text:
\section{Introduction}
Kinetic theory
is useful to describe systems out of equilibrium for a wide range of many-particle systems at classical and quantum
level. Its quantum version is based on Wigner functions,  which are quantum analogs of classical distribution functions\footnote{Quantum kinetic theory has been thoroughly reviewed in Ref.\cite{Hidaka:2022dmn}.}.
 At the quantum level they appear
from the Wigner transform   of the density matrix operator $\hat \rho(t)$, which for a single particle reads \cite{Wigner:1932eb}
\begin{align}
W(x, p; t):=\int \mathrm{d} u \  e^{-\ii p u}
\braket{x+\frac{1}{2}u| \hat \rho(t)|
x-\frac{1}{2}u},
\end{align}
where $x, p$ are phase-space variables.
 A suitable classical limit renders it as the classical distribution function $f(x, p )$. In the collision-less case the distribution function $f(x, p)$ satisfies
Liouville's equation
$\frac{\mathrm{d} f}{\mathrm{d} \tau}=0$,
which gives rise to the Boltzmann equation.
 The Wigner transform
can be generalized to describe quantum fields. Then, from QFT it is possible to deduce an equation for the Wigner function which satisfies classical kinetic equations thus confirming the analogy between Wigner functions and classical phase-space distributions.

Another alternative is to maintain the point-particle picture and bring together Wigner transformations and the Schwinger-Keldysh formalism in a
first-quantized worldline approach\footnote{The worldline approach has been reviewed in Ref.\cite{Schubert:2001he}}, thus allowing the construction of phase-space distributions and a derivation of classical kinetic equations of Vlasov type, which may include color and spin degrees of freedom \cite{Mueller:2019gjj}. Following the Schwinger-Keldysh strategy, the action  expressed in terms of retarded and advanced fields contains boundary terms that one can formally relate to the Wigner transform of the density operator, which is  typical in finite temperature QFT \cite{Jeon:2004dh, Jeon:2013zga}.
Also typical in this context is that
 the path integral over advanced variables imposes classical equations of motion--- which is also the conclusion one obtains in the worldline approach--- and hence the remaining path integrals are  over retarded variables.

 In Ref.~\cite{delaCruz:2020cpc},  iterative solutions of
kinetic equations were mapped  to certain off-shell currents in the classical limit, understood as the limit $\hbar \to 0$. There, we applied the  Kosower-Maybee-O'Connell (KMOC) \cite{Kosower:2018adc} formalism to deal with off-shell currents rather than amplitudes in the forward limit. KMOC  was originally proposed in the context of ongoing
efforts to extract classical information from scattering amplitudes (see Refs.\cite{Bjerrum-Bohr:2022blt,Kosower:2022yvp} and references therein for reviews). These efforts are concerned with matching solutions of classical equations of motion and scattering amplitudes, which is  similar to the scenario we  discuss here,  the additional ingredient being that equations of motion are now coupled to the Boltzmann equation for the distribution function.
These solutions were constructed in  the approximation where  the distribution function $f$ can be perturbatively expanded in powers of the coupling constant. This situation arises e.g., in the description of the so-called hard thermal loops\cite{Braaten:1989mz,  Frenkel:1989br, Braaten:1990az, Taylor:1990ia,
	Frenkel:1991ts}, which are relevant in the high temperature limit of QCD and  can be mapped to solutions of kinetic equations \cite{Kelly:1994ig}.

In the context of scattering amplitudes \cite{Elvang:2013cua, Weinzierl:2016bus}, bi-adjoint scalars are important ingredients to understand the relationship between QCD and gravity through the so called "double copy" \cite{Bern:2019prr}. At tree-level closed formulas for massless and massive amplitudes are available from the Cachazo-He-Yuan representation \cite{Cachazo:2013hca}. Bi-adjoint scalars  have been explored from many angles  ranging from  more mathematically-inspired ones \cite{Cachazo:2022vuo,Cachazo:2019apa}  to studies of its basic properties such as renormalization
\cite{Gracey:2020baa,Gracey:2020tkk} or its exact solutions \cite{White:2016jzc,Bahjat-Abbas:2018vgo} to name just a few.  Double-copy-type  relations also exist classically
\cite{Monteiro:2014cda} and, more importantly, they   also exist  for  particles carrying spin \cite{Goldberger:2017ogt}. Since  bi-adjoint scalar amplitudes encode the kinematic structure of QCD and gravity it raises the question of whether this is also the case at finite temperature even for particles carrying spin.

In this paper, we  will extend the approach proposed in Ref.\cite{delaCruz:2020cpc} to include bi-adjoint scalars to start exploring the above questions. In addition, we will consider classical spin from amplitudes which we will model through a generalization of scalar electrodynamics in which the scalar fields are  higher-integer-spin fields\cite{Bern:2020buy}. As we will see, the amplitudes-based approach  leads to consistent results that match perturbative solutions of their kinetic equations.

The remainder of the paper is organized as follows. In Section \ref{color-less}, we review semi-classical kinetic theory with spin and color and outline the amplitude-like approach to thermal currents.
 In Section \ref{Bi-adjoint}, we introduce color in the context of the bi-adjoint scalar model. In Section  \ref{gauge-spin}, we consider classical spin. Our conclusions are presented in Section \ref{conclusions}.

\section{Amplitudes approach to thermal currents}
\label{color-less}

\subsection{Thermal currents and the classical limit}
Let us consider a  distribution function $f(x,p,c,s)$, which in addition to the coordinate $x^\mu$ and momentum $p^\mu$ depends on two continuous variables $c^a$ and $s^\mu$, describing color and spin degrees of freedom, respectively. The Liouville's equation in the collision-less case can be expressed as \cite{Heinz:1984yq}
 \begin{align}
 \frac{\mathrm{d}}{\mathrm{d}  \tau} f(x,p,c,s) =
 \left(\dot x^\mu
 \frac{\partial }{\partial x^\mu}+ \dot p^\mu
 \frac{\partial }{\partial p^\mu}+\dot c^a\frac {\partial}{\partial c^a}+\dot s^\mu \frac{\partial}{\partial s^\mu} \right) f(x,p,c,s) =0,
 \end{align}
 where the Vlasov-Boltzmann equation can be obtained from the above after inserting the equations  of motion  of the phase-space variables, which are generalized versions of Wong equations \cite{Wong:1970fu}. In the color-less case the spin vector satisfies the BMT equation
 \cite{Bargmann:1959gz}.
  The associated color currents of the particles are obtained from
 %
 \begin{align}
 J_a^\mu(x)=\int\mathrm{d}\Phi (p) \,\int\mathrm{d} c \,\int\mathrm{d} s\, c_a\, (p^\mu +S^{\mu\nu}\partial_\nu) f(x,p,c,s),
 \label{current-1}
 \end{align}
 %
 where $S^{\mu\nu}$ is the spin tensor and  $\mathrm{d}\Phi(p)$ is the usual Lorentz Invariant Phase space. The invariant measures for color and spin space are
 $\mathrm{d} c$ and $\mathrm{d} s$,  respectively. The invariant measures are given by
\cite{Elze:1989un}
\begin{align}
\mathrm{d} \Phi (p):=& \frac{\mathrm{d} ^4 p}{(2\pi)^3}
\, \Theta(p_0) \delta(p^2-m^2),\\
 \mathrm{d} c:=&  \mathrm{d}^8 c\, c_R \delta(c^a c^b \delta^{ab}-q_2 ) \delta(d^{abc}c^a c^bc^c-q_3 ), \label{DIPSC} \\
 \mathrm{d} s:=& \mathrm{d}^4 s\, c_S \delta(s^\mu s_\mu+2\mathfrak 	s^2) \delta(p^\mu s_\mu),
\end{align} %
 where $q_2$, $q_3$, $\mathfrak s^2$ are Casimir invariants\footnote{We assume that classical kinetic theory to be valid in the approximation where color and spin representations are large so that color factors and spin vectors commute. }. The factors $c_S$ and $c_R$ ensure that spin and color measures are normalized to unity. We have set the gauge group to be $SU(3)$ and for spin to be $SU(2)$.  The integration over phase space can be done using the relation between the spin vector and spin tensor
 \begin{align}
 S^{\mu\nu}=-\frac 1 {m} \epsilon^{\mu\nu\rho \sigma}p_\rho s_\sigma,
 \label{spin-tensor-to-vector}
 \end{align}
 %
 where $\epsilon^{0123}=1$.    There is a one to one correspondence between  classical color factors $c^a$ and the generators of the gauge group and similarly for spin. Therefore the invariants $q_2$, $q_3$ and $\mathfrak s^2$	are defined through the traces of the generators of the gauge group
 \begin{align}
 \text{Tr}(T^a T^b)= C_2 \delta^{a b},
 \label{basic-trace}
 \end{align}
 where for $SU(N)$ the quadratic Casimir $C_2=N$ in the adjoint representation and $C_2=1/2$ in the fundamental.   Thus, properties of integration over classical phase spaces are equivalent to traces of color-factors, namely
 %
 \begin{equation}
 \begin{aligned}
 \int \mathrm{d} c\, c^a =& \int \mathrm{d} s\, s^\mu =0,\\
 \int \mathrm{d}  c\,  c^a c^b=&  C_2 \delta^{ab},\\
 \int \mathrm{d}  s\, \mathit s^\mu \mathit s^\nu=& - \frac {2} 3 \mathfrak s^2 \left(\eta^{\mu\nu}-\frac{p^\mu p^\nu}{m^2} \right).
 \end{aligned}
 \label{identities-phase-space}
 \end{equation}
 %
The last  equalities express the fact that $c^2$ and $s^2$ are constants of the motion. For spin, the last equality is equivalent to the trace  of the spin tensor and $\mathfrak s^2$ is proportional to the eigenvalue of the Casimir operator \cite{Bhadury:2020puc,Weickgenannt:2020aaf}. More generally, the projectors in parenthesis will depend on the matter content \cite{Hidaka:2022dmn}.

 We may evaluate the currents of Eq.\eqref{current-1} iteratively by assuming a perturbative expansion of $f(x,p,c,s)$ in powers of the coupling constant around the equilibrium state $f(p_0)$, where the distribution depends only on the energy $p_0$ (see Ref.\cite{Litim:2001db} for review of this approach). We allow the equilibrium distribution to be either Bose-Einstein ($f_ -(p_0)$)  or Fermi-Dirac  ($f_+(p_0)$)  so strictly speaking this approach is semi-classical.  For QCD
  the relation between color currents and thermal currents in the high temperature approximation (hard thermal loops) is conveyed by
 %
\begin{align}
J^a_\mu(x) = \Pi^{ab}_{\mu\nu}A^{b\nu}
+\Pi^{abc}_{\mu\nu\rho}A^{b\nu}A^{c\rho} +\dots,
\end{align}
where the currents $\Pi^{a_1 \cdots a_n}_{\mu_1 \cdots \mu_n}$ thus obtained  match those obtained
in the high temperature limit of QCD\cite{Kelly:1994ig, Kelly:1994dh}.
\begin{figure}
	\centering
\begin{tikzpicture}[thick, transform shape]
\node[circle, fill=lightgray, draw] (c) at (0,0){$\mathcal A$};
\draw[draw, snake it] (c.north west)--(-1.,1)node[left]
{$\qquad k_1$};
\draw[draw, snake it] (c.north east)--(1.,1)node[right]
{$k_n$};
\draw[draw] (c.south east)--(1.5,-0.30) node[right,     currarrow,
pos=0.5,
xscale=1,
sloped,
scale=1]
{$\quad \ \, p$};
\draw[draw] (c.south west)--(-1.5,-0.30)node[left,    currarrow,
pos=0.5,
xscale=1,
sloped,
scale=1]
{$\mkern-36mu p$};
\node(A) at (0,1){$\dots$};
\end{tikzpicture}
	\caption{Regulated off-shell current in the forward limit.  The blob represents a sum over tree-level Feynman diagrams where diagrams generating a zero-momentum internal edge are suppressed.  Wavy lines represent off-shell outgoing soft particles.}
	\label{current-off-shell}
\end{figure}

In Ref.\cite{delaCruz:2020cpc}, we proposed to  obtain thermal currents by taking the classical limit of a regulated off-shell current in the forward limit adapting the KMOC formalism. Let us briefly recap the main points. The forward limit is, in general, singular so a regularization scheme is required. Here, it  is understood as discarding  diagrams with a zero momentum internal edge \cite{Runkel:2019zbm}. Let
 $\mathcal S$ be the set of those diagrams and let $\mathcal F$ be the set of all diagrams contributing to the amplitude, then  the regularized current is defined by
\begin{align}
\mathcal A^n (p, k_1, ,\dots, k_n,p):=\sum_{G\in \mathcal F   \backslash \mathcal S}
d(G),
\end{align}
where $d(G)$ is a rational  expression of the form $N(G)/D(G)$. Denoting by $k:=(k_1, \dots, k_n)$ the $n$-tuple of external off-shell momenta and, with the understanding that the amplitude is computed in the forward limit, we sometimes write $\mathcal A(p,k)$. The classical limit is obtained as follows: Compute the
 Laurent expansion in powers of $\hbar$ of the $n+2$ current $\mathcal A (p, k_1, \dots, k_n, p)$ where the momenta of the particles $1, \dots, n $ is considered soft and off-shell  ($k_i^2\ne 0$), while  massive particles carrying momenta $p$ are on-shell ($p^2=m^2$). Soft momenta scale with $\hbar$, i.e,  $k\to \hbar k$, implementing the distinction between  momentum and  wavenumber of a particle, which is an important step in the KMOC formalism.
  In practice
the current can be computed e.g., from Feynman graphs (See Fig.\ref{current-off-shell}).
 Recall that we are using units in which $k_B=c=1$ but keeping $\hbar \ne 1$ since we are interested in the classical limit.
 We adopt the convention that our classical results  depend on the
 dimensionless coupling $g=\bar{g} \sqrt{\hbar}$ and $e=\bar{e} /\sqrt{\hbar}$, and that the external momenta is associated to wavenumbers. Suppressing	 color and Lorentz indices
 the classical limit of the current is given by
 %
 \begin{align}
 \bar {\mathcal A}^n (p,k)= \widehat{\text{Tr}}\left(\lim_{\hbar \to 0}
 \mathcal A^n (p, \hbar k)\right), \end{align}
 where the trace  is defined by
\begin{align}
\widehat{\text{Tr}} (\bullet) := \begin{cases}
\hbar^{n-2} \text{Tr} (\bullet) & \text{QCD}, \\
\text{Id} (\bullet) & \text{QED} \ \text{and} \ \text{gravity},
\end{cases}
\label{trace-tilde-def}
\end{align}
where $\text{Id}$ is the identity operator and the $\hbar^{n-2}$ is required on dimensional grounds. We then have a simple relation between thermal currents and the classical limit\footnote{Our conventions for the Fourier transform are
	$g(x)=\int  \mathrm{d} ^d k/(2\pi)^d \, e^{-\ii k\cdot x} \tilde g (k)
	$
}
\begin{align}
\Pi^n (k)= \int \mathrm{d} \Phi (p)  \,f(p_0) \ \bar{\mathcal A}^n (p, k)
\label{off-shell-current}
\end{align}
where $f(p_0)$ is the distribution function at equilibrium. This relation represents a map between the classical limit and the high temperature in the forward limit\cite{Frenkel:1991ts}.  In Sections \ref{Bi-adjoint} and \ref{gauge-spin} we will add two more cases where Eq.\eqref{off-shell-current} holds.

Let us briefly
mention that color factors in calculations are defined through
\begin{align}
\braket{p_i| \mathbb C^a|p_j}:= (C^a)_i^{\,j}=\hbar (T^a)_{i}^{\, j}, \label{color-expectation}
\end{align}
where $\mathbb C^a$ are operators that realize the Lie algebra of the gauge group
\begin{align}
[\mathbb C^a, \mathbb C^b]=\ii  \hbar f^{abc}\mathbb C^c.
\end{align}
Color charge operators can be explicitly derived from the Noether procedure (see Sections 2 and 4 of Ref.\cite{delaCruz:2020bbn}).
Classical color charges $c^a:=\braket{\psi|\mathbb C^a|\psi}$ are obtained as expectation values of those operators taken from appropriate coherent states $\ket{\psi}$. Using these estates, color charge operators satisfy the important property (shown in Appendix A of  Ref.\cite{delaCruz:2020bbn}) of factorization
\begin{align}
\braket{\psi|\mathbb C^a \mathbb C^b|\psi}=\braket{\psi|\mathbb C^a|\psi}
\braket{\psi|\mathbb C^b|\psi}+\mathcal O (\hbar),
\label{color-factorization}
\end{align}
so classical color charges commute.
\section{Color: the bi-adjoint scalar}\label{Bi-adjoint}
The field theory Lagrangian of the model is given by \cite{Luna:2015paa,White:2016jzc}
\begin{align}
\mathcal L_{\text {BA}}:={\cal L _{\text {BA}}}( \varphi, \partial \varphi ) = \frac12 \partial_\mu \varphi^{a \alpha} \partial^\mu \varphi^{a \alpha}
-\frac{m^2}{2} \varphi^{a \alpha}\varphi^{a \alpha}
+\frac{y}{3!} f^{abc} \tilde f^{\alpha \beta \gamma}
\varphi^{a \alpha} \varphi^{b \beta} \varphi^{c \gamma} \:,
\label{lag-phi3}
\end{align}
where $m$ is the mass and $y$ the coupling constant. The bi-adjoint scalar field  $\varphi^{a\alpha}$ transforms under the adjoint representation for each factor of its globally symmetry group $G \times \tilde G$. The Lie algebra for each factor has the form $[T^a, T^b]=\ii f^{abc} T^c$ and the adjoint representation is given by its structure constants, i.e.
$(T_A^a)^{b}_{\ c}=- \ii f^{abc}$. Throughout we  use Greek indices   for the group $\tilde G$. Before developing a kinetic theory for bi-adjoint scalars let us briefly discuss how these are computed from the KMOC formalism.

\subsection{Off-shell currents for bi-adjoint scalars}

 We consider the theory in $d=6-2\epsilon$  dimensions, where the bi-adjoint scalar can be renormalized with a dimension-less coupling. We will keep the same definition of color factors in the adjoint
representation as in Eq.\eqref{color-expectation}
so color factors have dimensions of $\hbar$
and, as it should be, they are  absent in the classical limit. Since the action has units\footnote{$M$ and $L$ denote mass and length, respectively.} of $ML$,   in $d=6$ the field
$\varphi^{a\alpha}$ has dimensions $\sqrt{M/L^3}$ so the dimensionful coupling $\bar y$ has units of $1/\sqrt{ML}$, and therefore  the dimensionless coupling scales as $\sqrt{\hbar} \bar y$. Then,  thermal currents are obtained through Eq.\eqref{off-shell-current} with the trace $\widehat{\text{Tr}} (\bullet)$ now defined as:
\begin{align}
\widehat{\text{Tr}} (\bullet):=\text{Tr} (\bullet) \
 \widetilde{\text{Tr}} (\bullet),
\label{trace-bi-adjoint}
\end{align}
where it is understood that first and second traces are referred to the untilded and tilded color factors, respectively. This adds another instance of Eq.\eqref{trace-tilde-def}. Notice however that for a general representation $R$
\cite{vanRitbergen:1998pn}
\begin{align}
\text{Tr} (T_R^{a} T_R^{b}T_R^{c})=d_R^{abc}+ \frac \ii 2 f^{abc} I_2 (R),
\end{align}
where $I_2(R)$ is the index of the representation. Since in the adjoint representation $d_R^{abc}=0$, classical phase space integration is reproduced through the replacement $f^{abc} \to d^{abc}$ at the end of the calculation (see Appendix \ref{3-point-bi-adjoint}).

\begin{figure}[t]
	\centering	\lotop{$\varphi^{a_1\alpha_1}$}{$\varphi^{a_2\alpha_2}$}{black}
	\caption{Diagram that contribute for 2-point current. }
	\label{diags-bi-adjoint}
\end{figure}

\subsubsection*{Example}
 The 2-point off-shell current in the regulated forward limit can be obtained from the trivalent diagrams shown in Fig.\ref{diags-bi-adjoint} and their permutations. The contributing diagrams coincide with the half-ladders that appear in the forward approach to thermal currents \cite{Brandt:2006aj}. Using momentum conservation and renaming the independent momentum by $k$, a simple calculation gives
\begin{align}
\ii \mathcal A^{\delta d, \alpha_1 a_1,\alpha_2 a_2, \epsilon e}
 (p, \hbar  k ) &= \ii y^2\Bigg[ \left(
f^{\delta  \alpha _1 \alpha _3} f^{\epsilon  \alpha _2 \alpha _3} f^{c d a_1} f^{c e a_2}-f^{\delta  \alpha _2 \alpha _3} f^{\epsilon  \alpha _1 \alpha _3} f^{c d a_2} f^{c e a_1}
\right)\frac{1}{\hbar p\cdot k}\\
&  +\frac 14 \left(f^{\delta  \alpha _2 \alpha _3} f^{\epsilon  \alpha _1 \alpha _3} f^{c d a_2} f^{c e a_1}+f^{\delta  \alpha _1 \alpha _3} f^{\epsilon  \alpha _2 \alpha _3} f^{c d a_1} f^{c e a_2}\right) \frac{k^2}{(p\cdot  k)^2}
\Bigg] +\mathcal O (\hbar), \nonumber
\end{align}
where the seemingly singular term in the first line vanishes upon computing the trace. Hence, calculating the trace  we obtain
\begin{align}
 \bar {\mathcal A}^{a_1 a_2, \alpha_1 \alpha_2}(p,k)= y^2C_2^2 \delta^{a_1a_2}\delta^{\alpha_1 \alpha_2} \frac 1 2 \frac{ k^2}{(p\cdot k)^2}.
\label{2-point-bi-adjoint}
\end{align}
The squared retarded propagator has been left implicit and can be recovered by the analytic continuation  $k_0 \to k_0 +\ii o$, where $o$ is a small positive number. Using now Eq.\eqref{off-shell-current} and the integrals computed in Appendix \ref{integrals} (with $f_-(p_0)$) we find that in the
the high temperature limit the 2-point current reads
\begin{align}
\Pi^{a_1 a_2, \alpha_1 \alpha_2}(k)
=\delta^{a_1a_2}\delta^{\alpha_1 \alpha_2}\frac{C_2^2 y^2}{2}\frac{T^2}{96 \pi}\left(\frac{k_0^2}{|\mathbf{k}|^2}-1\right)
\left[
-2+\frac{k_0}{|\bm k|}\log \left(\frac{|\bm k|+k_0 + \ii o}{-|\bm k|+k_0 + \ii o}\right)\right],
\end{align}
which agrees with cubic theory  in Ref.\cite{Brandt:2012cp}. In the limit where $k_0 \ll |\mathbf k|$ we have
\begin{align}
\Pi^{a_1 a_2, \alpha_1 \alpha_2}(k)
 \approx \delta^{a_1a_2}\delta^{\alpha_1 \alpha_2}\frac{C_2^2 y^2}{2}\frac{T^2}{96 \pi} \left(2-\ii \pi \frac{k_0}{|\bm k|}  \right),
\end{align}
where the imaginary part corresponds to Landau damping.

\subsection{Schwinger-Keldysh for bi-adjoints and kinetic theory}
We proceed now to develop a classical kinetic theory for bi-adjoint scalars along the lines of the Schwinger-Keldysh worldline approach by Mueller-Venugopalan \cite{Mueller:2019gjj}.   Suppose that we are interested in the description of the time evolution of the matrix density operator
\begin{align}
\rho (t^f)= U(t^f, t^i) \rho (t^i) U (t^i,t^f),
\end{align}
where $U(t,t')$ is the evolution operator with $t^i$ and $t^f$ denoting the initial and final times, respectively.
Following the usual Schwinger-Keldysh approach \cite{Das:1997gg} we double the degrees of freedom and
consider the difference
\begin{equation}
\mathcal L = \mathcal L_{\text {BA},1} (\varphi_1, \partial \varphi_1 ) -\mathcal L_{\text {BA},2} (\varphi_2, \partial \varphi_2),
\end{equation}
 where the Lagrangians $\mathcal L_{\text{BA},1} (\varphi_1, \partial \varphi_1 )$ and $\mathcal L_{\text{BA},2}(\varphi_2, \partial \varphi_2)$ are associated with the upper and lower branches of the contour shown in Fig.\ref{SK-contour}. Now,
using a quantum/background ($\varphi_i
\to\bar \varphi_i+\varphi_i$) expansion leads to a quadratic action of the form
\begin{equation}
\begin{aligned}
\mathcal L^{(2)} = \frac 1 2
\bar\varphi_1^{a\alpha} ( -\delta^{a\alpha}\delta^{b\beta}(\partial^2 + m^2)+y \hat \varphi_1^{a\alpha,b\beta} ) \bar\varphi_1^{b\beta}-
\frac 1 2
\bar\varphi_2^{a\alpha} ( -\delta^{a\alpha}\delta^{b\beta}(\partial^2 + m^2)+y \hat \varphi_2^{a\alpha,b\beta} ) \bar\varphi_2^{b\beta},
\label{Schwinger-Keldysh-fields}
\end{aligned}
\end{equation}
where
$ \hat\varphi^{a\alpha,b\beta}_i:=
f^{abc}\tilde f^{\alpha\beta\gamma} \varphi^{c\gamma}_i.
$
 Hence the particle matrix-valued Hamiltonian associated with each Lagrangian can be read-off from Eq.\eqref{Schwinger-Keldysh-fields}
 \begin{align}
 H_i= \frac 1 2  \left[ \delta^{a\alpha}\delta^{b\beta}(-p_i^2+m^2)-y\hat \varphi_i^{a\alpha,b\beta} \right].
 \end{align}
  Notice that the total Lagrangian \eqref{Schwinger-Keldysh-fields} has a  diagonal matrix structure w.r.t.  to the  indices of $\varphi_i^{a\alpha}$ in addition to the matrix structure due to color indices.

     Its diagonal form implies that the path integral representation of the dressed propagator\footnote{Dressed propagators have been developed in a worldline representation for a variety of models, see Ref.\cite{Edwards:2019eby} and references therein for a review.} on the Schwinger-Keldysh contour is simply
 %
 \begin{align}
G_\Omega:= G_\Omega\big(x_1^i,x_1^f; x_2^i, x_2^f;  \varphi_1,\varphi_2\big) =G_{1} (x_1^i, x_1^f; \varphi_1)
 G_{2} (x_2^i, x_2^f; \varphi_2)\Big|_{x_1^f=x_2^f}\ ,
 \end{align}
 %
where phase space variables $w$ evaluated at initial and final times are written as  $w(t^i)=w^i$, $w(t^f)=w^f$, respectively. Here the variables $x_1$, $x_2$ describe  upper and lower contours with the  boundary condition $x_1^f=x_2^f$.   Inserting a worldline path integral representation for each dressed propagator then leads to
\begin{align}
G_{\Omega}
=
\int \frac{\mathcal D [e_1, e_2]}{\text{vol}(\text {Gauge})}
\int \mathcal D [x_1,x_2]
\int \mathcal D [p_1,p_2]
\ \mathbf T  e^{\ii (S_1-S_2)},
\end{align}
where $\mathbf T$ denotes time ordering and   $\mathcal D[w_1,
w_2]:=\mathcal D w_1 \mathcal D w_2$.
Hence, after introducing complete sets of (initial) states, we find that the evolution of the matrix density operator $\rho^f:=\braket{x_1^f |\rho (t^f)| x_2^f}$ is given by
\begin{align}
\rho^f_\Omega
:=\int\mathrm{d}^4 x_1^i
\int\mathrm{d}^4 x_2^i \ \rho (x_1^i, x_2^i) G_\Omega\,,
\label{def-density-final}
\end{align}
which is to be understood as the dressed propagator evaluated over the Schwinger-Keldysh contour $\Omega$ (see Fig.\ref{SK-contour}) weighted by the density matrix. Closing the contour leads to the real time effective action in Ref.~\cite{Mueller:2019gjj}. This definition is a specialization of the QFT case applied to worldlines, see e.g.,\cite{Gozzi:2010iu, Jeon:2004dh, Jeon:2013zga} .

In the following we will
introduce the auxiliary variables $\psi^a$ and $\phi^\alpha$ to replace time ordering in path integrals following  Ref.\cite{Bastianelli:2021rbt}, where the interested reader can find details.
 The worldline action  in phase-space  for the bi-adjoint scalar in real time $\tau$ is
\begin{align}
S_{\text{BA}} =\int \mathrm{d}\tau
\Big [ p_\mu \dot x^\mu + \ii \bar \psi_a \dot \psi^a  + \ii \bar \phi_\alpha \dot \phi^\alpha
-e H - a J -  {\tilde a}{\tilde  J}  \Big ]  \;,
\label{action1}
\end{align}
where $H, J,\tilde J$ denote first class constraints

\begin{align}
H &  = \frac12 ( -p^2+ m^2 +y\, C^a  \varphi^{a\alpha}(x) {\tilde C}^{\alpha} ) \;, \qquad
C^a= -\ii f^{abc} \bar \psi ^b \psi^c \;,
\quad {\tilde C}^\alpha = -\ii \tilde f^{\alpha\beta\gamma} \bar \phi^\beta \phi^\gamma \;,
\label{ham}\\
J &  =  \bar \psi_a \psi^a -s \;, \qquad
\tilde J   = \bar \phi_{\alpha} \phi^\alpha -\tilde s    \;,
\label{cons2}
\end{align}
which we identify with the particle Hamiltonian, the color charges and the particle current, respectively.
To avoid cluttered expressions we  define  $\mathsf P:=(p_\mu, \ii \bar \psi_a, \ii \bar \phi_\alpha )$, $\mathsf X:= (x^\mu, \psi^a, \phi^\alpha)$ and  group together the first class constraints  into $\mathsf e \mathsf H^t $, where $\mathsf e  := (e,  a , \tilde a)$ and
 $\mathsf H:= (H, J, \tilde J)$.  Here $\mathsf A^t$ denotes transpose of $\mathsf A$\footnote{Bold  letters $\mathsf A$, $\mathsf B$,  denote  matrices.}. In condensed notation the worldline action is then
 %
 \begin{align}
 S_{\text{BA}}:=\int \mathrm{d}\tau\ \left(\mathsf P \dot {\mathsf X}^t -\mathsf e \mathsf H^t\right).
 \end{align}

  \begin{figure}
  	\centering
  	\begin{tikzpicture}[thick,decoration={%
	markings,%
	mark=at position 2.5cm with {\arrow[black]{stealth};}}]
\draw[->] (-2.,0) -- (2,0)  node[anchor=north west] {$t$};
\draw[->] (0,-.5) -- (0,1) node[anchor=south east] {};
\draw(-1.,0.1) -- (1.,0.1)  node[anchor=south west ] {$\Omega_1$};
\draw(-1.,-0.1)node[anchor=north west] {$\Omega_2$} -- (1.,-0.1);
\draw(1.,0.1) -- (1.,-0.1)node[below ]{$t^f$};
\filldraw [black] (-1,0.1) circle [radius=.6pt]node[above ]{$t^i$} ;
\filldraw [black] (-1,-0.1) circle [radius=.6pt];
\end{tikzpicture}
  	\caption{Representation of the Schwinger-Keldysh path. The small horizontal line at $t_f$ does not contribute to the path integral. Phase-space variables in the upper contour and lower contour  are labeled as $w_1$ and $w_2$, respectively.}
  	\label{SK-contour}
  \end{figure}
 %
 We may now introduce color variables that
exchange time-ordering by an additional integration over auxiliary variables $x,\psi,\phi$. The dressed propagator then
reads\footnote{In the path integral measures it is understood that only the variables in $\mathsf P$ are considering omitting factors of $\ii$.}
\begin{align}
G_\Omega[\mathsf X_1^i, \mathsf X_2^f;\varphi_1, \varphi_2] =
\int \frac{\mathcal D [\mathsf e_1, \mathsf e _2]}{\text{vol}(\text {Gauge})}
\int \mathcal D [\mathsf X_1,\mathsf X_2]
\int \mathcal D [\mathsf P_1, \mathsf P_2]
\ e^{\ii (S_{\text{BA},1}-S_{\text{BA},2})}\, .
\end{align}

   Setting $\mathrm{e}=2T$, were $T$ is the so-called Schwinger proper time, produces the usual representation of the dressed propagator but the message of Ref.\cite{Mueller:2019gjj} is to keep the einben unfixed to perform the path integral, which we will also do here.
 Hence the
path integral representation of the effective action is
 %
 \begin{align}
\Gamma_\Omega[\varphi_1, \varphi_2]= \int\mathrm{d} \mathsf X_1^i
 \int\mathrm{d} \mathsf X_2^i \ \rho (\mathsf X_1^i, \mathsf X_2^i) \int \frac{\mathcal D [\mathsf e_1, \mathsf e _2]}{\text{vol}(\text {Gauge})}
 \int \mathcal D [\mathsf X_1,\mathsf X_2]
 \int \mathcal D [\mathsf P_1, \mathsf P_2]
 \ e^{\ii (S_{\text{BA},1}-S_{\text{BA},2})},
 \end{align}
 where $\mathrm{d} \mathsf X:= \mathrm{d}^4 x\, \mathrm{d} \psi \mathrm{d} \bar \psi \,\mathrm{d} \phi \mathrm{d}\bar \phi$.
 Now we introduce
 Schwinger-Keldysh (SK) coordinates
 %
 \begin{align}
 z_R:= \frac{1}{2}(z_1+z_2),
 \qquad
 z_A:= z_1-z_2,
 \end{align}
 for all phase-space coordinates. We interpret $z_A$ as a quantum degree of freedom measured in units of $\hbar$. We can then obtain the so-called  truncated Wigner approximation by expanding in powers of $\hbar$
 %
 \begin{align}
 S_{\Omega, \text{BA}}=
 \int \mathrm{d} \tau \left(\mathsf P_R \dot {\mathsf X}_A^t +\mathsf P_A\dot {\mathsf X}_R^t -\mathsf e_R \mathsf H_A^t-\mathsf e_A \mathsf H_R^t\right),
 \end{align}
 where we have kept  orders up to $\mathcal O (z_A)$ and we have defined the following quantities:
 \begin{equation}
\begin{aligned}
\mathsf H_R=&\left(  \frac 12(-p_R^2+m^2+ y
C_R^a \varphi_R^{a\alpha} \tilde C_R^\alpha),\quad J_R, \quad \tilde J_R\right), \\
\mathsf H_A=&\left( ( - p_A \cdot p_R+y \frac 14 x_A^\mu
C_R^a \partial_\mu\varphi_R^{a\alpha}
 \tilde C_R^\alpha), \quad \bar \psi_{Aa}   \psi_R^a+ \bar \psi_{Ra}   \psi_A^a,\quad
\bar \phi_{A\alpha}  \phi_R^\alpha+ \bar \phi_{R\alpha} \phi_A^\alpha \right).
\end{aligned}
 \end{equation}
Notice the absence of mixing terms involving $\bar \psi^b_R \psi^c_A$ or
$\bar \phi^\beta_R \phi^\gamma_A$, which vanish due to the antisymmetry of the structure constants. Using the e.o.m we can rewrite the action as:
\begin{align}
 S_{\Omega, \text{BA}}=
 \mathsf P_R^i\cdot  {\mathsf X}_A^{i,t}-\mathsf e_A \mathsf H_R^t-\int \mathrm{d}\tau \, \left(\dot{\mathsf P}_R+ \frac {\partial H}{ \partial \mathsf X_A^t}\right) \mathsf X_A^t+
 \int \mathrm{d}\tau\,  \mathsf P_A\left(\dot {\mathsf X}_R^t-\frac{\partial H}{ \partial {\mathsf P}_A}\right),
\end{align}
where we notice that the boundary terms obtained through integration by parts  produce the
Wigner function
 %
\begin{align}
W(\mathsf X^i_R,\mathsf P^i_R):
=
 \int \mathrm{d} \mathsf X_A^i  e^{\ii \mathsf  P_R^i \cdot {\mathsf X}_A^i}
 \rho (\mathsf X_R^i+\frac 1 2\mathsf X_A^i,
 \mathsf X_R^i-\frac 1 2\mathsf X_A^i).
\end{align}
Finally, the Schwinger-Keldysh real-time effective action for the bi-adjoint scalars is
\begin{equation}
\begin{aligned}
\Gamma_\Omega= \int &
\mathrm{d} \mathsf X^i_R \int \mathrm{d} \mathsf P^i_R \,
W(\mathsf X^i_R,\mathsf P^i_R) \\
&\times \int \mathcal D \mathsf e_R \mathcal  \int  \mathcal  D \mathsf X_R
\int
\mathcal  D \mathsf P_R
\prod\limits_\tau
\delta(P_R^2-m^2)
\delta (J_R) \delta (\tilde J_R)
\delta( \dot {\mathsf P}_R
- \dot {\bar {\mathsf P}}
)
\delta(\dot {\mathsf X}_R-\dot {\bar {\mathsf X}}),
\end{aligned}
\end{equation}
where $\bar {\mathsf X}$ and $\bar {\mathsf P}$ satisfy classical equation of motion.
 Following \cite{Mueller:2019gjj},
the Liuoville's equation then reads
\begin{align}
\frac{\mathrm{d} }{\mathrm{d} \tau} W (\mathsf X, \mathsf P)= \left(\dot {x}^\mu \frac{\partial}{\partial x^\mu} +\dot {p}^\mu \frac{\partial}{\partial {  p}^\mu}+\dot { \psi}^a \frac{\partial}{\partial {\bar  \psi}^a}+
\dot { \bar \psi}^a \frac{\partial}{\partial {\bar  \psi}^a}+\dot {\phi}^a \frac{\partial}{\partial {\phi}^a}+
\dot {\bar \phi}^a \frac{\partial}{\partial {\bar  \phi}^a}\right)  W (\mathsf X, \mathsf P)=0
\end{align}
for some given initial condition $W(\mathsf X_R^i, \mathsf P_R^i)$. Let us mention that in
the field theory case,
the appearance of the on-shell delta function is required to conserve momentum at each vertex
\cite{Jeon:2004dh}.

\subsection{Vlasov-type equation for bi-adjoint scalars}
For our purposes it would be more convenient to work directly with the classical limits of the color charges $C^a$ and $\tilde C^\alpha$
instead of the auxiliary ones. In the classical limit the charges defined in Eq.\eqref{ham}  correspond to classical color charges for large representations in a coherent state basis. These are controlled by $s$ and $\tilde s$ in \eqref{cons2} as can be seen by performing explicit path integration over auxiliary variables after gauge fixing (See appendix A of \cite{Ahmadiniaz:2015xoa}).   Therefore in the classical limit we may simply set
\begin{align}
 C^a \to c^a, \qquad \tilde C^\alpha \to c^\alpha.
 \label{Ctoc}
\end{align}
We can derive classical kinetic equations from the action \eqref{action1} setting $a=\tilde a=0$ and  $e=-1$ (so $p=\dot x$). We obtain
\begin{align}
-\dot p^\mu + \frac{ y }{2} c^a \tilde c ^\alpha \partial^\mu \varphi^{a \alpha}=0, \\
\ii  \dot \psi ^a +  \frac y {2} \frac{\partial C ^{c}}{\partial \bar \psi _a} \varphi^{c \alpha}
\tilde C^\alpha=0, \\
\ii  \dot {\bar \psi} ^a -\frac y {2} \frac{\partial C ^{c}}{\partial \psi _a} \varphi^{c \alpha}
\tilde C^\alpha=0,
\end{align}
with an additional pair of equations for $\phi^\alpha$.
Since $\dot C^a= -\ii f^{abc}(\dot{\bar \psi}^b \psi^c +\bar \psi^b \dot{\psi}^c)$, we may pack together the last two equations leading to
\begin{align}
\dot c^a=&   \frac y 2 f^{abc} c^b \tilde c^{\alpha}
\varphi^{c \alpha},
\end{align}
where we have used the the Jacobi identity. The additional equation for $\tilde c^\alpha$ can be obtained similarly.
Therefore the Liouville's equation
$\mathrm{d} f/\mathrm{d}\tau=0$ for the
semi-classical distribution function $f(x, p, c, \tilde c)$ leads to the the Vlasov-type equation for bi-adjoint scalars
\begin{align}
\left(p^\mu \frac{\partial }{\partial x^\mu}+ \frac 12
y c^a \tilde c ^\alpha \partial^\mu \varphi^{a \alpha}
\frac{\partial }{\partial p^\mu}
+ \frac 12 y f^{abc}  c^b \tilde c^{\alpha}
\varphi^{c \alpha} \frac{\partial }{\partial  c^a}
+\frac 12 y f^{\alpha \beta \gamma } \tilde c^\beta c^{a}
\varphi^{ a \gamma} \frac{\partial }{\partial \tilde c^{\alpha}}\right)f=0.
\label{Vlasov-bi-adjoint}
\end{align}
Perturbative thermal currents can be obtained from
\begin{align}
J^{a \alpha}(x)=y \int \mathrm{d} \Phi (p)\int
\mathrm{d} c  \int \mathrm{d} \tilde c \ c^a \tilde c^\alpha
f(x, p, c, \tilde c),
\label{Vlasov-current}
\end{align}
where the color phase space invariant measure is defined in Eq.\eqref{DIPSC}.

\subsubsection*{Example}
Let us  check the consistency of the semi-classical kinetic theory just constructed for  the 2-point thermal current. Expanding the distribution function around equilibrium we have
\begin{align}
f(x,p, c, \tilde c)= f^{(0)}(p_0)+ y f^{(1)}(x,p,c,\tilde c)
+ \dots,
\label{expansion-bi-adjoint}
\end{align}
where $f^{(0)}(p_0)=1/(e^{\beta p_0}-1)$ is the Bose-Einstein distribution function and $\beta=1/T$. Moving to Fourier space and
 plugging in Eq.\eqref{expansion-bi-adjoint} into \eqref{Vlasov-bi-adjoint} we find
\begin{align}
\tilde f^{(1)}(k,p, c, \tilde c )=- c^a \tilde c^\alpha
\frac{k^\mu}{k\cdot p}
\frac{\partial f^{(0)}}{\partial p^\mu} \tilde \varphi^{a \alpha} (k),
\end{align}
where $\tilde f^{(1)}(k,p, c, \tilde c )$ denotes the Fourier transform of $f(x,p, c, \tilde c)$. Inserting this equation into the definition of the current  we obtain
\begin{align}
\tilde J^{a\alpha}= \frac{y^2}{2}  \int \mathrm{d} \Phi (p)  \int
\mathrm{d} c \int \mathrm{d} \tilde c \ c^a \tilde c ^\alpha
c^b \tilde c ^\beta
\ f^{(0)} (p_0) \frac{k^2}{ (k\cdot p)^2} \varphi^{b \beta },
\label{current-LO}
\end{align}
which leads to
\begin{align}
\Pi^{ab, \alpha\beta}(k)=\delta^{ab} \delta^{\alpha\beta}\frac{ C_2^2 y^2}{2}  \int \mathrm{d} \Phi (p)
\  f^{(0)} (p_0) \frac{k^2}{ (k\cdot p)^2},
\end{align}
where we have used the identities in Eq.\eqref{identities-phase-space}. This is reproduces exactly the result we obtained from QFT  in Eq.\eqref{2-point-bi-adjoint}.

\section{Spin}
\label{gauge-spin}
Classical color charges  and  spin vectors have in common that they may be associated with expectation values of certain operators  with respect to coherent states. Indeed, provided one is able to build coherent states that furnish an irreducible representation of the Lie group, say $SU(N)$, then
the following properties   hold \cite{Yaffe:1981vf}
\begin{align}
\braket{\psi|\mathbb A|\psi}&= \text{finite},\\
\braket{\psi|\mathbb A \mathbb B|\psi}&= \braket{\psi|\mathbb A|\psi} \braket{\psi|\mathbb B|\psi} + \dots \,,
\end{align}
where the ellipsis means terms that do not contribute in the classical limit.
In particular, one may use the Schwinger-boson formalism for $SU(N)$ to build
explicit realizations
 of these states and  show the above properties. For
color this construction was used in Ref.\cite{delaCruz:2020bbn} to describe colored observables  while for spin in Ref.~\cite{Aoude:2021oqj}.
Therefore one expects that the dynamics of color charges and spin share  some similarities as studied in Refs.~\cite{delaCruz:2021gjp,Maybee:2019jus, Cristofoli:2021jas}.

The description of classical spin may also be done by
introducing (integer) higher-spin massive particles  described by symmetric traceless rank-$s$ tensor fields $\varphi_s^{a_1\cdots a_s}$ \cite{Singh:1974qz}, where for brevity we will suppress its indices henceforth. Lorentz generators also carry (symmetrized) sets of indices, which we will also suppress  but use  matrix notation
$\mathsf M^{\mu\nu}$ to indicate that the contraction of indices is  understood as matrix multiplication.
Let $\varepsilon(p):=\varepsilon (s,p)$ be the   polarization tensors of the massive particle with momentum $p$. Denoting by $\varepsilon (\tilde p)\cdot
\varepsilon (p)$ the contraction of the tensor indices of the polarization tensors, the relation between classical spin tensors and Lorentz generators $\mathsf M^{\mu\nu}$ is given by  the identities
\begin{equation}
\begin{aligned}
\varepsilon (\tilde p)
\mathsf M^{\mu\nu} \varepsilon ( p)=&
S^{\mu\nu}
\varepsilon (\tilde p)\cdot
 \varepsilon (p) +\dots,\\
\varepsilon (\tilde p) \{ \mathsf M^{\mu\nu}, \mathsf M^{\rho \sigma}\}  \varepsilon ( p)&=S^{\mu\nu}S^{\rho\sigma}\varepsilon (\tilde p)\cdot
\varepsilon (p) + \dots,
\end{aligned}
\label{spin-identities}
\end{equation}
where $\{\mathsf A, \mathsf B\}:= \frac 12(\mathsf A \mathsf B+\mathsf A \mathsf B)$ and $\tilde p:=p-q$. We refer the interested reader to Ref.\cite{Bern:2020buy}  for details on the spin formalism.
The ellipsis denote terms that do not contribute in the classical limit. The spin vector and spin tensor are related through Eq.\eqref{spin-tensor-to-vector}. The spin vector is constrained by the so-called covariant spin supplementary condition
\begin{align}
p_\mu S^{\mu\nu}=0.
\label{spin-supplementary-condition}
\end{align}
\begin{figure}[tb]
	\centering	\lotop{$A^\mu$}{$A^\nu$}{black}
	\lotopB{$A^\nu$}{$A^\mu$}{black}
	\lotopC{$A^\mu$}{$A^\nu$}{black}
		\caption{2-point graphs that contribute to the current. Besides the usual interactions in scalar QED, the higher-spin model allows interactions proportional to Lorenz generators. }
	\label{topos-spin}
\end{figure}
\subsection{Spinning off-shell currents}

Let $F^{\mu\nu}$ be the usual Maxwell field strength and $D^\mu$ the corresponding covariant derivative. The Lagrangian density of the higher-spin scalar electrodynamics is given by
\cite{Bern:2020buy}
\begin{align}
\mathcal L_{\text{EM}} =-\frac 1 4 F^{\mu\nu} F_{\mu\nu}+ D^\dagger_\mu \bar \varphi_s D^\mu \varphi_s-m^2\bar \varphi_s \varphi_s+
e  F_{\mu\nu} \bar \varphi_s \mathsf M^{\mu\nu}\varphi_s,
\label{lagrangian-spin}
\end{align}
where $\varphi_s$ are higher-spin fields.
 Using the double copy, this theory has been used recently in the context of scattering amplitudes to describe the Post-Minkowskian two-body problem of spinning particles in gravity \cite{Kosmopoulos:2021zoq, Bern:2022kto}.
Classical currents including spin contributions may be  computed from Eq.\eqref{off-shell-current} with $\widehat {\text{Tr}}(\bullet)$ now defined as
\begin{align}
\widehat{\text{Tr}}(\bullet):= \text{Tr}_s (\bullet),
\end{align}
where by construction satisfies the same properties of phase-space integration over classical spin given in Eq.\eqref{identities-phase-space}. E.g., $\text{Tr}_0(\bullet)=\text{Id} (\bullet)$ is equivalent to $\int \mathrm{d} s=1 $ classically.
We can now use Eq.\eqref{off-shell-current} to perform calculations. Notice that in the forward limit we may set  $\varepsilon (p) \cdot \varepsilon (p)=1$.

For instance, consider the 2-point current (See Fig.\ref{topos-spin}). It receives contributions from the same diagrams as in scalar QED but has, in addition, contributions  coming from the last term in Eq.\eqref{lagrangian-spin}.
The off-shell current has the structure
\begin{align}
 \mathcal A^{\mu\nu}(p,k)=\varepsilon_s (p)\cdot   A^{\mu\nu} (p,k) \cdot \varepsilon_s (p)
\end{align}
 However from the properties of the trace (or equivalently from the phase integration) we see that
only those  terms appearing quadratically may lead to meaningful contributions. Keeping this in mind,
 the current now can be written as follows
\begin{align}
 A^{\mu\nu}(p,k)= A^{\mu\nu}_{0} (p, k) + A _{s}^{\mu\nu}(p, k),
 \label{def-current}
\end{align}
where the scalar part  is the current in scalar QED
\begin{align}
\ii A^{\mu\nu}_{0} (p, k)= 2 \ii e^2 \left[ \eta^{\mu\nu} -\frac{\left(k^{\mu }+2 p^{\mu }\right) \left(k^{\nu }+2 p^{\nu }\right)}{2 \left(2 k\cdot p+k^2\right)}-\frac{\left(k^{\mu }-2 p^{\mu }\right) \left(k^{\nu }-2 p^{\nu }\right)}{2 \left(k^2-2 k\cdot p\right)}\right],
\end{align}
which in the classical limit leads to
\begin{align}
A_0^{\mu \nu}(k,p):= 2e^2\overline {\Pi}^{\mu\nu}_0= 2e^2\left( \eta^{\mu\nu}-\frac{p^\mu k^\nu+p^\nu k^\mu}{k\cdot p}+\frac{k^2 p^\mu p^\nu}{(k\cdot p)^2}\right).
\label{current-QED-spin}
\end{align}
On the other hand, the spin-dependent contribution is
\begin{equation}
\begin{aligned}
\ii A _{s}^{\mu\nu}(p, k)=-2\ii e^2\Bigg\{ \left[
\frac{(2p+k)^\mu}{2p\cdot k+k^2}+
\frac{(2p-k)^\mu}{-2p\cdot k+k^2}\right]
\ii (k\cdot \mathsf M)^\nu-\mu\leftrightarrow\nu
\\
+k_\alpha k_\beta  \left(
\frac{\mathsf  M^{\alpha\mu}\mathsf M^{\beta\nu}}{2p\cdot k+k^2}+
\frac{\mathsf M^{\beta\nu}\mathsf M^{\alpha\mu}}{-2p\cdot k+k^2}\right)
\Bigg\},
\end{aligned}
\end{equation}
where $k_\mu \mathsf M^{\mu\nu}:=(k\cdot \mathsf M)^\nu$.
Now, keeping only contributions quadratic in spin and using Eq.\eqref{spin-identities}, we find that in the classical limit\footnote{In KMOC the spin vectors are of $\mathcal O(\hbar)$ and so spin-squared terms appear as terms of $\mathcal O (\hbar^2)$. The classical result shown here  it is understood as the expectation value w.r.t coherent states \cite{Aoude:2021oqj}.}
\begin{align}
 A _{s}^{\mu\nu}(p, k)|_{S^2}=2  e^2 \frac{k^2 }{(p\cdot k)^2}(k \cdot S)^\mu
(k \cdot S)^\nu,
\end{align}
which satisfies the Ward identity $k_\mu A_s^{\mu\nu} (p, k)|_{S^2}=0$. To reach this form we have also used the usual algebra  of $\mathsf M^{\mu\nu}$.
We will see in Section \ref{classical-comps} that this matches the result computed from a classical  perspective.  Computing the trace, or equivalently the phase space integration, we obtain
 %
 \begin{align}
 \text{Tr}_s( A_s^{\mu\nu}(p,k))=-\frac{4 e^2 \mathfrak s^2}{3}k^2 \left( \frac{1}{m^2}\overline{\Pi}^{\mu \nu}_0-
 \frac{k^2\eta^{\mu\nu}-k^\mu k^\nu}{(k\cdot p)^2}\right).
 \end{align}
Therefore the off-shell current in the classical limit including spin contributions is
\begin{align}
\bar {\mathcal A}^{\mu\nu} (p,k)
=2 e^2 \left[\left( 1-\frac{2\mathfrak s^2 k^2}{3m^2}\right)\overline{\Pi}_0^{\mu\nu}+  \left( \frac{2\mathfrak s^2 k^2}{3}\right) \frac{\overline{N}^{\mu\nu}}{(k\cdot p)^2}
\right],
\label{res-spin-2-point}
\end{align}
where $\overline{ N}^{\mu\nu}:=k^2 \eta^{\mu\nu}-k^\mu k^\nu$.

A few comments are in order. Recall that we still need to be integrate over phase space. Notice that the numerator $\overline{N}^{\mu\nu}$ in  second term in parenthesis also appears in the renormalization of QED.
Let's perform  the analytic continuation $k_0 \to k_0+\ii o$  as usual and  consider the high temperature regime. Since $\overline{N}^{\mu\nu}$ is independent of the momentum $p$ we require the integral
\begin{align}
\int \mathrm{d} \Phi(p)\ \frac{f_{+}^{(0)}(p_0)}{(k\cdot p)^2}=\dfrac{1}{16\pi^2 k^2} \left[-\dfrac{1}{\epsilon}+\dfrac{k_0}{|\bm k|} \log \left(\frac{|\bm k|+k_0+\ii o}{-|\bm k|+k_0+\ii o} \right)+\log(\beta^2 \mu^2)\right]+\mathcal{O}(\epsilon),
\end{align}
which has been evaluated in $d=4-2\epsilon$ (see Appendix \ref{integrals}). The temperature-independent divergence cancels after subtracting the zero temperature contribution while the temperature dependent log  may be combined with the $\log (m^2/\mu^2)$  in the renormalization of $\Pi^{\mu\nu}$. Notice also that Eq.\eqref{res-spin-2-point} is strictly valid only where $m\ne 0$. The reason is that the spin supplementary condition $p_\mu S^{\mu\nu}=0$ can no longer be used as a condition that fixes $S^{\mu\nu}$ uniquely as the spin tensor in the rest frame of the particle since there is no such frame for massless particles. Instead, for massless particles  one can choose a frame characterized by a
time-like vector $u^\mu$  and set $p^\mu \to u^\mu$
in Eq.\eqref{spin-supplementary-condition}  \cite{Weickgenannt:2019dks, Chen:2015gta}.
\subsection{Comparison with semi-classical kinetic theory}
\label{spin}
\label{classical-comps}
In order to check the validity of our approach let us now consider a  classical perspective.
The generic form of the collisionless relativistic Boltzmann-Vlasov equation reads
\cite{Israel:1978up}
\begin{align}
p^\mu \frac{\partial f}{\partial x^\mu}
+e \frac{\partial ({F}^\mu f)}{\partial p^\mu}=0,\quad
F^\mu=
F^{\mu\nu}p_\nu+ \frac{\partial F^{\nu\rho}}{
	\partial x_\mu} S_{\nu\rho} \,.
\label{vlasov-spin-1}
\end{align}
This equation can be derived within Wigner function formalism applied for spin $\frac 12$ particles \cite{Weickgenannt:2019dks, Gao:2019znl}.
Now, as in the spin-less case, let us perturb  the distribution function $f$ around equilibrium
\begin{align}
f= f^{(0)}(p_0)+e f^{(1)} (x,p, S)+
e^2 f^{(2)}(x,p, S)+\dots \, ,
\end{align}
where $f^{(0)} (p_0)$ is the Fermi-Dirac distribution function. We are interested in computing the associated current
\begin{align}
J^\mu(x)= e
\int \mathrm{d} \Phi(p) \int \mathrm{d} s\  (p^\mu +S^{\mu\nu}\partial_\nu) f(x, p, S) \,.
\label{current-spin-QED}
\end{align}
Solving for the coupled system of Eq.\eqref{vlasov-spin-1} and Eq.\eqref{current-spin-QED} in momentum space, we obtain
\begin{align}
\tilde f^{(1)}(k,p,S)=
\frac{\ii }{k\cdot p}\left[ k^\mu (k^\nu \tilde A ^\rho-k^\rho \tilde A^\nu ) \frac{\partial ( f^{(0)} S_{\nu\rho}) }{\partial p^\mu }+\ii p_\nu (k^\mu \tilde A^\nu-k^\mu \tilde A^\nu)\frac{\partial  f^{(0)} }{\partial p^\mu }
\right].
\end{align}
Plugging in this into the Eq.\eqref{current-spin-QED} and using integration by parts we can bring the current into the form
\begin{align}
\tilde J^{\mu,(1)} (k)= e^2 \int \mathrm{d} \Phi(p)\mathrm{d} s \, \tilde{f}^{(0)} (p_0) \left[\overline{\Pi}^{\mu \nu}_0(k,p)+\overline{\Pi}_s^{\mu \nu}(k,p)\right]\tilde A_{\nu} \,,
\end{align}
where the spin-dependent contribution is
\begin{align}
\overline{\Pi}^{\mu \nu}_s(k,p)=&\frac{2k^2}{(k\cdot p)^2} (k\cdot S)^\mu(k\cdot S)^\nu,
\end{align}
and $\overline \Pi_0^{\mu \nu}(k,p)$ is given in Eq.\eqref{current-QED-spin}. This matches exactly our results obtained using classical limits of off-shell currents.

\section{Conclusions}
\label{conclusions}
We have extended the scattering amplitudes approach of Ref.\cite{delaCruz:2020cpc} in two ways: we have considered thermal currents for bi-adjoint particles and  spin.  To test its  validity, we have compared against computations based on iterative solutions of classical kinetic equations finding  agreement.
The semi-classical kinetic equations for bi-adjoints were derived from the worldline approach in Sec.\ref{Bi-adjoint} following   Mueller-Venugopalan \cite{Mueller:2019gjj}, who have shown that in the classical limit kinetic equations follow from the Schwinger-Keldysh effective action adapted to worldlines. For the case of spin we have used the well-known Boltzmann-Vlasov equations for spinning particles. On the amplitudes side,  we have modeled classical spin through a higher-spin generalization of scalar QED.

Hard thermal loop actions can
be computed from other methods including the worldline approach as found long time ago \cite{Venugopalan:2001hp,Jalilian-Marian:1999uob}. Moreover,
the worldline setting of Ref.\cite{Mueller:2019gjj} suggests that these  may  also be derived from the recently proposed Worldline QFT approach \cite{Mogull:2020sak,Jakobsen:2021smu,Jakobsen:2021lvp, Jakobsen:2021zvh,Shi:2021qsb, Jakobsen:2022psy}. Therefore it would be interesting to investigate whether both thermal and classical observables  can be  unified in a worldline framework.
 Having introduced spin into the formalism of  Ref.\cite{delaCruz:2020cpc} the remaining task is to include collision functions. These have been studied in the context of classical transport in Ref.\cite{Litim:1999ns, Litim:1999id, Litim:1999ca}. We leave this for future work.

\addsec{Acknowledgements}
 We thank financial support from the Open Physics Hub at the Physics and Astronomy Department in Bologna. Some of the
calculations in this paper were done with Feyncalc \cite{Mertig:1990an, Shtabovenko:2016sxi, Shtabovenko:2020gxv}.
We thank Fiorenzo Bastianelli and Francesco Comberiati for discussions and for collaboration on related projects.
We thank Niklas Mueller for clarifications regarding Ref.\cite{Mueller:2019gjj}. We also thank Andr\'es Luna and Maria Elena Tejeda-Yeomans for comments on a previous version of this manuscript.

redpajama_set_name: RedPajamaArXiv


text:
Sep 2005 / expatriates :: email this story to a friend
The Reach of Home
By Aaron Shumaker
"Where are you from?" It's a question that I'm frequently asked. For one thing, it's one of the few questions that most of my Japanese students know how to ask. In addition, people are in the habit of making inferences about character based on one's origins, so it's only natural that there is a fair amount of curiosity about one's home town. For me, however, the answer to that question is not so simple. I've lived in Minneapolis, Seattle, Grand Rapids and Austin in addition to St. Louis, and now I live in a beautiful, small town in Japan.
When I first arrived here about a year ago, fresh from graduating from Washington University, I would answer the question with the long list of places I've lived. I quickly realized, however, that most of my day would be spent explaining my life to people if I continued in this manner. So, instead of having a tri-fold brochure printed up, I decided I should pick a place as my home. Now, when people ask me where I'm from, I definitively say, "St. Louis."
Instinctively, it was easy to choose St. Louis as my home, but I started wondering why I felt that way. St. Louis certainly isn't the best city I've lived in. Seattle has a much more vibrant downtown area, Austin's live music scene is more prolific, and Grand Rapids....well, St. Louis is definitely better than Grand Rapids (my apologies to any Grand Rapidians out there). But the truth is, St. Louis isn't necessarily the city I like the most. In fact, I don't think I would tell any of my students to visit St. Louis if they were going to visit America. Don't misunderstand, there are many things I love about St. Louis, and I would show someone a good time if they came, but it wouldn't be on my U.S. highlights tour.
Still, I can think of no other place that I would call home. I lived in St. Louis during two very different periods of my life. From age 7 to 13 I lived in Ellisville out in West County. It was very positive place to grow up and I still carry with me very fond memories of spending summers in Bluebird Park and watching my first baseball game at Busch Stadium. Following a move to Grand Rapids, MI, where I went to high school, I returned to St. Louis as a student at Washington University. The campus was in a very different neighborhood than West County but it still had a profound, and positive, effect on me. Wash U is where I became an adult, where I learned to live on my own and explore the world as I saw fit. I learned to be myself on the streets of The Loop, and learned how to maintain my inner child at the City Museum.
And that is why St. Louis is my home. It defines who I am. It is the place, more than any other, that has served as the backdrop for the most important and most character-shaping moments in my life. It's part of who I am now, whether I like it or not. It's like a family member you may not always be proud of, but whom you love nonetheless. I have an inescapable connection with the city, because I see myself as a St. Louisan. The other places I've lived have never put as much of an imprint on my character as St. Louis has. It was a subtle connection that was all around me, and I didn't notice it until I was gone.
In fact, when I left St. Louis for Japan, I didn't necessarily think of St. Louis as my home. However, I found myself in bars here looking proudly at the Budweiser they were pouring, and being overjoyed to see the Cardinals in the sports section of the Japanese newspaper. It wasn't until I was in another country that I really started to feel like St. Louis was home. In a place where I was inherently different, it suddenly became much more important to be able to define who I was. Here in Japan, I found that I see myself as being from St. Louis, being of the culture of the Gateway to the West.
Here in Japan, I've also found that this connection means something. True, many people look at me with blank faces when I say I'm from St. Louis, but you can be sure they know about Budweiser. Japan is a country that loves to drink, after all, and though Bud is like tap water in St. Louis, it's foreign and exotic over here. Also, I live in Nagano prefecture, which happens to be Missouri's sister state, so many people here have actually met people from St. Louis while they were on an exchange program. In addition, there seems to be general interest in their sister state here, as I was asked to give a presentation about St. Louis to a group of tour guides in a nearby city. Though at first the thought of giving a 2-hour-long presentation on St. Louis seemed a bit much, I found myself truly enjoying telling people about my home. Just talking about it seemed to make it feel a little closer.
I love St. Louis not because of its superlative excellence, but because of what it has meant to me. It's shaped and defined me in ways that I never realized until I was so far away. It's the basis of comparison I use for everything in Japan. When I go into a grocery store here, I compare it to Schnucks, not a grocery store from Austin. I get excited about seeing Budweiser much more than other American beers. I am much happier talking about the Arch than the Space Needle. It's because St. Louis is the place that connects me to the States and to home. I feel it reaching out to me even so far away. It's a connection I will carry with me always, and a connection I am happy to have.
Aaron Shumaker is a graduate of Washington University in St. Louis and is currently enjoying his time teaching English in Toyoshina, Japan.

redpajama_set_name: RedPajamaCommonCrawl


"The lamp stands out with a design that is both filigree and clear. Letting the light glow through its pattern structure, it creates a gentle, stylish atmosphere in any room."
A new look for flower pots - using a modular system of concrete rings and a steel frame (top).


text:
Q: Using virtualenv with Sublime Text 3 and SublimeREPL I'm trying to setup ST3 to work with Python's virtualenv, running on Windows 8.1. I usually use SublimeREPL with my global Python install to run files. Now that I'm using venvs, though, I'm having trouble getting anything to run. Here's what I've been trying:
I have a parent directory with a folder virtualenvs, then one scripts for my .py files that I muck around with. I usually just navigate to \virtualenvs\venv\scripts\activate and do my work using the python interpreter, but I'd like to be able to build the files without needing to go through command line stuff, using ST3 and SublimeREPL.
I made a build system that looks like this:
{
    "shell_cmd": ["\code\virtualenvs\venv\scripts\python.exe", "-u", "$file"],
    "file_regex": "^[ ]*File \"(...*?)\", line ([0-9]*)",
    "selector": "source.python",
}

But I'm not even sure if this is what I need to do to use SublimeREPL, as I've never had to worry about Tools > Build Systems before, just Tools > SublimeREPL > Python > Python - RUN current file.
How can I change the build system or customize SublimeREPL's RUN? For a test case, I have the requests module installed in the venv but not in my global python install. Importing it from command line works fine but a .py file with import requests returns an ImportError.

A: *

*install 'virtualenv' package in sublime editor

*Then goto to build system and set it as 'python + virtualenv'

3.Now go to the sublime project file and add path to the virtualenv
"virtualenv":"D:/my_projects/scrapping_env",

*To add Test build system add the following in the .sublime-project settings
"build_systems": [
  {
   "name": "Test",
   "shell_cmd": "D:/my_projects/scrapping_env/Scripts/python py.test"
  },
],

*Select Test build sytem from the tools as shown

A: The ST3 build system and the SublimeREPL provide two different pieces of functionality. The build system is something you would typically use when working with a compiled language like C. It allows you to define how you want to build your program from the source files.
You can also use the build system with Python. In this case, it would compile for you the file which is currently open.
The SublimeREPL on the other hand, allows you to run a terminal session inside Sublime and also (among other things) evaluate the files within the context of that session. I would suggest you take a look at the documentation for SublimeREPL and Python:

Both stock Python and Execnet integrations support virtualenv. Various ways to work with Python, including PDB and IPython, are supported.
For virtualenv created environments to be discoverable by SublimeREPL they should be created or symlinked in one of the following:

*

*~/.virtualenvs default for virtualenvwrapper

*~/.venvs default for venv

  Alternatively, more paths can be added to "python_virtualenv_paths" in the SublimeREPL configuration file.

redpajama_set_name: RedPajamaStackExchange


Renowned for her expertise in all things home design, the journalist and stylist Mieke ten Have, formerly of Vogue and Elle Décor, turns her editor's eye to the world of jewelry. Here, the writer, who contributes to such publications as the Wall Street Journal and British House & Garden, and the co-founder of Panier Home, shares her jewelry favorites – from the pieces she wears every day to her ultimate Memo picks.
What is exciting you at the moment in the home and design industry? I've been noticing a return to layers, color, history, and beautiful objects. Design went through a very long anemic period where neutrals and the purging of things were considered fresh and contemporary. I found it terribly boring. I think an appreciation for maximalism is reemerging.
Jewelry and home products both fall under the category of design objects. What relationship do you see between the two? A garment is predicated on how it is worn and how it fits on the subject, in many ways – I think jewelry and design pieces often exist with their own sense of integrity, regardless of the owner. There is a permanence and gravitas to both.
How would you describe your personal style (and what jewelry do you like to wear)? I tend to wear the same pieces day in and out – the four items I wear everyday are a chunky, hammered gold pink spinel ring from Elizabeth Locke, a gold arm band, and a long gold box chain with three pendants – a Roman coin I purchased in Jordan which I had mounted in gold, and two Florentine charms from the 50s, one depicting the god Mercury, the other a little basket of coral pomegranates. And, of course, my wedding ring, which is Edwardian.
As a design and home editor, you have an expertly trained eye when it comes to objects and design. How did your "editor's eye" influence your jewelry selections on Memo? I can't separate my personal design sensibilities from my jewelry predilections. It didn't surprise me that I selected two Neoclassical styles on Memo, the Lalaounis Lions Head Bracelet and the Bahina Amethyst Iolite and Venetian Glass Cameo Earrings. Neoclassicism and some of its offshoots like Federal and Biedermeier are periods I really gravitate towards decoratively and in my own home. I also seem to always return to the sense of proportion in classical design. It is ultimately modern and impervious to trends because it speaks to the way our eyes see harmony.
Tell us about your overall Memo experience. It was a treat to have these treasures arrive at my door and to have an intimate moment with them. The Memo service offers a much more personal experience with jewelry.


text:
See 9,222 scholarships worth $208M
What should every freshman at your school know before they start? (105) Describe the students at your school. (51) What is the stereotype of students at your school? (39) What are the academics like at your school? (37) Here's your chance: Say anything about your college! (36) Is the stereotype of students at your school accurate? (36) What are the most popular student activities/groups? (32) Describe how your school looks to someone who's never seen it. (29) What's unique about your campus? (26) What kind of person should attend this school? (21) Describe your favorite campus traditions. (19) What do you consider the worst thing about your school? Why? (19) What kind of person should not attend this school? (17) What's the one thing you wish someone had told you about freshman year? (15) What's the most frustrating thing about your school? (15) What do you brag about most when you tell your friends about your school? (12) What is the stereotype of students at your school? Is this stereotype accurate? (9) What are your classes like? (4) Why did you decide to go to this school? (4) When you step off campus what do you see? (3) Describe the dorms. (2) Describe a typical weekend. (1) Describe the best and worst parts of the social scene on campus. (1) Tell us about the food and dining options. (1) Where is the best place to get work done on campus? (1)
« See more reviews of Yale University
What is your overall opinion of this school?
it is very hard academically but the effort it takes to do all the academic work and to partake in a varsity sport is worth the large amount it takes
Yale is a wonderful place to go to school- I honestly could not be happier. There is definitely a lot of pressure and a lot of work, but it is also incredibly fun. The people are the best thing about the school- I never stop meeting fascinating people who have accomplished amazing things and yet are very down-to-earth. New Haven gets a bad reputation, but personally, I love it (I'm actually staying here over the summer). The clothing stores are mostly expensive, but there are tons of great restaurants, coffee shops, clubs, galleries, theater, etc. It is a city with a lot of history and character. Yalies are all basically in love with their school- it's just a very happy place to be. I would describe it as a "work hard, play hard" kind of place:" people are driven and devoted to their schoolwork (obviously...we all got into Yale after all), but there is also a huge party scene and for those not into that, tons of other fun things to do on weekends. Sports may not be as big as at state schools, but most people go out to support the football team and we are very proud of our bulldogs. The residential college system is one of the most unique and special things about Yale. It provides the perfect balance- you get all the resources of a huge university, but also get the familial feel of a smaller community. I am constantly amazed by the opportunities available at Yale, in everything from speakers coming here to visit to professors I can study with to summer programs/internships.
Class: Alum
Yale is forever. It becomes part of your identity, your sense of self. And you only have to pay for those first four years. If alumni are quick to mention they went to Yale, it's usually out of a lingering awe, like the ultimate brush with greatness, like being able to say you went to the original Woodstock or the party at Ben Franklin's house right after the signing of the Declaration of Independence. You get a seat at the table. Any table you want. Often, it's a dining hall table. There, dinnertime conversation can be like a capstone seminar in any topic - especially if you join different tables and even eat in different dining halls. One night the folks at the table could include a playwright beginning his journey toward a Pulitzer prize; a football player/theater studies major destined to metamorphose into a quadriplegic motivational speaker; an art major later listed and thriving in New York; a Hollywood insider trying on the very large shoes of her literary giant father (and soon walking very gracefully in her own shoes); and some soft-spoken history major eager to see her dad - your childhood idol - when he visits later that week. To stay in earshot of these conversations, next thing you know, you've offered to sew together little spermlike dance costumes for an upcoming production of Cabaret whose costume designer, too, goes into the business.
Yale has no real campus but there is a Yale bubble within New Haven so if you only like the idea of a gated community, the campus is not for you. There is a lot of school pride against Harvard and among the sports people, but other than that there is more residential college pride that is merely light hearted fun. This is not a school full of spirit like the other very large athletic schools. There are no real advisors during the first year which makes the experience difficult, especially as a freshman faces his or her first "shopping period." Although all of the food is sustainable and Yale tries to offer vegetarian options at every meal, the students certainly do get tired of the food and the only option is to go to one of New Haven's many fine restaurants that cost an arm and a leg. The school is a university focused school so if you do not plan to go into a profession in which research is important, Yale may not be the place for you.
I love Yale. It's the perfect size (about 5 thousand undergrads and 11 thousand total), and it's made to feel like home with the residential college system (think Hogwarts, except 12 colleges and there's no magic). New Haven is a wonderful city to live in because there are so many cool things to do and see (night clubs, restaurants, parks, museums, etc.), but it is also very close to New York City (easy train-ride away). There are definitely impoverished areas, but these provide great ways to help out and improve the world around you- I love it! When you are on the Yale campus or in New Haven it is great, but when you're somewhere else and somebody asks you where you go to school, I often feel like a tool when I say "Yale". Many people tell strangers that they go to Pierson College, or Calhoun College, etc. - whichever residential college they are in. That way, people don't assume anything about you for going to Yale.
The people. Generally really chill. All amazing. Actually, the best thing are the opportunities and resources that are available. If you want to do something cool, ANYTHING, Yale will probably let you do it and FUND it for you. You just need to pay attention to the things that are offered, the projects other students are doing, and make a little effort to get whatever you want started. Seriously any random cool projects are very possible. There are ALWAYS things to do. In fact, there are so many things to do, it's overwhelming. It is such an accepting school--on many levels. New Haven is really not bad. I love it actually. Lots of different kinds of restaurants. Just don't go out too late alone or else you might get mugged. You know, common sense. I'd say there is a good amount of school pride. Yale itself is unusual. Unsually fantastic. And the people who go there are mind-blowingly unusual in the best way possible.
The best thing about Yale is, to put it broadly, the people. Going to Yale was hands-down the best choice I've ever made. All of my closest friends are people that I met during college, and while I get a range of responses from "I'm impressed" to "So you think you're so great, huh?" when I tell people I went to Yale, I can't possibly imagine myself having gone anywhere else and being as happy. It was perfectly sized, the "shopping period" method of classes was a really interesting and effective way to make sure that you were going to take classes that you like and allowed you exposure to all sorts of academic topics. Honestly, one of the best things about Yale when I was there was the administration's lax approach to drinking on campus. The residential college system worked better on paper than it did in reality, especially for those of us who were stuck in Morse.
The best thing about Yale are the people you meet and the opportunities presented, both always things you are enriched by. Telling people one goes to Yale is context specific - for the most part, it is a well recognized name, but not always something easy or useful to give away. While on campus, I like to split my time among class areas, the many wonderful libraries, the dining halls, my college Berkeley and running errands around the remainder of the place. As much as Yale is split among different social circles, it is surprising and comforting to see how these overlap through several people, and how no matter what, Yalies look out for and love other Yalies. Unusual to people who visit, are the many resources we have available to us here.
Yale is an unparalleled institution. For the first time in my life, I feel as if school administrators are actually on my side and trying to make all of our lives better. The school is willing to help you accomplish almost anything you can dream of. They will back you on almost any issue. It is really amazing to see the things that students create at Yale. The mindset on campus is one of sheer possibility. Although there is a tendency for Yalies to overload themselves, it is part of the fun. As soon as I get on campus, I can feel the energy of the place. It is not just a place for learning in classrooms; it is a place where you are constantly learning.
Most people have the idea that Yale is some sort of exclusive gentleman's club. This is completely true, in a way: Mory's, the Yale Club, secret societies – they all exist. And of course, when you graduate, there are tons of successful Yalies who would be happy to give you a job or put you in contact with a colleague. Yet people seem to think that only rich white legacy kids have access to this, when the truth is that it is available to anyone who works hard enough to get in. Yale goes out of its way to let in a completely diverse group of students, both racially and economically. Now, anyone can be part of the Yale legacy.


text:
Q: I want to use the following web method from my android app using soap. Can anyone help me? WebMethod
    public XmlElement findUserInfoById(int Id)
    {
        //string name = ""; string email = ""; string phone = "";
        con.Open();
        SqlCommand cmd = new SqlCommand("select*from tblUsers where Id ="+ Id, con);
        cmd.ExecuteNonQuery();
        SqlDataAdapter da = new SqlDataAdapter(cmd);
        DataSet ds = new DataSet();
        da.Fill(ds);
        con.Close();
        XmlDataDocument xmldata = new XmlDataDocument(ds);
        XmlElement xmlElement = xmldata.DocumentElement;
        return xmlElement;
    }

A: try {
        androidHttpTransport.call(SOAP_ACTION, envelope);
        SoapObject response = (SoapObject) envelope.getResponse();
        SoapObject s = (SoapObject) response.getProperty(0);
        for (int i = 0;i<s.getPropertyCount();i++){
            Object property = s.getProperty(i);
            if(property instanceof SoapObject){
                SoapObject category_list = (SoapObject) property;
                String strName = category_list.getProperty("name").toString();
                String strEmail = category_list.getProperty("email").toString();
                String strPhone = category_list.getProperty("phone").toString();
                Result = "Name : " + strName + "\nEmail : " + strEmail + "\nPhone : " + strPhone;
            }
        }
    } catch (Exception e) {
        e.printStackTrace();
    }


text:
{0} Home - Barry University
Barry University Home
Monsignor Bryan O. Walsh Papers
Edouard Duval-Carrie Papers
William Lehman Papers
Operation Pedro Pan - Cuban Children Program Records
Subject Keyword
SubCollections
Edouard Duval-Carrié Papers
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About Barry University
Founded in 1940 by the Adrian Dominican Sisters, Barry University's growth throughout the years has been shaped by their core values. Originally founded as Barry College for Women, Barry has grown into a coeducational university that is one of the leading centers for education in South Florida.Grounded in the liberal arts tradition, Barry University is a scholarly community committed to the highest academic standards in undergraduate, graduate and professional education.
In the Catholic intellectual tradition, integration of study, reflection and action inform the intellectual life. Faithful to this tradition, a Barry education and university experience foster individual and communal transformation where learning leads to knowledge and truth, reflection leads to informed action, and a commitment to social justice leads to collaborative service.
Monsignor William Barry Memorial Library
The Monsignor William Barry Memorial Library is located at the center of the Miami Shores campus of Barry University. The Library is named in loving memory of Monsignor William Barry, one of the founders of Barry University and an inspirational figure in the Catholic Church within the Archdiocese of Miami.
The extensive Library research collections include more than 950,000 physical items, over 40,000 print and online journal subscriptions, and several thousand online books and streaming video resources. The collection is balanced to meet the research needs of the wide variety of undergraduate through doctoral programs offered at Barry University. The collection of Roman Catholic research materials is especially significant, and is one of the largest collections of its kind in the area.
Acknowledgement: Optical Character Recognition of this collection has been funded under the provisions of the Library Cooperative Grant program, administered by the Florida Department of State's Division of Library and Information Services.
This collection documents the work of Haitian-American artist Edouard Duval-Carrié since 1989 through the present. Materials include biographical information, exhibit catalogs, exhibit programs, lobby cards, articles, and clippings. Edouard Duval-Carrié Papers
William "Bill" Marx Lehman was born in Selma, Alabama on October 5, 1913, to Corinne Leva and Maurice M. Lehman. He received his bachelor's degree from the University of Alabama at Tuscaloosa. He married Joan Feibelman of Miami, who became a well-known sculptress. Together they raised two sons, William, Jr. and Thomas, and a daughter, Kathryn (Kathy). In 1972, he campaigned as a Democrat for the 93rd Congress and won Florida's newly created 13th Congressional seat. In 1982, after redistricting, he won the 17th District Congressional seat. During his early years in office Lehman served on the Committee on Education and Labor. In 1975, Lehman gained a seat on the Select Committee on Intelligence, which was charged with investigating Central Intelligence Agency (CIA) and Federal Bureau of Investigation (FBI) abuses. He later gained a seat on the Budget Committee and was appointed to serve on the President's Commission on the Holocaust. Congressman Lehman became a member of th William Lehman Papers
This collection documents the private and public life of Monsignor Bryan O. Walsh between 1948 through 2001. Materials include personal & family papers, subject files, works, photographs, scrapbooks and audiovisual materials. For the purpose of this project we will include materials within Series Works and Photographs. Works contains published & unpublished articles and papers by BOW-including speeches, homilies, and testimonies- and also includes works by other authors. Monsignor Bryan O. Walsh Papers
Between December 1960 and October 1962, some fourteen thousand unaccompanied Cuban minors entered the United States as refugees under the sponsoring of the Catholic Welfare Bureau of the Diocese of Miami, Florida. Operation Pedro Pan Cuban Children's Programs Records documents this children's exodus. Operation Pedro Pan - Cuban Children Program Records


Actually from last night; actual posting was interrupted by small child.
The Transcriptional Repressor DEC2 Regulates Sleep Length in Mammals: We have identified a mutation in a transcriptional repressor (hDEC2-P385R) that is associated with a human short sleep phenotype. Activity profiles and sleep recordings of transgenic mice carrying this mutation showed increased vigilance time and less sleep time than control mice in a zeitgeber time– and sleep deprivation–dependent manner.
This completely made my day; I think we both had a good laugh over it.
Ok, lunch over; deliverable-crankin' time... and where is that nail-polish remover?


Microbial populations in indoor environments, where we spent our maximum time are indeed essential for public health. Several microbial species found in the kitchens of rural and urban area, can be a prominent source of air borne diseases. During present study, we have selected several samples from kitchens of district Meerut (UP-India). The results suggested that kitchens have a higher level of bacterial growth represented by total 117 positive samples for bacterial contamination out of 200 samples from different sites. Our results are agreed with Shruti et al., (2011) and we have also found the contamination more pronounced in rural region than in urban.


text:
Winners and Losers of Cubs-A's Blockbuster Jeff Samardzija, Jason Hammel Trade
Rick Weiner@RickWeinerNYTwitter LogoFeatured ColumnistJuly 5, 2014
Gene J. Puskar/Associated Press
Chicago general manager Jed Hoyer and his counterpart in Oakland, Billy Beane, proved on Friday night that fireworks on the Fourth of July don't only explode overhead, completing a massive six-player swap that has wide-sweeping ramifications on the playoff picture in both leagues.
The first-place Athletics strengthened their rotation by adding two of the best pitchers available, Jason Hammel and Jeff Samardzija, while the rebuilding Cubs landed a trio of youngsters—shortstop Addison Russell, outfielder Billy McKinney and starting pitcher Dan Straily (along with a player to be named later)—to build a future contender around.
It's a deal that, on paper, seems to benefit both clubs. While it will be a few years before we can truly grasp which club came out on top, it's never too early to take a look at the immediate winners and losers in the aftermath of the first major trade of the regular season.
Winner: Dan Straily
From the perspective of an individual player, there may not be a bigger winner in this deal than Straily.
It was only two years ago that he led the minor leagues with 190 strikeouts and was named Oakland's second-best prospect by MLB.com's Jonathan Mayo. Last year, Straily won 10 games for the A's and finished fourth in the AL Rookie of the Year voting, so it's not like there's a lack of talent here.
But Straily struggled this season when called upon by the A's, pitching to a 4.93 ERA and allowing nine home runs in only 38.1 innings, and he wasn't much better with Triple-A Sacramento. He had furthermore clearly fallen out of favor with the current regime.
In Chicago, Straily will get a chance immediately to be a part of a now-depleted Cubs rotation.
Loser: Starlin Castro
Brian Kersey/Getty Images
Starlin Castro has silenced his doubters and put together a terrific season, one that finds him worthy of inclusion on the National League All-Star squad and one that has left his woeful 2013 in the past where it belongs.
For all of his efforts, Castro may eventually find himself wearing a different uniform.
Chicago now has three of the best young middle infielders under the age of 25 in baseball: Castro, 21-year-old Javier Baez and the 20-year-old Russell. While unproven at the major league level, the latter two are projected to be, just like Castro, All-Star-caliber shortstops.
The Cubs have three players for one position—that never works out well.
Even if Baez slides over to second base, that still leaves Russell nipping at Castro's heels. Not only is Russell a superior defensive player to Castro (who continues to struggle with the glove), but he's going to cost the Cubs significantly less than Castro will.
Castro is due at least $44 million through the 2019 season, $60 million if the Cubs were to pick up the $16 million team option that they hold on him for 2020. Russell wouldn't even become arbitration eligible until 2018 or so, depending on when his MLB service-time clock begins to tick.
The money the Cubs could save by trading Castro (when Baez and Russell are ready) could be used to bolster other areas of the roster—like the starting rotation.
Winner: Jeff Samardzija
Despite pitching to the National League's 10th-lowest ERA (2.83), Samardzija sits with a 2-8 record on the season, thanks largely in part to receiving less run support than any other pitcher, according to ESPN.
That's about to change drastically, as Samardzija joins an Oakland club that leads baseball with 430 runs scored, 100 more than the 330 that the Cubs have put on the board.
While he'll be facing some tougher lineups in the American League, Samardzija's win total figures to skyrocket—and with every additional win, the price on his next contract only increases.
Losers: Los Angeles Angels and Seattle Mariners
If you're looking for the two people whose Fourth of July celebrations were ruined more than anyone else's by this trade, look no further than a pair of general managers in the AL West, Los Angeles' Jerry DiPoto and Seattle's Jack Zduriencik.
Both of their clubs have stayed within shouting distance of the A's atop the division, with the Angels 3.5 games out and the Mariners six. Oakland GM Billy Beane told reporters, including John Hickey of the San Jose Mercury News, that the division foes certainly played a part in his thinking leading up to the deal:
We've got a tremendous amount of respect for the talent that the Angels and the Mariners have. Our priority is to try to win the division first, and then worry about the postseason.
Any way you can strengthen your team helps you if you can get there, but I think the narrative that all this is (for the postseason) is a bit arrogant. We need to make sure we get there first. This is a move to give us every chance to win the division, first and foremost.
By adding Hammel and Samardzija, the balance of power in the division has tilted significantly in Oakland's favor. Now, both the Angels and Mariners have no choice but to try to make significant additions in order to keep pace with Oakland.
Winners: The Rest of the NL Central
Dilip Vishwanat/Getty Images
With four of the NL Central's five teams—everyone but the Cubs—in contention for playoff spots, removing two of the better pitchers from the division is only going to help those contending clubs.
Three of the four contenders—Cincinnati (11), Milwaukee (10) and St. Louis (10)—still have at least 10 games to play against the Cubs. Each of those clubs figured to have faced Hammel and Samardzija twice more during the regular season. Pittsburgh, with six games left against the Cubs, would have faced them each at least once.
That's no longer the case.
Instead of going up against established veterans, they'll be going up against younger, less experienced arms. With all four clubs boasting veteran lineups, it's hard to see how that won't result in a few more wins than originally could have been hoped for.
Losers: Tampa Bay Rays
Tampa Bay has been dangling David Price on the trade market since the winter, asking for a significant package of talent in exchange for the 2012 AL Cy Young Award winner.
According to CBS Sports' Jon Heyman, the Rays recently told the Los Angeles Dodgers that, of the team's top prospects—right-handed pitcher Chris Anderson, outfielder Joc Pederson, right-handed pitcher Zach Lee, left-handed pitcher Julio Urias and shortstop/third baseman Corey Seager—they'd like "most of them" in a potential deal for Price.
Considering that Oakland just surrendered its two best prospects for two established starting pitchers, including Samardzija, who many put on even footing with Price in terms of being the more valuable trade chip, it's hard to see the Dodgers (or any team) meeting Tampa Bay's exorbitant asking price.
If you think about it, the Cubs really put the screws to Tampa Bay here. Chicago didn't have to trade Samardzija—it could have spent the rest of this season, the winter and the first half of 2015 trying to work out a contract extension before trading him.
Tampa Bay doesn't have that luxury with Price. The club has no chance of signing him to a long-term deal, and everyone knows it.
Price, earning $14 million, accounts for roughly 20 percent of the team's total payroll this season. His salary is only going to go up in his final year of arbitration, taking up even more of the financially strapped franchise's budget.
If the Rays are serious about moving Price, they're going to have to lower their demands.


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redpajama_set_name: RedPajamaGithub


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Brian Terrill's 100 Film Favorites – #31: "Walk Hard: The Dewey Cox Story"
Brian T. / September 2, 2013
100 Film Favorites – #31: Walk Hard: The Dewey Cox Story
(Jake Kasdan, 2007)
"Life's a race, and I'm in it to win it. And I'll walk as damn hard as I please."
In recent years, it has often been said that the art of parody, at least in film, is dead. People frequently cite the films of Jason Friedberg and Aaron Seltzer (Date Movie, Epic Movie, Disaster Movie, etc.) as evidence of the declining quality of film spoofs, with mere allusions to recent pop culture phenomena and "humorously" renamed characters taking the place of actual jokes, wit, or satire.
A disaster in every sense but the one they meant.
But parody is not dead, and a few filmmakers still utilize the form expertly. Previous Countdown selection The Cabin in the Woods thoughtfully skewers the conventions of the slasher genre, as does the similarly-themed Tucker and Dale vs. Evil (starring Countdown favorite Alan Tudyk, and highly recommended, though it won't be appearing on this list). Rather than simply referencing popular films (though this does occur), these movies poke fun at commonalities which apply across the genre itself. We as viewers have grown used to these tropes, and seeing them turned on their head not only makes us laugh, but makes us more aware of the conventions of contemporary storytelling, and more conscious of our film-going practices. Truly great parodies are not only funny, but also genuinely clever and often thought-provoking.
Which brings me to today's film. Most specifically, Walk Hard is a spoof of the 2005 Johnny Cash biopic Walk the Line, roughly mirroring the plot of that film. However, Walk Hard is a parody more broadly of the biopic genre in general, as well as the history of rock and roll in the second half of the 20th century.
The film follows Dewey Cox (John C. Reilly), a Johnny Cash analog whose music career has spanned both decades and genres. The movie begins just before an aged Cox is scheduled to take the stage for a monumental performance. However, Dewey remains backstage, leaning pensively against a wall, his head on one arm. His bandmates caution the anxious stage manager not to rush him, as "Dewey Cox needs to think about his entire life before he plays."
Cue a flashback to Dewey's childhood. Dewey and his younger brother, a multi-talented prodigy, gambol around the family farm, musing about the "long, long, lives" ahead of them while playing a series of outrageously dangerous games (rattlesnake toss, among others). When his brother challenges him to a good old-fashioned machete duel, the young Dewey inadvertently cleaves his sibling in two.
With a cry of "I'm halved!" the younger Cox expires, but not before reminding Dewey that he will now have to be "double great" for the both of them. Distraught by the loss of his Wunderkind son, Dewey's father adopts an oft-repeated mantra: "The wrong kid died!" The trauma of it all renders Dewey "smell-blind," unable to sense scents. Amidst all this heartache and smell-loss, Dewey takes up blues guitar and proves a natural.
Years later, a 14-year old Dewey (now played by Reilly) debuts a new song, "Take My Hand," at the high school talent show. Though the lyrics are seemingly innocuous, the song infuriates the town's older generation, while simultaneously sparking a fiery passion in the spirits of the young (girls strip to their bras, and a boy punches a priest). To escape threats from people labeling his rock and roll "devil's music," Dewey leaves town with a girl who soon becomes his first wife.
And just as soon is saddled with numerous children.
A few years pass, and Dewey is busing tables at a nightclub with primarily African-American clientele. One night, the star singer and guitarist (Craig Robinson) is unable to perform, and Cox takes the stage in his place. Dewey re-creates the singer's entire act, including a flawless rendition of "You've Got to Love Your Negro Man." The performance enchants the audience, especially the trio of Jewish executives who "control the industry" and are listening intently in the crowd.
Dewey is whisked away to a recording studio. After an ill-received cover of "That's Amore," Dewey decides to play one of his own songs…"Walk Hard." The song expresses his willingness to strive toward his goals, no matter what obstacles he may encounter. A montage shows the single quickly climbing to massive popularity nationwide (VERY quickly – a young DJ (Jack McBrayer) informs his listeners that the montage took place over just 35 minutes).
I could go on and on recounting this film, as the whole movie is chock-full of gags that make me laugh every time I watch it. Suffice it to say that the "Walk Hard" song kick-starts a storied career, with Dewey and his band releasing one hit album after another. However, fame comes with temptations: Dewey grows distant from his wife and an ever-expanding number of children, and begins experimenting with drugs…pot, cocaine, uppers, downers, LSD, PCP, and virtually every other drug known to man. Cox's tumultuous life leads him to create innovative music, ranging from punk rock, to Bob Dylan-style protest songs, to bizarre experimental pieces featuring goats, Maori tribesmen, and "a whole army of didgeridoos."
Dewey experiments with goats…
…and PCP.
Dewey's career likewise brings him into contact with many of rock & roll's most famous figures, including Buddy Holly, Elvis, and the Beatles. All are played by hilariously miscast celebrity cameos, with Jack Black as Paul McCartney a particular standout (you know he's Paul McCartney because he keeps telling everyone so).
After Dewey's relationship with his first wife falls apart, he meets the lovely Darlene Madison (Jenna Fischer). Over many years, Darlene and Dewey slowly build a strong relationship (popping out many an additional child in the process). With Darlene's help, Dewey is gradually able to overcome his addictions, reconnect with his many offspring, and even regain his sense of smell. In his twilight years, Cox sets about reuniting his band for a farewell concert to repair his image and leave a lasting legacy.
Darlene and Dewey duet.
That's all I'll say here. Far be it from me to spoil some of the movie's most hilarious bits. I would rank Walk Hard easily among the funniest films of the past decade. It skillfully mocks aspects of the biopic form (the same actor playing a character at a wide range of ages, Oscar-baity speeches, etc.), and lampoons some of the most legendary figures in pop culture history. On top of that, the film sports an ensemble cast, including numerous cameos by a wide variety of well-known comics.
One final note: The music in the film is honestly very good throughout. Though the lyrics are often humorous ("Let Me Hold You, Midget-Man"), the many songs are all catchy and well-written, with some actually being serious and touching in tone (the final number, "Beautiful Ride," features Dewey "accepting his mortality," and makes for a sentimental final chapter of the old rocker's life.
Walk Hard is endlessly quotable, full of great laughs and equally great music.
See this movie.
Tidbits:
-Director Jake Kasdan is the son of Lawrence Kasdan, screenwriter of The Empire Strikes Back, Return of the Jedi, and Raiders of the Lost Ark.
Brian Terrill is the host of television show Count Gauntly's Horrors from the Public Domain. You can keep up with Brian's 100 Film Favorites countdown here.
September 2, 2013 in Movies. Tags: 100 film favorites, 31, brian, brian terrill movie night, countdown, dewey cox, john c. reilly, let's duet, life's a race, walk hard, you don't want none of this shit
← Brian Terrill's 100 Film Favorites – #32: "The Mask of Zorro"
Brian Terrill's 100 Film Favorites – #30: "WALL-E" →


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My wonderful boyfriend Keith surprised me with a trip to Catalina for Valentines day. We hiked, biked, ate everything we could get our hands on and hung out with a few bison. Catalina is lovely and it's such a short trip for Southern California locals; hop on a boat and go explore in your own backyard!


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Q: Correct phrasing for describing a plot I am describing a plot (in a figure in a thesis I am currently writing) as follows:
Figure of [names of some mathematical functions]. Function A is covered by Function B for x < 10. (...)
Is this correct?
Basically I want to describe that Function B has the same values for x < 10 compared to Function A. That is why Function B is only plotted for x >= 10.

A: I can suggest that you use the verbs coincide and overlap. Consider:

As you can see, output (result) of the function A coincide (overlap) perfectly with the result of function B for input variables lower than 10.

You can also say:

The Figure illustrates the excellent agreement between the result of the function A and function B for input variables lower than 10.