Phys. Rev. D 86, 085037 (2012).
  * [43] A. Laddha and A. Sen, Logarithmic Terms in the Soft Expansion in Four Dimensions, JHEP 2018, 56 (2018).
  * [44] S. Choi and R. Akhoury, Subleading soft dressings of asymptotic states in QED and perturbative quantum gravity, JHEP 2019, 31 (2019).
  * [45] D. Neuenfeld, Infrared-safe scattering without photon vacuum transitions and time-dependent decoherence, JHEP 2021, 189 (2021).
  * [46] J.D. Jackson, Classical Electrodynamics, 3rd ed., Wiley (1999).
  * [47] V. Del Duca, High-energy Bremsstrahlung Theorems for Soft Photons, Nucl. Phys. B 345, 369 (1990).
  * [48] Z. Bern, S. Davies, and J. Nohle, On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons, Phys. Rev. D 90, 085015 (2014).
  * [49] S. He, Y.-t. Huang, and C. Wen, Loop Corrections to Soft Theorems in Gauge Theories and Gravity, JHEP 2014, 115 (2014).
  * [50] B. Sahoo and A. Sen, Classical and Quantum Results on Logarithmic Terms in the Soft Theorem in Four Dimensions, JHEP 2019, 86 (2019).
  * [51] D. Kapec, M. Perry, A.-M. Raclariu, and A. Strominger, Infrared Divergences in QED, Revisited, Phys. Rev. D 96, 085002, (2017).
  * [52] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York (1980).


# Fermat’s Last theorem and its another proof

Kim YangGon, Kim SooGon,Kim SeungKon, Kim ChangKon  emeritus professor,
Department of Mathematics, Jeonbuk National University, 567 Baekje-daero,
Deokjin-gu, Jeonju-si, Jeollabuk-do, 54896, Republic of Korea.
<EMAIL_ADDRESS><EMAIL_ADDRESS>former president of Jeonbuk
National University.  \-  emeritus professor, Department of Physics, Jeonbuk
National University<EMAIL_ADDRESS>doctor of the clinic ’Kim AnGwa’, Iksan
city, Jeollabuk-do<EMAIL_ADDRESS>
###### Abstract.

We announce here that Fermat’s Last theorem was solved, but there is an easy
proof of it on the basis of elemetary undergraduate mathematics. We shall
disclose such an easy proof and discuss more about it.

###### 2010 Mathematics Subject Classification:

Primary 12E99,13-02, secondary 11T99

##  1\. introduction

It is a well known fact that Fermat’s Last theorem was solved by Andrew
Wiles(1953.4.11 - ), but his paper is lengthy and hard. So probably many
mathematicians still want an easy one even for undergraduates or civilians to
understand without difficulty.

This paper could be a sort of an answer to their request. Hence it is hoped
that this paper could be really helpful for them.

Of course there could exist papers easier than this paper,

for example the readers might surf on the arxiv. org to view many uptodate
papers including even 1-page paper proving the Fermat Last Theorem.

But we found out therein some serious flaws or gaps before reaching the final
goal.

In this paper we shall proceed in the following order.

First, we are going to explain what the Fermat’s Last theorem is.

Secondly, we shall give an easy solution even for undergraduates or ordinary
people to comprehend.

Finally we finish this paper with a concluding remark dealing with some
Diophantine problem which is equivalent to the Fermat’s Last theorem .

##  2\. What is the Fermat’s Last theorem?

Let us consider the quadratic equation $X^{2}+Y^{2}=Z^{2}$ defined in the set
of natural numbers. It has a solution $X=3,Y=4,Z=5$ as an example. As is well
known,this equation is obtained from the Pythagorus theorem.

Here we ask a question. What about the equation

(2.1) $X^{n}+Y^{n}=Z^{n}$

for any integer value $n>2$ defined in the ring of integers $\mathbb{Z}$?

Does it have a solution at all with $X\neq 0,Y\neq 0,Z\neq 0?$ The answer is
’No’ according to the French mathematician Pierre de Fermat(1601.8.17-
1665.1.12).

But his proof was not given except that he said

” I have discovered a truly remarkable proof of this theorem which this margin
is too small to contain.”

In 1995 the first proof was given by Andrew Wiles. His proof uses the
Taniyama-Shimura conjecture relating to elliptic curves and modular forms of
them, and his paper is thought to be hard and long.

We intend to give a short and easy proof for this problem by making use of
reduction to absurdity.

It suffices to consider only positive integer solutions for this equation.

We may also assume that $n$ is any fixed odd prime number because the exponent
has a unique factorization of prime numbers and $X^{4}+Y^{4}=Z^{4}$ was solved
by Ferma himself. Furthermore we may assume that $X,Y$, and $Z$ are relatively
prime .

If $n$ is a prime number , then one of $X$ and $Y$ cannot be a multiple of $n$
by our assumption.

Suppose that we have a solution for the equation (2.1). If we divide both
sides by $X^{n}$ which may be assumed to be no multiple of $n$ without loss of
generality,we get the equation of the form $1+(Y/X)^{n}=(Z/X)^{n}$.

Now if we put

$A=Y/X,1+B=Z/X$, where $A$ and $B$ are rational numbers in the field
$\mathbb{Q}$, then there arises an equation of the form

(2.2) $1+A^{n}=(1+B)^{n}$,

where $A$ and $B$ must be positive rational numbers. Of course $A$ must be
exactly greater than $B$ for the equality of (2.2) to hold,i.e., $A$$>$$B>0$.

The converse also holds, i.e.,if there is a solution of (2.2) in $\mathbb{Q}$
, then there must be a solution of integers for the equation (2.1).

There are infinitely many solutions for $n=1,2$.

For $n=1$, there are obviously many solutions.

For $n=2$, it says nothing but the Pythagorean theorem ,and so we consider the
equation of the form $1+A^{n}=(A+\alpha)^{n}$. We then have
$1+A^{2}=A^{2}+2A\alpha+\alpha^{2}$, so that we may get solutions
$A=(1-\alpha^{2})/2\alpha$ for any rational number $\alpha$.

For $n\geq 3$, however, there is no solution at all. We shall prove this fact
in the next section 3 .

##  3\. Proof of the Fermat’s last theorem

Now we assume henceforth once and for all that there is a solution in
$\mathbb{Q}$ for the equation

(3.1) $1+A^{n}=(1+B)^{n}=B^{n}+\binom{n}{1}B^{n-1}+\cdots+\binom{n}{n-1}B+1$,

where $A$ and $B$ are positive rational numbers as manifested just above.

######  Theorem 3.1.

We have no solution of the designated equation (2.1).

###### Proof.

We note first thing that the solution of (2.1), if any, may be obtained
roughly from the congruence equation

(3.2) $1+A^{n}\equiv(1+A)^{n}\equiv 1+A\equiv(1+B)^{n}\equiv 1+B^{n}\equiv
1+B$ modulo $(n)$

as follows.

(3.3) $A=$$t\over s$$,B=(bt+ln)/(bs+kn)$ for some integers $b,l,k,s,$ and $t$
because $A^{n}\equiv A$ and $B^{n}\equiv B$ in the multiplicative group
$\mathbb{Z}_{n}^{\times}$ and so in $\mathbb{Z}_{n}$.

In particular we may assume without loss of generality that $b,s,t$ are
positive integers.We consider $b=1$ at first.

Here the fractions of $A$ and $B$ are assumed to be irreducible ones and we
would like to know the relationship of these in (3.3) with $n$ in more detail.

As a matter of fact we are dealing here with the localization
$S^{-1}\mathbb{Z}$ of the ring of integers $\mathbb{Z}$ at a prime ideal
$(n)$, where $S$ denotes a multiplicative set $\mathbb{Z}-(n)$.

At the same time we may consider by the universal mapping property the
composite map: $\mathbb{Z}\rightarrow
S^{-1}\mathbb{Z}\rightarrow\mathbb{Z}_{n}$ of ring homomorphisms,where
$\mathbb{Z}_{n}$ denotes the finite field consisting of $n$ numbers
$\\{0,1,2,\cdots,(n-1)\\}$.

From (2.2), we see that $s+nk$ must be a multiple of $s$ and the numerator $t$
of $A$ may happen to be a multiple of the numerator $t+nl$ of $B$, Thus we may
put

(3.4) $v(t+ln)=t,s+kn=ws$

for some integers $v$ and $w$. So we may have that both $t+ln$ and $s+kn$ are
negative integers. We put $c=vw$ for brevity.

On the other hand we have the following equations successively;

$1+A^{n}=(1+$$A\over c$$)^{n}$ $\Rightarrow
c^{n}(1+A^{n})=(c+A)^{n}\Rightarrow
c^{n}\\{1+$$(v(t+ln$)$/s$$)^{n}\\}=\\{c+(v(t+ln)/s)\\}^{n}$ $\Rightarrow
c^{n}\\{s^{n}+(v(t+ln))^{n}\\}=(cs+v(t+ln))^{n}\Rightarrow$ $c|v(t+ln)$ ,which
means that $c$ must divide $v(t+ln)=t$ by (3.4).

It follows that
$B=(t+ln)/(s+kn)=v(t+ln)/v(s+kn)=v(t+ln)/vws=v(t+ln)/cs=t/cs=d/s$,where
$cd=v(t+ln)$ for some positive integer $d$.

Hence we get $s+kn=-s$ and $t+ln=-d$, so that $2s=-kn$ and $t+d=-ln$ are
obtained respectively.

It follows that

(3.5) $n|s$

holds since $n\geq 3$,

which is a contradiction to our assumption that $X=s$ should not be a multiple
of $n$.

We may still have another contradiction granting that (3.5) is right.

If so, then we thus obtain that $s^{n}\neq 1,(s+kn)\neq 1$ along with the fact
$s^{n}\neq(s+kn)^{n}$ ( if $n$ is odd).

But then from (3.1) we get at the equation
$(s+kn)^{n}(s^{n}+t^{n})=s^{n}\\{(s+kn)+(t+ln)\\}^{n}$ by substitution of the
fractions , so that $(-s)^{n}+(-t)^{n}=(-s+t+ln)^{n}$. Hence in the field
$\mathbb{Z}_{n}$ we have $-s-t\equiv-s+t+ln$, which reults in $2t\equiv 0$
modulo $(n)$, and hence

(3.6) $n|t$

because $n\geq 3$.

Note that we should have $k\neq 0,l\neq 0$, $w=-1$ and $v\leq-2$ as integers.
At the same time from the congruence relation (3.2) , we have $c\equiv 1$
modulo $(n)$ as a byproduct.

After all we are led to contradictions (3.5) and (3.6) because we assumed that
the fraction $t/s=A$ is irreducible.

Now we consider the general case $B=(bt+ln)/(bs+kn)$= ($t+$$\ln\over
b$)/($s+$$\ kn\over b$),whose denominator and numerator are negative numbers.

If $s+$$\ kn\over b$= $-s$, then we are led to contradiction likewise as
above.

Otherwise we have an equation of the form
$1+(t/s)^{n}=[1+(d/s^{\prime})]^{n}$, where $s^{\prime}$ is a divisor of $s$ ;
so $s=ks^{\prime}$ for some $k$ with $d$ and $s^{\prime}$ relatively prime.

We may figure out a general proof as follows.

Consider the equality $1=$$[(s^{\prime}+d)/s^{\prime}]^{n}$\-
$(t/s)^{n}$=$[(s^{\prime}+d)/s^{\prime}-(t/s)][(s^{\prime}+d)/s^{\prime})^{n-1}+((s^{\prime}+d)/s^{\prime})^{n-2}(t/s)\cdots+(t/s)^{n-1}]$
and the factorization of the right hand side of this equation gives
$0<(ks^{\prime}+kd-t)/s<1$ and $(s+kd-t)|s$.

If we take the reverse of both sides of the last equation, then we have

(3.7) 1=
$[s/(ks^{\prime}+kd-t)][s^{n-1}/((ks^{\prime}+kd)^{n-1}+(ks^{\prime}+kd)^{n-2}t+\cdots+t^{n-1})]$.
If we assume $(ks^{\prime}+kd-t)\geq 2$, then we meet a contradiction $1\equiv
0$ modulo $(s+kd-t)$ since $(s+kd-t)|s$.

Hence we obtain $s+kd-t=1$, and so $kd\neq 1$. We thus have

(3.8)1= $s^{n}/[(s+kd)^{n-1}+(s+kd)^{n-2}t+\cdots+t^{n-1}]$,

which gives rise to

(3.9)1=$s^{n}/[(1+t)^{n}-t^{n}$

by dint of the identity
$[(1+t)^{n-1}+(1+t)^{n-2}t+\cdots+t^{n-1}][(1+t)-t]=(1+t)^{n}-t^{n}$.

From (3.9) it follows that
$s^{n}=(1+t-kd)^{n}=(!+t)^{n}+n(1+t)^{n-1}(-kd)+\cdots+(-kd)^{n}=(1+t)^{n}-t^{n}$
holds.

Hence we get

(4.0)
$\binom{n}{1}(1+t)^{n-1}(-kd)+\binom{n}{2}(1+t)^{n-2}(-kd)^{2}+\cdots+(-kd)^{n}=-t^{n}$.
Dividing both sides of (4.0) by $-nkd$, we obtain

(4.1)
$(1+t)^{n-1}[\binom{n}{2}/n](1+t)^{n-2}(-kd)+\cdots+[(-kd)^{n-1}/n]=(t/kd)(t^{n-1}/n)$.

Beware of the fact $kd\geq 2$. So if we take both sides modulo some prime
divisor of $d$, we are led to a contradiction

$1\equiv 0$ after all because of the assumption $n\geq 3$.

So we see that the equality $s+kd-t=1$ is impossible,

which implies that there is no solution of (2.1) at all.

Hence we have completely proved in this way the Fermat’s Last theorem.

∎

######  Corollary 3.2.

We have no solution of the designated equation (2.1) even in the field
$\mathbb{Q}$ of rational numbers.

###### Proof.

Immediate consequence of the above theorem. ∎

######  Corollary 3.3.

$(1-a^{n})^{1\over n}$ for any rational number $a$ with $0<a<1$ must be an
irrational number. Likewise $(1+a^{n})^{1\over n}$ for any nonzero rational
number $a$ must be an irrational number.

###### Proof.

If we divide both sides of (2.1) by $Z^{n}$, then we get

(4.2) $x^{n}+y^{n}=1$, where $0\leq x=X/Z\leq 1$ and $0\leq y=Y/Z\leq 1$.

Because there is no solution of rational number for the equation
$x^{n}=1-y^{n}$, the first assertion for $(1-a^{n})^{1\over n}$ is evident on
the one hand.

On the other hand if such a curve on the $xy$-plane met with the line $y=ax$
for any $a\in\mathbb{Q}$, then we would have an equation of the form
$x^{n}(1+a^{n})=1$ , or equivalently $x^{n}=(1+a^{n})^{-1}$.

But then there is no rational solution $x$ of this last equation by virtue of
the theorem 3.1. Hence the latter assertion is also evident.

∎

######  Corollary 3.4.

For any nonzero rational numbers $x,y$, we have $x^{n}+y^{n}\neq x^{n}y^{n}$.

###### Proof.

If we suppose $x^{n}+y^{n}=x^{n}y^{n}$, then we have
$(x^{n}+y^{n})/x^{n}y^{n}=1$ for any nonzero rational numbers $x,y$.

Thus we get $(1/x)^{n}+(1/y)^{n}=1$ ,which is nothing but of the form (4.2).
It is a contradiction.

∎

##  4\. concluding remark

We believe that there may be other ways to solve the Fermat’s Last theorem.
But this paper’s way is considered to be the easiest among them. Furthermore
we might wonder if Fermat himself solved his theorem in this way.

Also we think that most undergraduate students majoring in mathematics might
understand this paper without difficulty.

Moreover It is a marvelous fact that (2.1) has no solution even in the field
$\mathbb{Q}$ of rational numbers, whose fact is just nothing but a corollary
mentioned above of the Fermat’s last theorem.

As is well known, the equation of (2.1) is one thing of many sorts of the so
called ’Diophantine equations’.

Let’s take a look at some Diophantine equation for a moment such as

(4.3) $X^{n}\pm Y^{n}\pm Z^{n}=0$ ,

where $X\neq 0,Y\neq 0,Z\neq 0$.

It is not difficult to see that we can transform this equation into an
equivalent Diophantine equation with (2.1).

Even if we are not luminaries in this area , we could absorb ourselves in
Diophantine equations.

Because the proofs hereby up to now are elementary,we didn’t need to cite
bibliographies much except for [KY] in that it shows sort of localization of a
ring in the background material chapter.

Acknowledgement.

We’d like to give many sincere thanks to all persons whom we are indebted to
or who led us to good prosperity.

## References

  * [KY] Y.G.Kim, S-theory (1), KyungMoon Publishers(2001)