# From Kadanoff–Baym to Boltzmann equations for massive spin-1/2 fermions

Xin-Li Sheng  Key Laboratory of Quark and Lepton Physics (MOE) and Institute
of Particle Physics, Central China Normal University, Wuhan, 430079, China
Nora Weickgenannt  Institute for Theoretical Physics, Goethe University, Max-
von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany  Enrico Speranza  Illinois
Center for Advanced Studies of the Universe and Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA  Helmholtz
Research Academy Hesse for FAIR, Campus Riedberg, Max-von-Laue-Str. 12,
D-60438 Frankfurt am Main, Germany  Dirk H. Rischke  Institute for Theoretical
Physics, Goethe University, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main,
Germany  Helmholtz Research Academy Hesse for FAIR, Campus Riedberg, Max-von-
Laue-Str. 12, D-60438 Frankfurt am Main, Germany  Interdisciplinary Center for
Theoretical Study and Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China  Qun Wang  Interdisciplinary
Center for Theoretical Study and Department of Modern Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China  Peng Huanwu
Center for Fundamental Theory, Hefei, Anhui 230026, China

###### Abstract

We derive Boltzmann equations for massive spin-1/2 fermions with local and
nonlocal collision terms from the Kadanoff–Baym equation in the
Schwinger–Keldysh formalism, properly accounting for the spin degrees of
freedom. The Boltzmann equations are expressed in terms of matrix-valued spin
distribution functions, which are the building blocks for the quasi-classical
parts of the Wigner functions. Nonlocal collision terms appear at next-to-
leading order in $\hbar$ and are sources for the polarization part of the
matrix-valued spin distribution functions. The Boltzmann equations for the
matrix-valued spin distribution functions pave the way for simulating spin-
transport processes involving spin-vorticity couplings from first principles.

††preprint: USTC-ICTS/PCFT-21-12

##  I Introduction

The goal of high-energy heavy-ion collisions is to create a new state of
strong-interaction matter, called the quark-gluon plasma (QGP), at extreme
conditions of temperature and density, and to study its properties
(Rischke:2003mt, ; Gyulassy:2004zy, ; Shuryak:2004cy, ). In this state, the
fundamental degrees of freedom of quantum chromodynamics (QCD), quarks and
gluons, are deconfined. The collective flow observed in experiments
(Ackermann:2000tr, ) reveals that the QGP is a nearly perfect fluid with a
very low ratio of shear viscosity to entropy density (Kovtun:2004de, ). The
collective flow can be quantitatively described by relativistic hydrodynamical
models (Kolb:2003dz, ; Heinz:2013th, ; Florkowski:2017olj, ;
Romatschke:2017ejr, ), which are by now a well-established tool to describe
the evolution of QCD matter in heavy-ion collisions. However, until very
recently these models did not account for the dynamics of the spin degrees of
freedom.

Noncentral heavy-ion collisions have a large orbital angular momentum (OAM),
which may polarize the spin of particles in strong-interaction matter in a way
similar to the well-known Barnett effect (Barnett:1935, ). A global
polarization of particles created in high-energy heavy-ion collisions was
first proposed as a result of the spin-orbit coupling (Liang:2004ph, ;
Liang:2004xn, ) [see also Refs. (Voloshin:2004ha, ; Betz:2007kg, ;
Becattini:2007sr, )]. In 2007, the STAR collaboration measured the global
polarization of $\Lambda$ hyperons in Au+Au collisions at 200 GeV, but the
result was zero within errors (Abelev:2007zk, ). A nonvanishing global
polarization of $\Lambda$ hyperons was measured by the STAR collaboration in
Au+Au collisions at lower energies (STAR:2017ckg, ) and at 200 GeV with high
precision (Adam:2018ivw, ) [see, e.g., Refs. (Wang:2017jpl, ;
Becattini:2020ngo, ; Gao:2020vbh, ) for recent reviews].

The experimental data for the global polarization can be described by
theoretical models (Karpenko:2016jyx, ; Xie:2017upb, ; Li:2017slc, ;
Sun:2017xhx, ; Wei:2018zfb, ). All these models feature a spin-vorticity
coupling resulting from microscopic spin-orbit coupling. Here the vorticity is
the result of the local rotation of the fluid (Baznat:2013zx, ;
Csernai:2013bqa, ; Csernai:2014ywa, ; Becattini:2015ska, ; Teryaev:2015gxa, ;
Jiang:2016woz, ; Deng:2016gyh, ; Ivanov:2017dff, ; Shi:2017wpk, ). Statistical
models for relativistic fluids consisting of particles with spin
(Becattini:2007nd, ; Becattini:2007sr, ; Becattini:2013fla, ) can be
constructed based on the maximum-entropy principle (Zubarev_tmp1979_zps, ;
Weert_ap1982, ; Becattini:2014yxa, ). A lot of progress has been recently made
in theoretical studies of spin polarization and vorticity formation in heavy-
ion collisions. The theoretical models can be grouped into two main
categories: microscopic and macroscopic models. Microscopic models are based
on scatterings of particles with spin-orbit coupling (Gao:2007bc, ;
Chen:2008wh, ; Huang:2011ru, ). Macroscopic models mainly include statistical
models (Becattini:2007nd, ; Becattini:2007sr, ; Becattini:2013fla, ) for
fluids consisting of particles with spin in global equilibrium
(Becattini:2009wh, ; Becattini:2012tc, ; Becattini:2015nva, ; Hayata:2015lga,
) and spin-hydrodynamical models (Florkowski:2017ruc, ; Florkowski:2017dyn, ;
Montenegro:2017lvf, ; Montenegro:2017rbu, ; Hattori:2019lfp, ;
Gallegos:2021bzp, ; Li:2020eon, ; Bhadury:2020puc, ; Fukushima:2020ucl, ) [see
Ref. (Florkowski:2018fap, ; Speranza:2020ilk, ) for recent reviews]. Kinetic
theory based on the Wigner-function formalism (Gao:2012ix, ; Chen:2012ca, ;
Hidaka:2016yjf, ; Gao:2017gfq, ; Gao:2018wmr, ; Huang:2018wdl, ;
Carignano:2018gqt, ; Liu:2018xip, ; Gao:2019zhk, ; Yang:2020mtz, ;
Hou:2020mqp, ) can also describe the spin polarization for massive fermions
determined by the axial-vector or tensor component of the Wigner function
(Fang:2016vpj, ; Weickgenannt:2019dks, ; Gao:2019znl, ; Hattori:2019ahi, ;
Wang:2019moi, ; Liu:2020flb, ). However, the above works do not include
particle collisions, which are, however, necessary in order to describe the
spin dynamics of massive fermions.

In statistical models and spin-hydrodynamical models the spin-polarization
effect is described by the spin potential $\Omega_{\mu\nu}$ coupled to the
spin tensor $\Sigma^{\mu\nu}$ in the density matrix. However, unless one
considers a global-equilibrium state or a special choice for the energy-
momentum tensor, the spin potential is a priori not related to the thermal
vorticity
$\varpi_{\mu\nu}\equiv-\frac{1}{2}\left(\partial_{\mu}\beta_{\nu}-\partial_{\nu}\beta_{\mu}\right)$,
where $\beta^{\mu}\equiv u^{\mu}/T$, with $u^{\mu}$ the 4-velocity of matter
and $T$ the temperature (Florkowski:2017ruc, ; Becattini:2018duy, ;
Florkowski:2018ahw, ). Therefore, the main question is how the spin potential
evolves to its global-equilibrium value, given by a constant thermal
vorticity. A Boltzmann equation with a nonlocal collision term is essential to
describe such a dynamical process. A local collision term with spin degrees of
freedom has been studied under some approximations in Refs. (Li:2019qkf, ;
Kapusta:2020npk, ). In Ref. (Zhang:2019xya, ), a microscopic model for spin
polarization through spin-orbit coupling in particle collisions was
considered, which is based on collisions of partons as wave packets. The
method of wave packets is an effective way of dealing with particle
scatterings at nonvanishing impact parameter. The spin-vorticity coupling
naturally emerges from the spin-orbit one encoded via polarized scattering
amplitudes in the collision integrals. Such a microscopic model provides a
transparent picture for the way how spin polarization can arise from
vorticity. There is only one missing piece in the model: the back reaction is
not considered, which converts spin into vorticity. Thus, this model does not
reach a state where spin is equilibrated. A first attempt to systematically
incorporate nonlocal collisions in a kinetic framework based on quantum field
theory was recently made in Ref. (Yang:2020hri, ), however, without giving an
explicit expression for the collision term at order $\mathcal{O}(\hbar)$. In
previous work (Weickgenannt:2020aaf, ; Weickgenannt:2021cuo, ), we derived the
collision term to order $\mathcal{O}(\hbar)$ in the Boltzmann equation for
massive spin-1/2 particles in the Wigner-function approach (DeGroot:1980dk, ).
The nonlocality of the collision term allows for the conversion of orbital
into spin angular momentum. We showed that the collision term vanishes in
global equilibrium and that the spin potential is then equal to a constant
value of the thermal vorticity.

In this paper, we derive Boltzmann equations for massive spin-1/2 fermions in
the Schwinger-Keldysh, or closed-time-path (CTP), formalism (Martin:1959jp, ;
Keldysh:1964ud, ) with collision terms of leading and next-to-leading order in
$\hbar$ [for reviews of the CTP formalism, see, e.g., Refs. (Chou:1984es, ;
Blaizot:2001nr, ; Berges:2004yj, ), for a recent application of the CTP
formalism to dissipative hydrodynamics, see Ref. (Crossley:2015evo, )]. Our
goal is to derive Boltzmann equations that are suitable for the simulation of
spin transport processes involving the spin-vorticity coupling.

We consider a system of massive spin-1/2 fermions interacting via generic one-
boson exchange. We will assume the interaction range to be much smaller than
all other scales in the problem, which effectively reduces the interaction to
a four-fermion vertex, similar to the time-honored Nambu–Jona-Lasinio (NJL)
model (Nambu:1961tp, ; Nambu:1961fr, ).

The paper is organized as follows. In Sec. II the model system is introduced
and a short introduction to the CTP formalism is given. In Sec. III we derive
the Kadanoff–Baym (KB) equation (Kadanoff:1962, ) in quasi-particle
approximation for the Wigner function from the Dyson-Schwinger equation on the
CTP contour. We decompose the Wigner function in terms of its Clifford
components and derive a system of equations of motion for the latter. We also
derive mass-shell conditions and Boltzmann-type equations for the latter and
show that off-shell contributions cancel to lowest order in the coupling
constant. In Sec. IV we perform an expansion of the KB equation in powers of
Planck’s constant $\hbar$. The structure of the matrix-valued spin
distribution functions and the quasi-classical parts of the Wigner functions
are discussed in Sec. V. The Boltzmann equations for the matrix-valued spin
distribution functions at leading and next-to-leading order in $\hbar$ are
derived in Secs. VI and VII, respectively. We close this work with a summary
of the results in Sec. VIII.

We adopt the following notation and conventions: $a\cdot b=a^{\mu}b_{\mu}$,
$a_{[\mu}b_{\nu]}\equiv a_{\mu}b_{\nu}-a_{\nu}b_{\mu}$,
$a_{\\{\mu}b_{\nu\\}}\equiv a_{\mu}b_{\nu}+a_{\nu}b_{\mu}$,
$g_{\mu\nu}=\mathrm{diag}(+,-,-,-)$, $\epsilon^{0123}=-\epsilon_{0123}=1$, and
summation over repeated indices is implied if not stated explicitly. Natural
units are chosen, $c=\varepsilon_{0}=\mu_{0}=k_{B}=1$, but the reduced Planck
constant $\hbar$ is shown explicitly in order to perform the power counting.

##  II Fermions in the CTP formalism

###  II.1 Lagrangian

In our units, Planck’s constant has dimension [energy $\times$ length]. Not
setting $\hbar$ equal to one, as one usually does in natural units, entails
that units of energy are not identical to units of inverse length, which, in
turn, introduces certain ambiguities in the dimensions of fields and coupling
constants, which need to be removed by making additional definitions for the
action of the system. In order to make this explicit for our case, consider
the action of a system of spin-1/2 fermions with mass $m$ interacting with
scalar bosons with mass $M$ via a Yukawa coupling,

$S[\overline{\psi},\psi,\phi]=\int d^{4}x\,\mathcal{L}=\int
d^{4}x\left[\overline{\psi}(x)\left(i\hbar\gamma\cdot\partial_{x}-m\right)\psi(x)+g\,\overline{\psi}(x)\phi(x)\psi(x)-\frac{1}{2}\phi(x)\left(\hbar^{2}\partial_{x}^{2}+M^{2}\right)\phi(x)\right]\;.$
(1)

Since the action has the same dimension as $\hbar$, we deduce that the fermion
fields $\psi,\overline{\psi}$ have dimension [length]-3/2, while the boson
field $\phi$ has dimension [energy $\times$ length3]-1/2. The dimension of the
Yukawa coupling $g$ is [energy $\times$ length]3/2. Note that, while our unit
convention differs from that of Ref. Itzykson:1980rh , it is still true that,
in momentum space, each propagator comes with a factor of $\hbar$, while each
boson-fermion vertex comes with a factor of $\hbar^{-1}$.

Integrating out the boson field, we arrive at an action where a scalar fermion
current, $\overline{\psi}(x)\psi(x)$, interacts with another current,
$\overline{\psi}(y)\psi(y)$, via one-boson exchange mediated by the boson
propagator $\Delta(x,y)$. Such a model can be readily generalized to
incorporate one-boson exchange interactions with other quantum numbers,

$S[\overline{\psi},\psi]=\int
d^{4}x\,\overline{\psi}(x)\left(i\hbar\gamma\cdot\partial_{x}-m\right)\psi(x)+\sum_{c}\frac{g_{c}^{2}}{\hbar}\int
d^{4}xd^{4}y\,\overline{\psi}(x)\Gamma^{(c)}_{a}\psi(x)\,\Delta^{(c)}_{ab}(x,y)\,\overline{\psi}(y)\Gamma^{(c)}_{b}\psi(y)\;,$
(2)

where $\Delta^{(c)}_{ab}(x,y)$ is the propagator of the boson of type $c$
(which can be scalar, pseudoscalar, vector, axial-vector, tensor, etc. in
space-time, color space, flavor space etc.),
$\Gamma^{(c)}_{a},\Gamma^{(c)}_{b}$ are certain products of Dirac, color, and
flavor matrices, properly chosen to reflect the coupling of the boson to the
fermions, and $g_{c}$ denotes the coupling constant for the coupling of
fermions to bosons of type $c$.

The fermion action (2) is generic for fermions in all theories with Yukawa
interactions between fermions and bosons, and emerges naturally after
integrating out the boson degrees of freedom (when neglecting boson self-
interactions). It is also generic for certain gauge theories, such as QED, and
even for QCD when neglecting gluon self-interactions.

Assuming that the range of the boson-exchange interaction is much smaller than
any other scale in the problem, we may replace
$(g_{c}^{2}/\hbar)\Delta^{(c)}_{ab}(x,y)\simeq
G_{c}\,\delta_{ab}\delta^{(4)}(x-y)$, where the 4-fermion coupling constant
$G_{c}$ has dimension [energy $\times$ length3], and arrive at the Lagrangian
of the time-honored NJL model (Nambu:1961tp, ; Nambu:1961fr, ),

$\mathcal{L}=\overline{\psi}(x)\left(i\hbar\gamma\cdot\partial_{x}-m\right)\psi(x)+\sum_{c}G_{c}\,\left[\overline{\psi}(x)\Gamma^{(c)}_{a}\psi(x)\right]^{2}\;.$
(3)

Note that, in momentum space, the four-fermion vertex also carries a factor of
$\hbar^{-1}$.

###  II.2 CTP formalism

All information on the non-equilibrium dynamics of the system is provided by
the generating functional for correlation functions on the CTP

$\displaystyle Z\left[\eta,\overline{\eta},\rho\right]$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left\\{\rho(t_{0})T_{C}\exp\left[\frac{i}{\hbar}\int_{C}d^{4}x\left(\mathcal{L}+\overline{\psi}\eta+\overline{\eta}\psi\right)\right]\right\\}$
(4) $\displaystyle=$
$\displaystyle\mathrm{Tr}\left\\{\rho(t_{0})T_{C}\exp\left[\frac{i}{\hbar}\int_{t_{0}}^{\infty}dt_{+}d^{3}x\left(\mathcal{L}_{+}+\overline{\psi}_{+}\eta_{+}+\overline{\eta}_{+}\psi_{+}\right)-\frac{i}{\hbar}\int_{t_{0}}^{\infty}dt_{-}d^{3}x\left(\mathcal{L}_{-}+\overline{\psi}_{-}\eta_{-}+\overline{\eta}_{-}\psi_{-}\right)\right]\right\\}\;,$

where $\rho(t_{0})$ is the density matrix at the initial time, $C$ denotes the
CTP as shown in Fig. 1, $T_{C}$ is the time-ordering operator on the CTP, and
$\eta$ and $\overline{\eta}$ are sources for $\overline{\psi}$ and $\psi$,
respectively. The subscript $\pm$ denotes quantities on the positive/negative
time branch.

The two-point function of the theory can be put into matrix form,

$G(x_{1},x_{2})=\left(\begin{array}[]{cc}G^{++}(x_{1},x_{2})&G^{+-}(x_{1},x_{2})\\\
G^{-+}(x_{1},x_{2})&G^{--}(x_{1},x_{2})\end{array}\right)=\left(\begin{array}[]{cc}G^{F}(x_{1},x_{2})&G^{<}(x_{1},x_{2})\\\
G^{>}(x_{1},x_{2})&G^{\bar{F}}(x_{1},x_{2})\end{array}\right)\;.$ (5)

where $G^{ij}(x_{1},x_{2})$ (with $i,j=+,-$) means that the first time
argument $t_{1}=x_{1}^{0}$ lives on the time branch $i$ and the second time
argument $t_{2}=x_{2}^{0}$ lives on the time branch $j$. For $G^{++}$, both
$t_{1}$ and $t_{2}$ are on the positive time branch. Then, the ordering on the
CTP is just the standard time ordering of quantum field theory, so that
$G^{++}$ is simply the Feynman propagator $G^{F}$. On the other hand, $G^{-+}$
means that $t_{1}$ lives on the negative and $t_{2}$ on the positive time
branch, respectively, such that $t_{1}$ is later than $t_{2}$ considering the
ordering on the CTP. Consequently, this two-point function is denoted as
$G^{>}$. Analogously, $G^{+-}\equiv G^{<}$. When both $t_{1}$ and $t_{2}$ live
on the negative time branch, $t_{1}>t_{2}$ actually means that, on the CTP,
$t_{2}$ is later than $t_{1}$, so the ordering on the CTP is actually
equivalent to the standard anti-time ordering, and hence we denote $G^{--}$ as
$G^{\bar{F}}$. The definitions of the various Green’s functions are

$\displaystyle G_{\alpha\beta}^{F}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\left\langle
T\psi_{\alpha}(x_{1})\overline{\psi}_{\beta}(x_{2})\right\rangle\;,$ (6)
$\displaystyle G_{\alpha\beta}^{\bar{F}}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\left\langle
T_{A}\psi_{\alpha}(x_{1})\overline{\psi}_{\beta}(x_{2})\right\rangle\;,$ (7)
$\displaystyle G_{\alpha\beta}^{<}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle-\left\langle\overline{\psi}_{\beta}(x_{2})\psi_{\alpha}(x_{1})\right\rangle\;,$
(8) $\displaystyle G_{\alpha\beta}^{>}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\left\langle\psi_{\alpha}(x_{1})\overline{\psi}_{\beta}(x_{2})\right\rangle\;,$
(9)

where $T$ and $T_{A}$ denote the time-ordering and anti-time-ordering
operators, respectively, and angular brackets denote averages weighted by
$\rho(t_{0})$. Not all of the four types of two-point functions appearing in
Eq. (5), or Eqs. (6) – (9), respectively, are independent, for example they
satisfy

$G^{F}+G^{\bar{F}}=G^{<}+G^{>}\;,$ (10)

which is a direct consequence of the anticommutation relations for fermion
field operators. Equivalently we can use the following two-point functions,
which are linear combinations of those in Eqs. (6) – (9),

$\displaystyle G^{R}$ $\displaystyle=$ $\displaystyle
G^{F}-G^{<}=-G^{\bar{F}}+G^{>}\;,$ $\displaystyle G^{A}$ $\displaystyle=$
$\displaystyle G^{F}-G^{>}=-G^{\bar{F}}+G^{<}\;,$ (11) $\displaystyle G^{C}$
$\displaystyle=$ $\displaystyle G^{F}+G^{\bar{F}}=G^{>}+G^{<}\;,$

where $G^{R}$ and $G^{A}$ are the retarded and advanced two-point Green’s
functions, respectively, the explicit forms of which are given by

$\displaystyle G^{R}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\theta(t_{1}-t_{2})\left[G^{>}(x_{1},x_{2})-G^{<}(x_{1},x_{2})\right]\;,$
(12) $\displaystyle G^{A}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle-\theta(t_{2}-t_{1})\left[G^{>}(x_{1},x_{2})-G^{<}(x_{1},x_{2})\right]\;.$
(13)

We can actually express all two-point functions in terms of $G^{>}$ and
$G^{<}$ with the help of $\theta(t_{1}-t_{2})$ and $\theta(t_{2}-t_{1})$. We
note that Eqs. (10) – (13) are also valid for the self-energy
$\Sigma(x_{1},x_{2})$, which is a two-point vertex function.

Figure 1: The closed-time path.

##  III Kadanoff–Baym equation for fermions

###  III.1 Kadanoff-Baym equation in quasi-particle approximation

The Kadanoff–Baym (KB) equation (Kadanoff:1962, ) can be derived from the
Dyson–Schwinger equation on the CTP contour. For $G^{<}(x_{1},x_{2})$, this
equation reads Mrowczynski:1992hq

$\left(i\hbar\gamma\cdot\partial_{x_{1}}-m\right)G^{<}(x_{1},x_{2})=i\hbar\int
d^{4}x^{\prime}\left[\Sigma^{R}(x_{1},x^{\prime})\,G^{<}(x^{\prime},x_{2})+\Sigma^{<}(x_{1},x^{\prime})G^{A}(x^{\prime},x_{2})\right]\;.$
(14)

The two-point Wigner function is defined by a Fourier transform of the two-
point function (8) with respect to the difference $y=x_{1}-x_{2}$ of the two
space-time points $x_{1}$ and $x_{2}$,

$G_{\alpha\beta}^{<}(x,p)\equiv-\int d^{4}y\,e^{ip\cdot
y/\hbar}\left\langle\overline{\psi}_{\beta}\left(x-\frac{y}{2}\right)\psi_{\alpha}\left(x+\frac{y}{2}\right)\right\rangle\;,$
(15)

where $x=(x_{1}+x_{2})/2$ is the arithmetic mean (or center) of the two space-
time points $x_{1}$ and $x_{2}$. Up to second order in an expansion in
$\hbar$, the KB equation for the Wigner function $G^{<}(x,p)$ reads
Mrowczynski:1992hq

$\displaystyle\left(\gamma\cdot K-m\right)G^{<}(x,p)$ $\displaystyle=$
$\displaystyle
i\hbar\left[\Sigma^{R}(x,p)G^{<}(x,p)+\Sigma^{<}(x,p)G^{A}(x,p)\right]$ (16)
$\displaystyle+\,\frac{\hbar^{2}}{2}\left[\left\\{\Sigma^{R}(x,p),G^{<}(x,p)\right\\}_{\mathrm{PB}}+\left\\{\Sigma^{<}(x,p),G^{A}(x,p)\right\\}_{\mathrm{PB}}\right]\;,$

where we defined the operator

$K^{\mu}\equiv p^{\mu}+\frac{i\hbar}{2}\partial_{x}^{\mu}\;,$ (17)

and where $\\{A,B\\}_{\mathrm{PB}}$ denotes the Poisson bracket

$\left\\{A,B\right\\}_{\mathrm{PB}}\equiv(\partial_{x}A)\cdot(\partial_{p}B)-(\partial_{p}A)\cdot(\partial_{x}B)\;.$
(18)

Note that $G^{i}(x,p),\,i=<,>,R,A$ is formally of order $\hbar$ [cf. remarks
after Eq. (1)]. We will tacitly factor out one power of $\hbar$ from the
$G^{i}(x,p)$ on both sides of Eq. (16), such that an expansion of $G^{i}(x,p)$
in powers of $\hbar$ starts at order $\hbar^{0}$.

We will work to lowest diagrammatic order in the $T$-matrix approximation for
the self-energies $\Sigma^{i},\,i=<,>,R,A$. The $T$-matrix approximation is
appropriate for the description of binary collisions, because to lowest order
in this approximation, the self-energy contains two loops, so that upon
cutting the respective diagram, one obtains two on-shell particles in the in-
and the outgoing channel. Such a two-loop diagram is of order
$\mathcal{O}(G_{c}^{2})$ in the coupling constant of the NJL Lagrangian (3).
We will neglect all diagrams of higher order in loops, or equivalently, of
higher order in the coupling constant $G_{c}$. Following the standard Feynman
rules in momentum space, where each propagator comes with a factor of $\hbar$
and a four-fermion vertex comes with a factor of $\hbar^{-1}$, such a diagram
is of order $\hbar$. This factor is explicitly pulled out from $\Sigma^{i}$ in
Eq. (16), such that the $\hbar$ expansion of $\Sigma^{i}$ starts at order
$\hbar^{0}$, just as that of $G^{i}$. The second term in Eq. (16) has an extra
factor of $\hbar$, since the Poisson bracket involves the product of a spatial
and a momentum derivative.

Equation (16) is consistent with Ref. (Schonhofen:1994zf, ), except for
factors of $i$, since we absorbed an $i$ into the definition of the two-point
Green’s functions in Eqs. (6) – (9). The adjoint KB equation can be obtained
from Eq. (16) by exchanging $\Sigma\leftrightarrow G$ on the right-hand side
and by pulling $G^{<}$ to the left of the differential operator, changing
$\partial_{x}\rightarrow-\overleftarrow{\partial}_{x}$ on the left-hand side,

$\displaystyle G^{<}(x,p)\left(\gamma\cdot\overleftarrow{K}^{*}-m\right)$
$\displaystyle=$ $\displaystyle
i\hbar\left[G^{R}(x,p)\Sigma^{<}(x,p)+G^{<}(x,p)\Sigma^{A}(x,p)\right]$ (19)
$\displaystyle+\,\frac{\hbar^{2}}{2}\left[\left\\{G^{R}(x,p),\Sigma^{<}(x,p)\right\\}_{\mathrm{PB}}+\left\\{G^{<}(x,p),\Sigma^{A}(x,p)\right\\}_{\mathrm{PB}}\right]\;.$

Similarly one can derive the KB equations for $G^{>}$ from Eqs. (16), (19) by
replacing $G^{<}\rightarrow G^{>}$ and $\Sigma^{<}\rightarrow\Sigma^{>}$ but
keeping all retarded and advanced quantities unchanged.

Taking the Fourier transform of Eqs. (12), (13) and the equivalent relation
for $\Sigma$, the retarded and advanced two-point (vertex) function in
momentum space can be expressed as

$\displaystyle O^{R/A}(x,p)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int dk_{0}\frac{1}{k_{0}-p_{0}\mp
i\epsilon}\left[O^{>}(x,k_{0},\mathbf{p})-O^{<}(x,k_{0},\mathbf{p})\right]$
(20) $\displaystyle=$
$\displaystyle\pm\frac{1}{2}\left[O^{>}(x,p)-O^{<}(x,p)\right]+\frac{1}{2\pi
i}\mathscr{P}\int
dk_{0}\frac{1}{k_{0}-p_{0}}\left[O^{>}(x,k_{0},\mathbf{p})-O^{<}(x,k_{0},\mathbf{p})\right]\;,$

where $O=\Sigma$ or $G$. The principal value of the integral is denoted by
$\mathscr{P}$. The second term represents off-shell contributions, including
either off-mass-shell or energy non-conservation effects. In the quasiparticle
approximation, we can neglect the principal-value part of Eq. (20), and Eq.
(16) can be put into the form

$\left(\gamma\cdot K-m\right)G^{<}(x,p)=I_{\mathrm{coll}}\,,$ (21)

where the collision term $I_{\mathrm{coll}}$ is given by

$\displaystyle I_{\mathrm{coll}}$ $\displaystyle\equiv$
$\displaystyle-\frac{i\hbar}{2}\left[\Sigma^{<}(x,p)G^{>}(x,p)-\Sigma^{>}(x,p)G^{<}(x,p)\right]$
(22)
$\displaystyle-\frac{\hbar^{2}}{4}\left[\left\\{\Sigma^{<}(x,p),G^{>}(x,p)\right\\}_{\mathrm{PB}}-\left\\{\Sigma^{>}(x,p),G^{<}(x,p)\right\\}_{\mathrm{PB}}\right]\;.$

The above equation is our starting point for deriving the Boltzmann equation
for spin-1/2 particles, with a collision term including contributions up to
next-to-leading order in $\hbar$. Similarly one can derive the KB equation for
$G^{>}$ from Eq. (21) by replacing $G^{<}\rightarrow G^{>}$ on the left-hand
side, while the collision term remains unchanged.

###  III.2 Clifford decomposition

We can expand $G^{<}(p,X)$ in terms of the 16 independent generators of the
Clifford algebra, $\Gamma_{a}$, $a=1,\ldots,16$, with

$\Gamma_{a}\in\left\\{1,\,\gamma^{\mu},\,i\gamma^{5}=-\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3},\,\gamma^{5}\gamma^{\mu},\,\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]\right\\}\;,$
(23)

such that

$G^{<}(x,p)=\frac{1}{4}\left(\mathcal{F}+i\gamma^{5}\mathcal{P}+\gamma^{\mu}\mathcal{V}_{\mu}+\gamma^{5}\gamma^{\mu}\mathcal{A}_{\mu}+\frac{1}{2}\sigma^{\mu\nu}\mathcal{S}_{\mu\nu}\right)\;.$
(24)

The real-valued coefficient functions $\mathcal{F}$, $\mathcal{P}$,
$\mathcal{V}_{\mu}$, $\mathcal{A}_{\mu}$, and $\mathcal{S}_{\mu\nu}$ are the
scalar, pseudo-scalar, vector, axial-vector, and tensor components of
$G^{<}(x,p)$, respectively, which can be obtained by taking the trace of
$G^{<}(x,p)$ multiplied with the appropriate generator $\Gamma_{a}$ of the
Clifford algebra. Note the sign convention of
$\mathcal{A}^{\mu}=\mathrm{Tr}\left(\gamma^{\mu}\gamma^{5}G^{<}\right)$, which
is the same as in Ref. (Gao:2012ix, ; Weickgenannt:2019dks, ), but different
from Ref. (Vasak:1987um, ).

We can rewrite Eq. (21) by inserting Eq. (24) into the left-hand side,
multiplying with the appropriate $\Gamma_{a}$, and taking the trace. Then we
obtain a system of coupled equations for the components of the Wigner
function. The real parts of these equations read

$\displaystyle p^{\mu}\mathcal{V}_{\mu}-m\mathcal{F}$ $\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(I_{\mathrm{coll}}\right)\;,$ (25)
$\displaystyle
m\mathcal{P}+\frac{\hbar}{2}\,\partial_{x}^{\mu}\mathcal{A}_{\mu}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(i\gamma^{5}I_{\mathrm{coll}}\right)\;,$
(26) $\displaystyle
p_{\mu}\mathcal{F}-m\mathcal{V}_{\mu}+\frac{\hbar}{2}\,\partial_{x}^{\nu}\mathcal{S}_{\mu\nu}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(\gamma_{\mu}I_{\mathrm{coll}}\right)\;,$
(27)
$\displaystyle\frac{1}{2}\,\epsilon_{\mu\nu\alpha\beta}p^{\nu}\mathcal{S}^{\alpha\beta}+m\mathcal{A}_{\mu}-\frac{\hbar}{2}\,\partial_{x,\mu}\mathcal{P}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(\gamma^{5}\gamma_{\mu}I_{\mathrm{coll}}\right)\;,$
(28)
$\displaystyle\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}^{\beta}+m\mathcal{S}_{\mu\nu}-\frac{\hbar}{2}\partial_{x[\mu}\mathcal{V}_{\nu]}$
$\displaystyle=$
$\displaystyle-\mathrm{Re\,Tr}\left(\sigma_{\mu\nu}I_{\mathrm{coll}}\right)\;,$
(29)

while the imaginary parts are

$\displaystyle\frac{\hbar}{2}\,\partial_{x}^{\mu}\mathcal{V}_{\mu}$
$\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(I_{\mathrm{coll}}\right)\;,$ (30)
$\displaystyle p^{\mu}\mathcal{A}_{\mu}$ $\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(-i\gamma^{5}I_{\mathrm{coll}}\right)\;,$
(31) $\displaystyle
p^{\nu}\mathcal{S}_{\nu\mu}+\frac{\hbar}{2}\,\partial_{x,\mu}\mathcal{F}$
$\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(\gamma_{\mu}I_{\mathrm{coll}}\right)\;,$
(32) $\displaystyle
p_{\mu}\mathcal{P}+\frac{\hbar}{4}\,\epsilon_{\mu\nu\alpha\beta}\partial_{x}^{\nu}\mathcal{S}^{\alpha\beta}$
$\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(\gamma^{5}\gamma_{\mu}I_{\mathrm{coll}}\right)\;,$
(33) $\displaystyle
p_{[\mu}\mathcal{V}_{\nu]}+\frac{\hbar}{2}\,\epsilon_{\mu\nu\alpha\beta}\partial_{x}^{\alpha}\mathcal{A}^{\beta}$
$\displaystyle=$
$\displaystyle-\mathrm{Im\,Tr}\left(\sigma_{\mu\nu}I_{\mathrm{coll}}\right)\;,$
(34)

In the absence of collisions, Eqs. (25) – (34) are equivalent to Eqs. (11) –
(20) of Ref. Weickgenannt:2019dks  without electromagnetic fields. Equations
(26), (27), (29), (30), (31), and (34) correspond to Eqs. (6) – (11) of Ref.
Weickgenannt:2020aaf , where the collision term was derived using a different
method. In the following, we will write Eqs. (25) – (34) perturbatively order
by order in $\hbar$.

There is an equivalent way of expressing the Dirac-like KB equation (21) in
terms of components of the Wigner function. Acting with the operator
$\gamma\cdot K+m$ onto Eq. (21) and combining the resulting equation with its
Hermitian conjugate, multiplied from the left- and the right-hand side with
$\gamma^{0}$, we can derive a Klein-Gordon-type equation, which gives an on-
shell condition, and a Boltzmann-type equation,

$\displaystyle\left(p^{2}-\frac{\hbar^{2}}{4}\partial_{x}^{2}-m^{2}\right)G^{<}(x,p)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left\\{(\gamma\cdot
K+m)I_{\mathrm{coll}}+\gamma^{0}\left[(\gamma\cdot
K+m)I_{\mathrm{coll}}\right]^{\dagger}\gamma^{0}\right\\}\,;$ (35)
$\displaystyle\hbar\,p\cdot\partial_{x}G^{<}(x,p)$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\left\\{(\gamma\cdot
K+m)I_{\mathrm{coll}}-\gamma^{0}\left[(\gamma\cdot
K+m)I_{\mathrm{coll}}\right]^{\dagger}\gamma^{0}\right\\}\;,$ (36)

where we have used $\gamma^{0}(G^{<})^{\dagger}\gamma^{0}\equiv G^{<}$, which
can be derived from Eq. (24). Taking the trace with the appropriate basis
elements of the Clifford decomposition, we obtain for the components of the
Wigner function

$\displaystyle\left(p^{2}-\frac{\hbar^{2}}{4}\partial_{x}^{2}-m^{2}\right)\,\mathrm{Tr}\left(\Gamma_{a}G^{<}\right)$
$\displaystyle=$
$\displaystyle\mathrm{Re}\mathrm{Tr}\left[\Gamma_{a}(\gamma\cdot
K+m)I_{\mathrm{coll}}\right]\;,$ (37)
$\displaystyle\hbar\,p\cdot\partial_{x}\,\mathrm{Tr}\left(\Gamma_{a}G^{<}\right)$
$\displaystyle=$
$\displaystyle\mathrm{Im}\mathrm{Tr}\left[\Gamma_{a}(\gamma\cdot
K+m)I_{\mathrm{coll}}\right]\;.$ (38)

Equations (37) and (38) are equivalent to Eqs. (25) – (34).

###  III.3 Cancellation of off-shell terms

We now show that off-shell terms cancel in the Boltzmann-type equations (38)
to lowest order in the coupling constant. We can decompose the Wigner function
into an on-shell and off-shell part as

$G^{<}(x,p)=G_{\mathrm{on}}^{<}(x,p)+G_{\mathrm{off}}^{<}(x,p)\;,$ (39)

where the on-shell part fulfills $(p^{2}-m^{2})G_{\mathrm{on}}^{<}(x,p)\equiv
0$ up to order $\mathcal{O}(\hbar)$. Thus we obtain from the on-shell
condition (37)

$\mathrm{Tr}\left(\Gamma_{a}G_{\mathrm{off}}^{<}\right)=\frac{1}{p^{2}-m^{2}}\,\mathrm{Re}\mathrm{Tr}\left[\Gamma_{a}(\gamma\cdot
K+m)I_{\mathrm{coll}}\right]+\mathcal{O}(\hbar^{2})\;.$ (40)

Inserting this into the Boltzmann equation (38), we obtain

$\displaystyle\hbar\,p\cdot\partial_{x}\mathrm{Tr}\left(\Gamma_{a}G_{\mathrm{on}}^{<}\right)$
$\displaystyle=$
$\displaystyle\mathrm{Im}\mathrm{Tr}\left[\Gamma_{a}(\gamma\cdot
K+m)I_{\mathrm{coll}}\right]-\frac{\hbar}{p^{2}-m^{2}}\,p\cdot\partial_{x}\,\mathrm{Re}\mathrm{Tr}\left[\Gamma_{a}(\gamma\cdot
K+m)I_{\mathrm{coll}}\right]+\mathcal{O}(\hbar^{3})\;.$ (41)

We can also decompose the collision term into an on-shell and off-shell part
as

$I_{\mathrm{coll}}=I_{\mathrm{coll}}^{(\mathrm{on})}+I_{\mathrm{coll}}^{(\mathrm{off})}\;,$
(42)

such that Eq. (41) becomes

$\hbar\,p\cdot\partial_{x}\,\mathrm{Tr}\left(\Gamma_{a}G_{\mathrm{on}}^{<}\right)=\mathrm{Im}\mathrm{Tr}\left[\Gamma_{a}(\gamma\cdot
K+m)I_{\mathrm{coll}}^{(\mathrm{on})}\right]+\mathrm{Im}\mathrm{Tr}\left[\Gamma_{a}(\gamma\cdot
K+m)\left(I_{\mathrm{coll}}^{(\mathrm{off})}-i\frac{\hbar}{p^{2}-m^{2}}\,p\cdot\partial_{x}\,I_{\mathrm{coll}}\right)\right]+\mathcal{O}(\hbar^{3})\;.$
(43)

From Eqs. (19) and (22) we conclude that the collision term satisfies

$I_{\mathrm{coll}}\left(\gamma\cdot
p-m-\frac{i\hbar}{2}\gamma\cdot\overleftarrow{\partial}_{x}\right)=\mathcal{O}(\hbar^{2}G_{c}^{4})\;.$
(44)

This is of higher order in the coupling constant and will thus be set to zero
in the following. Note that the derivative $\partial_{x}$ acts only on
$G^{\lessgtr}$ but not on $\Sigma^{\lessgtr}$ in $I_{\mathrm{coll}}$, cf. Eq.
(22), since it comes from $\partial_{x_{2}}$ before Fourier transformation of
$y=x_{1}-x_{2}$. We act with the operator $\left(\gamma\cdot
p+m-\frac{i\hbar}{2}\gamma\cdot\overleftarrow{\partial}_{x}\right)$ from the
right-hand side onto the above equation to obtain

$I_{\mathrm{coll}}\left[\left(p^{2}-\frac{\hbar^{2}}{4}\overleftarrow{\partial}_{x}^{2}-m^{2}\right)-i\hbar\,p\cdot\overleftarrow{\partial}_{x}\right]=\mathcal{O}(\hbar^{2}G_{c}^{4})\;.$
(45)

With Eq. (42), Eq. (45) leads to

$I_{\mathrm{coll}}^{(\mathrm{off})}=i\frac{\hbar}{p^{2}-m^{2}}\,p\cdot\partial_{x}\,I_{\mathrm{coll}}+\mathcal{O}(\hbar^{2}G_{c}^{4})\;.$
(46)

Thus, Eq. (43) leads to Boltzmann equations for the on-shell components of the
Wigner function

$\hbar\,p\cdot\partial_{x}\,\mathrm{Tr}\left(\Gamma_{a}G_{\mathrm{on}}^{<}\right)=\mathrm{Im}\mathrm{Tr}\left[\Gamma_{a}(\gamma\cdot
K+m)I_{\mathrm{coll}}^{(\mathrm{on})}\right]+\mathcal{O}(\hbar^{2}G_{c}^{4})\;.$
(47)

This concludes the proof that, to lowest order in the coupling constant, off-
shell contributions cancel in the Boltzmann equation. A similar result using a
different method has been obtained in Refs. Weickgenannt:2020aaf ;
Weickgenannt:2021cuo .

##  IV Semiclassical expansion of KB equation

In this section we derive the system of equations (25) – (34) order by order
in powers of $\hbar$. To this end, we expand every quantity (functions as well
as operators) as

$F=\sum_{n=0}^{\infty}\hbar^{n}F^{(n)}\;,$ (48)

and truncate this expansion at a given order $n$.

###  IV.1 Zeroth order in $\hbar$

At $\mathcal{O}(\hbar^{0})$, the collision term (22) vanishes,
$I_{\mathrm{coll}}^{(0)}=0$, since it is at least of order
$\mathcal{O}(\hbar)$. Equations (25) – (34) become

$\displaystyle p^{\mu}\mathcal{V}_{\mu}^{(0)}-m\mathcal{F}^{(0)}$
$\displaystyle=$ $\displaystyle 0\;,$ (49) $\displaystyle\mathcal{P}^{(0)}$
$\displaystyle=$ $\displaystyle 0\;,$ (50) $\displaystyle
p_{\mu}\mathcal{F}^{(0)}-m\mathcal{V}_{\mu}^{(0)}$ $\displaystyle=$
$\displaystyle 0\;,$ (51)
$\displaystyle\frac{1}{2}\,\epsilon_{\mu\nu\alpha\beta}p^{\nu}\mathcal{S}^{(0)\alpha\beta}+m\mathcal{A}_{\mu}^{(0)}$
$\displaystyle=$ $\displaystyle 0\;,$ (52)
$\displaystyle\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}^{(0)\beta}+m\mathcal{S}^{(0)}_{\mu\nu}$
$\displaystyle=$ $\displaystyle 0\;,$ (53)

and

$\displaystyle p^{\mu}\mathcal{A}^{(0)}_{\mu}$ $\displaystyle=$ $\displaystyle
0\;,$ (54) $\displaystyle p^{\nu}\mathcal{S}^{(0)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle 0\;,$ (55) $\displaystyle p_{[\mu}\mathcal{V}^{(0)}_{\nu]}$
$\displaystyle=$ $\displaystyle 0\;,$ (56)

respectively. In the following, we choose the scalar and axial-vector
components $\mathcal{F}$ and $\mathcal{A}_{\mu}$ as independent ones. Then we
can express the other components in terms of $\mathcal{F}$ and
$\mathcal{A}_{\mu}$ using Eqs. (50), (51), and (53), respectively,

$\displaystyle\mathcal{P}^{(0)}$ $\displaystyle=$ $\displaystyle 0\;,$ (57)
$\displaystyle\mathcal{V}_{\mu}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{m}p_{\mu}\mathcal{F}^{(0)}\;,$ (58)
$\displaystyle\mathcal{S}^{(0)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{m}\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}^{(0)\beta}\;.$
(59)

The solutions for all components at $\mathcal{O}(\hbar^{0})$ are known and
given in Eq. (28) of Ref. (Weickgenannt:2019dks, ). Following Eqs. (37) and
(38), all components at $\mathcal{O}(\hbar^{0})$ satisfy on-shell conditions
and kinetic equations.

###  IV.2 First order in $\hbar$

At $\mathcal{O}(\hbar)$, the collision term (22) is given by

$I_{\mathrm{coll}}^{(1)}=-\frac{i}{2}\left[\Sigma^{<(0)}(x,p)G^{>(0)}(x,p)-\Sigma^{>(0)}(x,p)G^{<(0)}(x,p)\right]\;,$
(60)

where all propagators appearing in $\Sigma^{\gtrless(0)}(x,p)$ are taken at
zeroth order in $\hbar$. We note that the Wigner functions
$G^{\gtrless(0)}(x,p)$ are on-shell since they satisfy Eq. (36) with
$I_{\mathrm{coll}}^{(0)}=0$, so $I_{\mathrm{coll}}^{(1)}$ in Eq. (60) is also
on-shell.

Equations (25) – (29) read to first order in $\hbar$:

$\displaystyle p^{\mu}\mathcal{V}_{\mu}^{(1)}-m\mathcal{F}^{(1)}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(I_{\mathrm{coll}}^{(1)}\right)\;,$ (61)
$\displaystyle\frac{1}{2}\,\partial^{\mu}_{x}\mathcal{A}^{(0)}_{\mu}+m\mathcal{P}^{(1)}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(i\gamma^{5}I_{\mathrm{coll}}^{(1)}\right)\;,$
(62)
$\displaystyle\frac{1}{2}\,\partial_{x}^{\nu}\mathcal{S}_{\nu\mu}^{(0)}-p_{\mu}\mathcal{F}^{(1)}+m\mathcal{V}_{\mu}^{(1)}$
$\displaystyle=$
$\displaystyle-\mathrm{Re\,Tr}\left(\gamma_{\mu}I_{\mathrm{coll}}^{(1)}\right)\;,$
(63)
$\displaystyle\frac{1}{2}\,\epsilon_{\mu\nu\alpha\beta}p^{\nu}\mathcal{S}^{(1)\alpha\beta}+m\mathcal{A}_{\mu}^{(1)}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(\gamma^{5}\gamma_{\mu}I_{\mathrm{coll}}^{(1)}\right)\;,$
(64)
$\displaystyle\frac{1}{2}\,\partial_{x[\mu}\mathcal{V}_{\nu]}^{(0)}-\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}^{(1)\beta}-m\mathcal{S}_{\mu\nu}^{(1)}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(\sigma_{\mu\nu}I_{\mathrm{coll}}^{(1)}\right)\;,$
(65)

where we have used Eq. (50) to simplify Eq. (64). On the other hand, Eqs. (30)
– (34) read to first order in $\hbar$:

$\displaystyle\frac{1}{2}\,\partial^{\mu}_{x}\mathcal{V}^{(0)}_{\mu}$
$\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(I_{\mathrm{coll}}^{(1)}\right)\;,$ (66)
$\displaystyle p^{\mu}\mathcal{A}^{(1)}_{\mu}$ $\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(-i\gamma^{5}I_{\mathrm{coll}}^{(1)}\right)\;,$
(67)
$\displaystyle\frac{1}{2}\,\partial_{x,\mu}\mathcal{F}^{(0)}+p^{\nu}\mathcal{S}_{\nu\mu}^{(1)}$
$\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(\gamma_{\mu}I_{\mathrm{coll}}^{(1)}\right)\;,$
(68)
$\displaystyle\frac{1}{4}\,\epsilon_{\mu\nu\alpha\beta}\partial_{x}^{\nu}\mathcal{S}^{(0)\alpha\beta}+p_{\mu}\mathcal{P}^{(1)}$
$\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(\gamma^{5}\gamma_{\mu}I_{\mathrm{coll}}^{(1)}\right)\;,$
(69)
$\displaystyle\frac{1}{2}\,\epsilon_{\mu\nu\alpha\beta}\partial_{x}^{\alpha}\mathcal{A}^{(0)\beta}+p_{[\mu}\mathcal{V}_{\nu]}^{(1)}$
$\displaystyle=$
$\displaystyle-\mathrm{Im\,Tr}\left(\sigma_{\mu\nu}I_{\mathrm{coll}}^{(1)}\right)\;.$
(70)

Contracting Eq. (51) with $\partial^{\mu}_{x}$ and then using Eq. (66), we
derive the Boltzmann equation for the zeroth-order scalar component at
$\mathcal{O}(\hbar)$,

$p\cdot\partial_{x}\mathcal{F}^{(0)}=2m\,\mathrm{Im\,Tr}\left(I_{\mathrm{coll}}^{(1)}\right)\;.$
(71)

Alternatively, we can derive this equation by contracting Eq. (68) with
$p^{\mu}$ and using the fact that $I_{\mathrm{coll}}^{(1)}$ is on-shell, cf.
discussion after Eq. (60), which allows to replace $p\cdot\gamma\rightarrow m$
under the trace.

On the other hand, multiplying Eq. (70) with
$p_{\sigma}\epsilon^{\rho\sigma\mu\nu}$ and using Eq. (54), we derive the
Boltzmann equation for the zeroth-order axial-vector component at
$\mathcal{O}(\hbar)$,

$p\cdot\partial_{x}\mathcal{A}^{(0)\mu}=-\epsilon^{\mu\nu\alpha\beta}p_{\nu}\,\mathrm{Im\,Tr}\left(\sigma_{\alpha\beta}I_{\mathrm{coll}}^{(1)}\right)\;.$
(72)

Another way of deriving this equation is by multiplying Eq. (53) with
$\epsilon^{\mu\nu\rho\sigma}\partial_{x,\sigma}$ and using Eqs. (62) and (69),
as well as the well-known identity
$\gamma_{5}\sigma^{\mu\nu}=\frac{i}{2}\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta}$
and using the fact that $I_{\mathrm{coll}}^{(1)}$ is on-shell, which allows to
replace
$\mathrm{Tr}\left(p\cdot\gamma\gamma_{5}I_{\mathrm{coll}}^{(1)}\right)\rightarrow
m\,\mathrm{Tr}\left(\gamma_{5}I_{\mathrm{coll}}^{(1)}\right)$.

Multiplying Eq. (61) with $m$, Eq. (63) with $p^{\mu}$, subtracting both
equations, and using Eq. (55) as well as the on-shell condition for
$I_{\mathrm{coll}}^{(1)}$, we derive a mass-shell constraint for
$\mathcal{F}^{(1)}$,

$(p^{2}-m^{2})\mathcal{F}^{(1)}=2m\,\mathrm{Re\,Tr}\left(I_{\mathrm{coll}}^{(1)}\right)\;.$
(73)

The mass-shell constraint for $\mathcal{A}^{(1)}_{\mu}$ follows from Eqs. (64)
and (65), using Eqs. (56) and (67), as well as the on-shell condition for
$I_{\mathrm{coll}}^{(1)}$,

$(p^{2}-m^{2})\mathcal{A}^{(1)}_{\mu}=-\epsilon_{\mu\nu\alpha\beta}p^{\nu}\,\mathrm{Re\,Tr}\left(\sigma^{\alpha\beta}I_{\mathrm{coll}}^{(1)}\right)\;.$
(74)

Multiplying Eq. (64) with $p_{\lambda}\epsilon^{\lambda\rho\sigma\mu}$ and Eq.
(65) with $m$, adding both equations, and using Eqs. (51), (68), we obtain a
mass-shell constraint for $\mathcal{S}^{(1)}_{\mu\nu}$,

$(p^{2}-m^{2})\mathcal{S}^{(1)}_{\mu\nu}=2p_{[\mu}\,\mathrm{Im\,Tr}\left(\gamma_{\nu]}I_{\mathrm{coll}}^{(1)}\right)+2m\,\mathrm{Re\,Tr}\left(\sigma_{\mu\nu}I_{\mathrm{coll}}^{(1)}\right)\;,$
(75)

where we again used the on-shell condition for $I_{\mathrm{coll}}^{(1)}$. The
mass-shell constraint for $\mathcal{P}^{(1)}$ can be obtained from Eqs. (62)
and (69), using Eq. (52) and the on-shell condition for
$I_{\mathrm{coll}}^{(1)}$,

$(p^{2}-m^{2})\mathcal{P}^{(1)}=0\;.$ (76)

The mass-shell constraint for $\mathcal{V}^{(1)}_{\mu}$ can be derived from
Eqs. (63) and (70), using Eqs. (53) and (61), as well as the on-shell
condition for $I_{\mathrm{coll}}^{(1)}$,

$(p^{2}-m^{2})\mathcal{V}_{\mu}^{(1)}=2p_{\mu}\,\mathrm{Re\,Tr}\left(I_{\mathrm{coll}}^{(1)}\right)\;.$
(77)

Once $\mathcal{F}^{(1)}$ and $\mathcal{A}_{\mu}^{(1)}$ are known, we can
obtain all other components at order $\mathcal{O}(\hbar)$ via Eqs. (62), (63),
and (65),

$\displaystyle\mathcal{P}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2m}\partial_{x}^{\mu}\mathcal{A}^{(0)}_{\mu}+\frac{1}{m}\mathrm{Re\,Tr}\left(i\gamma^{5}I_{\mathrm{coll}}^{(1)}\right)\;,$
(78) $\displaystyle\mathcal{V}_{\mu}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{m}p_{\mu}\mathcal{F}^{(1)}-\frac{1}{2m}\partial_{x}^{\nu}\mathcal{S}_{\nu\mu}^{(0)}-\frac{1}{m}\mathrm{Re\,Tr}\left(\gamma_{\mu}I_{\mathrm{coll}}^{(1)}\right)\;,$
(79) $\displaystyle\mathcal{S}_{\mu\nu}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{m}\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}^{(1)\beta}+\frac{1}{2m}\,\partial_{x[\mu}\mathcal{V}_{\nu]}^{(0)}-\frac{1}{m}\mathrm{Re\,Tr}\left(\sigma_{\mu\nu}I_{\mathrm{coll}}^{(1)}\right)\;.$
(80)

Note that the terms $\mathcal{A}_{\mu}^{(0)}$, $\mathcal{S}_{\nu\mu}^{(0)}$,
$\mathcal{V}_{\nu}^{(0)}$, and
$\mathrm{Tr}\left(\Gamma_{a}I_{\mathrm{coll}}^{(1)}\right)$ are all on-shell.
We can act with $(p^{2}-m^{2})$ onto the above equations and verify that the
resulting equations are consistent with Eqs. (73) – (77) for the on-shell
conditions.

From Eqs. (73), (74), (75), and (77) we observe that $\mathcal{F}^{(1)}$,
$\mathcal{V}_{\mu}^{(1)}$, $\mathcal{A}^{(1)}_{\mu}$, and
$\mathcal{S}_{\mu\nu}^{(1)}$ in principle have off-shell contributions at
$\mathcal{O}(\hbar)$, while $\mathcal{P}^{(1)}$ remains on-shell at
$\mathcal{O}(\hbar)$ as shown in Eq. (76), i.e., off-shell effects should
enter at most at $\mathcal{O}(\hbar^{2})$.

###  IV.3 Second order in $\hbar$

At order $\mathcal{O}(\hbar^{2})$ the collision term (22) reads

$I_{\mathrm{coll}}^{(2)}\equiv\Delta
I_{\mathrm{coll}}^{(1)}+I_{\mathrm{coll,PB}}^{(0)}\;,$ (81)

where

$\Delta
I_{\mathrm{coll}}^{(1)}\equiv-\frac{i}{2}\left[\Sigma^{<(1)}(x,p)G^{>(0)}(x,p)-\Sigma^{>(1)}(x,p)G^{<(0)}(x,p)+\Sigma^{<(0)}(x,p)G^{>(1)}(x,p)-\Sigma^{>(0)}(x,p)G^{<(1)}(x,p)\right]$
(82)

arises from an expansion to first order in $\hbar$ of the first term in Eq.
(22), while

$I_{\mathrm{coll,PB}}^{(0)}\equiv-\frac{1}{4}\left[\left\\{\Sigma^{<(0)}(x,p),G^{>(0)}(x,p)\right\\}_{\mathrm{PB}}-\left\\{\Sigma^{>(0)}(x,p),G^{<(0)}(x,p)\right\\}_{\mathrm{PB}}\right]\;,$
(83)

is the leading-order contribution from the second term in Eq. (22). Note that
all two-point functions appearing in $\Sigma^{\gtrless(1)}(x,p)$ are taken at
zeroth order in $\hbar$, $G^{\gtrless}\equiv G^{\gtrless(0)}$, except for one,
for which one has to take the first-order contribution $G^{\gtrless(1)}$.

Equations (25) – (29) read to second order in $\hbar$:

$\displaystyle p^{\mu}\mathcal{V}_{\mu}^{(2)}-m\mathcal{F}^{(2)}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(I_{\mathrm{coll}}^{(2)}\right)\;,$ (84)
$\displaystyle\frac{1}{2}\,\partial^{\mu}_{x}\mathcal{A}^{(1)}_{\mu}+m\mathcal{P}^{(2)}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(i\gamma^{5}I_{\mathrm{coll}}^{(2)}\right)\;,$
(85)
$\displaystyle\frac{1}{2}\,\partial_{x}^{\nu}\mathcal{S}_{\nu\mu}^{(1)}-p_{\mu}\mathcal{F}^{(2)}+m\mathcal{V}_{\mu}^{(2)}$
$\displaystyle=$
$\displaystyle-\mathrm{Re\,Tr}\left(\gamma_{\mu}I_{\mathrm{coll}}^{(2)}\right)\;,$
(86)
$\displaystyle-\frac{1}{2}\,\partial_{x,\mu}\mathcal{P}^{(1)}+\frac{1}{2}\,\epsilon_{\mu\nu\alpha\beta}p^{\nu}\mathcal{S}^{(2)\alpha\beta}+m\mathcal{A}_{\mu}^{(2)}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(\gamma^{5}\gamma_{\mu}I_{\mathrm{coll}}^{(2)}\right)\;,$
(87)
$\displaystyle\frac{1}{2}\,\partial_{x[\mu}\mathcal{V}_{\nu]}^{(1)}-\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}^{(2)\beta}-m\mathcal{S}_{\mu\nu}^{(2)}$
$\displaystyle=$
$\displaystyle\mathrm{Re\,Tr}\left(\sigma_{\mu\nu}I_{\mathrm{coll}}^{(2)}\right)\;.$
(88)

On the other hand, Eqs. (30) – (34) read to second order in $\hbar$:

$\displaystyle\frac{1}{2}\,\partial^{\mu}_{x}\mathcal{V}^{(1)}_{\mu}$
$\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(I_{\mathrm{coll}}^{(2)}\right)\;,$ (89)
$\displaystyle p^{\mu}\mathcal{A}^{(2)}_{\mu}$ $\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(-i\gamma^{5}I_{\mathrm{coll}}^{(2)}\right)\;,$
(90)
$\displaystyle\frac{1}{2}\,\partial_{x,\mu}\mathcal{F}^{(1)}+p^{\nu}\mathcal{S}_{\nu\mu}^{(2)}$
$\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(\gamma_{\mu}I_{\mathrm{coll}}^{(2)}\right)\;,$
(91)
$\displaystyle\frac{1}{4}\,\epsilon_{\mu\nu\alpha\beta}\partial_{x}^{\nu}\mathcal{S}^{(1)\alpha\beta}+p_{\mu}\mathcal{P}^{(2)}$
$\displaystyle=$
$\displaystyle\mathrm{Im\,Tr}\left(\gamma^{5}\gamma_{\mu}I_{\mathrm{coll}}^{(2)}\right)\;,$
(92)
$\displaystyle\frac{1}{2}\,\epsilon_{\mu\nu\alpha\beta}\partial_{x}^{\alpha}\mathcal{A}^{(1)\beta}+p_{[\mu}\mathcal{V}_{\nu]}^{(2)}$
$\displaystyle=$
$\displaystyle-\mathrm{Im\,Tr}\left(\sigma_{\mu\nu}I_{\mathrm{coll}}^{(2)}\right)\;.$
(93)

Taking the four-divergence of Eq. (63) and using Eq. (89) we derive a
Boltzmann equation for the first-order scalar component at
$\mathcal{O}(\hbar^{2})$,

$p\cdot\partial_{x}\mathcal{F}^{(1)}=2m\,\mathrm{Im\,Tr}\left(I_{\mathrm{coll}}^{(2)}\right)+\mathrm{Re\,Tr}\left(\gamma\cdot\partial_{x}I_{\mathrm{coll}}^{(1)}\right)\;.$
(94)

Alternatively, we can obtain a Boltzmann equation for $\mathcal{F}^{(1)}$ by
multiplying Eq. (91) with $p^{\mu}$,

$p\cdot\partial_{x}\mathcal{F}^{(1)}=2p^{\mu}\,\mathrm{Im\,Tr}\left(\gamma_{\mu}I_{\mathrm{coll}}^{(2)}\right)\;.$
(95)

Comparing the right-hand sides of Eqs. (94) and (95), we derive a constraint
equation for the collision terms at first and second order,

$\mathrm{Re\,Tr}\left(\gamma\cdot\partial_{x}I_{\mathrm{coll}}^{(1)}\right)=2\,\mathrm{Im\,Tr}\left[(\gamma\cdot
p-m)I_{\mathrm{coll}}^{(2)}\right]\;.$ (96)

Multiplying Eq. (93) with $p_{\sigma}\epsilon^{\rho\sigma\mu\nu}$ and using
Eq. (67) we derive a Boltzmann equation for the first-order axial-vector
component at $\mathcal{O}(\hbar^{2})$,

$\displaystyle p\cdot\partial_{x}\mathcal{A}^{(1)\mu}$ $\displaystyle=$
$\displaystyle-\epsilon^{\mu\nu\alpha\beta}p_{\nu}\,\mathrm{Im\,Tr}\left(\sigma_{\alpha\beta}I_{\mathrm{coll}}^{(2)}\right)-\mathrm{Re\,Tr}\left(\gamma^{5}\partial_{x}^{\mu}I_{\mathrm{coll}}^{(1)}\right)$
(97) $\displaystyle=$
$\displaystyle-2p^{\mu}\,\mathrm{Im\,Tr}\left(\gamma^{5}I_{\mathrm{coll}}^{(2)}\right)-2\,\mathrm{Im\,Tr}\left(\gamma\cdot
p\,\gamma^{5}\gamma^{\mu}I_{\mathrm{coll}}^{(2)}\right)-\mathrm{Re\,Tr}\left(\gamma^{5}\partial_{x}^{\mu}I_{\mathrm{coll}}^{(1)}\right)\;.$

An alternative way of deriving a Boltzmann equation for $\mathcal{A}^{(1)\mu}$
is by multiplying Eq. (65) with
$\epsilon^{\rho\sigma\mu\nu}\partial_{x,\sigma}$ and using Eqs. (85) and (92),

$p\cdot\partial_{x}\mathcal{A}^{(1)\mu}=-2p^{\mu}\,\mathrm{Im\,Tr}\left(\gamma^{5}I_{\mathrm{coll}}^{(2)}\right)-2m\,\mathrm{Im\,Tr}\left(\gamma^{5}\gamma^{\mu}I_{\mathrm{coll}}^{(2)}\right)-\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}\mathrm{Re\,Tr}\left(\sigma_{\alpha\beta}\partial_{x,\nu}I_{\mathrm{coll}}^{(1)}\right)\;.$
(98)

Comparing the right-hand sides of Eqs. (97) and (98) we derive another
constraint equation for the collision terms at first and second order,

$\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}\,\mathrm{Re\,Tr}\left(\sigma_{\alpha\beta}\partial_{x,\nu}I_{\mathrm{coll}}^{(1)}\right)-\mathrm{Re\,Tr}\left(\gamma^{5}\partial_{x}^{\mu}I_{\mathrm{coll}}^{(1)}\right)=2\,\mathrm{Im\,Tr}\left[(\gamma\cdot
p-m)\gamma^{5}\gamma^{\mu}I_{\mathrm{coll}}^{(2)}\right]\;.$ (99)

With the help of Eq. (44), one can prove that Eqs. (96), (99) are satisfied up
to terms of order $\mathcal{O}(\hbar G_{c}^{4})$.

One can also derive Boltzmann-type equations similar to Eqs. (94), (97), or
Eqs. (95), (98), respectively, for the other components of the Wigner
functions, as well as additional constraints similar to Eqs. (96), (99).
However, we aim at deriving a Boltzmann-type equation for the matrix-valued
spin distribution function, and for this purpose we will only make use of Eqs.
(94), (97).

##  V Matrix-valued spin distribution functions and quasi-classical
contributions to Wigner functions

To the order we are computing, the Wigner functions $G^{\gtrless(0)}$ and
$G^{\gtrless(1)}$ enter the collision terms $I_{\mathrm{coll}}^{(1)}$,
$I_{\mathrm{coll}}^{(2)}$ in the Boltzmann equations (71), (72), (94), (95),
(97), and (98), cf. Eqs. (60), (82), and (83). The zeroth-order quantities
$G^{\gtrless(0)}$ and their components are all on-shell (see Sec. IV.1 for
$G^{<(0)}$; $G^{>(0)}$ has the same properties) and take the same form as in
kinetic theory in the absence of collisions Weickgenannt:2019dks . In the
following, we call a contribution to the $n$-th order Wigner function
$G^{\gtrless(n)}$ a _quasi-classical_ (qc) contribution, if its components
satisfy Eqs. (57) – (59). However, the Clifford components of the first-order
quantities $G^{\gtrless(1)}$ satisfy Eqs. (78) – (80), which contain
additional terms besides the (first-order) quasi-classical contributions.
These are _gradients_ of zeroth-order quantities as well as _collision terms_.
We denote these additional contributions with an index $\nabla$. In general,
the Wigner function at $\mathcal{O}(\hbar)$ can be written as follows:

$G^{\lessgtr(1)}=G_{\mathrm{qc}}^{\lessgtr(1)}+G_{\nabla}^{\lessgtr(1)}+G_{\mathrm{off}}^{\lessgtr(1)}\;,$
(100)

where $G_{\mathrm{qc}}^{\lessgtr(1)}$, $G_{\nabla}^{\lessgtr(1)}$, and
$G_{\mathrm{off}}^{\lessgtr(1)}$ denote the quasi-classical term, the on-shell
gradient and collision term, and the off-shell term, respectively. Note that
$G_{\mathrm{qc}}^{\lessgtr(1)}$ and $G_{\nabla}^{\lessgtr(1)}$ are both on-
shell. By definition, the Clifford components of the quasi-classical
contribution $G_{\mathrm{qc}}^{<(1)}$ satisfy

$\displaystyle\mathcal{P}_{\mathrm{qc}}^{(1)}$ $\displaystyle=$ $\displaystyle
0\;,$ (101) $\displaystyle\mathcal{V}_{{\mathrm{qc}},\mu}^{(1)}$
$\displaystyle=$
$\displaystyle\frac{1}{m}p_{\mu}\mathcal{F}^{(1)}_{\mathrm{qc}}\;,$ (102)
$\displaystyle\mathcal{S}_{{\mathrm{qc}},\mu\nu}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{m}\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}^{(1)\beta}_{\mathrm{qc}}\;,$
(103)

where $\mathcal{F}^{(1)}_{\mathrm{qc}}$ and
$\mathcal{A}^{(1)\beta}_{\mathrm{qc}}$ are the quasi-classical contributions
of the respective Clifford components (which are on-shell). On the other hand,
the gradient and collision contributions contain space-time derivatives of the
zeroth-order Wigner functions and the collision term
$I_{\mathrm{coll}}^{(1)}$,

$\displaystyle\mathcal{P}_{\nabla}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2m}\partial_{x}^{\mu}\mathcal{A}_{\mu}^{(0)}+\frac{1}{m}\mathrm{Re\,Tr}\left(i\gamma^{5}I_{\mathrm{coll}}^{(1)}\right)\;,$
(104) $\displaystyle\mathcal{V}_{\nabla,\mu}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2m}\partial_{x}^{\nu}\mathcal{S}_{\nu\mu}^{(0)}-\frac{1}{m}\mathrm{Re\,Tr}\left(\gamma_{\mu}I_{\mathrm{coll}}^{(1)}\right)\;,$
(105) $\displaystyle\mathcal{S}_{\nabla,\mu\nu}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2m}\,\partial_{x[\mu}\mathcal{V}_{\nu]}^{(0)}-\frac{1}{m}\mathrm{Re\,Tr}\left(\sigma_{\mu\nu}I_{\mathrm{coll}}^{(1)}\right)\;.$
(106)

Note that we have set $\mathcal{F}_{\nabla}^{(1)}=0$ and
$\mathcal{A}_{\nabla}^{(1)\mu}=0$. If $\mathcal{F}_{\nabla}^{(1)}$ and
$\mathcal{A}_{\nabla}^{(1)\mu}$ are nonvanishing, it can be proved that they
can be absorbed into the quasi-classical contributions of the Wigner functions
by a redefinition of the first-order matrix-valued spin distribution functions
(see discussion below). Finally, the _off-shell contributions_ are defined as

$\displaystyle\mathcal{P}_{\mathrm{off}}^{(1)}$ $\displaystyle=$
$\displaystyle 0\;,$ (107) $\displaystyle\mathcal{V}_{\mathrm{off},\mu}^{(1)}$
$\displaystyle=$
$\displaystyle\frac{1}{m}p_{\mu}\,\mathcal{F}^{(1)}_{\mathrm{off}}\;,$ (108)
$\displaystyle\mathcal{S}_{\mathrm{off},\mu\nu}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{m}\epsilon_{\mu\nu\alpha\beta}p^{\alpha}\mathcal{A}^{(1)\beta}_{\mathrm{off}}\;.$
(109)

The sum of quasi-classical, gradient and collision, and off-shell
contributions (101) – (109) satisfy Eqs. (78) – (80). Since $G^{<(0)}$
contains only quasi-classical contributions, there is no need to specify
zeroth-order quantities with the subscript “qc”.

In the remainder of this section we will analyze the structure of the quasi-
classical as well as gradient and collision contributions in terms of single-
particle distribution functions, which are $(2\times 2)$ matrices in the space
of spin indices, the so-called matrix-valued single-particle distribution
functions (MVSDs). Since these contributions exist also in the absence of
collisions, the starting point is the solution of the free Dirac equation,
which can be expanded in the standard way as

$\displaystyle\psi(x)$ $\displaystyle=$
$\displaystyle\sum_{s}\int\frac{d^{3}\mathbf{k}}{(2\pi\hbar)^{3}}\,\frac{1}{2E_{k}}\left[a(s,\mathbf{k})u(s,\mathbf{k})e^{-ik\cdot
x/\hbar}+b^{\dagger}(s,\mathbf{k})v(s,\mathbf{k})e^{ik\cdot
x/\hbar}\right]\;,$ (110) $\displaystyle\overline{\psi}(x)$ $\displaystyle=$
$\displaystyle\sum_{s}\int\frac{d^{3}\mathbf{k}}{(2\pi\hbar)^{3}}\,\frac{1}{2E_{k}}\left[a^{\dagger}(s,\mathbf{k})\overline{u}(s,\mathbf{k})e^{ik\cdot
x/\hbar}+b(s,\mathbf{k})\overline{v}(s,\mathbf{k})e^{-ik\cdot
x/\hbar}\right]\;,$ (111)

where $k^{0}\equiv E_{k}\equiv E_{\mathbf{k}}=\sqrt{\mathbf{k}^{2}+m^{2}}$,
the index $s=\pm 1$ denotes the spin state parallel or anti-parallel to the
quantization direction $\mathbf{n}$, and $a(s,\mathbf{k})$,
$a^{\dagger}(s,\mathbf{k})$, $b(s,\mathbf{k})$, and
$b^{\dagger}(s,\textbf{$\mathbf{k}$})$ are creation and annihilation operators
for fermions and antifermions (of dimension [energy $\times$ length3]1/2),
respectively. The spinors for fermions and antifermions are $u(\mathbf{k},s)$
and $v(\mathbf{k},s)$, respectively. In Dirac representation, the spinors take
the explicit form

$\displaystyle u(s,\mathbf{p})$ $\displaystyle=$
$\displaystyle\sqrt{E_{p}+m}\left(\begin{array}[]{c}\chi_{s}\\\
\frac{\boldsymbol{\sigma}\cdot\mathbf{p}}{E_{p}+m}\chi_{s}\end{array}\right)\;,$
(114) $\displaystyle v(s,\mathbf{p})$ $\displaystyle=$
$\displaystyle-i\sqrt{E_{p}+m}\left(\begin{array}[]{c}\frac{\boldsymbol{\sigma}\cdot\mathbf{p}}{E_{p}+m}\sigma_{2}\chi_{s}^{*}\\\
\sigma_{2}\chi_{s}^{*}\end{array}\right)\;.$ (117)

They satisfy the relation $v(s,\mathbf{p})=i\gamma^{2}u^{*}(s,\mathbf{p})$,
which one readily proves using
$\sigma_{2}\boldsymbol{\sigma}\sigma_{2}=-\boldsymbol{\sigma}^{*}$. Here,
$\chi_{s}$ is the Pauli spinor with spin projection $s$ quantized along the
direction $\mathbf{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{T}$,
so we have $(\mathbf{n}\cdot\boldsymbol{\sigma})\chi_{s}=s\chi_{s}$. One can
verify that the spin orientation of the state $\sigma_{2}\chi_{s}^{*}$ is
opposite to that of $\chi_{s}$.

We now insert Eqs. (110) and (111) into Eq. (15) and, neglecting contributions
from space-like momenta $p^{2}<0$ DeGroot:1980dk , obtain the Wigner function,

$\displaystyle G_{\alpha\beta}^{<}(x,p)$ $\displaystyle=$
$\displaystyle-(2\pi\hbar)^{2}\sum_{r,s}\int\frac{d^{4}q}{(2\pi\hbar)^{4}}\,\frac{1}{4E_{\mathbf{p}+\mathbf{q}/2}E_{\mathbf{p}-\mathbf{q}/2}}$
(118)
$\displaystyle\times\left[\delta\left(p_{0}-\frac{E_{\mathbf{p}+\mathbf{q}/2}+E_{\mathbf{p}-\mathbf{q}/2}}{2}\right)\,\delta\left(q^{0}-E_{\mathbf{p}+\mathbf{q}/2}+E_{\mathbf{p}-\mathbf{q}/2}\right)\,e^{-iq\cdot
x/\hbar}\right.$
$\displaystyle\times\;u_{\alpha}\left(r,\mathbf{p}+\frac{\mathbf{q}}{2}\right)\,\overline{u}_{\beta}\left(s,\mathbf{p}-\frac{\mathbf{q}}{2}\right)\left\langle
a^{\dagger}\left(s,\mathbf{p}-\frac{\mathbf{q}}{2}\right)\,a\left(r,\mathbf{p}+\frac{\mathbf{q}}{2}\right)\right\rangle$
$\displaystyle+\delta\left(p_{0}+\frac{E_{\mathbf{p}+\mathbf{q}/2}+E_{\mathbf{p}-\mathbf{q}/2}}{2}\right)\,\delta\left(q^{0}+E_{\mathbf{p}+\mathbf{q}/2}-E_{\mathbf{p}-\mathbf{q}/2}\right)\,e^{iq\cdot
x/\hbar}$
$\displaystyle\times\left.v_{\alpha}\left(s,-\mathbf{p}+\frac{\mathbf{q}}{2}\right)\,\overline{v}_{\beta}\left(r,-\mathbf{p}-\frac{\mathbf{q}}{2}\right)\left\langle
b\left(r,-\mathbf{p}-\frac{\mathbf{q}}{2}\right)b^{\dagger}\left(s,-\mathbf{p}+\frac{\mathbf{q}}{2}\right)\right\rangle\right]\;.$

We see that the Wigner function depends on $\left\langle
a^{\dagger}\left(s,\mathbf{p}-\mathbf{q}/2\right)\,a\left(r,\mathbf{p}+\mathbf{q}/2\right)\right\rangle$
and $\left\langle
b\left(r,-\mathbf{p}-\mathbf{q}/2\right)b^{\dagger}\left(s,-\mathbf{p}+\mathbf{q}/2\right)\right\rangle$,
a kind of density-matrix elements, which are in general nondiagonal in spin
and momentum space. The delta functions in Eq. (118) show that
$G_{\alpha\beta}^{<}(x,p)$ has both on-shell and off-shell contributions.
While strictly valid only in the noninteracting case (with the collision term
$I_{\mathrm{coll}}$ set to zero), the result (118) still contains all orders
of $\hbar$. For the collision term, we require $G^{<}(x,p)$ up to first order
in $\hbar$, i.e., we are allowed to expand Eq. (118) up to this order.

In order to exhibit the structure of the on-shell part of
$G_{\alpha\beta}^{<}(x,p)$ in Eq. (118), we expand the integrand in powers of
$\mathbf{q}$, with the exception of the creation and annihilation operators.
Such an expansion is equivalent to an expansion in $\hbar$, because
$\mathbf{q}$, accompanied by a factor $e^{i\mathbf{q}\cdot\mathbf{x}/\hbar}$,
is equivalent to $-i\hbar\nabla_{x}$ acting on the latter exponential. At
leading order $\mathcal{O}(\hbar^{0})$, it can be shown that the Wigner
function can be written as

$\displaystyle G_{\alpha\beta}^{<(0)}(x,p)$ $\displaystyle=$
$\displaystyle-2\pi\hbar\,\theta(p_{0})\delta\left(p^{2}-m^{2}\right)\sum_{r,s}u_{\alpha}\left(r,p\right)\overline{u}_{\beta}\left(s,p\right)\,f_{sr}^{(+,0)}\left(x,p\right)$
(119)
$\displaystyle-2\pi\hbar\,\theta(-p_{0})\delta\left(p^{2}-m^{2}\right)\sum_{r,s}v_{\alpha}\left(s,\overline{p}\right)\overline{v}_{\beta}\left(r,\overline{p}\right)\left[\delta_{sr}-f_{sr}^{(-,0)}\left(x,\overline{p}\right)\right]\;,$

where the MVSDs for fermions and antifermions are defined as

$\displaystyle f_{sr}^{(+,0)}\left(x,p\right)$ $\displaystyle\equiv$
$\displaystyle\int\frac{d^{4}q}{2(2\pi\hbar)^{4}}\,2\pi\hbar\,\delta(p\cdot
q)\,e^{-iq\cdot x/\hbar}\,\left\langle
a^{\dagger}\left(s,\mathbf{p}-\frac{\mathbf{q}}{2}\right)\,a\left(r,\mathbf{p}+\frac{\mathbf{q}}{2}\right)\right\rangle\;,$
(120) $\displaystyle f_{sr}^{(-,0)}\left(x,\overline{p}\right)$
$\displaystyle\equiv$
$\displaystyle\int\frac{d^{4}q}{2(2\pi\hbar)^{4}}\,2\pi\hbar\,\delta(\overline{p}\cdot
q)\,e^{-iq\cdot x/\hbar}\,\left\langle
b^{\dagger}\left(s,-\mathbf{p}-\frac{\mathbf{q}}{2}\right)b\left(r,-\mathbf{p}+\frac{\mathbf{q}}{2}\right)\right\rangle\;,$
(121)

with $p\equiv(E_{\mathbf{p}},\mathbf{p})$ and
$\overline{p}^{\mu}\equiv(E_{\mathbf{p}},-\mathbf{p})$. Note that the overall
factor $\hbar$ in Eq. (119) is convention and does not participate in the
power counting in $\hbar$ (it is the same factor that is factored out on both
sides of the KB equation, see discussion in Sec. III.1). One can verify that
$G^{<(0)}$ in Eq. (119) takes the same form as in kinetic theory without
collisions. Note that even in the noninteracting case without collisions,
there are contributions of $\mathcal{O}(\hbar)$ to the Wigner function (118)
in the $\mathbf{q}$-expansion, which result in space-time derivatives of
$f_{sr}^{(\pm,0)}$. With collisions, the Wigner function of
$\mathcal{O}(\hbar)$ has to be determined by solving the respective equation
of motion at the same order.

We assume that the quasi-classical contribution to the Wigner function at
$\mathcal{O}(\hbar)$ arises from the $\mathcal{O}(\hbar)$ corrections of the
MVSDs to Eq. (119),

$\displaystyle G_{\mathrm{qc},\alpha\beta}^{<(1)}(x,p)$ $\displaystyle=$
$\displaystyle-2\pi\hbar\,\theta(p_{0})\delta\left(p^{2}-m^{2}\right)\sum_{r,s}u_{\alpha}\left(r,p\right)\overline{u}_{\beta}\left(s,p\right)\,f_{sr}^{(+,1)}\left(x,p\right)$
(122)
$\displaystyle+2\pi\hbar\,\theta(-p_{0})\delta\left(p^{2}-m^{2}\right)\sum_{r,s}v_{\alpha}\left(s,\overline{p}\right)\overline{v}_{\beta}\left(r,\overline{p}\right)f_{sr}^{(-,1)}\left(x,\overline{p}\right)\;.$

Note that $f_{sr}^{(\pm,0)}$ and $f_{sr}^{(\pm,1)}$ are to be determined by
solving the Boltzmann equation at $\mathcal{O}(\hbar)$ and
$\mathcal{O}(\hbar^{2})$, respectively. One can verify that the Clifford
components of $G^{<(1)}_{\mathrm{qc}}$ in Eq. (122) satisfy Eqs. (101) –
(103). Therefore, up to $\mathcal{O}(\hbar)$ the quasi-classical contribution
to the Wigner function can be written as

$\displaystyle G_{\mathrm{qc},\alpha\beta}^{<}(x,p)$ $\displaystyle=$
$\displaystyle G_{\alpha\beta}^{<(0)}(x,p)+\hbar
G_{\mathrm{qc},\alpha\beta}^{<(1)}(x,p)+\mathcal{O}(\hbar^{2})$ (123)
$\displaystyle=$
$\displaystyle-2\pi\hbar\,\theta(p_{0})\delta\left(p^{2}-m^{2}\right)\sum_{r,s}u_{\alpha}\left(r,p\right)\overline{u}_{\beta}\left(s,p\right)\,f_{sr}^{(+)}\left(x,p\right)$
$\displaystyle-2\pi\hbar\,\theta(-p_{0})\delta\left(p^{2}-m^{2}\right)\sum_{r,s}v_{\alpha}\left(s,\overline{p}\right)\overline{v}_{\beta}\left(r,\overline{p}\right)\left[\delta_{sr}-f_{sr}^{(-)}\left(x,\overline{p}\right)\right]+\mathcal{O}(\hbar^{2})\;,$

where

$f_{sr}^{(\pm)}\left(x,p\right)=f_{sr}^{(\pm,0)}\left(x,p\right)+\hbar
f_{sr}^{(\pm,1)}\left(x,p\right)+\mathcal{O}(\hbar^{2})\;.$ (124)

Equation (123) indicates that, up to $\mathcal{O}(\hbar)$, the quasi-classical
part of the Wigner function can be expressed in terms of the MVSDs.

On the other hand, the Wigner function $G^{>(0)}(x,p)$ can be obtained from
$G^{<(0)}(x,p)$ by replacing
$f_{sr}^{(+,0)}\rightarrow-(\delta_{sr}-f_{sr}^{(+,0)})$ and
$\delta_{sr}-f_{sr}^{(-,0)}\rightarrow-f_{sr}^{(-,0)}$ in Eq. (119). At
$\mathcal{O}(\hbar)$ one can convince oneself by an explicit calculation that
$G_{\mathrm{qc}}^{>(1)}(x,p)=G_{\mathrm{qc}}^{<(1)}(x,p)$, i.e., it is also
given by Eq. (122).

We can extract the MVSDs $f_{sr}^{(\pm)}$ up to $\mathcal{O}(\hbar)$ by
contracting $G^{<}_{\mathrm{qc}}(x,p)$ in Eq. (123) with Dirac spinors and an
integration over $p^{0}$:

$\displaystyle f_{sr}^{(+)}(x,p)$ $\displaystyle=$
$\displaystyle-\frac{E_{p}}{4\pi\hbar m^{2}}\int
dp^{0}\,\theta(p^{0})\,\overline{u}(r,p)G_{\mathrm{qc}}^{<}(x,p)u(s,p)+\mathcal{O}(\hbar^{2})\;,$
(125) $\displaystyle f_{sr}^{(-)}(x,\overline{p})$ $\displaystyle=$
$\displaystyle\frac{E_{p}}{4\pi\hbar m^{2}}\int
dp^{0}\,\theta(-p^{0})\,\overline{v}(s,\overline{p})G_{\mathrm{qc}}^{<}(x,p)v(r,\overline{p})+\delta_{rs}+\mathcal{O}(\hbar^{2})\;.$
(126)

We can also relate the MVSDs to the components of the Clifford decomposition
of $G^{<}(x,p)$. From $\mathcal{F}(x,p)=\mathrm{Tr}[G^{<}(x,p)]$ and Eqs.
(119), (122) we obtain the quasi-classical part of the scalar component up to
$\mathcal{O}(\hbar)$

$\displaystyle\mathcal{F}_{\mathrm{qc}}(x,p)$ $\displaystyle=$
$\displaystyle-2\pi\hbar\frac{m}{E_{p}}\left\\{\delta(p_{0}-E_{p})\,\mathrm{tr}\left[f^{(+)}(x,p)\right]+\delta(p_{0}+E_{p})\,\mathrm{tr}\left[f^{(-)}(x,\overline{p})-1\right]\right\\}+\mathcal{O}(\hbar^{2})\;,$
(127)

where “tr” denotes the trace in the two-dimensional space of spin indices and
$f^{(+)}$ denotes the MVSD in $(2\times 2)$ matrix form. The $\hbar$ in the
prefactor of the first term again does not participate in the power counting
in $\hbar$. On the other hand, from
$\mathcal{A}^{\mu}(x,p)=\mathrm{Tr}[\gamma^{\mu}\gamma^{5}G^{<}(x,p)]$ and
Eqs. (119), (122) we obtain the quasi-classical part of the axial-vector
component up to $\mathcal{O}(\hbar)$

$\displaystyle\mathcal{A}_{\mathrm{qc}}^{\mu}$ $\displaystyle=$
$\displaystyle-2\pi\hbar\frac{m}{E_{p}}\left\\{\delta(p_{0}-E_{p})\,n_{j}^{(+)\mu}\,\mathrm{tr}\left[\tau_{j}^{T}\,f^{(+)}(x,p)\right]+\delta(p_{0}+E_{p})\,n_{j}^{(-)\mu}\,\mathrm{tr}\left[\tau_{j}^{T}\,f^{(-)}(x,\overline{p})\right]\right\\}+\mathcal{O}(\hbar^{2})\;,$
(128)

where $n_{j}^{(\pm)\mu}$, with $j=1,2,3$, are polarization four-vectors
defined by

$n_{j}^{(\pm)\mu}\equiv
n^{\mu}(\pm\mathbf{p},\mathbf{n}_{j})=\left(\pm\frac{\mathbf{n}_{j}\cdot\mathbf{p}}{m},\mathbf{n}_{j}+\frac{(\mathbf{n}_{j}\cdot\mathbf{p})\mathbf{p}}{m(E_{p}+m)}\right)^{T}\;.$
(129)

Here, $\mathbf{n}_{j}$ ($j=1,2,3$) are the set of orthonormal vectors (which
form a right-handed basis in three spatial dimensions)

$\mathbf{n}_{1}\equiv\left(\begin{array}[]{c}\cos\phi\cos\theta\\\
\sin\phi\cos\theta\\\
-\sin\theta\end{array}\right)\;,\;\;\mathbf{n}_{2}\equiv\left(\begin{array}[]{c}-\sin\phi\\\
\cos\phi\\\
0\end{array}\right)\;,\;\;\mathbf{n}_{3}\equiv\left(\begin{array}[]{c}\cos\phi\sin\theta\\\
\sin\phi\sin\theta\\\ \cos\theta\end{array}\right)\;,$ (130)

where $\mathbf{n}_{3}\equiv\mathbf{n}$ equals the spin quantization direction
in the rest frame of the particle. These vectors are constant, i.e., the
angles $\phi$ and $\theta$ point into a fixed direction in space. With the
Pauli spinors

$\displaystyle\chi_{+}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}e^{-i\phi}\cos(\theta/2)\\\
\sin(\theta/2)\end{array}\right),$ (133) $\displaystyle\chi_{-}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-e^{-i\phi}\sin(\theta/2)\\\
\cos(\theta/2)\end{array}\right),$ (136)

we have the identity

$\chi_{r}^{\dagger}\boldsymbol{\sigma}\chi_{s}=\left(\begin{array}[]{cc}\mathbf{n}_{3}&\mathbf{n}_{1}-i\mathbf{n}_{2}\\\
\mathbf{n}_{1}+i\mathbf{n}_{2}&-\mathbf{n}_{3}\end{array}\right)_{rs}=\mathbf{n}_{j}(\tau_{j})_{rs}=\mathbf{n}_{j}(\tau_{j}^{T})_{sr}\;,$
(137)

where $r,s=\pm$ denotes the spin state and
$\boldsymbol{\tau}=(\tau_{1},\tau_{2},\tau_{3})$ is the vector of Pauli
matrices in spin index space. One can verify $n_{j}^{+}\cdot
n_{j}^{+}=n_{j}^{-}\cdot n_{j}^{-}=-1$ (no summation over $j$) with
$n_{j}^{\pm\mu}$ given by Eq. (129). The order $\mathcal{O}(\hbar)$ quasi-
classical contributions to the other components $\mathcal{P}(x,p)$,
$\mathcal{V}^{\mu}(x,p)$, and $\mathcal{S}^{\mu\nu}(x,p)$ can be obtained from
Eqs. (101) – (103).

To conclude this section, we summarize the main result: We define the quasi-
classical contribution to the Wigner function as that whose Clifford
components satisfy Eqs. (101) – (103). The quasi-classical scalar and axial-
vector components are treated as independent variables. All components that
satisfy Eqs. (101) – (103) can be assembled into $G^{\lessgtr}$ in terms of
MVSDs in the same way as in the free-streaming or collisionless case. This
does not mean that the MVSDs have the same values as in the collisionless
case: they are the solutions of Boltzmann equations including collisions. In
other words, in terms of the MVSDs, the quasi-classical contribution
$G^{\lessgtr}_{\mathrm{qc}}$ takes the same form as in the collisionless case,
but the MVSDs assume different values. We remark that, as an alternative to
the MVSDs, we can also define a scalar spin-dependent distribution by
extending the phase space and introducing a continuous spin variable. The
scalar spin-dependent distribution is equivalent to the MVSDs and allows to
combine the Boltzmann equations for the scalar and axial-vector components
into a single equation, for details see Refs. Weickgenannt:2020aaf ;
Weickgenannt:2021cuo .

##  VI Boltzmann equations with collisions at leading order

In this section we will derive Boltzmann equations in terms of the leading-
order MVSDs, based on Eqs. (71) and (72) for the scalar and axial-vector
components, respectively. For the sake of simplicity, we will neglect the
contribution of antiparticles and consequently suppress the “+” superscripts
at the MVSDs. As shown in Eq. (119), the Wigner function and its components at
$\mathcal{O}(\hbar^{0})$ are on-shell and take the same form as in kinetic
theory. Also the collision term $I_{\mathrm{coll}}^{(1)}$ appearing in
Boltzmann equations, i.e., Eqs. (71), (72), involves the Wigner functions only
at $\mathcal{O}(\hbar^{0})$, as shown in Eq. (60). Another feature of the
leading-order Boltzmann equations is that the collision term
$I_{\mathrm{coll}}^{(1)}$ is local in space-time, namely, space-time
derivatives are absent, which means that the particles collide at a single
point in space-time.

In order to isolate the fermion contribution, we integrate Eq. (71) over
$p_{0}$ from $0$ to $\infty$ and obtain, using Eq. (127) to lowest order in
$\hbar$, the Boltzmann equation for the trace of the MVSD,

$\frac{1}{E_{p}}\,p\cdot\partial_{x}\mathrm{tr}\left[f^{(0)}(x,p)\right]=-\frac{1}{\pi\hbar}\int_{0}^{\infty}dp_{0}\,\mathrm{Im}\mathrm{Tr}\left(I_{\mathrm{coll}}^{(1)}\right)\;.$
(138)

Similarly, we integrate Eq. (72) over $p_{0}$ from $0$ to $\infty$ to obtain
the Boltzmann equation for the polarization part of the MVSD

$\frac{1}{E_{p}}\,p\cdot\partial_{x}\mathrm{tr}\left[n_{j}^{(+)\mu}\tau_{j}^{T}f^{(0)}(x,p)\right]=\frac{1}{2\pi\hbar
m}\epsilon^{\mu\nu\alpha\beta}\int_{0}^{\infty}dp_{0}\,p_{\nu}\mathrm{Im}\mathrm{Tr}\left(\sigma_{\alpha\beta}I_{\mathrm{coll}}^{(1)}\right)\;.$
(139)

In the above two equations $f^{(0)}$ denotes the matrix form of the zeroth-
order MVSD, which can be expanded in a basis of unit matrix and Pauli matrices
as

$f^{(0)}=\frac{1}{2}\mathrm{tr}\left(f^{(0)}\right)+\frac{1}{2}\boldsymbol{\tau}^{T}\cdot\mathrm{tr}\left(\boldsymbol{\tau}^{T}f^{(0)}\right)\;.$
(140)

We observe that the trace and the polarization part of the MVSD are equivalent
to the MVSD itself. Equations (138) and (139) constitute the Boltzmann
equations at leading order in $\hbar$.

The collision term $I_{\mathrm{coll}}^{(1)}$ in Eq. (60) depends on the self-
energies $\Sigma^{\lessgtr(0)}$. Corresponding to the Feynman diagrams in Fig.
2, we have

$\displaystyle\Sigma^{>}(x,p)$ $\displaystyle=$ $\displaystyle
4G_{a}G_{b}\int\frac{d^{4}p_{1}}{(2\pi\hbar)^{4}}\frac{d^{4}p_{2}}{(2\pi\hbar)^{4}}\frac{d^{4}p_{3}}{(2\pi\hbar)^{4}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})\mathrm{Tr}\left[\Gamma_{a}G^{<}(x,p_{3})\Gamma_{b}G^{>}(x,p_{1})\right]\Gamma_{b}G^{>}(x,p_{2})\Gamma_{a}$
$\displaystyle-$ $\displaystyle
4G_{a}G_{b}\int\frac{d^{4}p_{1}}{(2\pi\hbar)^{4}}\frac{d^{4}p_{2}}{(2\pi\hbar)^{4}}\frac{d^{4}p_{3}}{(2\pi\hbar)^{4}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})\Gamma_{b}G^{>}(x,p_{1})\Gamma_{a}G^{<}(x,p_{3})\Gamma_{b}G^{>}(x,p_{2})\Gamma_{a}\;,$
$\displaystyle\Sigma^{<}(x,p)$ $\displaystyle=$
$\displaystyle\Sigma^{>}[G^{>}\leftrightarrow G^{<}]\;.$ (141)

The zeroth-order self-energies $\Sigma^{\lessgtr(0)}$ appearing in
$I_{\mathrm{coll}}^{(1)}$, Eq. (60), are obtained from Eq. (141) by replacing
$G^{\lessgtr}$ by their zeroth-order values as e.g. given in Eq. (119). One
can obtain $\Sigma^{<}$ from $\Sigma^{>}$ by interchanging
$G^{>}\leftrightarrow G^{<}$ and vice versa. Note that there is an overall
sign difference between the two terms in $\Sigma^{>}(x,p)$ [and similar for
$\Sigma^{<}(x,p)$]. The first terms correspond to Figs. 2(a) and (b), while
the second terms correspond to Figs. 2(c) and (d), respectively.

We insert Eq. (119) for $G^{<(0)}$ and the corresponding formula for
$G^{>(0)}$ into Eq. (141) to obtain $\Sigma_{(0)}^{\lessgtr}$, and then insert
$\Sigma_{(0)}^{\lessgtr}$ and $G_{(0)}^{\lessgtr}$ into Eq. (60) to obtain
$I_{\mathrm{coll}}^{(1)}$, to finally obtain the Boltzmann equation (138) for
the scalar part in the form

$\displaystyle\frac{1}{E_{p}}p\cdot\partial_{x}\mathrm{tr}\left[f^{(0)}(x,p)\right]=\int_{0}^{\infty}\frac{dp_{0}}{2\pi\hbar}\,\mathrm{Re}\mathrm{Tr}\left[\Sigma^{<(0)}(x,p)G^{>(0)}(x,p)-\Sigma^{>(0)}(x,p)G^{<(0)}(x,p)\right]$
(142) $\displaystyle=$
$\displaystyle\frac{1}{2E_{p}}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}2E_{1}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}2E_{2}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}2E_{3}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})$
$\displaystyle\times\mathrm{Re}\left\\{\left[f_{1}^{(0)}f_{2}^{(0)}\left(1-f_{3}^{(0)}\right)\left(1-f^{(0)}\right)-f_{3}^{(0)}f^{(0)}\left(1-f_{1}^{(0)}\right)\left(1-f_{2}^{(0)}\right)\right]\left(M_{a}^{\mathrm{scalar}}+M_{b}^{\mathrm{scalar}}\right)\right\\}\;,$

where $f_{i}^{(0)}\equiv f_{s_{i}r_{i}}^{(0)}(x,p_{i})$,
$p_{i}^{\mu}=(E_{i},\mathbf{p}_{i})\equiv(E_{p_{i}},\mathbf{p}_{i})$
($i=1,2,3$) and $f^{(0)}\equiv f_{sr}^{(0)}(x,p)$,
$p^{\mu}=(E_{p},\mathbf{p})$, and we denoted unit matrices in spin space as 1,
e.g., $\delta_{rs}\rightarrow 1$. The scalar spin-dependent matrix elements
$M_{a}^{\mathrm{scalar}}$ and $M_{b}^{\mathrm{scalar}}$ in the above equation
are defined as

$\displaystyle M_{a}^{\mathrm{scalar}}$ $\displaystyle\equiv$ $\displaystyle
4G_{a}G_{b}\mathrm{Tr}\left[\Gamma_{a}u(r_{3},p_{3})\overline{u}(s_{3},p_{3})\Gamma_{b}u(r_{1},p_{1})\overline{u}(s_{1},p_{1})\right]\mathrm{Tr}\left[\Gamma_{b}u(r_{2},p_{2})\overline{u}(s_{2},p_{2})\Gamma_{a}u(r,p)\overline{u}(s,p)\right],$
$\displaystyle M_{b}^{\mathrm{scalar}}$ $\displaystyle\equiv$
$\displaystyle-4G_{a}G_{b}\mathrm{Tr}\left[\Gamma_{b}u(r_{1},p_{1})\overline{u}(s_{1},p_{1})\Gamma_{a}u(r_{3},p_{3})\overline{u}(s_{3},p_{3})\Gamma_{b}u(r_{2},p_{2})\overline{u}(s_{2},p_{2})\Gamma_{a}u(r,p)\overline{u}(s,p)\right]\;.$
(143)

Since these matrix elements depend on all spin indices, sums over
$s_{i},r_{i}$ ($i=1,2,3$) and $s,r$ are implied on the right-hand side of Eq.
(142).

Following the same procedure as in the derivation of Eq. (142), the Boltzmann
equation (139) for the polarization part can be written as

$\displaystyle\frac{1}{E_{p}}p\cdot\partial_{x}\mathrm{tr}\left[n_{j}^{(+)\mu}\tau_{j}^{T}f^{(0)}(x,p)\right]=-\frac{1}{2m}\epsilon^{\mu\nu\alpha\beta}\int_{0}^{\infty}\frac{dp_{0}}{2\pi\hbar}p_{\nu}\mathrm{Re}\mathrm{Tr}\left[\sigma_{\alpha\beta}\left(\Sigma^{<(0)}(x,p)G^{>(0)}(x,p)-\Sigma^{>(0)}(x,p)G^{<(0)}(x,p)\right)\right]$
(144) $\displaystyle=$
$\displaystyle-\frac{1}{4E_{p}m}\epsilon^{\mu\nu\alpha\beta}p_{\nu}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}2E_{1}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}2E_{2}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}2E_{3}}(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})$
$\displaystyle\times\mathrm{Re}\left\\{\left[f_{1}^{(0)}f_{2}^{(0)}\left(1-f_{3}^{(0)}\right)\left(1-f^{(0)}\right)-f_{3}^{(0)}f^{(0)}\left(1-f_{1}^{(0)}\right)\left(1-f_{2}^{(0)}\right)\right]\left(M_{a,\alpha\beta}^{\mathrm{pol}}+M_{b,\alpha\beta}^{\mathrm{pol}}\right)\right\\}\;,$

where the polarization parts of the spin-dependent matrix elements
$M_{a,\alpha\beta}^{\mathrm{pol}}$ and $M_{b,\alpha\beta}^{\mathrm{pol}}$ are
defined as

$\displaystyle M_{a,\alpha\beta}^{\mathrm{pol}}$ $\displaystyle=$
$\displaystyle
4G_{a}G_{b}\mathrm{Tr}\left[\Gamma_{a}u(r_{3},p_{3})\overline{u}(s_{3},p_{3})\Gamma_{b}u(r_{1},p_{1})\overline{u}(s_{1},p_{1})\right]\mathrm{Tr}\left[\sigma_{\alpha\beta}\Gamma_{b}u(r_{2},p_{2})\overline{u}(s_{2},p_{2})\Gamma_{a}u(r,p)\overline{u}(s,p)\right]\;,$
$\displaystyle M_{b,\alpha\beta}^{\mathrm{pol}}$ $\displaystyle=$
$\displaystyle-4G_{a}G_{b}\mathrm{Tr}\left[\sigma_{\alpha\beta}\Gamma_{b}u(r_{1},p_{1})\overline{u}(s_{1},p_{1})\Gamma_{a}u(r_{3},p_{3})\overline{u}(s_{3},p_{3})\Gamma_{b}u(r_{2},p_{2})\overline{u}(s_{2},p_{2})\Gamma_{a}u(r,p)\overline{u}(s,p)\right]\;.$
(145)

An interesting observation is that, if the MVSD is proportional to the unit
matrix, i.e., only the first term in Eq. (140) is nonzero, the left-hand side
of Eq. (144) vanishes. Thus, also the collison term on the right-hand side
must vanish. In general, this is not obvious, unless one is in global
equilibrium, where already the term in brackets vanishes on account of Eq.
(142).

Since we considered only fermions in deriving Eq. (142) from Eq. (138), we
have integrated over $p_{i0}$ ($i=1,2,3$) in the range $[0,+\infty]$. For
processes involving antifermions, we integrate over any $p_{i0}$ ($i=1,2,3$)
or $p_{0}$ in the range $[-\infty,0]$. For an antifermion, the four-momentum
in the distribution function and the spinors becomes
$\overline{p}^{\mu}=(E_{p},-\mathbf{p})$.

In summary, we have shown in this section that the MVSDs play the central role
in the Boltzmann equations for the scalar and polarization part, which are
derived from the corresponding equations for the scalar and axial-vector
components of the Wigner function. These equations are closed and describe the
time evolution of the MVSDs at leading order in $\hbar$.

Figure 2: Feynman diagrams for (a,c) $\Sigma^{>}(x,p)$ and (b,d)
$\Sigma^{<}(x,p)$. Solid lines represent fermion propagators, wavy lines
represent boson propagators, solid circles represent the vertices $g_{a}$ with
$a$ labeling coupling types. In the NJL model wavy lines attached with two
solid circles represent vertices $2G_{a}$.

##  VII Boltzmann equations with collisions at next-to-leading order

In this section we derive Boltzmann equations at next-to-leading order
$\mathcal{O}(\hbar^{2})$ from Eqs. (94) and (97), respectively. The Boltzmann
equation (94) for $\mathcal{F}^{(1)}$ involves the collision terms
$I_{\mathrm{coll}}^{(2)}=\Delta
I_{\mathrm{coll}}^{(1)}+I_{\mathrm{coll},\mathrm{PB}}^{(0)}$ and
$\partial_{x}I_{\mathrm{coll}}^{(1)}$, where the collision term $\Delta
I_{\mathrm{coll}}^{(1)}$, cf. Eq. (82), depends on the first-order Wigner
functions $G^{\lessgtr(1)}$ as well as the leading-order ones
$G^{\lessgtr(0)}$, while the collision term
$I_{\mathrm{coll},\mathrm{PB}}^{(0)}$, cf. Eq. (83), and
$\partial_{x}I_{\mathrm{coll}}^{(1)}$ depend on $G^{\lessgtr(0)}$ only. The
same holds for the Boltzmann equation (97) for $\mathcal{A}^{(1)}_{\mu}$.
Since these equations and the Wigner functions are rather complicated at this
order [$\mathcal{O}(\hbar^{2})$ for the equations and $\mathcal{O}(\hbar)$ for
the Wigner functions], it is helpful to decompose the full Wigner function at
$\mathcal{O}(\hbar)$ into quasi-classical, gradient and collision, as well as
off-shell contributions, as shown in Eq. (100). The quasi-classical
contributions (101) – (103) and the gradient and collision contributions (104)
– (106) are on-shell. The off-shell part $G_{\mathrm{off}}^{\lessgtr(1)}$, cf.
Eqs. (107) – (109), will be neglected, since it is of higher order in the
coupling constant as discussed in Sec. III.3.

Now consider the collision term $\Delta I_{\mathrm{coll}}^{(1)}$ in Eq. (82),
which involves $\Sigma^{\lessgtr(1)}(x,p)$. This function can be obtained from
Eq. (141) by replacing one $G^{\lessgtr}$ by $G^{\lessgtr(1)}$ and the other
two by $G^{\lessgtr(0)}$. The quasi-classical contributions $G^{\lessgtr(0)}$
and $G^{\lessgtr(1)}_{\mathrm{qc}}$ are given by Eqs. (119) and (122),
respectively, while the gradient and collision contributions
$G_{\nabla}^{\lessgtr(1)}$ can be simplified as

$\displaystyle G_{\nabla}^{<(1)}=G_{\nabla}^{>(1)}$ $\displaystyle\approx$
$\displaystyle-\frac{1}{8m}i\gamma^{5}\partial_{x}\cdot\mathcal{A}^{(0)}-\frac{1}{8m}\gamma^{\nu}\partial_{x}^{\mu}\mathcal{S}_{\mu\nu}^{(0)}+\frac{1}{8m}\sigma^{\mu\nu}\partial_{x,\mu}\mathcal{V}_{\nu}^{(0)}=\frac{i}{4m}\left[\gamma_{\mu},\partial_{x}^{\mu}G^{<(0)}\right]\;.$
(146)

Here, we neglected terms that involve $I_{\mathrm{coll}}^{(1)}$, since they
are of higher order in the coupling constant $G_{c}$, when we insert
$G_{\nabla}^{\lessgtr(1)}$ into $\Delta I_{\mathrm{coll}}^{(1)}$ on the right-
hand side of the Boltzmann equations. We note that $G_{\nabla}^{\lessgtr(1)}$
contains only pseudoscalar, vector, and tensor components as in Eqs. (104) –
(106). Therefore we can write $\Delta I_{\mathrm{coll}}^{(1)}=\Delta
I_{\mathrm{coll,\,qc}}^{(1)}+\Delta I_{\mathrm{coll,}\,\nabla}^{(1)}$ as the
sum of quasi-classical and gradient plus collision contributions $\Delta
I_{\mathrm{coll,\,qc}}^{(1)}$ and $\Delta I_{\mathrm{coll,}\,\nabla}^{(1)}$,
where Eq. (146) is used for the first-order Wigner functions in $\Delta
I_{\mathrm{coll,}\,\nabla}^{(1)}$.

As an example for the first-order self-energy, we consider the first term in
Eq. (141), which corresponds to Figs. 2 (a,b). This term involves the
following Dirac part in the integrand,

$\displaystyle\mathrm{Tr}\left[\Gamma_{a}G^{<(1)}(x,p_{3})\Gamma_{b}G^{>(0)}(x,p_{1})\right]\Gamma_{b}G^{>(0)}(x,p_{2})\Gamma_{a}$
$\displaystyle+$
$\displaystyle\mathrm{Tr}\left[\Gamma_{a}G^{<(0)}(x,p_{3})\Gamma_{b}G^{>(1)}(x,p_{1})\right]\Gamma_{b}G^{>(0)}(x,p_{2})\Gamma_{a}$
$\displaystyle+$
$\displaystyle\mathrm{Tr}\left[\Gamma_{a}G^{<(0)}(x,p_{3})\Gamma_{b}G^{>(0)}(x,p_{1})\right]\Gamma_{b}G^{>(1)}(x,p_{2})\Gamma_{a}\;.$
(147)

On the other hand, the second term corresponding to Figs. 2 (c,d) involves the
following Dirac part in the integrand,

$\displaystyle\Gamma_{b}G^{>(1)}(x,p_{1})\Gamma_{a}G^{<(0)}(x,p_{3})\Gamma_{b}G^{>(0)}(x,p_{2})\Gamma_{a}$
$\displaystyle+$
$\displaystyle\Gamma_{b}G^{>(0)}(x,p_{1})\Gamma_{a}G^{<(1)}(x,p_{3})\Gamma_{b}G^{>(0)}(x,p_{2})\Gamma_{a}$
$\displaystyle+$
$\displaystyle\Gamma_{b}G^{>(0)}(x,p_{1})\Gamma_{a}G^{<(0)}(x,p_{3})\Gamma_{b}G^{>(1)}(x,p_{2})\Gamma_{a}\;.$
(148)

One can obtain $\Sigma^{<(1)}(x,p)$ from $\Sigma^{>(1)}(x,p)$ by exchanging
$>\;\leftrightarrow\;<$. Note that $G^{\lessgtr(1)}$ in Eqs. (147), (148) are
the full Wigner functions and contain both quasi-classical as well as gradient
and collision contributions.

Finally, we look at the nonlocal collision term
$I_{\mathrm{coll},\mathrm{PB}}^{(0)}$ defined in Eq. (83). It is nonlocal due
to the space-time derivatives appearing in the Poisson brackets, so it
represents collisions taking place at different space-time points. We can
write $I_{\mathrm{coll},\mathrm{PB}}^{(0)}$ explicitly by expanding Poisson
brackets

$\displaystyle I_{\mathrm{coll,PB}}^{(0)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\left\\{\left[\partial_{x}^{\mu}\Sigma^{<(0)}(x,p)\right]\left[\partial_{\mu}^{p}G^{>(0)}(x,p)\right]-\left[\partial_{p}^{\mu}\Sigma^{<(0)}(x,p)\right]\left[\partial_{\mu}^{x}G^{>(0)}(x,p)\right]\right.$
(149)
$\displaystyle\left.-\left[\partial_{x}^{\mu}\Sigma^{>(0)}(x,p)\right]\left[\partial_{\mu}^{p}G^{<(0)}(x,p)\right]+\left[\partial_{p}^{\mu}\Sigma^{>(0)}(x,p)\right]\left[\partial_{\mu}^{x}G^{<(0)}(x,p)\right]\right\\}\;.$

We observe that only the leading-order Wigner functions are relevant:
$I_{\mathrm{coll,PB}}^{(0)}$ is fully determined once the leading-order MVSDs
are known by solving the Boltzmann equations (142) and (144). The derivatives
$\partial_{\mu}^{p}$ with respect to four-momentum have to be treated
carefully, since they act on delta functions. There are two kinds of delta
functions: one corresponds to energy-momentum conservation in
$\Sigma^{\lessgtr(0)}(x,p)$ and the other corresponds to on-shell conditions
in $G^{\lessgtr(0)}(x,p)$. The derivatives of delta functions for energy-
momentum conservation are from $\partial_{p}^{\mu}\Sigma^{\lessgtr(0)}(x,p)$
and may be connected with effects of energy non-conservation, while the
derivatives of mass-shell delta functions come from
$\partial_{p_{0}}G^{\lessgtr(0)}(x,p)$ and give off-mass-shell contributions.
A further and careful treatment of four-momentum derivatives is essential for
a numerical simulation of the spin Boltzmann equations, but will not be made
in this paper and will be addressed in a future work.

Following the same procedure as in the derivation of Eqs. (138) and (139), we
obtain Boltzmann equations at next-to-leading order from Eqs. (94) and (97),

$\displaystyle\frac{1}{E_{p}}p\cdot\partial_{x}\mathrm{tr}\left[f^{(1)}(x,p)\right]=-\frac{1}{\pi\hbar}\int_{0}^{\infty}dp_{0}\,\mathrm{Im}\mathrm{Tr}\left(I_{\mathrm{coll}}^{(2)}\right)-\frac{1}{2\pi\hbar
m}\,\mathrm{Re\,Tr}\left(\gamma\cdot\partial_{x}I_{\mathrm{coll}}^{(1)}\right)$
(150) $\displaystyle\equiv$
$\displaystyle\mathscr{C}_{\mathrm{scalar}}\left(\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}\right)+\mathscr{C}_{\mathrm{scalar}}\left(\Delta
I_{\mathrm{coll,}\,\nabla}^{(1)}\right)+\mathscr{C}_{\mathrm{scalar}}\left(I_{\mathrm{coll},\mathrm{PB}}^{(0)}\right)+\mathscr{C}_{\mathrm{scalar}}\left(\partial_{x}I_{\mathrm{coll}}^{(1)}\right)\;,$
$\displaystyle\frac{1}{E_{p}}p\cdot\partial_{x}\mathrm{tr}\left[n_{j}^{(+)\mu}\tau_{j}^{T}f^{(1)}(x,p)\right]=\frac{1}{2\pi\hbar
m}\int_{0}^{\infty}dp_{0}\,\left[\epsilon^{\mu\nu\alpha\beta}p_{\nu}\mathrm{Im}\mathrm{Tr}\left(\sigma_{\alpha\beta}I_{\mathrm{coll}}^{(2)}\right)+\mathrm{Re\,Tr}\left(\gamma^{5}\partial_{x}^{\mu}I_{\mathrm{coll}}^{(1)}\right)\right]$
(151) $\displaystyle\equiv$
$\displaystyle\mathscr{C}^{\mu}_{\mathrm{pol}}\left(\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}\right)+\mathscr{C}^{\mu}_{\mathrm{pol}}\left(\Delta
I_{\mathrm{coll,}\,\nabla}^{(1)}\right)+\mathscr{C}^{\mu}_{\mathrm{pol}}\left(I_{\mathrm{coll},\mathrm{PB}}^{(0)}\right)+\mathscr{C}^{\mu}_{\mathrm{pol}}\left(\partial_{x}I_{\mathrm{coll}}^{(1)}\right)\;.$

The first Boltzmann equation is for the scalar part of the first-order MVSD,
while the second one is for the polarization part of the first-order MVSD. Due
to the fact that $I_{\mathrm{coll}}^{(2)}$ consists of $\Delta
I_{\mathrm{coll}}^{(1)}$ and $I_{\mathrm{coll},\mathrm{PB}}^{(0)}$, we define
the collision terms on the right-hand sides of Eqs. (150), (151) according to
their origins. Note that the collision terms involving $\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}$ depend on $f^{(1)}(x,p)$ as well as on
$f^{(0)}(x,p)$ in an algebraic way, while those involving $\Delta
I_{\mathrm{coll,}\,\nabla}^{(1)}$, $I_{\mathrm{coll},\mathrm{PB}}^{(0)}$ and
$\partial_{x}I_{\mathrm{coll}}^{(1)}$ contain space-time derivatives of
$f^{(0)}(x,p)$, which are determined by solving the leading-order Boltzmann
equations (142), (144). Therefore, the collision terms involving $\Delta
I_{\mathrm{coll,}\,\nabla}^{(1)}$, $I_{\mathrm{coll},\mathrm{PB}}^{(0)}$, and
$\partial_{x}I_{\mathrm{coll}}^{(1)}$ containing space-time derivatives of
$f^{(0)}(x,p)$ can be regarded as source terms for the scalar and polarization
parts of $f^{(1)}(x,p)$.

The explicit form of the local collision term
$\mathscr{C}_{\mathrm{scalar}}\left(\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}\right)$ is given by

$\displaystyle\mathscr{C}_{\mathrm{scalar}}\left(\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi\hbar}\int_{0}^{\infty}dp_{0}\,\mathrm{Im\,Tr}\left(\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}\right)$ (152) $\displaystyle=$
$\displaystyle\frac{1}{2E_{p}}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}2E_{1}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}2E_{2}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}2E_{3}}\,(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})$
$\displaystyle\times\mathrm{Re}\left[\left\\{f^{(1)}_{1}\left[f^{(0)}_{2}\left(1-f^{(0)}-f^{(0)}_{3}\right)+f^{(0)}_{3}f^{(0)}\right]+f^{(1)}_{2}\left[f^{(0)}_{1}\left(1-f^{(0)}_{3}-f^{(0)}\right)+f^{(0)}_{3}f^{(0)}\right]\right.\right.$
$\displaystyle\hskip
22.76228pt-f^{(1)}\left[f^{(0)}_{3}\left(1-f^{(0)}_{1}-f^{(0)}_{2}\right)+f^{(0)}_{1}f^{(0)}_{2}\right]\left.-f^{(1)}_{3}\left[f^{(0)}\left(1-f^{(0)}_{1}-f^{(0)}_{2}\right)+f^{(0)}_{1}f^{(0)}_{2}\right]\right\\}$
$\displaystyle\hskip
19.91684pt\times\left.\left(M_{a}^{\mathrm{scalar}}+M_{b}^{\mathrm{scalar}}\right)\right]\;,$

where similar to the notation employed above for $f^{(0)}_{i}$ we have defined
$f^{(1)}_{i}\equiv f_{s_{i}r_{i}}^{(1)}(x,p_{i})$ ($i=1,2,3$) and
$f^{(1)}\equiv f_{sr}^{(1)}(x,p)$. The scalar parts of the matrix elements
$M_{a,b}^{\mathrm{scalar}}$ are given by Eq. (143). Note that
$\mathscr{C}_{\mathrm{scalar}}\left(\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}\right)$ is actually the first-order
perturbation to the collision term of the leading-order Boltzmann equation
(142). The explicit form of the local collision term
$\mathscr{C}_{\mathrm{pol}}\left(\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}\right)$ is given by

$\displaystyle\mathscr{C}^{\mu}_{\mathrm{pol}}\left(\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\hbar
m}\int_{0}^{\infty}dp_{0}\,\epsilon^{\mu\nu\alpha\beta}p_{\nu}\mathrm{Im}\mathrm{Tr}\left(\sigma_{\alpha\beta}\Delta
I_{\mathrm{coll},\,\mathrm{qc}}^{(1)}\right)$ (153) $\displaystyle=$
$\displaystyle-\frac{1}{4E_{p}m}\,\epsilon^{\mu\nu\alpha\beta}p_{\nu}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}2E_{1}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}2E_{2}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}2E_{3}}\,(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})$
$\displaystyle\times\mathrm{Re}\left[\left\\{f^{(1)}_{1}\left[f^{(0)}_{2}\left(1-f^{(0)}-f^{(0)}_{3}\right)+f^{(0)}_{3}f^{(0)}\right]+f^{(1)}_{2}\left[f^{(0)}_{1}\left(1-f^{(0)}_{3}-f^{(0)}\right)+f^{(0)}_{3}f^{(0)}\right]\right.\right.$
$\displaystyle\hskip
22.76228pt-f^{(1)}\left[f^{(0)}_{3}\left(1-f^{(0)}_{1}-f^{(0)}_{2}\right)+f^{(0)}_{1}f^{(0)}_{2}\right]\left.-f^{(1)}_{3}\left[f^{(0)}\left(1-f^{(0)}_{1}-f^{(0)}_{2}\right)+f^{(0)}_{1}f^{(0)}_{2}\right]\right\\}$
$\displaystyle\hskip
19.91684pt\times\left.\left(M_{a,\alpha\beta}^{\mathrm{pol}}+M_{b,\alpha\beta}^{\mathrm{pol}}\right)\right]\;,$

where the polarization parts of the matrix elements
$M_{a,b;\alpha\beta}^{\mathrm{pol}}$ are given by Eq. (145). The polarization
part of the collision term $\mathscr{C}^{\mu}_{\mathrm{pol}}\left(\Delta
I_{\mathrm{coll,\,qc}}^{(1)}\right)$ is actually the first-order perturbation
to the collision term of the leading-order Boltzmann equation (144). From Eqs.
(152), (153) we observe that the quasi-classical parts of the collision terms
depend on $f^{(1)}$ as well as on $f^{(0)}$ and do not contain space-time
derivatives.

The gradient parts of the collision terms are defined as

$\displaystyle\mathscr{C}_{\mathrm{scalar}}\left(\Delta
I_{\mathrm{coll,}\,\nabla}^{(1)}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi\hbar
m}\int_{0}^{\infty}dp_{0}\,\mathrm{Im\,Tr}\left(p\cdot\gamma\Delta
I_{\mathrm{coll,}\,\nabla}^{(1)}\right)\;,$
$\displaystyle\mathscr{C}^{\mu}_{\mathrm{pol}}\left(\Delta
I_{\mathrm{coll,}\,\nabla}^{(1)}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\hbar
m}\int_{0}^{\infty}dp_{0}\epsilon^{\mu\nu\alpha\beta}p_{\nu}\mathrm{Im}\mathrm{Tr}\left(\sigma_{\alpha\beta}\Delta
I_{\mathrm{coll},\,\nabla}^{(1)}\right)\;,$ (154)

which are nonlocal since they involve space-time derivatives of $f^{(0)}$ as
shown in Eq. (146) for $G_{\nabla}^{\lessgtr(1)}$.

The explicit form of the nonlocal collision term
$\mathscr{C}_{\mathrm{scalar}}\left(\partial_{x}I_{\mathrm{coll}}^{(1)}\right)$
reads

$\displaystyle\mathscr{C}_{\mathrm{scalar}}\left(\partial_{x}I_{\mathrm{coll}}^{(1)}\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi\hbar
m}\mathrm{Re\,Tr}\left(\gamma\cdot\partial_{x}I_{\mathrm{coll}}^{(1)}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{4mE_{p}}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}2E_{1}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}2E_{2}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}2E_{3}}\,(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})$
$\displaystyle\times\mathrm{Im}\left\\{\partial_{x}^{\mu}\left[f^{(0)}_{1}f^{(0)}_{2}\left(1-f^{(0)}_{3}\right)\left(1-f^{(0)}\right)-f^{(0)}_{3}f^{(0)}\left(1-f^{(0)}_{1}\right)\left(1-f^{(0)}_{2}\right)\right]\left(M_{a,\mu}^{\mathrm{scalar}}+M_{b,\mu}^{\mathrm{scalar}}\right)\right\\}\;,$

where the matrix elements
$M_{a,\mu}^{\mathrm{scalar}},\,M_{b,\mu}^{\mathrm{scalar}}$ are defined by

$\displaystyle M_{a,\mu}^{\mathrm{scalar}}$ $\displaystyle=$ $\displaystyle
4G_{a}G_{b}\mathrm{Tr}\left[\Gamma_{a}u(r_{3},p_{3})\overline{u}(s_{3},p_{3})\Gamma_{b}u(r_{1},p_{1})\overline{u}(s_{1},p_{1})\right]\mathrm{Tr}\left[\gamma_{\mu}\Gamma_{b}u(r_{2},p_{2})\overline{u}(s_{2},p_{2})\Gamma_{a}u(r,p)\overline{u}(s,p)\right]\;,$
$\displaystyle M_{b,\mu}^{\mathrm{scalar}}$ $\displaystyle=$
$\displaystyle-4G_{a}G_{b}\mathrm{Tr}\left[\gamma_{\mu}\Gamma_{b}u(r_{1},p_{1})\overline{u}(s_{1},p_{1})\Gamma_{a}u(r_{3},p_{3})\overline{u}(s_{3},p_{3})\Gamma_{b}u(r_{2},p_{2})\overline{u}(s_{2},p_{2})\Gamma_{a}u(r,p)\overline{u}(s,p)\right]\;.$
(155)

The explicit form of the nonlocal collision term
$\mathscr{C}^{\mu}_{\mathrm{pol}}\left(\partial_{x}I_{\mathrm{coll}}^{(1)}\right)$
reads

$\displaystyle\mathscr{C}^{\mu}_{\mathrm{pol}}\left(\partial_{x}I_{\mathrm{coll}}^{(1)}\right)=\frac{1}{2\pi\hbar
m}\int_{0}^{\infty}dp_{0}\,\mathrm{Re\,Tr}\left(\gamma^{5}\partial_{x}^{\mu}I_{\mathrm{coll}}^{(1)}\right)$
(156) $\displaystyle=$
$\displaystyle\frac{1}{4mE_{p}}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi\hbar)^{3}2E_{1}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi\hbar)^{3}2E_{2}}\frac{d^{3}\mathbf{p}_{3}}{(2\pi\hbar)^{3}2E_{3}}\,(2\pi\hbar)^{4}\delta^{(4)}(p+p_{3}-p_{1}-p_{2})$
$\displaystyle\times\mathrm{Im}\left\\{\partial_{x}^{\mu}\left[f^{(0)}_{1}f^{(0)}_{2}\left(1-f^{(0)}_{3}\right)\
\left(1-f^{(0)}\right)-f^{(0)}_{3}f^{(0)}\left(1-f^{(0)}_{1}\right)\left(1-f^{(0)}_{2}\right)\right]\left(M_{a,5}^{\mathrm{pol}}+M_{b,5}^{\mathrm{pol}}\right)\right\\}\;,$

where the matrix elements $M_{a,5}^{\mathrm{pol}},\,M_{b,5}^{\mathrm{pol}}$
are defined by

$\displaystyle M_{a,5}^{\mathrm{pol}}$ $\displaystyle=$ $\displaystyle
4G_{a}G_{b}\mathrm{Tr}\left[\Gamma_{a}u(r_{3},p_{3})\overline{u}(s_{3},p_{3})\Gamma_{b}u(r_{1},p_{1})\overline{u}(s_{1},p_{1})\right]\mathrm{Tr}\left[\gamma_{5}\Gamma_{b}u(r_{2},p_{2})\overline{u}(s_{2},p_{2})\Gamma_{a}u(r,p)\overline{u}(s,p)\right]\,,$
$\displaystyle M_{b,5}^{\mathrm{pol}}$ $\displaystyle=$
$\displaystyle-4G_{a}G_{b}\mathrm{Tr}\left[\gamma_{5}\Gamma_{b}u(r_{1},p_{1})\overline{u}(s_{1},p_{1})\Gamma_{a}u(r_{3},p_{3})\overline{u}(s_{3},p_{3})\Gamma_{b}u(r_{2},p_{2})\overline{u}(s_{2},p_{2})\Gamma_{a}u(r,p)\overline{u}(s,p)\right]\;.$
(157)

The nonlocal collision terms involving Poisson brackets are

$\displaystyle\mathscr{C}_{\mathrm{scalar}}\left(I_{\mathrm{coll},\mathrm{PB}}^{(0)}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{\pi\hbar}\int_{0}^{\infty}dp_{0}\,\mathrm{Im}\mathrm{Tr}\left(I_{\mathrm{coll},\mathrm{PB}}^{(0)}\right)\,,$
$\displaystyle\mathscr{C}^{\mu}_{\mathrm{pol}}\left(I_{\mathrm{coll},\mathrm{PB}}^{(0)}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi\hbar
m}\int_{0}^{\infty}dp_{0}\,\epsilon^{\mu\nu\alpha\beta}p_{\nu}\mathrm{Im}\mathrm{Tr}\left(\sigma_{\alpha\beta}I_{\mathrm{coll},\mathrm{PB}}^{(0)}\right)\,,$
(158)

where $I_{\mathrm{coll},\mathrm{PB}}^{(0)}$ is given by Eq. (149). We refrain
from giving the explicit forms of the above collision terms, as they are too
lengthy.

We conclude this section with some remarks about the results: (a) For the
gradient and collision contributions of the Wigner functions, we assume Eq.
(146), i.e., the scalar component $\mathcal{F}_{\nabla}^{(1)}$ and the axial-
vector component $\mathcal{A}_{\nabla}^{(1)\mu}$ are set to zero. This is the
most general solution to $G_{\nabla}^{\lessgtr(1)}$ that satisfies Eqs. (78) –
(80). If $\mathcal{F}_{\nabla}^{(1)}$ and $\mathcal{A}_{\nabla}^{(1)\mu}$ are
nonvanishing, it can be proved that they can be absorbed into the kinetic
contributions of the Wigner functions by a redefinition of the first-order
MVSD $f^{(1)}$. (b) The set of Boltzmann equations (150) and (151) for the
MVSD $f^{(1)}$ at $\mathcal{O}(\hbar)$ can be solved once the Boltzmann
equations (142) and (144) have been solved for $f^{(0)}$ at
$\mathcal{O}(\hbar^{0})$. Once $f^{(0)}$ is known, the nonlocal terms arising
from space-time derivatives of $f^{(0)}$ play the role of source terms for the
collisions terms in the Boltzmann equations (150) and (151). Therefore, the
polarization part of $f^{(1)}$ is driven by space-time derivatives of
$f^{(0)}$, which are proportional to the thermal vorticity in global
equilibrium.

##  VIII Summary

The closed-time-path formalism is an effective method to deal with non-
equilibrium problems. In this paper, we derived the Kadanoff–Baym equation
from the Dyson-Schwinger equation on the CTP contour for massive spin-1/2
fermions, where collisions are provided by the irreducible self-energy. We
focussed on the Kadanoff–Baym equation for the two-point function
$G^{<}(x_{1},x_{2})$, which is a $4\times 4$ matrix in Dirac space. We
performed a gradient expansion for the self-energy and took the Fourier
transform of the resulting equation with respect to the distance between two
space-time points. Thus we derived the Kadanoff–Baym equation for the Wigner
function $G^{<}(x,p)$. The self-energy is expressed in terms of the Wigner
functions $G^{<}(x,p)$ and $G^{>}(x,p)$. We employed a semi-classical
expansion of the Kadanoff–Baym equation for the two-point function in powers
of $\hbar$. By projecting the matrix form of the Kadanoff–Baym equation onto
the Dirac matrices of the Clifford decomposition, we derived a set of
equations in terms of the Clifford components of the Wigner function up to
$\hbar^{2}$. The scalar component of the Wigner function corresponds to the
phase-space distribution function, while the axial-vector component carries
the information for the phase-space distribution of the spin polarization.

At leading order $\mathcal{O}(\hbar^{0})$, the Wigner functions
$G^{<(0)}(x,p)$ and $G^{>(0)}(x,p)$ can be expressed in terms of matrix-valued
spin distribution functions (MVSDs). The form of $G^{\lessgtr(0)}(x,p)$ is the
same as in kinetic theory without collisions. At $\mathcal{O}(\hbar)$ or next-
to-leading order, the Wigner functions $G^{<(1)}(x,p)$ and $G^{>(1)}(x,p)$ can
be separated into (on-shell) quasi-classical, (on-shell) gradient and
collision, as well as off-shell contributions. The quasi-classical
contributions to $G^{<(1)}(x,p)$ and $G^{>(1)}(x,p)$ can be obtained by
considering the first-order correction $f^{(1)}(x,p)$ to the leading-order
MVSD $f^{(0)}(x,p)$. The gradient and collision contributions contain, besides
collision terms, space-time derivatives of zeroth-order Wigner functions in
the equations of motions. The off-shell contributions are the ones that
violate the on-shell conditions. These belong to higher-order contributions in
the coupling constant and can be decoupled from the on-shell parts of the
Boltzmann equations. The Boltzmann equations for the scalar and axial-vector
(i.e., polarization) parts can be expressed in terms of the MVSDs. At
$\mathcal{O}(\hbar)$ or leading order, only local collision terms appear in
the Boltzmann equations, without space-time derivatives, meaning that
collisions take place at the same space-time point. At
$\mathcal{O}(\hbar^{2})$ or next-to-leading order, the Boltzmann equations
describe how $f^{(1)}(x,p)$ evolves under the influence of local as well as
nonlocal collision terms with space-time derivatives. Nonlocal collision terms
depend on the leading-order MVSD $f^{(0)}(x,p)$ and its space-time derivative
(generating the vorticity in equilibrium), which are determined by solving the
leading-order Boltzmann equations, while the local collision terms depend on
$f^{(1)}(x,p)$ as well as $f^{(0)}(x,p)$. Therefore, the nonlocal collision
terms can be regarded as sources for the polarization part of $f^{(1)}(x,p)$
in the Boltzmann equations at next-to-leading order. The system of Boltzmann
equations in terms of the MVSDs paves the way for simulating spin transport
processes from first principles.

###### Acknowledgements.

The authors thank F. Becattini, W. Florkowski, X. Guo, U. Heinz, Y.-C. Liu, R.
Ryblewski, L. Tinti, and G. Torrieri for enlightening discussions. Q.W. is
supported in part by the National Natural Science Foundation of China (NSFC)
under Grant No. 11890713 (a sub-grant of 11890710), and 11947301, and by the
Strategic Priority Research Program of Chinese Academy of Sciences under Grant
No. XDB34030102. X.L.S. is supported in part by the National Natural Science
Foundation of China (NSFC) under Grant No. 12047528 and 11890714. The work of
D.H.R., E.S., and N.W. is supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through the Collaborative Research Center
CRC-TR 211 “Strong-interaction matter under extreme conditions” \- project
number 315477589 - TRR 211. E.S. acknowledges support by BMBF
“Forschungsprojekt: 05P2018 - Ausbau von ALICE am LHC (05P18RFCA1)”.

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