text:
\section{Introduction} \label{sec:intro}

Our subject in this paper is the well-known ``2-path" problem in quantum mechanics \cite{feynman3}, 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 \cite{2class}, 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" \cite{JZ,joos,breuerPet,stamp}; 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 \cite{imry,ford,breuer}. 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 \cite{scatt}, and the latter using a ``spin bath" theory \cite{RPP00}. 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 \cite{blochN37}. Early work clarified the ways in which the leading divergences coming from radiative corrections to scattering processes \cite{blochN37,sudakov56,abrikosov} were cancelled by divergences from virtual soft particles, leading to the derivation of multiplicative  ``soft factors" (see, eg., refs. \cite{konoshita50,nakanishi58,murota60}, and also refs. \cite{yennie61,konoshita62,yennie73,weinberg}) 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. \ref{fig:softTh}), 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
\begin{equation}
\Lambda(p,q) \;=\; K_2(p) \, \left[{\cal S}_{(0)}(q) + {\cal S}_{(1)}(q)\right] \;+\;  \mathcal{O}(|{\bf q}|)
 \label{softTh}
\end{equation}
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 ${\cal S}_{(1)}(q)) \sim \mathcal{O}(1)$.

\begin{figure}[ht]
\centering
\includegraphics[scale=0.25]{Lambda.pdf}
\vspace{-7mm}
\caption{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. (\ref{softTh}) in the text. }
 \label{fig:softTh}
\end{figure}

Later work has shown that this result can also be understood by using either an eikonal analysis \cite{fradkin63,fradkin66,jordan18}, or a coherent state representation for the coupled electron-photon system \cite{chung,kibble1,kibble2,kibble3,kibble4,FK}.  The latter representation, in which the system is represented by ``dressed" electron states, is quite illuminating, for two reasons.

First, it highlights that the na\"ive 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 \cite{gordon1,gordon4}. 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 \cite{low1,low2,BK,GG} (this separation is also done in a very clear way in the eikonal expansion \cite{fradkin63,fradkin66}). 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 \cite{me}. 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 \cite{breuer}. 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.

\subsection{Summary \& Structure of the Paper}

The rest of the paper proceeds as follows. First, in Section \ref{sec:model} 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 \ref{sec:currents} 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 \cite{me}, 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 \ref{sec:dressing} 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 \ref{sec:dressing} 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 \ref{sec:eval} 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 \ref{sec:discussion} 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.

\section{Interferometer Model} \label{sec: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 \cite{aspect}. The spatial trajectories taken by the superposed particle are illustrated in Figure \ref{fig:2path}. 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.

\begin{figure}[ht]
\centering
\begin{tikzpicture} [scale=2]

\coordinate (I) at (-.3,.5);
\coordinate (BS_1) at (.5,.5);
\coordinate (M_L) at (.5, 2.5);
\coordinate (M_R) at (2.5,.5);
\coordinate (F) at (2.5,2.5);

\draw [middlearrow={>},thick,double] (I) -- (BS_1);
\draw [middlearrow={>},very thick] (BS_1) -- node[left, scale=1] {$\dot{X}^a_1$} (M_L);
\draw [middlearrow={>},very thick] [dashed] (BS_1) -- node[above, scale=1] {$\dot{X}^a_2$} (M_R);
\draw [middlearrow={>},very thick] (M_L) -- node[below, scale=1] {$\dot{X}^a_2$} (F);
\draw [middlearrow={>},very thick] [dashed] (M_R) -- node[left, scale=1] {$\dot{X}^a_1$} (F);

\draw [->] (-.5,2) -- (-.1,2) node [right, scale=0.8] (xlabel) {$x$};
\draw [->] (-.5,2) -- (-.5,2.4) node [above, scale=0.8] (ylabel) {$y$};

\draw [<->] ($(BS_1) + (.25,0.08)$) -- node[right, scale=1] {$l$} ($(M_L) + (.25,-.08)$);

\node[circle, draw=teal, fill=lime, scale=.8] (c) at (BS_1){$X^a_i$};
\node[circle, draw=teal, fill=lime, scale=.8] (c) at (M_L){$X^a_L$};
\node[circle, draw=teal, fill=lime, scale=.8] (c) at (M_R){$X^a_R$};

\node [draw=purp_line,
    fill=purp_bg,
    diamond,
    minimum width=1.3cm,
    minimum height=1.3cm
]  (detector) at ($(F) + (0.1,0.1)$){$\mathcal{D}$};

\end{tikzpicture}
\caption{Top-down view of the two-path geometry traversed by the charged particle.}
 \label{fig:2path}
\end{figure}

As the system then evolves, that branch of the superposition following the trajectory $X_L^a(s)$ (the solid line in Figure \ref{fig:2path}) 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. \ref{fig:2path} is schematic in the sense that in reality, a {\it single} path shown going along the left or right trajectory actually represents the set of {\it 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 \cite{feynman3}).

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
\begin{equation} \label{eq:Current}
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) ),
\end{equation}
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:
\begin{equation} \label{eq:coherent_states}
|L/R \rangle \equiv \mathcal{T} e^{-i\int d^4x \; j_{L/R}^a(x) \hat{A}_a(x)} |0\rangle
\end{equation}
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 \eqref{eq:Current} 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 $\int d^4x$ 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
\begin{equation}
\frac{1}{\sqrt{2}} \left[ |\psi_L\rangle |L\rangle + |\psi_R\rangle |R\rangle  \right] ,
\end{equation}
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
\begin{align} \label{eq:prob_dist}
P(x) &= \frac{1}{2} |\psi_L(x)|^2 + \frac{1}{2}|\psi_R(x)|^2 \\
&+ \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)  \nonumber ,
\end{align}
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,
\begin{equation} \label{eq:LR_inner_product}
\langle R | L \rangle \equiv e^{i\Theta - \Gamma}; \hspace{1em} \Theta,\Gamma \in \mathbb{R} , \hspace{1em} \Gamma \ge 0.
\end{equation}
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 \eqref{eq:LR_inner_product} as a functional of these currents,
\begin{equation}
\langle R | L \rangle \equiv \langle R | L \rangle[j_L, j_R].
\end{equation}
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 {\it decoherence functional} \cite{imry,breuerPet,decoF,feynmanV63}. 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
\begin{equation}
|\langle R | L \rangle| = e^{-\Gamma}.
\end{equation}

The electromagnetic decoherence functional was first derived long ago \cite{feynmanV63,FeynmanHibbs}, and has been discussed more recently e.g. by Ford \cite{ford}, and by Breuer and Petruccione \cite{breuer}. At zero temperature it takes the form
\begin{equation} \label{eq:gamma_m}
\Gamma[j_L, j_R] =   {1\over2}\int \frac{d^3q}{(2\pi)^3 2 \omega} \; P_{ab} \, \delta  j^a(q) \delta  j^b(-q),
\end{equation}
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^3q}{(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 \eqref{eq:Current}.

\section{Form of Electromagnetic Currents \& Decoherence Functional} \label{sec:currents}

Let us now further discuss the matter currents we will insert into the decoherence functional \eqref{eq:gamma_m}; 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, $j^a(q) \equiv \int d^4x \, e^{iq\cdot x} j^a(x)$, which using \eqref{eq:Current} is
\begin{equation} \label{eq:j_first_expression}
j^a(q) \;=\;  e \int ds\,\dot{X}^a(s) \, e^{iq\cdot X(s)} .
\end{equation}
This current can be rewritten as
\begin{equation}
j^a(q) \;=\;  e \int ds\, \dot{X}^a \left( \frac{1}{iq\cdot \dot{X}} \right) \frac{d}{ds} e^{iq\cdot X} ,
\end{equation}
so that integrating by parts in $s$ gives
\begin{equation} \label{eq:j_with_bdry_term}
\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}
\end{equation}
Discarding the boundary term appearing here \cite{ilderton,laddha1,me} 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:
\begin{equation} \label{eq:jq_final}
j^a(q) =  ie \int ds\, e^{iq\cdot X} \frac{d}{ds} \left( \frac{\dot{X}^a}{q\cdot \dot{X}} \right)
\end{equation}

Notice that the form \eqref{eq:jq_final} 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 \eqref{eq:gamma_m}, we note immediately the important fact that \textit{there is no decoherence without acceleration.}

\subsection{Soft Electromagnetic Currents}

Recently, we showed \cite{me} that the electromagnetic current of a point particle \eqref{eq:jq_final} 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 \eqref{eq:jq_final} in powers of $q$, we get
\begin{equation} \label{eq:Jboundary}
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).
\end{equation}
If the proper time integral in \eqref{eq:jq_final} runs from $s_i$ to $s_f$, then the $\Delta$ in this expression has the meaning
\begin{equation}
\Delta f(s) \equiv f(s_f) - f(s_i) = \int_{s_i}^{s_f} \partial_s f(s),
\end{equation}
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''
\begin{equation} \label{eq:soft_factor_defn}
\mathcal{S}_{(0)}^a(q,p) \equiv   \frac{p^a}{q\cdot p} ,\hspace{1em}
\mathcal{S}_{(1)}^a(q,x,p) \equiv  i \frac{q_b{J}^{ba}}{q\cdot p} .
\end{equation}
Here, ${J}^{ab} \equiv 2 p^{[a}x^{b]}$ is the angular momentum tensor. Ref. \cite{me} 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 \textit{not} necessarily located at asymptotic coordinate times $t\rightarrow\pm\infty$. They exist and take the form \eqref{eq:Jboundary} for any choice of bounds on the proper time integral in \eqref{eq:jq_final}.

From eq. \eqref{eq:Jboundary} it is then clear that we can always split the point particle current into three pieces,
\begin{equation} \label{eq:soft_current}
j^a(q) \equiv j_{div}^a(q) + j_{sub}^a(q) + j_{hard}^a(q),
\end{equation}
with
\begin{equation} \label{eq:soft_current_parts}
\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}
\end{equation}
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 \textit{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 \ref{fig:currents}.

We note immediately that the leading soft current $j^a_{div}$ is \textit{infrared divergent}, as it goes like $1/\omega$ and blows up as $\omega\rightarrow0$. 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 $\omega\rightarrow0$ limit.  Additionally we note that each of these currents is individually conserved, i.e.,
\begin{equation}
q_a j_{div}^a(q) = q_a j_{sub}^a(q) = q_a j_{hard}^a(q) = 0 .
\end{equation}

The identification of these soft currents will allow us to rather directly assess the contribution to the decoherence functional \eqref{eq:gamma_m} from leading and sub-leading soft photon modes. In particular we will expand the decoherence functional \eqref{eq:gamma_m} in terms of the splitting \eqref{eq:soft_current} to get
\begin{equation} \label{eq:softDeco}
\begin{split}
\Gamma =   {e^2\over2} &\int \frac{d^3q}{(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}
\end{equation}
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. (\ref{eq:gamma_eval_1}) below.

\begin{figure}[t!]
\centering
    \begin{tikzpicture} [scale=2.9]

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    \coordinate (T) at ($(F) - (0,0.2)$);

    \draw [thick, rounded corners, middlearrow={>}] ($(I) + (0,0.2)$) .. controls (-.3,0)  .. ($(T) - (0.01,0.015)$);
    \draw [thick, rounded corners, dotted, middlearrow={>}] ($(I) + (0,0.2)$) .. controls (.3,0)  .. (T) ;

    \draw[snake it,segment aspect=0, segment length=2mm, purple,thick,->] (-.3,.1) -- (-.62,.42);
    \draw[snake it,segment aspect=0,purple, segment length=2mm, thick,dash pattern=on 1pt off .5pt,->] (.3,.1) -- (.62,.42);

    \draw[snake it,segment aspect=0, segment length=4.5mm, orange, thick,dash pattern=on 1pt off .5pt,->] (.2,.3) -- (.46,.56);
    \draw[snake it,segment aspect=0, segment length=4.5mm, thick, orange,->] (-.2,.3) -- (-.46,.56);

    \draw[snake it,segment aspect=0, segment length=7mm, red, thick,dash pattern=on 1pt off .5pt,->] (.1,.5) -- (.3,.7);
    \draw[snake it,segment aspect=0, segment length=7mm, thick, red,->] (-.1,.5) -- (-.3,.7);

    \draw [very thick, rounded corners, red] (I) -- ($(I) + (0,0.2)$);
    \draw [very thick, rounded corners, dotted, red]  (T) -- (FL);
    \draw [very thick, rounded corners, red]  ($(T) + (0.011,0.0165)$) -- (FR);

    \node [circle, draw=black, fill=orange, scale=0.5] at (I) {};
    \node [below] at (I) {$s_i$};
    \node [circle, draw=black, fill=orange, scale=0.5] at (FR) {};
    \node [circle, draw=black, pattern={Lines[angle=-57,distance=1pt]}, pattern color=orange, scale=0.5] at (FL) {};
    \node [above] at (F) {$s_f$};

    \end{tikzpicture}

    \caption{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.}
    \label{fig:currents}
\end{figure}

\subsection{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{equation}
\begin{split}
j^a_L(q) = ie \int ds \, e^{i q \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^{i q\cdot X_R(s)}  \partial_s
\left[\frac{\dot{X}_R^a(s)}{q\cdot \dot{X}_R(s)}\right]  .
\end{split}
\end{equation}
The difference between the two currents on the superposed paths,
\begin{equation}
\delta j^a(q) \equiv j^a_L(q) - j^a_R(q)
\end{equation}
can then be approximated in the geometry illustrated in Figure \ref{fig:2path} as
\begin{equation} \label{eq:dj1}
\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] .
\end{equation}
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
\begin{equation}
v \equiv {l\over \tau} \ll 1
\end{equation}
in units with $c=1$ -- then we can assume that the spatial parts of the phases appearing in \eqref{eq:dj1} can be neglected to leading order. This is just the well-known dipole approximation \cite{jackson}; 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
\begin{equation} \label{eq:dj_approx}
\delta j^a(q) \approx ie  \left( 1 - 2 e^{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]  .
\end{equation}

Plugging this current difference into the decoherence functional gives
\begin{align} \label{eq:fullgamma}
\Gamma &\approx {e^2 \over 4(2\pi)^3} \int_\lambda^\Omega {d\omega\over\omega}  \, \left(5 -  4\cos{\omega\tau}  \right) \times \\
&\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] . \nonumber
\end{align}
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 \cite{breuer}. The length scale $l_{0} \equiv 1 / \Omega$ is then the shortest length scale relevant to our analysis.

The result (\ref{eq:fullgamma}) for $\Gamma$ incorporates all the soft contributions -- what we will now show is that the contribution from the leading divergent contributions is actually zero.

\section{Contribution from Leading Divergent Term} \label{sec:dressing}

In order for the integral \eqref{eq:fullgamma} to be well-defined, we have also been forced to introduce an infrared cutoff $\lambda$. The integrand of \eqref{eq:fullgamma} 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 na\"ively, 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.

\subsection{Infrared Divergences}

Begin by splitting the current \eqref{eq:dj_approx} 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
\begin{align} \label{eq:dj_parts}
\delta j_{div}^a(q) &\approx -ie  \left[\frac{\dot{X}_1^a}{q\cdot \dot{X}_1} - \frac{\dot{X}_2^a}{q\cdot \dot{X}_2}\right]  \nonumber \\
\delta j_{sub}^a(q) &\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]  \\
\delta j_{hard}^a(q) &\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]  \nonumber
\end{align}
in the dipole approximation.

From the general form of the decoherence functional, eq. \eqref{eq:softDeco}, we see that $\Gamma$ contains a term involving only the leading soft current $j_{div}^a$:
\begin{align}
&\Gamma \supset   {1\over2}  \int \frac{d^3q}{(2\pi)^3 2\omega}\; P_{ab} \, \delta  j_{div}^a(q) \delta  j_{div}^b(-q) \nonumber \\
&= {e^2 \over 4(2\pi)^3} \int_\lambda^\Omega {d\omega\over\omega}  \times \\
&\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] . \nonumber
\end{align}
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.

\subsection{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 \ref{fig:2path}. This is obviously wrong: if it were the case, we would have $\Gamma \rightarrow \infty$ for \textit{any} choice of experimental parameters ($e, l, \tau$), and from eq. \eqref{eq:prob_dist} we see that there would \textit{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 \cite{blochN37}. That one needs to properly handle these divergences is also obvious in the eikonal formulation of QED \cite{fradkin63,fradkin66} and was a motivating factor in the coherent state formulation of QED in terms of dressed states \cite{chung,kibble1,kibble2,kibble3,kibble4,FK,gordon1}.
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 \ref{app:dressing} 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 \textit{dressed coherent states} in the same way as we defined the coherent radiation states in eq. \eqref{eq:coherent_states}, but now explicitly subtracting out the divergent current $j^a_{div}$, i.e., we write:
\begin{equation}
||L/R \rangle\rangle \equiv  \mathcal{T} e^{-i \int d^4x  \;  [j_{L/R}^a - j_{L/R,div}^a](x) \hat{A}_a(x)} |0\rangle
\end{equation}

We can then define a \textit{dressed decoherence functional} in the same way we defined $\Gamma$ in the ``undressed'' case, but using the dressed coherent states instead,
\begin{equation}
|\langle\langle R | L \rangle\rangle| \equiv e^{-\Gamma_{dressed}} .
\end{equation}
The dressed decoherence functional $\Gamma_{dressed}$ can be obtained from the undressed decoherence functional \eqref{eq:softDeco} by simply setting the divergent current $j_{div}^a$ to zero, giving
\begin{equation} \label{eq:gamma_dressed}
\begin{split}
\Gamma_{dressed} =  {e^2\over2} &\int \frac{d^3q}{(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}
\end{equation}

The expression \eqref{eq:gamma_dressed} 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.

\section{Sub-Leading Soft Photons} \label{sec:eval}

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 \eqref{eq:gamma_dressed}.

\subsection{True Dressed Decoherence Functional}

We begin by inserting the sub-leading and hard parts of the current difference from \eqref{eq:dj_parts} into \eqref{eq:gamma_dressed} gives
\begin{align} \label{eq:gamma_eval_1}
&\Gamma_{dressed} \approx {2e^2 \over (2\pi)^3} \int_0^\Omega {d\omega\over\omega}  \, \left(1 -  \cos{\omega\tau}  \right) \times \\
&\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] . \nonumber
\end{align}
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. (\ref{eq:gamma_eval_1}) 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 $\omega\rightarrow0$, 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 \eqref{eq:gamma_eval_1}, 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 \eqref{eq:gamma_eval_1} using these approximations in Appendix \ref{app:integrals_dressed}, with the result
\begin{equation} \label{eq:dressed_result}
\Gamma_{dressed} \approx {e^2v^2\over3\pi^2} \ln\Omega\tau .
\end{equation}
which agrees with previous calculations of electromagnetic decoherence \cite{breuer,dominik}.

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.

\subsection{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 (\ref{eq:gamma_eval_1}) for $\Gamma_{dressed}$.

In the general form \eqref{eq:gamma_dressed} 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
\begin{align} \label{eq:sub_gamma}
&\Gamma_{sub} \equiv   {1\over2}  \int \frac{d^3q}{(2\pi)^3 2\omega}\; P_{ab} \, \delta  j_{sub}^a(q) \delta  j_{sub}^b(-q) \nonumber \\
&= †{e^2 \over (2\pi)^3} \int_0^\Omega {d\omega} \; \omega \tau^2  \times \\
&\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] , \nonumber
\end{align}
where we have used the expression for $\delta j_{sub}^a$ found in eq. \eqref{eq:dj_parts}, and where we see that this result is nothing but the lowest order term in the expansion of the integrand of (\ref{eq:gamma_eval_1}) in powers of $\omega \tau$.

The quantity $\Gamma_{sub}$ has an intuitive interpretation. Imagine allowing the electromagnetic field to couple \textit{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
\begin{equation}
|| l/r \rangle\rangle \equiv  \mathcal{T} e^{-i \int d^4x  \;  j_{L/R, sub}^a(x) \hat{A}_a(x)} |0\rangle ,
\end{equation}
and $\Gamma_{sub}$ would encode the modulus of the inner product between these two states as
\begin{equation}
|\langle\langle r | l \rangle\rangle| \equiv e^{-\Gamma_{sub}} .
\end{equation}
$\Gamma_{sub}$ 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 \eqref{eq:sub_gamma} in Appendix \ref{app:integrals_sub}, with the result
\begin{equation} \label{eq:sub_result}
\Gamma_{sub} \approx {e^2 \over 3 \pi^2}\Omega^2 v^2 \tau^2.
\end{equation}
Because $v \equiv l/\tau$, this expression is equivalent to
\begin{equation} \label{eq:sub_result_final}
\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,
\end{equation}
where we have used the fundamental length scale $l_{0} \equiv 1 / \Omega$ defined by our ultraviolet cutoff.

The result \eqref{eq:sub_result_final} 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 \eqref{eq:dressed_result} 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.

\section{Discussion} \label{sec:discussion}

Interpreting the result \eqref{eq:sub_result_final} is mostly straightforward. The full electromagnetic field causes an amount of decoherence given by \eqref{eq:dressed_result}. 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 \eqref{eq:sub_result_final}, 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 \eqref{eq:sub_result_final} 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 \eqref{eq:sub_result_final}.

We could, if we wanted, also consider ``dressing away'' the sub-leading modes \cite{choi}. 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 \textit{can} do this, it isn't clear why we would \textit{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 \cite{me} to encode the leading and sub-leading soft photon theorems \textit{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 \cite{loops1,loops2,loops3,loops4}. 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.

\section{Acknowledgements}
\label{sec:discussion}

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

\vspace{40mm}

\begin{appendix}

\section{Dressing, in Detail}  \label{app:dressing}

In this appendix, we will illustrate the usual ``dressed'' formalism \cite{chung,FK,kibble1,kibble2,kibble3,kibble4} 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.

\subsection{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
\begin{equation}
\langle p_f, \beta | \hat{S} | p_i, \alpha \rangle,
\end{equation}
where $\hat{S}$ is the QED S-matrix. Applying infrared dressing to this formalism amounts to making the replacement
\begin{equation}
\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,
\end{equation}
where the ``dressing operator,'' which acts on the photon Hilbert space, is
\begin{equation} \label{eq:dress_op}
\hat{R}_{i/f} \equiv
 \sum_{h = +, -}  \int \frac{d^3q}{(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]   ,
\end{equation}
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
\begin{equation}
\left[ \hat{a}_h(q),   \hat{a}^\dagger_{h'}(q')  \right]  = \delta_{h h' }(2\pi)^3 \delta^{(3)}(\vec{q} - \vec{q}') .
\end{equation}
We have also defined
\begin{equation} \label{eq:formfactor}
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}) ,
\end{equation}
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
\begin{equation} \label{eq:min_dress}
\phi(q,p) = 1,
\end{equation}
a choice which has been referred to elsewhere as ``minimal'' dressing \cite{kapec}. With this choice, the frequency integral in \eqref{eq:dress_op} 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 \cite{me}.

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 \cite{gordon1}, 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. \eqref{eq:coherent_states}, as
\begin{equation}
\begin{split}
\mathcal{T} e^{-i \int d^4x \; j^a(x) \hat{A}_a(x)} \\
\rightarrow e^{-\hat{R}_{f}}  \, &\left[\mathcal{T} e^{-i \int d^4x \; j^a(x) \hat{A}_a(x)}  \right]  \,e^{\hat{R}_{i}} ,
\end{split}
\end{equation}
with $\hat{R}_{i/f}$ given by \eqref{eq:dress_op}, but now with the factors $F_{i/f}^h(q)$ of \eqref{eq:formfactor} 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 \eqref{eq:min_dress}, 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{equation}  \label{eq:app1}
\begin{split}
e^{-\hat{R}_{f}}  \, &\left[\mathcal{T} e^{-i \int d^4x \; j^a(x) \hat{A}_a(x)}  \right]  \,e^{\hat{R}_{i}}  \\
 &= \mathcal{T} e^{-i \int d^4x  \;  [j^a - j_{div}^a](x) \hat{A}_a(x)} .
\end{split}
\end{equation}
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 $j_{div}^a$.

\subsection{Minimal Dressing Removes the Divergent Soft Current}

To prove eq. \eqref{eq:app1}, let us begin with the second line and work our way backward. The second term in the exponent there is
\begin{equation} \label{eq:app2}
i   \int d^4x \; j_{div}^a(x) \hat{A}_a(x),
\end{equation}
with $j_{div}^a$ defined by eq. \eqref{eq:soft_current_parts}.
Using the usual mode expansion of $\hat{A}_a(x)$,
\begin{equation}
\hat{A}_a(x) = \sum_{h = +,-}  \int \frac{d^3q}{(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}   ],
\end{equation}
we can rewrite \eqref{eq:app2} as
\begin{align}
&i \int d^4x \; j_{div}^a(x) \sum_{h = +,-} \int \frac{d^3q}{(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}   ] \nonumber \\
&= i \sum_{h = +,-} \int \frac{d^3q}{(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)  ]  \\
&= \sum_{h = +,-} \int \frac{d^3q}{(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) \nonumber .
\end{align}
Comparing to eqs. \eqref{eq:dress_op} and \eqref{eq:formfactor}, this is nothing but
\begin{equation}
-\hat{R}_f + \hat{R}_i ,
\end{equation}
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{equation}
\begin{split}
&\mathcal{T} e^{-i  \int d^4x \;  [j^a - j_{div}^a](x) \hat{A}_a(x)} \\
=\; &\mathcal{T} e^{-i  \int d^4x \;  j^a(x) \hat{A}_a(x) - \hat{R}_f + \hat{R}_i}.
\end{split}
\end{equation}
We can rewrite this using the form \eqref{eq:Current} of the position space current $j^a(x)$ as
\begin{equation}
\mathcal{T} e^{-i e \int_{s_i}^{s_f} ds \;  \dot{X}^a(s) \hat{A}_a(X(s)) - \hat{R}_f + \hat{R}_i},
\end{equation}
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
\begin{equation}
e^{-\hat{R}_{f}}  \, \left[ \mathcal{T} e^{-i e \int_{s_i}^{s_f} ds \;  \dot{X}^a(s) \hat{A}_a(X(s)) } \right]  \,e^{\hat{R}_{i}}    ,
\end{equation}
or equivalently
\begin{equation}
e^{-\hat{R}_{f}}  \, \left[\mathcal{T} e^{-i \int d^4x \; j^a(x) \hat{A}_a(x)}  \right]  \,e^{\hat{R}_{i}}  ,
\end{equation}
proving eq. \eqref{eq:app1}.

\section{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 \cite{breuer}.

\subsection{The Dressed Decoherence Functional} \label{app:integrals_dressed}

First write $\Gamma_{dressed}$ (eq. \eqref{eq:gamma_eval_1}) as
\begin{equation} \label{eq:dressed_integrals}
\Gamma_{dressed} \equiv {2e^2 \over (2\pi)^3} \mathcal{I}_{\omega} \mathcal{I}_{\hat{n}}
\end{equation}
with
\begin{equation} \label{eq:Ifreq}
\mathcal{I}_{\omega} \equiv  \int_0^\Omega {d\omega}  \, \frac{\left(1 -  \cos{\omega\tau}  \right)}{\omega}
\end{equation}
and
\begin{align}\label{eq:Iangle}
&\mathcal{I}_{\hat{n}} \equiv \\
&\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]. \nonumber
\end{align}
We will evaluate each of these integrals one at a time.

The frequency integral $\mathcal{I}_\omega$ can be written as (using $\varpi \equiv \omega \tau$)
\begin{equation}
\int_0^{\Omega \tau} {d\varpi } \frac{(1 - \cos{\varpi})}{\varpi}  .
\end{equation}
We can then use the following identity \cite{GandR} involving the cosine integral,
\begin{equation}
\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}
\end{equation}
where $\gamma_{EM} \approx 0.577$ is the Euler-Mascheroni constant.
from which it follows that $\mathcal{I}_\omega$ is
\begin{equation}
 \int_0^{\Omega\tau} d\varpi {(1 - \cos\varpi)  \over \varpi} = \gamma + \ln{\Omega\tau} - \mathrm{Ci}(\Omega\tau) .
\end{equation}
In the limit $\Omega\tau \gg 1$ discussed in the main text, the logarithm dominates this expression, giving the approximate result
\begin{equation} \label{eq:freq_approx}
\mathcal{I}_\omega \approx   \ln{\Omega\tau} .
\end{equation}

The angular integral $\mathcal{I}_{\hat{n}}$ is a sum of three terms with integrals of the form
\begin{equation}
\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}
\end{equation}
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
\begin{equation} \label{chVI:eq:angle_integral}
\oint dS^2(\hat n) \, \frac{1}{(1 - \hat{n} \cdot \vec{v}_{AB})} = {4\pi \over  v_{AB}} \tanh^{-1}v_{AB},
\end{equation}
where $v_{AB}$ is the relative velocity,
\begin{equation}
v_{AB} \equiv \sqrt{1 - {1 \over (\dot{X}_A \cdot \dot{X}_B)^2}}.
\end{equation}
We can use the result \eqref{chVI:eq:angle_integral} to see that the angular integral \eqref{eq:Iangle} is
\begin{equation}
\mathcal{I}_{\hat n} = 8 \pi \left[ {1 \over  v_{12}} \tanh^{-1}v_{12} - 1 \right].
\end{equation}
In the geometry illustrated in Figure \ref{fig:2path}, the relative velocity $v_{12}$ is approximately
\begin{equation} \label{eq:relvel}
v_{12} \approx \sqrt{2} v,
\end{equation}
in the case of non-relativistic speeds, $v\ll1$,where again $v \equiv l / \tau$. We can further approximate
\begin{equation}
{1 \over v_{12}} \tanh^{-1}{v_{12}} = 1 + {1\over 3}v_{12}^2 + \dots  \approx 1 + {2\over 3}v^2
\end{equation}
In this limit, the angular integral is
\begin{equation} \label{eq:angle_approx}
\mathcal{I}_{\hat{n}} \approx {16\pi \over 3}v^2 .
\end{equation}

Plugging eq. \eqref{eq:freq_approx} and eq. \eqref{eq:angle_approx} into eq. \eqref{eq:dressed_integrals} yields eq. \eqref{eq:dressed_result},
\begin{equation}
\Gamma_{dressed} \approx {4e^2v^2\over3\pi^2} \ln\Omega\tau .
\end{equation}

\subsection{The Sub-Leading Soft Contribution to the Decoherence Functional}    \label{app:integrals_sub}

The sub-leading soft part of the dressed decoherence functional, given by equation \eqref{eq:sub_gamma}, can also be written in the form of a frequency integral multiplied by an angular integral:
\begin{equation} \label{eq:sub_integrals}
\Gamma_{sub} \equiv {e^2 \over (2\pi)^3} \mathcal{I}'_{\omega} \mathcal{I}_{\hat{n}}
\end{equation}
Here, the angular integral $\mathcal{I}_{\hat{n}}$ is the same one we have already evaluated, so we can simply use the result \eqref{eq:angle_approx}. The frequency integral is different, however; we have
\begin{equation}
\mathcal{I}'_{\omega} \equiv \int_0^\Omega {d\omega} \; \omega \tau^2 = \frac{1}{2} \Omega^2 \tau^2.
\end{equation}

Plugging these results into \eqref{eq:sub_integrals} gives the result \eqref{eq:sub_result}
\begin{equation}
\Gamma_{sub} \approx {e^2 \over 3 \pi^2}\Omega^2 v^2 \tau^2.
\end{equation}

\end{appendix}

redpajama_set_name: RedPajamaArXiv


TRIPLE HONOUR (g. by Honours List). 7 wins-3 at 2-from 1100m to 2200m, A$2,221,960, AJC Doncaster H., Gr.1, BRC Eagle Farm Cup, Gr.2, STC Premiere S., Gr.2, AJC Sydney Childrens Hospital Foundation Sprint H., STC De Bortoli Wines 2YO H., Retirement Show 2YO H., 2d AJC Randwick Guineas, Gr.1, Queen Elizabeth S., Gr.1, Chelmsford S., Gr.2, Warwick S., Gr.2, STC Hill S., Gr.2, AJC Liverpool City Cup, Gr.3, BRC Members' H., L, Coca-Cola Amatil H., 3d BRC Doomben Cup, Gr.1, ATC TJ Smith S., Gr.1, AJC Chelmsford S., Gr.2, STC Theo Marks H., Gr.2, AJC Craven P., Gr.3, 4th BTC Doomben Cup, Gr.1, VRC Patinack Classic, Gr.1, NSW Tatt's RC Chelmsford S., Gr.2, STC Golden Rose S., Gr.2, WATC CB Cox S., Gr.2.
HEROIC VALOUR (c. by Fastnet Rock). 4 wins. See below.
Enormous Honour (g. by Encosta de Lago). 6 wins from 1400m to 1650m, A$137,770, HK$4,640,850, to 2017-18, ATC Declaration of War at Coolmore H., Testa Rossa at Vinery P., Big Sports Breakfast H., HKJC Hung Hom H., Hibiscus H., Lancashire H., 2d ATC Vinery Yearlings H., HKJC Garden Road H., Fakei Cup, Chevalier Property Development H., Wang Chau H., Kranji H., Wangfujing H., Big Profit H., 3d ATC Membership on Sale P., HKJC JCI Hong Kong Senate Cup H., Lugard H., HKU Faculties of Dentistry & Education H., Serving the Community H., New York H.
Strong Gain (g. by Testa Rossa). 6 wins to 1550m, A$50,375, HK$2,008,700, HKJC Chek Lap Kok H., Kowloon City H., World Bar Conference Cup H., STC Jacob's Creek H., 2d HKJC Kwai Shing H., Songhua Jiang H., Mastermind H., STC Rosehill Gardens Event Centre H., 3d HKJC Victoria Racing Club Trophy H.
Disciple (g. by God's Own). 5 wins from 1200m to 1600m, A$345,130, ATC Blackwoods H., Catholic Trade Training Centres H., Turf Club H., Brad Tonks H., 2d Coffs Harbour Cup, ATC Membership on Sale H., Theraces.com.au H., Nova 969 Fitzy & Wippa H., Gaypak & United Fruit H., Pirtek Fluid Transfer Solutions H., 3d ATC Sydney Markets Filante H., Shoot Out H., Pluck at Vinery H., Membership on Sale Now H., McGrath Estate Agents H., Greg Carmody H., 4th ATC Civic S., L.
Bryony (f. by Fastnet Rock). Winner at 1300m.
Satellite Prince. 6 wins at 1100m, 1600m to 2017, Singapore TC Kranji D S.-twice, Class 5 H.-twice, Onceuponatime S., 3d Singapore TC World Airlines S., Class 4 H., War Affair S., Class 5 H.-twice.
The Bronx. 8 wins to 1100m, A$168,846, to 2016-17, 2d SAJC Qatar Airways H., Mittys H., Mac Drink Driving Grow Up H., 3d SAJC Toll Express H.
Heart of Courage (g. by Fastnet Rock). Placed at 3 in 2017-18 in NZ.
Cinnabon. Winner at 1100m in 2017-18.
Chic Sox. 2 wins at 1000m, 1200m to 2017-18.
Triple Effort. Winner at 1500m.
Astro Castro. 3 wins to 1500m, A$107,300, to 2017-18, VRC Recognition H., 2d MRC Mypunter.com H.
Beer Belly. 3 wins-1 at 2-from 1150m to 1650m, TRC James Boags P., 2d TTC Launceston Skoda H., Carrick P.
Tangier Sun. 2 wins at 1050m, 1200m, SAJC TDR Electrical H., 2d SAJC Drink Driving? Grow Up H., James Boag's H., 3d SAJC Medallion Homes H.
Timber Cat. 2 wins at 1600m, 2d MRC (Mornington) Slickpix H.
Neurotic Princess. Winner at 1500m, 3d VRC Tudor Peak 2YO H.
Absent Friend. Winner at 1350m, BTC Rose of Kingston H.
Race Record: 19 Starts 4-7-3 NZ$351,325, A$25,000.
4 wins-3 at 2-at 1000m, 1200m, NZ$351,325, A$25,000, to 2016-17, ARC Diamond S., Gr.1, Matamata RC Slipper S., L, Whangarei RC Northland Breeders S., L, 2d Gold Coast Guineas, Gr.3, ARC Darley P., Gr.3, Karaka 3YO Mile, RL, Ashburton RC John Grigg S., L, WRC Lightning H., L, ARC Ray White NZ H., John Deere Agrowquip S., 3d New Zealand Two Thousand Guineas, Gr.1, Hawke's Bay Guineas, Gr.2, Counties Bowl H., L, 4th Waikato Sprint, Gr.1, Manawatu RC Flying H., L.


Bioretention systems have been found to be effective for the removal of TSS, COD and heavy metals in rainfall runoff. However, in many cases, they fail to remove nitrogen (N) and phosphorus (P). Water treatment residual (WTR), a common by-product of drinking water treatment, has shown high P removal capability. In this work, to enhance P removal in bioretention, WTR (15 wt%) was employed as an amendment in a sandy loam (85 wt%) bioretention column. The pollutant/nutrient (especially N and P) removal efficiencies of the WTR-modified column were evaluated and compared with those of the unmodified control. The results showed that both systems achieved excellent TSS (92–97%) and COD (64–95%) removal. For P removal, however, the unmodified reactor only captured an average of 10–23% total P (TP) after 1-month of operation, while the WTR-modified column achieved a TP removal efficiency of >99%, with the effluent P concentration being ≤0.08 mg L−1. For N, while both reactors were able to remove >90% of NH4+-N, they were much less effective for the removal of TN and NO3−-N, with an average removal of ∼39% and ∼21%, respectively. To enhance N removal, a 40 cm internal water storage zone (IWS) was installed at the bottom of the columns, which increased the retention time and created an anoxic zone for denitrification. As a result, the average TN and NO3−-N removal increased to ∼75% and ∼85%, respectively, with a retention time of 2 h regardless of the WTR. While both reactors achieved remarkable removal of heavy metals or metalloids, which included Cu, Zn, Cd, Pb, and As, the WTR-modified column demonstrated much higher Cr(VI) removal. Overall, this study demonstrated that WTR amendment and IWS modification can substantially enhance the removal efficiencies of multiple contaminants in bioretention systems.


Maria Margaronis, journalist and former writing tutor in the Department of History, will be holding a second writing workshop for research students in December: students who attended these sessions with Maria in previous years found them to be immensely helpful and the department strongly encourages everyone to take advantage of this opportunity.
The workshop will take the form of a supportive and detailed discussion of one or two excerpts from students' work in progress, addressing issues of structure, argument, tone, style, clarity and anything else that comes up.
If you plan to attend and would like to be sent a copy of the text for discussion, please RSVP by email to Adam Shapiro (a.shapiro@bbk.ac.uk).


text:
Just good business
Solent's School of Business, Law and Communications is a bridge between the professional world and upcoming talent. The school's Gillian Saieva talks about some of the programmes helping maximise students' career opportunities.
Business and financeEnglish and journalismMarketing, communications and public relationsTravel and tourismAnnual review
From journalism to accountancy, the school actively manages a network of diverse business partners and connections. Close ties to organisations such as Hampshire Chamber of Commerce, Business South and Local Area Partnerships (LEPs) allows Solent to facilitate professional networking opportunities like our recent third Annual Business Conference, bringing together organisations from the Thames Valley right down to the central South Coast of England.
With a focus on 'UK PLC: Open for Business', the conference explored international trade support and emerging opportunities available to businesses in the wake of the Brexit referendum and brought together keynote speakers from Royal Bank of Scotland (RBS), the Department of International Trade's South East International Trade Team, leading management consultants Lee Hecht Harrison and more.
Our active networking has allowed students to benefit from connections as far afield as Australia and the US. Business management students recently worked with the Bundaberg Brewery in Queensland, Australia, researching how the company's products might fit in the UK market, while a group of Solent criminology students visited Washington in November 2016, hosted by the Department of Homeland Security and hearing from speakers in the US Marshal Service, the Secret Service and the FBI.
In an increasingly globalised world, these international elements to the Solent experience really help our students stand out. Our language department recently hosted visiting lecturers from Ukraine and Russia as part of a knowledge exchange programme; November's university-wide international enrichment week gave the school another opportunity to bring guest speakers from global organisations on-campus to supplement the curriculum.
Guests like Jen Groover, UN delegate and serial entrepreneur, who spoke on emotional intelligence, are shaping Solent's students into savvy, well-rounded and creative leaders of the future. This kind of experience pays off; Solent's PR courses have come top for overall student satisfaction in the National Student Survey for four years running, and other courses also score highly.
It pays off for our students, too. Two students recently excelled at the recent 2016 Professional Publishers Association (PPA) Magazine Academy Awards, with Iga Shlegal shortlisted for the BBC Worldwide Best New Magazine Brand and Alice Freeman winning the Felix Dennis Memorial Prize for Best New Magazine concept for her final-year project Wunderkammer, exploring the weird world of the macabre.
The school also continues to innovate on the academic side. In line with the government's apprenticeship agenda, in early 2017 we're rolling out employer-led chartered management degree apprenticeships with integrated Chartered Management Institute (CMI) recognition, while assessed strategic management simulations with Marwell Zoo (2015) and Paultons Park (2016) give students valuable experience under real-world pressures. The latest simulation was commended by academic external examiners as "exceptionally well-conceived" and "exemplary practice".
On the entrepreneurial side, the school has an ongoing collaboration with the Saints Foundation, a charitable organisation within local Premier League team (and Solent partner) Southampton FC. 'The Pitch' gives students the opportunity to put their entrepreneurial expertise to the test and raise vital funds for Saints Foundation, offering teams of students the opportunity to present their business idea to a panel of judges to secure one of ten pots of £1,000 seed funding.
At our first Pitch event, students came forward with a range of business ideas, from drive-in outdoor cinemas to bespoke book-keeping services, and were given three months to turn their ideas into a money-raising reality. It's this kind of creativity and innovation which really sets Solent apart, both for our students and for business at a local and international level. And we'll continue to adapt, to innovate – ensuring we deliver graduate talent that is precisely suited to the fast-changing needs of business.

redpajama_set_name: RedPajamaCommonCrawl


text:
Polygonum bistorta subsp. bistorta é uma subespécie de planta com flor pertencente à família Polygonaceae.

A autoridade científica da subespécie é L., tendo sido publicada em Sp. Pl. 1: 360 (1753).

Os seus nomes comuns são bistorta, colubrina ou serpentina-vermelha.

Portugal

Trata-se de uma subespécie presente no território português, nomeadamente em Portugal Continental.

Em termos de naturalidade é nativa da região atrás indicada.

Protecção

Não se encontra protegida por legislação portuguesa ou da Comunidade Europeia.

Bibliografia

 Sequeira M, Espírito-Santo D, Aguiar C, Capelo J & Honrado J (Coord.) (2010). Checklist da Flora de Portugal (Continental, Açores e Madeira). Associação Lusitana de Fitossociologia (ALFA).
 Checklist da Flora do Arquipélago da Madeira (Madeira, Porto Santo, Desertas e Selvagens) - Grupo de Botânica da Madeira
 Polygonum bistorta - Portal da Biodiversidade dos Açores
 Polygonum bistorta subsp. bistorta - The Plant List (2010). Version 1. Published on the Internet; http://www.theplantlist.org/ (consultado em 16 de novembro de 2014).
 Polygonum bistorta - International Plant Names Index

Ligações externas

 Polygonum bistorta subsp. bistorta - Flora-on
 Polygonum bistorta subsp. bistorta - The Euro+Med PlantBase
 Polygonum bistorta - Flora Vascular
 Polygonum bistorta - Biodiversity Heritage Library - Bibliografia
 Polygonum bistorta subsp. bistorta - JSTOR Global Plants
 Polygonum bistorta subsp. bistorta - Flora Europaea
 Polygonum bistorta subsp. bistorta - NCBI Taxonomy Database
 Polygonum bistorta subsp. bistorta - Global Biodiversity Information Facility
 Polygonum bistorta subsp. bistorta - Encyclopedia of Life

Flora de Portugal
b

redpajama_set_name: RedPajamaWikipedia


To do this field-ops, go the the Everyday Phoning Facility, and stand near the steps. It's a new puzzle, try to work it out for yourself.
To find the HD TV, click the cherry.
To buy a candy stash, click the pink section of the neapolitan lamp.
If you want to buy a marshmallow, click the center of the donut, circled below.
On the next page, click the coin after Wishing Well: 500 to buy a penguin gnome.
Click the wishing well to buy the Portal Box.
Click the cloud to purchase a Clover Balloon.
Finally, click the word Clover to buy a Clover Garland.
You can enter the St. Patrick's igloo contest from the 11th to the 14th. You'll have a good chance if you use the items on the page above!
A couple of glitches to report now, unfortunately I have no images. One is that when trying to enter the contest, it comes up with the Holiday Party igloo contest. Huh?
Another is that when my puffles go to play with the toys bought from the puffle catalog, they all turn white while playing! When they finish playing, they go back to normal. Weird, huh?
That's all for now, anyway. Bye!
This entry was posted on March 11, 2011 at 5:07 pm and is filed under Catalogs, Field-ops, New stuff, Pins.	You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.


Active Directory Domain Services provide multi-master update. Multi-master update means that all full replicas of a given partition are writable (the partial replicas on global catalog servers are not writable.) Multi-master update means that updates are not blocked even when some replicas are inoperable. The Active Directory server propagates the changes from the updated replica to all other replicas. Replication is automatic and transparent.


Microsoft PowerPoint® is a general-purpose presentation application. SongShow Plus™ is designed specifically for displaying song lyrics (or other media) in a worship service. The following list of SongShow Plus™ features are not available in PowerPoint®, Corel Presentations® or Lotus Freelance®.
SongShow Plus™ includes a searchable database of songs, which you can add to or modify.
SongShow Plus™ allows you to find and display songs in other compatible databases, including CCLI's database of over 8000 songs.
You can search for songs alphabetically, by key, by phrase or by keywords.
You can jump directly to any verse, chorus, or song in a worship list.
In a spontaneous worship setting, SongShow Plus™ lets you preview or display any song in your open databases with a single mouse click.
With SongShow Plus™, you can easily make global modifications to display properties that effect all songs.
SongShow Plus™ includes a handy Clipboard Capture Utility for transferring song text from other Windows® programs into a database.


Received Early March 2018 at 1:44 A.M.
"I AM the Alpha and Omega, the First and the Last. The time has come to show the world what I have been warning of through My Word and prophets. Great and terrible things are about to happen. You are going to experience things never imagined in your minds. Those of you who have mocked, abused, and tortured My children will drink from the cup of My wrath. For I AM a very loving Father. I AM long suffering and wish all of My children to come before Me on bent knee. How long should I warn thee while I watch My innocent children suffer horrible things at the hand of the wicked one? Satan knows his time is up as he was also forewarned and made his choice to defy Me. He will have his place in hell where he will spend eternity. I will not wait any longer for THIS IS IT. This is the final countdown. You are sealed. I wish to say I will never abandon any one of My children, but know the ark is now full. The light of the world is leaving.
Are you saying the Lord says its too late to repent?
Is this the same Brandy I wrote to last week?
This is true. The Father has given me knowledge as it happens. August 2017 Satan and all evil was cast out of heaven and hurled to earth. 7 year tribulation began. Sept 2017 Yeshua arrived on Mt of Olives. He rested there in preparation for the rapture. Jan 2019, the Father requested Michael to deliver the Ark of the Covenant to the temple in heaven. Jan 2019, the Father opened the Book of Life and accounted the names. Feb 2019 the Father sealed all who were named in the Book of Life. Feb 11, 2019 the Father gave me instructions for the sealed. Get on your knees and lift your face to heaven, put your right hand over your left hand palms up and raise them to heaven. This will show him that you are sealed by him and not the beast. Then recite, "Father I am a great sinner and not worthy of your forgiveness, but you sent the Messiah Yeshua to take my sins away. Please forgive me as I sinned greatly and hold no judgement within me, for I forgave those who sinned against me. I am but a sheep and I pray to you Father that you receive me. I love you Father and I am forever grateful of your mercy. Thank You." The Father urges compliance because all is in place and all is ready for his command. My name is Jorge. I am Angel Phanuel sent in the flesh as Enoch, and I came back to heaven when I completed my purpose. I was sent back to earth with one more purpose and that is the same purpose as Enoch. I'm protector of truth and urge repentance. I am a witness to the Father and will be here until the end. Evil struck me since I was in the womb, but death has no power over me, since I was sealed by the Father in heaven before I came down to earth. Have no fear, the Father has a plan and the righteous will be protected from harm. 2/3 will ascend with the Messiah Yeshua. This will happen soon. The last 1/3 will remain behind with me and we will work diligently to gather the lost sheep and return them to the Messiah. More and more chosen have awakened. Prepare in spirit and live in FAITH. Our blessed Father is with us and he brought the Kingdom of Heaven with him. Go out and rejoice. Spread the word that salvation is near. I love you all and I pray to the Father that I'm with you all again in his heavenly kingdom. All praise and glory be to the Father and thanks given to his heavenly kingdom.


For hundreds of years, brides have been given a horseshoe to carry on their wedding day and bring luck to their new marriage. But have you ever wondered how this tradition started?
Horseshoes have long been considered lucky. In fact the roots of the lucky horseshoe tradition can be traced all the way back to ancient Greece where it symbolised the crescent moon, which was regarded as a symbol of fertility.
In pre-Christian times it was also thought that iron could ward off evil spirits. Although horseshoes are now generally made from steel, they were originally forged in iron, which folklore dictates keeps away spirits and fairies. They were also held in place with seven nails, seven being the luckiest number.
The superstition acquired a further support from a Christian story, the legend surrounding the 10th century Saint Dunstan who worked as a blacksmith before becoming Archbishop of Canterbury. The story goes that Saint Dunstan was working in his blacksmith's forge when the Devil appeared. Saint Dunstan is said to have trapped the Devil with his blacksmith's tongs and as a result extracted a promise never to enter the house of a Christian, which he would recognise by a horseshoe hung above the door.
It is said that because the horseshoe is "U" shaped, to retain the good luck forever it is essential that the horseshoe is hung by the ribbons which are attached to the shoulders. The horseshoe should not be turned upside down or else all the good luck of the marriage may fall out.
Wedding Horseshoes make all of their gifts from real horseshoes, forged by a blacksmith, to ensure the most luck possible for the brides who carry them.
When you order your horseshoe from us, you'll get a genuine, traditional steel horseshoe that's completely personalised for you. Prices start from £23.90. We are especially popular with Scottish brides as we are able to source any tartan for the decorative ribbons on the horseshoes. Check this out loanora.co.uk offer help to your financing with lower interest and quick process.


text:
"Take a Seat…in the Orchestra"
Hudson Festival Orchestra
Gwen Gould
Organ Recital Press
A Concert to Celebrate
The 225th Anniversary of the Town of Claverack
Unpublished Review by Sheila Silver, composer
Gwen Gould's playing on the organ at the Claverack Reformed Dutch Church was clear and crisp. Everything you'd expect from a person whose first incarnation was as an organist and is now returning after having focused for many years as a conductor. Bach's Fantasy in g minor was dramatic and compelling and had the audience spellbound. Chorale Prelude "Kommst du nun" had charm and wit and Gwen's handling of the chorale tune in the pedals was deft (with trills included). Her lively rendition of Buxtehude's fugue in a minor made it a joy to follow the intricate dialogue. Well done Gwen!
Review/Music
by C.T. Barr
Weekend April 28-May 4, 1994
Taconic Newspapers Entertainment Weekly
New venue, artists are a nice surprise
There is nothing to compare with new pasture, is there? It is a tonic to find new cultural territory, new and enthusiastic artists of quality. So it was in Hudson, New York on Saturday afternoon at Christ Episcopal Church on Courthouse Square.
Gwen Gould, organist, and Jeffrey McClendon, baritone, presented a perfectly organized program in which four of J.S. Bach's Chorale Preludes in the first half were countered with four of Brahms' in the section after intermission. Mr. McClendon's Bach arias fit into the Baroque segment and Vaughan Williams' "Five Mystical Songs" played a similar role in the essentially Romantic part.
The Bach "Preludes" and "Arias" were framed with a Couperin "Chaconne" (cousin to the brass-oriented passacaglia form) and a stunning Buxtehude "Prelude and Fugue in A Minor." No wonder that Bach was enchanted with Buxtehude's works on his one extended visit outside the Leipzig area! The Prelude ends on an extended cadence over pedal-point and the Fugue is broken by an instrumental recitative which goes beyond tradition and provides elaborate flourishes adding only aesthetic advantages.
The cantata form was crucial to the Lutheran Service of Bach's time. In the aria from No. 56, "Ich will den Kreuzstab Gerne Tragen" couplets of the verse are treated as separate units, in each instance making a salient contribution to the whole. Mr. McClendon chose a part of the Credo from the "B Minor Mass" for his second contribution to the first half on the program. As a soloist he handled the long melodic lines with ease. His voice has an even scale without "breaks" and, in these pieces, evidencing uniform color.
Of the Bach Chorale Preludes, two come from well-known works, "Wachet Auf!" and "Christ lag in Totesbanden." the texture of the Brahms Preludes is quite different, attributable in part to the era in which they were written. In the final one, "O Gott, du frommer Gott," well-placed rests lend drama to the polished, refined harmonies.
Mr. McClendon is an undemonstrative singer who does everything through the voice. Even so, Vaughan Williams' "Mystical Songs" (on poetry of George Herbert), seemed more congenial to his temperament than the earlier Bach. In the first song there was more variety in the tone color, in the second, a beautiful opening lyric line. Of the fifth, "Antiphon," there are so many settings of one kind or another that one waits, anxiously, to hear what yet another composer has done with the text. The final heroic manner seemed very appropriate, fitting natural.
Ms. Gould is an accomplished organist, in this instance somewhat impeded by an electronic instrument. She has a strong rhythmic sense and clean articulation, but above all, sensitivity to the material.
Throughout the Baroque half of the recital one was struck by the beauty of the sanctuary and of how it complemented what one was hearing. The time of the recital between day and night seemed almost symbolic.
by Alan Hovhaness
The American Organist, June 1970
The organist, Gwen Gould, gave an outstanding program at St. Michael's Church [New York City]. Her playing was always clear and technically of a high excellence. The lovely Magnificat of Jean Francois Dandrieu was especially beautiful in her choice of registration and her expressive, beautifully legato performance. Outstanding was her magnificent performance of the Great St. Anne Prelude and Fugue in E flat by J.S. Bach. She played this towering work with great majesty, clear precision and splendid rhythmic vitality. It was wonderful to follow such a clearly delineated rendering of the polyphonic complexities of this great triple fugue with its solemn and ever ascending first section, its light fantastic development of the second subject combined with a dance-like version of the first subject in many inversions, followed by the great culmination of the third subject in majestic triple counterpoint, with the first and second subjects, all leading to a brilliant, triumphal ending. This great architectural work revealed her splendid mastery of all elements of organ playing and musicianship.
Following the intermission, she was joined by Louis Fletcher, trumpet, in a beautiful and moving performance of my Prayer of St. Gregory. Both organ and trumpet played with a Byzantine feeling of fervor. The Adagio and Allegro of G.F. Handel was beautifully played by two trumpets and organ, F. Joseph Docksey, joining the ensemble with excellent effect.
The Suite, opus 5, by Maurice Durufle, is a large organ work in the grand post-Franck style, with some Ravel and Widor influences. This is in the form of a Prelude, Sicilienne and Toccata. This work revealed the organist in all her skill and sense of color. I found the Prelude most moving from a purely musical standpoint in its beautiful harmony. Each piece was different in style but highly inventive. She played the Toccata with great virtuosity and brilliance.
by Alfreda Hays
the Summit Independent 10/21/1970
If ever a concert deserved to be described as "absolute perfection," it was the program of music for organ and brass presented at St. Mary's Abbey, Delbarton, last Sunday afternoon. All the elements that go to make up a memorable performance were present: highly skilled artists, an unusually well-planned program, excellent acoustics, and a good-sized, warmly appreciative audience. An added attraction was the presence of the gifted young composer Hubert Arnold, whose Sonata for two trumpets and organ appeared on the program.
Arrangements for this outstanding event – the third in the 1970-71 series of monthly concerts open to the public without charge – must again be credited to Father Gabriel M. Coless, who has an obvious knack for ferreting out exceptional talent. Gwen Gould, Assistant Organist at St. Luke's Chapel, Trinity Parish, New York, first attracted Father Gabriel's attention with her recital on St. Michael's new Beckerath, considered one of the country's finest tracker-action organs. In accepting the invitation to perform on the Abbey's own tracker organ, Miss Gould, in turn, invited two associates to be part of the program – trumpet players Louis Fletcher and F. Joseph Docksey II, now completing their Army service in the West Point Band.
Seated at the organ, dressed in a brightly colored mini-tunic over black tights, her dark curly hair cut short, the attractive Miss Gould looked rather like a young page in a play by Shakespeare. The first bars of Maurice Durufle's Prelude from his Suite Op. 5 (1930) which opened the program, however, left no doubts as to her qualifications as a concert organist, and an adventuresome one at that. To open the program with a contemporary French work was unusual enough, but to reverse the customary chronological order of programming and end with a work representative of the early North German School and make it work was a credit to the extraordinary skill of the performer.
The Sonata in D by Petronio Franceschini introduced Messrs. Fletcher and Docksey, as agile a pair of trumpet players as it has ever been my pleasure to hear. Franceschini, who died in 1681, at the age of 31, is rarely mentioned in music histories, and then only as a composer of operas, so this work was a revelation. In four movements with typically Baroque tempi (slow-fast-slow-fast), it had more musical interest than many works of the period. the antiphonal effects so fluidly rendered by the well-matched trumpets over the organ continuo were more characteristic of the music used in the great Italian churches of the Renaissance. St. Mary's, with its wonderful acoustic properties, was an ideal setting for this work, as well as the sparkling Sonata (1968) by Mr. Arnold, who is staff arranger for the West Point Band. In three movements – March, Intermezzo, and Finale – this well-written modern work showed off the brilliant technique of the two brass payers and their associate at the organ. The Sonata, incidentally, was written specifically for Mr. Fletcher and Mr. Docksey, who introduced the composer to share in the applause they received.
Miss Gould rounded out her program with two of Bach's Schubler Chorales: "Meine Seele erhebet den Herrn" and the joyous "Kommst du nun, Jesu, vom Himmel herunter", in which the Zimbelstern was used. Two Sketches by Robert Schumann were nicely registered to accentuate the romantic elements. Dietrich Buxtehude's mighty Prelude and Fugue in A Minor was the grand finale. Miss Gould, who received not one, but two floral tributes, was repeatedly recalled by the enthusiastic audience.


My friends Grand bundle of joy is a girl, So the decorations were full of with pinks and purples. They were all beautiful including a Unicorn themed cake with lights. One of the decorations that will also be a Memento for the parents-to-be was a board where we were to write down our words of wisdom for the new parents. It's funny how before I had a child I had no problem offering words of wisdom. Now that I'm a mom, who admittedly doesn't know what she's doing half the time, I had no clue what to write.
So I thought about it for a few more minutes and wrote down what's been working for me. Get sleep whenever you can and take advantage of your village." The only reason I was able to attend the shower without Baby Joy Chaser was because my Village, my brother and sister-in-law. We're watching him.
It truly does take a village to raise a child. It's not just a cliche. Parenting is hard. It is the most exhausting job but also one of the most fulfilling. I'm simultaneously overwhelmed and ecstatic to be a mom. But I cannot do it on my own. I have a village. Not the village that I imagined. As I am I am often very sad that my mom is not here helping me to raise my little boy. But I have awesome cousins and my brother and sister-in-law and aunts and Friends who've all been super supportive.
Now that I am a mom I think back on the advice that I've given other women over the years on Chasing Joy, at Casing Joy events, and through the Joy Makeover program and I 100% stand by the things that I said. I still believe that happy moms have happy children and happy families. However, I understand so much more the importance of relationships. Having healthy relationships with people that you can trust is Pparamount when it comes to being a happy mom.
To be a happy mom you need to have some time for yourself for self-care and healthy habits. You need time and encouragement to pursue your goals. You need people in your corner who will support you in your faith practices and raising your children in accordance with those traditions. In essence you need people to help you Chase Joy. I understand this so much more now. So my advice to any mom who hasn't done so already, and to any future mom, especially if it will be a solo project like my single mother by choice Journey, is to build your support system.
So here is a snapshot of what my support system looks like. The Inner Circle people who I actually have allowed to be alone with my child. These include my brother, sister-in-law, my nieces, my aunt, and my cousins. So basically that's family LOL. Next up are the people who've I've allowed to be around and interact with Baby J chaser when supervised that includes the daycare, other family, lifelong friends, and a woman from my church group who provides child care services. Next up are the people who provide me with encouraging words, advice, and a place to vent. These include co-workers, friends, women from the local single mother by choice group, and women from my church group. Last but certainly not least are social media friends and blog readers!
Yes, don't underestimate the value of social media especially for a single mother by choice. Oftentimes when the baby is doing the cutest things I'm the only one to see it. it's just me and him. I don't have a husband to say oh honey did you see that or my mom here to witness it with me. So being able to share those moments in my Instagram stories, on Facebook, Twitter and IG makes me feel not alone in those moments. Because even in the happiest times when the baby is doing the cutest things it can be sad to not have anyone to share it with. That's your support really comes through.
So that's my Village both in person and virtual. So if you want to be a happy mom get you a village!
What does your village look like? Who's there to give you much needed support? Who's there to help you get a break so you can have some time just for yourself?
If you feel like your support system is lacking and you're in the Philadelphia area check out the mama day off event that's coming up in March. There will be opportunities for you to connect with other moms and build that Village. Also you can meet me because I'll be speaking.


One of the best view homes in Granada Hills. If you're looking for views, look no further. Located in prestige Granada Hills neighborhood, this home offers 4 bedrooms, 3.5 bathrooms, just under 3600 sqft of living space, formal living room with high ceilings, large open chef's kitchen with island, huge family room, upstairs has a large loft or second den, master suite has balcony with incredible views all the way to downtown. The perfect entertainers house.
Sold by Afifa Rafiq BRE#01276434 of Century 21 Everest.
Listing provided courtesy of Gary Keshishyan DRE#01276434 of Pinnacle Estate Properties.
Based on information from CARETS as of 2019-04-22T23:44:03.44. The information being provided by CARETS is for the visitor's personal, noncommercial use and may not be used for any purpose other than to identify prospective properties visitor may be interested in purchasing. The data contained herein is copyrighted by CARETS, CLAW, CRISNet MLS, i-Tech MLS, PSRMLS and/or VCRDS and is protected by all applicable copyright laws. Any dissemination of this information is in violation of copyright laws and is strictly prohibited. Any property information referenced on this web site comes from the Internet Data Exchange (IDX) program of CARETS. This web site may reference real estate listing(s) held by a brokerage firm other than the broker and/or agent who owns this web site. The accuracy of all information, regardless of source, is deemed reliable but not guaranteed and should be personally verified through personal inspection by and/or with the appropriate professionals.


If you're just starting out in your career, and you're in a relatively low income tax bracket, but you think you might be in a higher one when you retire, you might want to consider the Roth 401(k). You'll be paying taxes now on the money you earn and contribute to your Roth account, but you'll avoid being taxed at the higher rate when you start taking withdrawals. Conversely, if you think your tax rate will be lower when you retire, you might be more inclined to go with the traditional 401(k), which allows you to avoid paying taxes on your contributions now, when your tax rate is high.
You may want to consult with your tax advisor before making any decisions about a Roth or traditional 401(k) – or Roth and traditional 401(k) – but in the final analysis, these are positive choices to make, because a 401(k), in whatever form, is a great way to save for retirement. Try to take full advantage of it.