annihilation operators was a non-trivial Bogolyubov transformation, therefore
the Rindler vacuum and the Minkowski vacuum did not coincide. The consequences
of this fact will be further explored in Subsection 5.2.2.

Now we turn to the density operator. It is important to stress that the
expression of the non-equilibrium density operator (5.21) is exactly the same
as the local thermodynamic equilibrium one (5.23) evaluated at $\tau_{0}$.
Whether one or the other is used makes indeed a difference when it comes to
calculating thermal expectation values, as exemplified by equations (5.20) and
(5.25). However, as far as the very structure of the density operator is
concerned, they really are equivalent. Therefore, without loss of generality,
we can let $\tau$ be whatever and focus on the local thermodynamic equilibrium
one for the rest of this part. Then, in Section 5.2 we will perform a local
thermodynamic equilibrium analysis, whereas in Section 5.3 a non-equilibrium
one, thus distinguishing explicitly the two cases. Moreover, the ${\sf
p}$-dependence will be omitted throughout for ease of notation, restoring it
only when necessary.

As for the energy-momentum tensor operator, we consider again the canonical
tensor

$\widehat{T}^{\mu\nu}=\frac{1}{2}\left(\partial^{\mu}\widehat{\psi}\partial^{\nu}\widehat{\psi}+\partial^{\nu}\widehat{\psi}\partial^{\mu}\widehat{\psi}\right)-g^{\mu\nu}\widehat{\cal
L},\qquad\widehat{\cal
L}=\frac{1}{2}\left(g^{\mu\nu}\partial_{\mu}\widehat{\psi}\partial_{\nu}\widehat{\psi}-m^{2}\widehat{\psi}^{2}\right).$
(5.38)

In order to work $\widehat{\Pi}(\tau)$ out, the above expression must be
contracted twice with the hydrodynamic velocity field, yielding

$\widehat{T}^{\mu\nu}u_{\mu}u_{\nu}=\frac{1}{2}\left[(\partial_{\tau}\widehat{\psi})^{2}+(\partial_{x}\widehat{\psi})^{2}+(\partial_{y}\widehat{\psi})^{2}+\frac{1}{\tau^{2}}(\partial_{\eta}\widehat{\psi})^{2}+m^{2}\widehat{\psi}^{2}\right].$
(5.39)

By using the quantum field expansion (5.27) together with the integral
representation of the Hankel functions (5.31) and exploiting the invariance of
$h(\tau)$ and $h^{*}(\tau)$ under reflections ${\sf p}\mapsto-{\sf p}$, the
following expression is obtained

$\begin{split}\widehat{\Pi}(\tau)=&\tau\int{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu\frac{\pi}{8}\left[\left(|\partial_{\tau}h(\tau)|^{2}+\omega^{2}(\tau)|h(\tau)|^{2}\right)\left(\widehat{b}_{\sf
p}\widehat{b}_{\sf p}^{\dagger}+\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{\sf
p}\right)\right.\\\
&\left.+\left((\partial_{\tau}h(\tau))^{2}+\omega^{2}(\tau)h^{2}(\tau)\right)\widehat{b}_{\sf
p}\widehat{b}_{-{\sf
p}}+\left((\partial_{\tau}h^{*}(\tau))^{2}+\omega^{2}(\tau)(h^{*}(\tau))^{2}\right)\widehat{b}_{\sf
p}^{\dagger}\widehat{b}_{-{\sf p}}^{\dagger}\right].\end{split}$ (5.40)

It is convenient to define the positive real function $K(\tau)$ and the
complex function $\Lambda(\tau)$ as

$K(\tau)\equiv\frac{\pi\tau}{4\omega(\tau)}\left[|\partial_{\tau}h(\tau)|^{2}+\omega^{2}(\tau)|h(\tau)|^{2}\right],$
(5.41)
$\Lambda(\tau)\equiv\frac{\pi\tau}{4\omega(\tau)}\left[(\partial_{\tau}h(\tau))^{2}+\omega^{2}(\tau)h^{2}(\tau)\right].$
(5.42)

Let us stress that the ${\sf p}$-dependence is intended. Being $h(\tau)$ and
$\omega(\tau)$ invariant under momentum reflections ${\sf p}\mapsto-{\sf p}$,
so are $K(\tau)$ and $\Lambda(\tau)$, but more interestingly they are related
by

$K^{2}(\tau)-|\Lambda(\tau)|^{2}=1.$ (5.43)

This occurs thanks to the left-hand side being proportional to the Wronskian
of the Hankel functions

$K^{2}(\tau)-|\Lambda(\tau)|^{2}=-\left(\frac{\pi m_{\rm
T}\tau}{4}\right)^{2}\left(W[{\rm H}^{(2)}_{i\mu}(m_{\rm T}\tau),{\rm
H}^{(1)}_{i\mu}(m_{\rm T}\tau)]\right)^{2},$ (5.44)

which not only is known, but it turns out to be a very simple function of the
argument [91]

$W[{\rm H}^{(2)}_{i\nu}(x),{\rm H}^{(1)}_{i\nu}(x)]={\rm
H}^{(2)^{\prime}}_{i\nu}(x){\rm H}^{(1)}_{i\nu}(x)-{\rm
H}^{(1)^{\prime}}_{i\nu}(x){\rm H}^{(2)}_{i\nu}(x)=\frac{4i}{\pi x}.$ (5.45)

This will greatly simplify the calculation of thermal expectation values at
local thermodynamic equilibrium in the next Section. The property (5.43) can
also be rephrased by expressing $K(\tau)$ and $\Lambda(\tau)$ in terms of some
hyperbolic angle $\Theta({\sf p},\tau)$ and phase $\chi({\sf p},\tau)$

$K(\tau)=\cosh(2\Theta(\tau)),\qquad\Lambda(\tau)={\rm
e}^{i\chi(\tau)}\sinh(2\Theta(\tau)),$ (5.46)

which will be useful later on. For now, $\widehat{\Pi}(\tau)$ is recast in
terms of $K(\tau)$ and $\Lambda(\tau)$ as

$\widehat{\Pi}(\tau)=\int{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu\frac{\omega(\tau)}{2}\left[K(\tau)\left(\widehat{b}_{\sf
p}\widehat{b}_{\sf p}^{\dagger}+\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{\sf
p}\right)+\Lambda(\tau)\widehat{b}_{\sf p}\widehat{b}_{-{\sf
p}}+\Lambda^{*}(\tau)\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{-{\sf
p}}^{\dagger}\right]$ (5.47)

At variance with equations (4.57) and (4.58) found in Chapter 4, this
expression is not diagonal in the creation and annihilation operators due to
the terms proportional to $\Lambda(\tau)$ and $\Lambda^{*}(\tau)$. This is
quite unfortunate, for if it were diagonal, thermal expectation values of
products of creation and annihilation operators, whence of operators quadratic
in the field, could be calculated by using standard methods. In the following,
we look for a suitable Bogolyubov transformation to diagonalize
$\widehat{\Pi}(\tau)$.

####  5.1.3 Diagonalization of the density operator

Let us define a new set of creation and annihilation operators by combining
$\widehat{b}_{\sf p}$ and $\widehat{b}_{\sf p}^{\dagger}$ to attempt a
diagonalization of $\widehat{\Pi}(\tau)$ in (5.47), whence
$\widehat{\rho}_{\rm LE}(\tau)$ in (5.23). Since the terms we wish to get rid
of mix $\widehat{b}_{\sf p}$ with $\widehat{b}_{-{\sf p}}$ and their Hermitean
conjugates, we start with a Bogolyubov transformation of the following form

$\begin{split}\widehat{\xi}_{\sf p}^{\dagger}(\tau)\equiv&A({\sf
p},\tau)\widehat{b}_{\sf p}^{\dagger}-B({\sf p},\tau)\widehat{b}_{-{\sf p}}\\\
\widehat{\xi}_{\sf p}(\tau)\equiv&A^{*}({\sf p},\tau)\widehat{b}_{\sf
p}-B^{*}({\sf p},\tau)\widehat{b}_{-{\sf p}}^{\dagger}\end{split}$ (5.48)

where $A$ and $B$ are complex functions to be determined. In order for
$\widehat{\xi}_{\sf p}^{\dagger}(\tau)$ and $\widehat{\xi}_{\sf p}(\tau)$ to
be creation and annihilation operators respectively, they are demanded to obey
the usual algebra

$\begin{split}[\widehat{\xi}_{\sf p}(\tau),\widehat{\xi}_{{\sf
p}^{\prime}}^{\dagger}(\tau)]=&\delta^{2}({\bf p}_{\rm T}-{\bf p}_{\rm
T}^{\prime})\,\delta(\mu-\mu^{\prime})\\\ [\widehat{\xi}_{\sf
p}(\tau),\widehat{\xi}_{{\sf p}^{\prime}}(\tau)]=&0=[\widehat{\xi}_{\sf
p}^{\dagger}(\tau),\widehat{\xi}_{{\sf
p}^{\prime}}^{\dagger}(\tau)].\end{split}$ (5.49)

By enforcing the commutation relations (5.33), these in turn imply

$\begin{split}&\left(|A({\sf p},\tau)|^{2}-|B({\sf
p},\tau)|^{2}\right)\delta^{2}({\bf p}_{\rm T}-{\bf p}_{\rm
T}^{\prime})\,\delta(\mu-\mu^{\prime})=\delta^{2}({\bf p}_{\rm T}-{\bf p}_{\rm
T}^{\prime})\,\delta(\mu-\mu^{\prime})\\\ &\left[A^{*}(-{\sf
p},\tau)B^{*}({\sf p},\tau)-A^{*}({\sf p},\tau)B^{*}(-{\sf
p},\tau)\right]\delta^{2}({\bf p}_{\rm T}+{\bf p}_{\rm
T}^{\prime})\,\delta(\mu+\mu^{\prime})=0\\\ &\left[A({\sf p},\tau)B(-{\sf
p},\tau)-A(-{\sf p},\tau)B({\sf p},\tau)\right]\delta^{2}({\bf p}_{\rm T}+{\bf
p}_{\rm T}^{\prime})\,\delta(\mu+\mu^{\prime})=0\end{split}$ (5.50)

whose a possible solution is

$A({\sf p},\tau)=A(-{\sf p},\tau),\qquad B({\sf p},\tau)=B(-{\sf
p},\tau),\qquad|A({\sf p},\tau)|^{2}-|B({\sf p},\tau)|^{2}=1$ (5.51)

hence the following hyperbolic expressions

$A({\sf p},\tau)=\cosh\theta({\sf p},\tau)\,{\rm e}^{i\chi_{A}({\sf
p},\tau)},\qquad B({\sf p},\tau)=\sinh\theta({\sf p},\tau)\,{\rm
e}^{i\chi_{B}({\sf p},\tau)}$ (5.52)

for some hyperbolic angle $\theta({\sf p},\tau)$ and phases $\chi_{A}({\sf
p},\tau)$ and $\chi_{B}({\sf p},\tau)$. This is not the only possible
solution, but we choose it as it makes it easy to invert the Bogolyubov
transformation (5.48) as

$\begin{split}\widehat{b}_{\sf p}=&A({\sf p},\tau)\widehat{\xi}_{\sf
p}(\tau)+B^{*}({\sf p},\tau)\widehat{\xi}_{-{\sf p}}^{\dagger}(\tau)\\\
\widehat{b}_{\sf p}^{\dagger}=&A^{*}({\sf p},\tau)\widehat{\xi}_{\sf
p}^{\dagger}(\tau)+B({\sf p},\tau)\widehat{\xi}_{-{\sf p}}(\tau),\end{split}$
(5.53)

plug it into (5.47) and obtain

$\begin{split}\widehat{\Pi}(\tau)=&\int{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu\,\frac{\omega}{2}\left\\{\left[K\left(|A|^{2}+|B|^{2}\right)+\Lambda
AB^{*}+\Lambda^{*}A^{*}B\right]\left(\widehat{\xi}_{\sf p}\widehat{\xi}_{\sf
p}^{\dagger}+\widehat{\xi}_{\sf p}^{\dagger}\widehat{\xi}_{\sf
p}\right)\right.\\\ &\left.+\left(2KAB+\Lambda
A^{2}+\Lambda^{*}B^{2}\right)\widehat{\xi}_{\sf p}\widehat{\xi}_{-{\sf
p}}+\left(2KA^{*}B^{*}+\Lambda^{*}{A^{*}}^{2}+\Lambda{B^{*}}^{2}\right)\widehat{\xi}_{\sf
p}^{\dagger}\widehat{\xi}_{-{\sf p}}^{\dagger}\right\\},\end{split}$ (5.54)

where the invariance under momentum reflections ${\sf p}\mapsto-{\sf p}$ was
used. In order to make $\widehat{\Pi}(\tau)$ diagonal, the second line must
vanish, so we look for $A$ and $B$ such that

$\displaystyle 2KAB+\Lambda A^{2}+\Lambda^{*}B^{2}=$ $\displaystyle 0$ (5.55a)
$\displaystyle 2KA^{*}B^{*}+\Lambda^{*}{A^{*}}^{2}+\Lambda{B^{*}}^{2}=$
$\displaystyle 0.$ (5.55b)

The two equations are the complex conjugate of each other, so they are in fact
one complex equation, that is two real equations. Apparently there are four
real unknowns, namely the moduli and phases of $A$ and $B$, but the moduli are
related by the last equation in (5.51). Moreover, the two equations are
invariant under an overall phase shift of both $A$ and $B$, so they actually
depend on the difference between the two phases instead of the phases
themselves. In summary, they are two real equations in two real unknowns, one
modulus and the phase difference, so we can try to solve the system. With the
hyperbolic expressions (5.52) and (5.46), equation (5.55a) can be rewritten as

$\cosh(2\Theta)\sinh(2\theta)\,{\rm
e}^{i(\chi_{A}+\chi_{B})}+\sinh(2\Theta)\left(\cosh^{2}\theta\,{\rm
e}^{i(\chi+2\chi_{A})}+\sinh^{2}\theta\,{\rm e}^{i(2\chi_{B}-\chi)}\right)=0$
(5.56)

whose solution is

$\chi_{B}({\sf p},\tau)-\chi_{A}({\sf p},\tau)=\chi({\sf
p},\tau),\qquad\theta({\sf p},\tau)=-\Theta({\sf p},\tau).$ (5.57)

By setting $\chi_{A}\equiv 0$ without loss of generality, the functions $A$
and $B$ fulfilling the Bogolyubov relations (5.48) are

$A({\sf p},\tau)=\cosh\Theta({\sf p},\tau),\qquad B({\sf
p},\tau)=-\sinh\Theta({\sf p},\tau)\,{\rm e}^{i\chi({\sf p},\tau)},$ (5.58)

so the inverse transformation (5.53) reads

$\begin{split}\widehat{b}_{\sf p}=&\cosh\Theta({\sf p},\tau)\widehat{\xi}_{\sf
p}(\tau)-\sinh\Theta({\sf p},\tau)\,{\rm e}^{-i\chi({\sf
p},\tau)}\widehat{\xi}_{-{\sf p}}^{\dagger}(\tau)\\\ \widehat{b}_{\sf
p}^{\dagger}=&\cosh\Theta({\sf p},\tau)\widehat{\xi}_{\sf
p}^{\dagger}(\tau)-\sinh\Theta({\sf p},\tau)\,{\rm e}^{i\chi({\sf
p},\tau)}\widehat{\xi}_{-{\sf p}}(\tau).\end{split}$ (5.59)

Plugging (5.58) and (5.46) into (5.54) and using the commutation relations
(5.49) we finally obtain the following result

$\begin{split}\widehat{\Pi}(\tau)=&\int{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu\,\frac{\omega(\tau)}{2}\left(\widehat{\xi}_{\sf
p}(\tau)\widehat{\xi}_{\sf p}^{\dagger}(\tau)+\widehat{\xi}_{\sf
p}^{\dagger}(\tau)\widehat{\xi}_{\sf p}(\tau)\right)\\\ =&\int{\rm d}^{2}{\rm
p_{T}}\,{\rm d}\mu\,\omega(\tau)\left(\widehat{\xi}_{\sf
p}^{\dagger}(\tau)\widehat{\xi}_{\sf p}(\tau)+\frac{1}{2}\right).\end{split}$
(5.60)

Since this is diagonal, so is the density operator; however, note that the
density operator being diagonal in a set of creation and annihilation
operators different from that appearing in the quantum field expansion is at
variance with the case of global thermodynamic equilibrium with acceleration
studied in Chapter 4, as can be checked by looking at equations (4.52) and
(4.55) together with (4.57) and (4.58). With a diagonal density operator, we
are now in a good spot to calculate thermal expectation values of operators
quadratic in the quantum field, such as the energy-momentum tensor, by using
standard methods.

###  5.2 Local thermodynamic equilibrium analysis

So far, we have been concerned only with the structure of the density
operator, therefore we could take either $\tau$ or $\tau_{0}$ equivalently
since $\widehat{\rho}_{\rm LE}(\tau)$ and $\widehat{\rho}=\widehat{\rho}_{\rm
LE}(\tau_{0})$ have the same properties. However, when it comes to thermal
expectation values it does indeed make a difference whether the density
operator is evaluated at $\tau$ or $\tau_{0}$. In this Section, we will
perform a local thermodynamic equilibrium analysis, therefore the
$\tau$-dependence will be highlighted in order to dispel any potential doubt.
On the other hand, for ease of notation, the dependence on the momentum ${\sf
p}$ will be omitted throughout whenever unnecessary. As we shall see, we will
be able to obtain analytical results, making this the first example of an
exactly solvable system at local thermodynamic equilibrium, to the best of our
knowledge.

####  5.2.1 Thermal expectation values

As a quick recap, the thermal expectation value of the energy-momentum tensor
calculated with the local thermodynamic equilibrium density operator is

$\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}=\mathrm{tr}(\widehat{\rho}_{\rm
LE}(\tau)\widehat{T}^{\mu\nu}(\tau)),$ (5.61)

and boost invariance constrains its form to be

$\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}=\rho_{\rm
LE}(\tau)u^{\mu}u^{\nu}+p^{\rm T}_{\rm
LE}(\tau)(\hat{i}^{\mu}\hat{i}^{\nu}+\hat{j}^{\mu}\hat{j}^{\nu})+p^{\rm
L}_{\rm LE}(\tau)\hat{\eta}^{\mu}\hat{\eta}^{\nu}.$ (5.62)

The local thermodynamic equilibrium density operator built with the canonical
energy-momentum tensor operator of a free real scalar quantum field is not
diagonal in the creation and annihilation operators $\widehat{b}_{\sf
p}^{\dagger}$ and $\widehat{b}_{\sf p}$, but it is so in the creation and
annihilation operators $\widehat{\xi}_{\sf p}^{\dagger}(\tau)$ and
$\widehat{\xi}_{\sf p}(\tau)$ obtained from the former through the Bogolyubov
transformation (5.48)

$\widehat{\rho}_{\rm LE}(\tau)=\frac{1}{Z_{\rm
LE}(\tau)}\exp\left[-\frac{\widehat{\Pi}(\tau)}{T(\tau)}\right],\qquad\widehat{\Pi}(\tau)=\int{\rm
d}^{2}{\rm p_{T}}\,{\rm d}\mu\,\omega(\tau)\left(\widehat{\xi}_{\sf
p}^{\dagger}(\tau)\widehat{\xi}_{\sf p}(\tau)+\frac{1}{2}\right).$ (5.63)

This diagonal expression allows for thermal expectation values of products of
$\widehat{\xi}_{\sf p}^{\dagger}(\tau)$ and $\widehat{\xi}_{\sf p}(\tau)$ to
be obtained by using the standard method shown through (4.61)–(4.65) in
Chapter 4, hence the result

$\displaystyle\langle\widehat{\xi}_{\sf p}^{\dagger}(\tau)\widehat{\xi}_{{\sf
p}^{\prime}}(\tau)\rangle_{\rm LE}=$ $\displaystyle n_{\rm
B}(\tau)\,\delta^{2}({\bf p}_{\rm T}-{\bf p}_{\rm
T}^{\prime})\,\delta(\mu-\mu^{\prime})$ (5.64a)
$\displaystyle\langle\widehat{\xi}_{\sf p}(\tau)\widehat{\xi}_{{\sf
p}^{\prime}}^{\dagger}(\tau)\rangle_{\rm LE}=$ $\displaystyle\left(n_{\rm
B}(\tau)+1\right)\delta^{2}({\bf p}_{\rm T}-{\bf p}_{\rm
T}^{\prime})\,\delta(\mu-\mu^{\prime})$ (5.64b)
$\displaystyle\langle\widehat{\xi}_{\sf p}(\tau)\widehat{\xi}_{{\sf
p}^{\prime}}(\tau)\rangle_{\rm LE}=$ $\displaystyle
0=\langle\widehat{\xi}_{\sf p}^{\dagger}(\tau)\widehat{\xi}_{{\sf
p}^{\prime}}^{\dagger}(\tau)\rangle_{\rm LE},$ (5.64c)

where $n_{\rm B}(\tau)$ is the Bose-Einstein distribution

$n_{\rm B}(\tau)=\frac{1}{{\rm e}^{\omega(\tau)/T(\tau)}-1}.$ (5.65)

with $\omega(\tau)$ given by (5.32). This is indeed the average number of
excitations of the operator $\widehat{\xi}_{\sf p}^{\dagger}(\tau)$, but by no
means a density of particles. Since the quantum field is expanded in terms of
the creation and annihilation operators $\widehat{b}_{\sf p}^{\dagger}$ and
$\widehat{b}_{\sf p}$, the particle density is given by
$\langle\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf
p}^{\prime}}\rangle_{\rm LE}$. Another reason to be interested in thermal
expectation values of products of $\widehat{b}_{\sf p}^{\dagger}$ and
$\widehat{b}_{\sf p}$ is the fact that the energy-momentum tensor operator is
quadratic in the quantum field, therefore it contains such terms. The thermal
expectation values of these products are most readily obtained from
(5.64a)–(5.64c) by means of the inverse Bogolyubov transformation (5.59)

$\displaystyle\langle\widehat{b}_{\sf p}\widehat{b}_{{\sf
p}^{\prime}}\rangle_{\rm LE}=$
$\displaystyle-\frac{1}{2}\sinh\left(2\Theta(\tau)\right){\rm
e}^{-i\chi(\tau)}\left(2n_{\rm B}(\tau)+1\right)\delta^{2}({\bf p}_{\rm
T}-{\bf p}_{\rm T}^{\prime})\,\delta(\mu-\mu^{\prime})$ (5.66a)
$\displaystyle\langle\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf
p}^{\prime}}^{\dagger}\rangle_{\rm LE}=$
$\displaystyle-\frac{1}{2}\sinh\left(2\Theta(\tau)\right){\rm
e}^{i\chi(\tau)}\left(2n_{\rm B}(\tau)+1\right)\delta^{2}({\bf p}_{\rm T}-{\bf
p}_{\rm T}^{\prime})\,\delta(\mu-\mu^{\prime})$ (5.66b)
$\displaystyle\langle\widehat{b}_{\sf p}\widehat{b}_{{\sf
p}^{\prime}}^{\dagger}\rangle_{\rm LE}=$
$\displaystyle\left[\cosh\left(2\Theta(\tau)\right)n_{\rm
B}(\tau)+\cosh^{2}\Theta(\tau)\right]\delta^{2}({\bf p}_{\rm T}-{\bf p}_{\rm
T}^{\prime})\,\delta(\mu-\mu^{\prime})$ (5.66c)
$\displaystyle\langle\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf
p}^{\prime}}\rangle_{\rm LE}=$
$\displaystyle\left[\cosh\left(2\Theta(\tau)\right)n_{\rm
B}(\tau)+\sinh^{2}\Theta(\tau)\right]\delta^{2}({\bf p}_{\rm T}-{\bf p}_{\rm
T}^{\prime})\,\delta(\mu-\mu^{\prime}).$ (5.66d)

These equations make it clear that the hyperbolic angle $\Theta(\tau)$, which
can be determined from (5.46), accounts for field vacuum effects. More
concerning the vacuum will be said shortly in Subsection 5.2.2.

Note that the $+1$ term in (5.64b), stemming from the commutation relations
(5.49), gives rise to divergencies. The details of renormalization will be
shortly discussed in Subsection 5.2.2. For the time being, we simply point out
that renormalizing with respect to the vacuum of the operators
$\widehat{\xi}_{\sf p}(\tau)$, whence of $\widehat{\Pi}(\tau)$, is but the
subtraction of the $T(\tau)=0$ contribution, which eventually amount to
canceling the $+1$ in (5.64b).

With equations (5.66a)–(5.66d), we are in a good position to calculate the
thermal expectation value of the energy-momentum tensor (5.62), that is the
energy density, transverse pressure and longitudinal pressure at local
thermodynamic equilibrium. First of all, these scalar functions are obtained
from the following contractions

$\rho_{\rm LE}(\tau)=\langle\widehat{T}^{\mu\nu}\rangle_{\rm
LE}u_{\mu}u_{\nu},$ (5.67) $p^{\rm T}_{\rm
LE}(\tau)=\langle\widehat{T}^{\mu\nu}\rangle_{\rm
LE}\hat{i}_{\mu}\hat{i}_{\nu}=\langle\widehat{T}^{\mu\nu}\rangle_{\rm
LE}\hat{j}_{\mu}\hat{j}_{\nu}=\frac{1}{2}\langle\widehat{T}^{\mu\nu}\rangle_{\rm
LE}\left(\hat{i}_{\mu}\hat{i}_{\nu}+\hat{j}_{\mu}\hat{j}_{\nu}\right),$ (5.68)
$p^{\rm L}_{\rm LE}(\tau)=\langle\widehat{T}^{\mu\nu}\rangle_{\rm
LE}\hat{\eta}_{\mu}\hat{\eta}_{\nu}.$ (5.69)

We take the energy density as an example, the pressures are worked out through
the same procedure. By plugging the quantum field expansion (5.27) into the
contraction (5.39), we have the following expression of the energy density in
terms of thermal expectation values of products of creation and annihilation
operators $\widehat{b}_{\sf p}^{\dagger}$ and $\widehat{b}_{\sf p}$

$\begin{split}\rho_{\rm LE}(\tau)=&\langle\widehat{T}^{\mu\nu}\rangle_{\rm
LE}u_{\mu}u_{\nu}=\int\frac{{\rm d^{2}p_{T}}\,{\rm d}\mu\,{\rm
d^{2}p_{T}^{\prime}}\,{\rm d}\mu^{\prime}}{4(4\pi)^{2}}\times\\\
&\times\left\\{\left[(\partial_{\tau}h({\sf p},\tau))(\partial_{\tau}h({\sf
p}^{\prime},\tau))-\left(p_{x}p_{x}^{\prime}+p_{y}p_{y}^{\prime}+\frac{1}{\tau^{2}}\mu\mu^{\prime}-m^{2}\right)h({\sf
p},\tau)h({\sf p}^{\prime},\tau)\right]\times\right.\\\ &\times{\rm
e}^{i[({\bf p}_{\rm T}+{\bf p}_{\rm T}^{\prime})\cdot{\bf x}_{\rm
T}+(\mu+\mu^{\prime})\eta]}\langle\widehat{b}_{\sf p}\widehat{b}_{{\sf
p}^{\prime}}\rangle_{\rm LE}+\\\ &+\left[(\partial_{\tau}h({\sf
p},\tau))(\partial_{\tau}h^{*}({\sf
p}^{\prime},\tau))+\left(p_{x}p_{x}^{\prime}+p_{y}p_{y}^{\prime}+\frac{1}{\tau^{2}}\mu\mu^{\prime}+m^{2}\right)h({\sf
p},\tau)h^{*}({\sf p}^{\prime},\tau)\right]\times\\\ &\times{\rm e}^{i[({\bf
p}_{\rm T}-{\bf p}_{\rm T}^{\prime})\cdot{\bf x}_{\rm
T}+(\mu-\mu^{\prime})\eta]}\langle\widehat{b}_{\sf p}\widehat{b}_{{\sf
p}^{\prime}}^{\dagger}\rangle_{\rm LE}+\\\ &+\left[(\partial_{\tau}h^{*}({\sf
p},\tau))(\partial_{\tau}h({\sf
p}^{\prime},\tau))+\left(p_{x}p_{x}^{\prime}+p_{y}p_{y}^{\prime}+\frac{1}{\tau^{2}}\mu\mu^{\prime}+m^{2}\right)h^{*}({\sf
p},\tau)h({\sf p}^{\prime},\tau)\right]\times\\\ &\times{\rm e}^{-i[({\bf
p}_{\rm T}-{\bf p}_{\rm T}^{\prime})\cdot{\bf x}_{\rm
T}+(\mu-\mu^{\prime})\eta]}\langle\widehat{b}_{\sf
p}^{\dagger}\widehat{b}_{{\sf p}^{\prime}}\rangle_{\rm LE}+\\\
&+\left[(\partial_{\tau}h^{*}({\sf p},\tau))(\partial_{\tau}h^{*}({\sf
p}^{\prime},\tau))-\left(p_{x}p_{x}^{\prime}+p_{y}p_{y}^{\prime}+\frac{1}{\tau^{2}}\mu\mu^{\prime}-m^{2}\right)h^{*}({\sf
p},\tau)h^{*}({\sf p}^{\prime},\tau)\right]\times\\\ &\left.\times{\rm
e}^{-i[({\bf p}_{\rm T}+{\bf p}_{\rm T}^{\prime})\cdot{\bf x}_{\rm
T}+(\mu+\mu^{\prime})\eta]}\langle\widehat{b}_{\sf
p}^{\dagger}\widehat{b}_{{\sf p}^{\prime}}^{\dagger}\rangle_{\rm
LE}\right\\}.\end{split}$ (5.70)

Then, by using equations (5.66a)–(5.66d), we obtain

$\begin{split}\rho_{\rm LE}(\tau)=&\int\frac{{\rm d^{2}p_{T}}\,{\rm
d}\mu}{4(4\pi)^{2}}\bigg{[}\left[|\partial_{\tau}h(\tau)|^{2}+\omega^{2}(\tau)|h(\tau)|^{2}\right]\left(2n_{\rm
B}(\tau)+1\right)\cosh(2\Theta(\tau))\\\
&-\frac{1}{2}\left[(\partial_{\tau}h(\tau))^{2}+\omega^{2}(\tau)h^{2}(\tau)\right]\left(2n_{\rm
B}(\tau)+1\right)\sinh(2\Theta(\tau))\,{\rm e}^{-i\chi(\tau)}\\\
&\left.-\frac{1}{2}\left[(\partial_{\tau}h^{*}(\tau))^{2}+\omega^{2}(\tau)(h^{*}(\tau))^{2}\right]\left(2n_{\rm
B}(\tau)+1\right)\sinh(2\Theta(\tau))\,{\rm
e}^{i\chi(\tau)}.\right]\end{split}$ (5.71)

In square brackets we recognize the functions $K$ and $\Lambda$ defined in
(5.41) and (5.42). These are expressed in terms of the hyperbolic angle
$\Theta(\tau)$ and the phase $\chi(\tau)$ as in (5.46), hence

$\begin{split}\rho_{\rm LE}(\tau)=&\int{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu\frac{\omega(\tau)}{16\pi^{3}\tau}\left[\cosh^{2}(2\Theta(\tau))-\sinh^{2}(2\Theta(\tau))\right]\left(2n_{\rm
B}(\tau)+1\right)\\\ =&\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)\left(n_{\rm
B}(\tau)+\frac{1}{2}\right).\end{split}$ (5.72)

The transverse and longitudinal pressures are worked out following the same
steps, and the final expressions read

$\rho_{\rm LE}(\tau)=\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\omega^{2}(\tau)\left(n_{\rm
B}(\tau)+\frac{1}{2}\right),$ (5.73) $p^{\rm T}_{\rm LE}(\tau)=\int\frac{{\rm
d}^{2}{\rm p_{T}}\,{\rm d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\frac{|{\bf p}_{\rm
T}|^{2}}{2}\left(n_{\rm B}(\tau)+\frac{1}{2}\right),$ (5.74) $p^{\rm L}_{\rm
LE}(\tau)=\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\frac{\mu^{2}}{\tau^{2}}\left(n_{\rm
B}(\tau)+\frac{1}{2}\right).$ (5.75)

These are the integral expressions of the thermal expectation value of the
canonical energy-momentum tensor of a free real scalar quantum field at local
thermodynamic equilibrium in the future light-cone. To the best of our
knowledge, this was first obtained in [49].

From the mathematical point of view, there is a key fact that greatly eases
the calculations to the point that simple analytical results such as
(5.73)–(5.75) can be obtained, and that is the following. The functions
$A(\tau)$ and $B(\tau)$ of the Bogolyubov transformation, appearing above in
the thermal expectation values of products of $\widehat{b}_{\sf p}$ and
$\widehat{b}_{\sf p}^{\dagger}$, are expressed in terms of the functions
$K(\tau)$ and $\Lambda(\tau)$ through equation (5.58). The functions $K(\tau)$
and $\Lambda(\tau)$, in turn, can be written in terms of the hyperbolic angle
$\Theta(\tau)$ and the phase $\chi(\tau)$ thanks to the relation (5.43), which
eventually is a consequence of the Wronskian of the Hankel functions being
reconstructed. In summary, the ultimate reason for the above analytical
results is the reconstruction of the Wronskian of the Hankel functions,
occurring because the density operator and the energy-momentum tensor operator
in (5.61) are evaluated at the same $\tau$. In the non-equilibrium analysis to
be carried out in Section 5.3, this will no longer hold true because of the
mixing of $\tau$ and $\tau_{0}$, as exemplified in (5.20). The difficulties we
will encounter there will make us appreciate the above exact analytical
results even more.

Equations (5.73)–(5.75) can be recast in a more compact form. Let us define
the following real function taking on different expressions for either the
energy density, the transverse pressure or the longitudinal pressure

$\gamma(\tau)\equiv\left\\{\begin{aligned}
\omega^{2}(\tau),\qquad\qquad&\rho_{\rm LE}(\tau)\\\
-\frac{\mu^{2}}{\tau^{2}}-m^{2},\qquad\qquad&p^{\rm T}_{\rm LE}(\tau)\\\
-m_{\rm T}^{2}+\frac{\mu^{2}}{\tau^{2}},\qquad\qquad&p^{\rm L}_{\rm
LE}(\tau)\end{aligned}\right.$ (5.76)

and a modification of the functions $K(\tau)$ and $\Lambda(\tau)$ defined in
(5.41) and (5.42) respectively

$K_{\gamma}(\tau)\equiv\frac{\pi\tau}{4\omega(\tau)}\left[|\partial_{\tau}h(\tau)|^{2}+\gamma(\tau)|h(\tau)|^{2}\right],$
(5.77)
$\Lambda_{\gamma}(\tau)\equiv\frac{\pi\tau}{4\omega(\tau)}\left[(\partial_{\tau}h(\tau))^{2}+\gamma(\tau)h^{2}(\tau)\right].$
(5.78)

Once again, the ${\sf p}$-dependence is omitted in $\gamma(\tau)$,
$K_{\gamma}(\tau)$ and $\Lambda_{\gamma}(\tau)$ both to ease the notation and
to highlight the $\tau$-dependence, which is the conceptually important one.
Much in the same way as $K(\tau)$ and $\Lambda(\tau)$ satisfy (5.43) thanks to
the Wrosnkian of the Hankel functions, $K_{\gamma}(\tau)$ and
$\Lambda_{\gamma}(\tau)$ are related by

$K^{2}_{\gamma}(\tau)-|\Lambda_{\gamma}(\tau)|^{2}=\frac{\gamma(\tau)}{\omega^{2}(\tau)}.$
(5.79)

In the case of the energy density, $K_{\gamma}(\tau)$ and
$\Lambda_{\gamma}(\tau)$ coincide with $K(\tau)$ and $\Lambda(\tau)$
respectively, thus (5.79) reduces correctly to (5.43). With these new
functions, we can further define $\Gamma^{\gamma}_{\rm LE}\equiv\\{\rho_{\rm
LE},p^{\rm T}_{\rm LE},p^{\rm L}_{\rm LE}\\}$ as the set of functions that
make up the thermal expectation value of the energy-momentum tensor
distinguished by the value of the function $\gamma(\tau)$, thus the
(5.73)–(5.75) can be compactly rearranged as

$\Gamma^{\gamma}_{\rm LE}(\tau)=\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\omega^{2}(\tau)\left[K_{\gamma}(\tau)K(\tau)-{\rm
Re}\left(\Lambda_{\gamma}(\tau)\Lambda^{*}(\tau)\right)\right]\left(n_{\rm
B}(\tau)+\frac{1}{2}\right).$ (5.80)

That is the same structure we will recover in the non-equilibrium analysis
too. However, in the present local thermodynamic equilibrium analysis we can
take it a step further by recasting the expression in square brackets in the
following form by using the definitions (5.41), (5.42), (5.77) and (5.78) and
exploiting again the Wronskian of the Hankel functions

$K_{\gamma}(\tau)K(\tau)-{\rm
Re}\left(\Lambda_{\gamma}(\tau)\Lambda^{*}(\tau)\right)=\frac{\omega^{2}(\tau)+\gamma(\tau)}{2\omega^{2}(\tau)},$
(5.81)

hence

$\Gamma^{\gamma}_{\rm LE}(\tau)=\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\frac{\omega^{2}(\tau)+\gamma(\tau)}{2}\left(n_{\rm
B}(\tau)+\frac{1}{2}\right).$ (5.82)

This can be further recast in a more familiar form. By changing the
integration variable $\mu$ to $p_{z}\equiv\mu/\tau$, the measure becomes

${\rm d}^{2}{\rm p_{T}}\frac{{\rm d}\mu}{\tau}={\rm d}^{2}{\rm p_{T}}\,{\rm
d}p_{z}={\rm d}^{3}{\rm p},$ (5.83)

and $\omega(\tau)$ the on-shell energy

$\omega(\tau)=\sqrt{m_{\rm T}^{2}+\frac{\mu^{2}}{\tau^{2}}}=\sqrt{m_{\rm
T}^{2}+p_{z}^{2}}=\sqrt{|{\bf p}_{\rm
T}|^{2}+p_{z}^{2}+m^{2}}\equiv\varepsilon.$ (5.84)

In turn, $n_{\rm B}(\tau)$ becomes the energy-dependent Bose-Einstein
distribution $n_{\rm B}(\varepsilon,T(\tau))$ in the phase space, hence the
the energy density (5.73) as well as the transverse and longitudinal pressures
(5.74) and (5.75) can be written as the familiar momentum integrals of the
relativistic neutral Bose gas. Altogether, the non-renormalized thermal
expectation value of the energy-momentum tensor at local equilibrium reads

$\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}=\int\frac{{\rm d}^{3}{\rm
p}}{\varepsilon}P^{\mu}P^{\nu}\left(\frac{1}{{\rm
e}^{\beta_{\lambda}P^{\lambda}}-1}+\frac{1}{2}\right),$ (5.85)

where $\beta^{\mu}$ is the four-temperature (5.7). Therefore, the
thermodynamic functions $\Gamma^{\gamma}_{\rm LE}(\tau)$ are just the familiar
functions of $T(\tau)$ as the usual ideal relativistic gas. In particular, the
transverse and the longitudinal pressures are in fact equal, namely

$p^{\rm T}_{\rm LE}(\tau)=p^{\rm L}_{\rm LE}(\tau)\equiv p_{\rm LE}(\tau).$
(5.86)

We can convince ourselves of this equality simply by noticing that the measure
(5.83), the energy (5.84) and the Bose-Einstein distribution $n_{\rm B}(\tau)$
are all invariant under rotations of the momentum ${\bf p}$, so it makes no
difference between $p_{x}$, $p_{y}$ and $p_{z}$. Thus, the transverse pressure
(5.74) and the longitudinal pressure (5.75) are equal. In other words, the
thermal expectation value of the energy-momentum tensor at local thermodynamic
equilibrium has the ideal form

$\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}=\rho_{\rm
LE}(\tau)u^{\mu}u^{\nu}-p_{\rm LE}(\tau)\Delta^{\mu\nu}.$ (5.87)

This is a somewhat unexpected feature of local thermodynamic equilibrium,
totally at variance with the case of global thermodynamic equilibrium with
acceleration studied in Chapter 4.

Of course, the plain thermal expectation values (5.80), or equivalently (5.82)
or (5.85), are divergent owing to the $1/2$ terms stemming from the
commutation relations (5.49), therefore they must be suitably renormalized.

####  5.2.2 Renormalization

Renormalization in free Quantum Field Theory is most readily established by
the subtraction of some vacuum expectation value. In the present case of a
relativistic quantum fluid with boost invariance, we face an ambiguity in the
choice of the vacuum which is somehow different from the one discussed in
Subsection 4.2.3 for a relativistic quantum fluid at global thermodynamic
equilibrium with acceleration. There, the quantum field was expanded in the
Rindler wedges in terms of the Rindler creation and annihilation operators,
and the density operator was diagonal with respect to them. Their vacuum was
then the Rindler vacuum. Those operators were related to the usual plane wave
ones by a non-trivial Bogolyubov transformation, therefore the Rindler vacuum
did not coincide with the Minkowski vacuum. Thus, the ambiguity back then was
choosing between the two natural vacua given by the Rindler and the Minkowski
vacuum. By taking the standpoint of the Rindler observer, the Minkowski vacuum
appeared not as empty of particles but populated by a thermal distribution of
them at a temperature proportional to the acceleration. Together with the
vacuum choice ambiguity, the key ingredient that let the Unruh effect emerge
was the fact that the four-temperature possessed a Killing horizon.

Here, the quantum field is expanded in the creation and annihilation operators
$\widehat{b}_{\sf p}^{\dagger}$ and $\widehat{b}_{\sf p}$, but the density
operator is diagonal in a different set of operators, the $\widehat{\xi}_{\sf
p}^{\dagger}(\tau)$ and $\widehat{\xi}_{\sf p}(\tau)$, related to the previous
ones by a non-trivial Bogolyubov transformation. Thus, the vacua of the two
set of operators do not coincide. The $\widehat{b}_{\sf p}^{\dagger}$ and
$\widehat{b}_{\sf p}$ can be obtained from the usual plane wave ones
$\widehat{a}_{\bf p}^{\dagger}$ and $\widehat{a}_{\bf p}$ by means of the
“Fourier-like” transformation (5.37), which does not involve any mixing of
creation and annihilation operators, therefore the vacuum of $\widehat{b}_{\sf
p}$ is in fact the Minkowski vacuum $|0_{\rm M}\rangle$. The Minkowski vacuum,
of course, is a static state. On the other hand, the vacuum of
$\widehat{\xi}_{\sf p}(\tau)$ is a $\tau$-dependent state, and we indicated it
as $|0_{\tau}\rangle$ to emphasize this fact

$\widehat{\xi}_{\sf p}(\tau)|0_{\tau}\rangle=0.$ (5.88)

By using standard methods [92], from the coefficients of the Bogolyubov
transformation (5.59), it can be shown that $|0_{\tau}\rangle$ is related to
$|0_{\rm M}\rangle$ by [49]

$|0_{\tau}\rangle=\prod_{{\sf p}}\frac{1}{|\cosh\Theta({\sf
p},\tau)|^{1/2}}\exp\left[-\frac{1}{2}\tanh\Theta({\sf p},\tau){\rm
e}^{-i\chi({\sf p},\tau)}\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{-{\sf
p}}^{\dagger}\right]|0_{\rm M}\rangle$ (5.89)

In summary, the ambiguity here lies in the choice of either $|0_{\tau}\rangle$
or $|0_{\rm M}\rangle$, leading to two different renormalization schemes

${\langle\widehat{T}^{\mu\nu}\rangle_{\rm
LE}}_{\tau}\equiv\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}-\langle
0_{\tau}|\widehat{T}^{\mu\nu}|0_{\tau}\rangle,$ (5.90)
${\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}}_{\rm
M}\equiv\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}-\langle 0_{\rm
M}|\widehat{T}^{\mu\nu}|0_{\rm M}\rangle.$ (5.91)

This situation is reminiscent of the Unruh effect, and indeed from the
velocity field (5.5) a non-vanishing acceleration field can be derived;
however, the relation (5.37) between the plane wave operators
$\widehat{a}_{\bf p}^{\dagger}$ and $\widehat{a}_{\bf p}$ and the Milne ones
$\widehat{b}_{\sf p}^{\dagger}$ and $\widehat{b}_{\sf p}$ is linear, making
the vacua of the two sets of operators coincide. Therefore, Milne observers
count the same particles as inertial observers, given by

$\langle\widehat{a}_{\bf p}^{\dagger}\widehat{a}_{\bf p}\rangle_{\rm
LE}=\frac{1}{2\pi m_{\rm T}\sqrt{\cosh{\rm y}\cosh{\rm
y}^{\prime}}}\int_{-\infty}^{+\infty}{\rm d}\mu\,{\rm e}^{-i\mu({\rm y}-{\rm
y}^{\prime})}\langle\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{\sf
p}\rangle_{\rm LE}.$ (5.92)

In other words, no analog of the Unruh effect is present between inertial and
Milne observers. The ultimate reason for this seems also to be the fact that
the future light-cone is not a stationary spacetime, and as such it does not
possess any global timelike Killing vector field. In fact, the four-
temperature (5.7), which is the only possible one compatible with boost
invariance, is not a Killing vector field, so it has no Killing horizon
whatsoever. Without a Killing horizon, no Unruh effect seems to emerge.

The next step is the comparison between the two renornalization schemes (5.90)
and (5.91). Let us start from the former, which is also the most
straightforward one. In the limit of vanishing proper temperature, the local
thermodynamic equilibrium density operator reduces to the vacuum
$|0_{\tau}\rangle$, that is

$\lim_{T(\tau)\to 0}\widehat{\rho}_{\rm LE}(\tau)=\lim_{T(\tau)\to
0}\frac{1}{Z_{\rm
LE}(\tau)}\exp\left[-\frac{\widehat{\Pi}(\tau)}{T(\tau)}\right]=|0_{\tau}\rangle\langle
0_{\tau}|.$ (5.93)

The first thing we can deduce from this fact is that the vacuum
$|0_{\tau}\rangle\langle 0_{\tau}|$ has the same symmetries as the local
thermodynamic equilibrium density operator $\widehat{\rho}_{\rm LE}(\tau)$
which, however, is less symmetric than the supposedly Poincaré-invariant
Minkowski vacuum. This does not imply that $|0_{\tau}\rangle$ is degenerate,
but that Poincaré transformations will give rise to non-vanishing components
of excited states. The second one is that renormalization with respect to the
vacuum $|0_{\tau}\rangle$ is essentially the subtraction of the $T(\tau)=0$
contribution. This could just as well be realized by looking at the thermal
expectation values of products of $\widehat{\xi}_{\sf p}(\tau)$ and
$\widehat{\xi}_{\sf p}^{\dagger}(\tau)$ (5.64a)–(5.64c). There, the
subtraction of the $T(\tau)=0$ contribution eventually amounts to canceling
the $+1$ term in (5.64b), which in turn stems from the commutation relations
(5.49). In this scheme, the energy density and the pressure at local
thermodynamic equilibrium become

${\rho_{\rm LE}}_{\tau}(\tau)\equiv\rho_{\rm LE}(\tau)-\langle
0_{\tau}|\widehat{T}^{\mu\nu}|0_{\tau}\rangle u_{\mu}u_{\nu}=\int\frac{{\rm
d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\omega^{2}(\tau)n_{\rm B}(\tau)$ (5.94)
${p_{\rm LE}}_{\tau}(\tau)\equiv p_{\rm LE}(\tau)-\langle
0_{\tau}|\widehat{T}^{\mu\nu}|0_{\tau}\rangle\hat{e}_{\mu}\hat{e}_{\nu}=\int\frac{{\rm
d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}p_{i}^{2}(\tau)n_{\rm B}(\tau)$ (5.95)

with $\hat{e}_{\mu}$ either $\hat{i}_{\mu}$, $\hat{j}_{\mu}$ or
$\hat{\eta}_{\mu}$ and $p_{i}$ either $p_{x}$, $p_{y}$ or $\mu/\tau$.
Equivalently, these can be recast in the more compact forms (5.80) and (5.82)
as

$\begin{split}{\Gamma^{\gamma}_{\rm LE}}_{\tau}(\tau)=&\int\frac{{\rm
d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\omega^{2}(\tau)\left[K_{\gamma}(\tau)K(\tau)-{\rm
Re}\left(\Lambda_{\gamma}(\tau)\Lambda^{*}(\tau)\right)\right]n_{\rm
B}(\tau)\\\ =&\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\frac{\omega^{2}(\tau)+\gamma(\tau)}{2}n_{\rm
B}(\tau).\end{split}$ (5.96)

These integrals are convergent due to the exponential suppression for large
values of the momentum ${\sf p}$ provided by the Bose-Einstein distribution.
In particular, in the massless limit they can be carried out analytically to
yield the familiar expressions of the energy density and pressure of a neutral
Bose gas at temperature $T(\tau)$

${\rho_{\rm LE}}_{\tau}(\tau)=\frac{\pi^{2}}{30}T^{4}(\tau),\qquad(m=0),$
(5.97) ${p_{\rm LE}}_{\tau}(\tau)=\frac{1}{3}{\rho_{\rm
LE}}_{\tau}(\tau)=\frac{\pi^{2}}{90}T^{4}(\tau),\qquad(m=0).$ (5.98)

Now we turn to the renormalization scheme (5.91), namely we take the
standpoint either of the inertial or Milne observer and we subtract the
Minkowski vacuum contribution. In order to calculate the expectation value of
the energy-momentum tensor operator in the Minkowski vacuum, we exploit the
fact that $\widehat{T}^{\mu\nu}$ is built with the quantum field as in (5.38),
and the quantum field is in turn expanded in Milne coordinates as in (5.27) in
terms of $\widehat{b}_{\sf p}$ and $b_{\sf p}^{\dagger}$, whose vacuum is
$|0_{\rm M}\rangle$. Much in the same way as previously done, we take the
energy density as a prototype, with the pressure to be worked out following
analogous steps. The energy density in the Minkowski vacuum is given by (5.70)
where the thermal expectation values of products of $\widehat{b}_{\sf p}$ and
$\widehat{b}_{\sf p}^{\dagger}$ are supposed to be calculated on $|0_{\rm
M}\rangle$ instead of using $\widehat{\rho}_{\rm LE}(\tau)$. Of course, the
products $\widehat{b}_{\sf p}\widehat{b}_{{\sf p}^{\prime}}$,
$\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf p}^{\prime}}^{\dagger}$ and
$\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf p}^{\prime}}$ have vanishing
expectation value on the Minkowski vacuum as $\widehat{b}_{\sf p}|0_{\rm
M}\rangle=0$. On the other hand, the expectation value of $\widehat{b}_{\sf
p}\widehat{b}^{\dagger}_{{\sf p}^{\prime}}$ is non-vanishing and worked out
with the commutation relations (5.33)

$\langle 0_{\rm M}|\widehat{b}_{\sf p}\widehat{b}^{\dagger}_{{\sf
p}^{\prime}}|0_{\rm M}\rangle=\langle 0_{\rm M}|\widehat{b}^{\dagger}_{{\sf
p}^{\prime}}\widehat{b}_{\sf p}|0_{\rm M}\rangle+\langle 0_{\rm
M}|[\widehat{b}_{\sf p},\widehat{b}^{\dagger}_{{\sf p}^{\prime}}]|0_{\rm
M}\rangle=\delta^{2}({\bf p}_{\rm T}-{\bf p}_{\rm
T}^{\prime})\,\delta(\mu-\mu^{\prime}).$ (5.99)

Thus, the energy density in the Minkowski vacuum is readily obtained from
(5.70)

$\begin{split}\langle 0_{\rm M}|\widehat{T}^{\mu\nu}|0_{\rm M}\rangle
u_{\mu}u_{\nu}=&\int\frac{{\rm d^{2}p_{T}}\,{\rm
d}\mu}{4(4\pi)^{2}}\left(|\partial_{\tau}h(\tau)|^{2}+\omega^{2}(\tau)|h(\tau)|^{2}\right)\\\
=&\int\frac{{\rm d^{2}p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\omega^{2}(\tau)\frac{K(\tau)}{2}.\end{split}$
(5.100)

Repeating the same steps for the pressure, the same result is found with
$K(\tau)$ replaced by $K_{\gamma}(\tau)$, the two of them coinciding in the
case of the energy density. We can therefore conclude that the thermal
expectation value of the energy-momentum tensor renormalized with respect to
the Minkowski vacuum reads

$\begin{split}{\Gamma^{\gamma}_{\rm LE}}_{\rm M}(\tau)=&\int\frac{{\rm
d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\omega^{2}(\tau)\times\\\
&\times\left\\{\left[K_{\gamma}(\tau)K(\tau)-{\rm
Re}\left(\Lambda_{\gamma}(\tau)\Lambda^{*}(\tau)\right)\right]\left(n_{\rm
B}(\tau)+\frac{1}{2}\right)-\frac{K_{\gamma}(\tau)}{2}\right\\}.\end{split}$
(5.101)

This expression, however, subtly hides some problem. Let us take once again
the energy density as an example and split it into two terms as

$\begin{split}{\rho_{\rm LE}}_{\rm M}(\tau)=&\int\frac{{\rm d}^{2}{\rm
p_{T}}\,{\rm d}\mu}{(2\pi)^{3}\tau}\omega(\tau)n_{\rm B}(\tau)-\int\frac{{\rm
d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)\frac{K(\tau)-1}{2}.\end{split}$ (5.102)

The first term coincides with (5.94) found in the former renormalization
scheme and is clearly convergent. The second term depends on the transverse
momentum ${\bf p}_{\rm T}$ only through its modulus ${\rm p_{T}}\equiv|{\bf
p}_{\rm T}|$, therefore we can change to polar variables in the transverse
momentum and factorize the angular part

$\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)\frac{K(\tau)-1}{2}=\frac{1}{(2\pi)^{2}\tau}\int_{-\infty}^{+\infty}{\rm
d}\mu\int_{0}^{+\infty}{\rm d}{\rm p_{T}}\,{\rm
p_{T}}\,\omega(\tau)\frac{K(\tau)-1}{2}.$ (5.103)

Changing variable from ${\rm p_{T}}$ to the transverse mass $m_{\rm T}$ and
then to $v\equiv m_{\rm T}\tau$ we have

$\frac{1}{(2\pi)^{2}\tau^{2}}\int_{-\infty}^{+\infty}{\rm
d}\mu\int_{m\tau}^{+\infty}{\rm d}v\,v\sqrt{v^{2}+\mu^{2}}\frac{K(v)-1}{2}.$
(5.104)

In [49], an asymptotic analysis was performed for $m\tau\gg 1$, or, restoring
the natural constants, $m\tau c^{2}\gg\hbar$. As the details of this analysis
are quite involved, they will be addressed in a dedicated space in Subsection
5.3.2. For the time being, we simply mention that the trend of $K(\tau)-1$ in
this limit is

$K(\tau)-1=\cosh\Theta(\tau)-1\simeq\frac{\Theta^{2}(\tau)}{2}\simeq\frac{m_{\rm
T}^{4}}{\tau^{2}\omega^{6}(\tau)}.$ (5.105)

In this limit we also have $v\gg 1$, and using (5.105) the integral in $v$
turns out to diverge

$\int_{m\tau}^{+\infty}{\rm
d}v\,v\sqrt{v^{2}+\mu^{2}}\frac{v^{4}}{\left(\sqrt{v^{4}+\mu^{2}}\right)^{6}}\sim\int_{m\tau}^{+\infty}{\rm
d}v\,v\frac{v^{4}}{v^{5}}\sim\infty.$ (5.106)

Of course this is not a formal proof, but a strong indication nonetheless that
renormalizing with respect to the Minkowski vacuum provides us with thermal
expectation values that are still divergent. This fact seems to automatically
single out $|0_{\tau}\rangle$ as a preferred choice, however, it must be
emphasized that $|0_{\tau}\rangle$ is a $\tau$-dependent vacuum, therefore
thermal expectation values renormalized accordingly might inherit some
eventually undesired $\tau$-dependence. These issues will be further discussed
in Subsection 5.3.3 in the context of the non-equilibrium analysis. For the
time being, we would like the takeaway to be the fact that the Minkowski
vacuum renormalization scheme should be discarded as it appears to provide us
with thermal expectation values that are still divergent. Moreover, the next
Subsection will make it clear why at some point we would have needed to study
the effects of the $|0_{\tau}\rangle$ contributions subtraction anyway.

####  5.2.3 Entropy current in the future light-cone

With the plain thermal expectation value of the energy-momentum tensor, we are
now in a good spot to calculate the entropy current at local thermodynamic
equilibrium by using the method put forward Section 3.3. That is in fact well-
suited for systems at local thermodynamic equilibrium, which is the underlying
hypothesis of relativistic hydrodynamics, and global thermodynamic equilibrium
is included as a particular case. However, no known expression for the entropy
current nor for the entropy production exists for a Quantum Field Theory out
of thermodynamic equilibrium, therefore in the non-equilibrium analysis to be
carried out in Section 5.3 we will not be able to re-apply our method.

As a quick recap, the method demands to be provided with two ingredients:

  1. 1.

the thermal expectation value the energy-momentum tensor
$\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}$, and

  2. 2.

the eigenvector $|0_{\Upsilon}\rangle$ corresponding to the lowest, non-
degenerate eigenvalue of $\widehat{\Upsilon}(\tau)$.

Then, the algorithm is the following:

  1. 1.

Take $\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}$ and subtract $\langle
0_{\Upsilon}|\widehat{T}^{\mu\nu}|0_{\Upsilon}\rangle$ by using the
$\lambda$-dependent density operator $\widehat{\rho}_{\rm LE}(\lambda)$
defined according to (3.54). This way, the result is $\lambda$-dependent.

  2. 2.

Contract with $\beta_{\nu}$, which is $\lambda$-independent, and integrate in
$\lambda$ from $\lambda=1$ to $\lambda=+\infty$ in order to obtain the
thermodynamic potential current $\phi^{\mu}$ defined in (3.67).

  3. 3.

Plug the result into (3.68) and obtain the entropy current $s^{\mu}$.

By comparing the definition (3.42) of the operator $\widehat{\Upsilon}(\tau)$
with the expression (5.23) of the local thermodynamic equilibrium density
operator with boost invariance in the future light-cone, we immediately
understand that

$\widehat{\Upsilon}(\tau)=\frac{\widehat{\Pi}(\tau)}{T(\tau)}=\frac{1}{T(\tau)}\int{\rm
d}^{2}{\rm p_{T}}\,{\rm d}\mu\,\omega(\tau)\left(\widehat{\xi}_{\sf
p}^{\dagger}(\tau)\widehat{\xi}_{\sf p}(\tau)+\frac{1}{2}\right).$ (5.107)

Thus, the lowest eigenvector of $\widehat{\Upsilon}(\tau)$ is the vacuum of
$\widehat{\Pi}(\tau)$, that is $|0_{\tau}\rangle$. As we discussed already in
Subsection 5.2.2, $|0_{\tau}\rangle$ is a $\tau$-dependent and non-degenerate
state which does not coincide with the Minkowski vacuum and possesses the same
symmetries as the local thermodynamic equilibrium density operator. Hence

$\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}-\langle
0_{\Upsilon}|\widehat{T}^{\mu\nu}|0_{\Upsilon}\rangle=\langle\widehat{T}^{\mu\nu}\rangle_{\rm
LE}-\langle 0_{\tau}|\widehat{T}^{\mu\nu}|0_{\tau}\rangle,$ (5.108)

which was calculated in (5.94) and (5.95) or equivalently (5.96), with
analytical results given by (5.97) and (5.98) for a massless field. So, we are
good to go for what concerns the ingredients.

According to the method, the above quantity should be calculated with the
modified density operator $\widehat{\rho}_{\rm LE}(\tau,\lambda)$ defined in
(3.54) in order to be $\lambda$-dependent. Once again, by comparing (3.54)
with (5.107), we can tell that this is but a rescaling of the proper
temperature as $T(\tau)\mapsto T(\tau)/\lambda$, in fact this transformation
gives precisely the sought result. Thus, we have

$\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}(\lambda)-\langle
0_{\Upsilon}|\widehat{T}^{\mu\nu}|0_{\Upsilon}\rangle(\lambda)={\rho_{\rm
LE}}_{\tau}(\tau,\lambda)u^{\mu}u^{\nu}-{p_{\rm
LE}}_{\tau}(\tau,\lambda)\Delta^{\mu\nu},$ (5.109)

where the rescaling $T(\tau)\mapsto T(\tau)/\lambda$ affects only the energy
density and the pressure because $u^{\mu}$ and $\Delta^{\mu\nu}$ are
independent of $T(\tau)$. This result should be contracted with the four-
temperature while taking care that $\beta_{\nu}$ does not undergo the
temperature rescaling, for it concerns only the thermal expectation value
$\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}-\langle
0_{\Upsilon}|\widehat{T}^{\mu\nu}|0_{\Upsilon}\rangle$, as can be clearly seen
in the derivation in Section 3.3. Whence

$\left(\langle\widehat{T}^{\mu\nu}\rangle_{\rm LE}(\lambda)-\langle
0_{\Upsilon}|\widehat{T}^{\mu\nu}|0_{\Upsilon}\rangle(\lambda)\right)\beta_{\nu}={\rho_{\rm
LE}}_{\tau}(\tau,\lambda)\beta^{\mu},$ (5.110)

therefore we do not need to know the thermal expectation value of the whole
energy-momentum tensor, only the energy density contributes to the
thermodynamic potential current and the entropy current. By looking at
equation (5.94), we see that the only dependence of the energy density on the
proper temperature is in the Bose-Einstein distribution, thus

${\rho_{\rm LE}}_{\tau}(\tau,\lambda)=\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\omega^{2}(\tau)\frac{1}{{\rm
e}^{\lambda\omega(\tau)/T(\tau)}-1}.$ (5.111)

For a massless field, the $\lambda$-integration can be carried out
analytically, yielding

$\int_{1}^{+\infty}{\rm d}\lambda\,{\rho_{\rm
LE}}_{\tau}(\tau,\lambda)=\int_{1}^{+\infty}{\rm
d}\lambda\,\frac{\pi^{2}}{30}\frac{T^{4}(\tau)}{\lambda^{4}}=\frac{\pi^{2}}{90}T^{4}(\tau),\qquad(m=0)$
(5.112)

and the following expression for the thermodynamic potential current is
readily obtained

$\phi^{\mu}=\frac{\pi^{2}}{90\beta^{4}}\beta^{\mu}=\frac{\pi^{2}}{90}T^{3}(\tau)u^{\mu},\qquad(m=0).$
(5.113)

Plugging this result into (3.68), we finally obtain the entropy current

$s^{\mu}=\frac{2\pi^{2}}{45\beta^{4}}\beta^{\mu}=\frac{2\pi^{2}}{45}T^{3}(\tau)u^{\mu},\qquad(m=0).$
(5.114)

These are the expressions of the thermodynamic potential current and the
entropy current of a free real massless scalar field at local thermodynamic
equilibrium with boost invariance in the future light-cone [49]. The entropy
current coincides with the classical expression of the usual relativistic
kinetic theory, therefore, no quantum corrections are apparently present,
which is somewhat unexpected.

###  5.3 Non-equilibrium analysis

So far, we have been concerned with an analysis at local thermodynamic
equilibrium, which is a non-equilibrium configurations but still close to
global thermodynamic equilibrium, in the sense that the volume term in (3.23)
containing the derivatives of the thermodynamic fields is small enough to be
neglected. In this Section, we perform a full non-equilibrium analysis
instead, therefore we consider the full non-equilibrium density operator
$\widehat{\rho}=\widehat{\rho}_{\rm LE}(\tau_{0})$. The calculation of the
thermal expectation value of the energy-momentum tensor goes on much in the
same way as we saw in the local thermodynamic equilibrium analysis. However,
there are now complications due to the mixing of terms calculated at $\tau$
and others at $\tau_{0}$. It is therefore mandatory to pay attention to these
dependencies, so the dependence on ${\sf p}$ will still be omitted whenever
possible.

####  5.3.1 Thermal expectation values

As previously mentioned, the non-equilibrium density operator and the local
thermodynamic equilibrium one have the same symmetries, so the former will
simply be the latter calculated at $\tau_{0}$, namely

$\widehat{\rho}=\widehat{\rho}_{\rm LE}(\tau_{0})=\frac{1}{Z_{\rm
LE}(\tau_{0})}\exp\left[-\frac{\widehat{\Pi}(\tau_{0})}{T(\tau_{0})}\right]$
(5.115)

with $\widehat{\Pi}(\tau_{0})$ given by

$\widehat{\Pi}(\tau_{0})=\int{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu\,\omega(\tau_{0})\left(\widehat{\xi}_{\sf
p}^{\dagger}(\tau_{0})\widehat{\xi}_{\sf p}(\tau_{0})+\frac{1}{2}\right).$
(5.116)

The operators $\widehat{\xi}_{\sf p}(\tau_{0})$ and $\widehat{\xi}_{\sf
p}^{\dagger}(\tau_{0})$ are those that diagonalize $\widehat{\Pi}(\tau_{0})$,
which is but $\widehat{\Pi}(\tau)$ at $\tau_{0}$, so they are just
$\widehat{\xi}_{\sf p}(\tau)$ and $\widehat{\xi}_{\sf p}^{\dagger}(\tau)$
respectively at $\tau_{0}$. In terms of $\widehat{b}_{\sf p}$ and
$\widehat{b}_{\sf p}^{\dagger}$, they are obtained from (5.59) at $\tau_{0}$,
namely

$\begin{split}\widehat{b}_{\sf p}=&\cosh\Theta({\sf
p},\tau_{0})\widehat{\xi}_{\sf p}(\tau_{0})-\sinh\Theta({\sf
p},\tau_{0})\,{\rm e}^{-i\chi({\sf p},\tau_{0})}\widehat{\xi}_{-{\sf
p}}^{\dagger}(\tau_{0}),\\\ \widehat{b}_{\sf p}^{\dagger}=&\cosh\Theta({\sf
p},\tau_{0})\widehat{\xi}_{\sf p}^{\dagger}(\tau_{0})-\sinh\Theta({\sf
p},\tau_{0})\,{\rm e}^{i\chi({\sf p},\tau_{0})}\widehat{\xi}_{-{\sf
p}}(\tau_{0}).\end{split}$ (5.117)

Thermal expectation values out of thermodynamic equilibrium are calculated
with this density operator as anticipated in (5.20), which we hereby report.
The non-equilibrium thermal expectation value of the energy-momentum tensor is

$T^{\mu\nu}=\mathrm{tr}(\widehat{\rho}\widehat{T}^{\mu\nu})=\mathrm{tr}(\widehat{\rho}_{\rm
LE}(\tau_{0})\widehat{T}^{\mu\nu}(\tau)).$ (5.118)

This is constrained by boost invariance to be of the form (5.19), that is

$T^{\mu\nu}=\rho(\tau)u^{\mu}u^{\nu}+p^{\rm
T}(\tau)(\hat{i}^{\mu}\hat{i}^{\nu}+\hat{j}^{\mu}\hat{j}^{\nu})+p^{\rm
L}(\tau)\hat{\eta}^{\mu}\hat{\eta}^{\nu}.$ (5.119)

In order to calculate the energy density and the two pressures, we need to
know the non-equilibrium thermal expectation values of products of
$\widehat{b}_{\sf p}$ and $\widehat{b}_{\sf p}^{\dagger}$ with
$\widehat{\rho}$, that is

$\displaystyle\langle\widehat{b}_{\sf p}\widehat{b}_{{\sf
p}^{\prime}}\rangle=$
$\displaystyle\mathrm{tr}(\widehat{\rho}\,\widehat{b}_{\sf p}\widehat{b}_{{\sf
p}^{\prime}})=\mathrm{tr}(\widehat{\rho}_{\rm LE}(\tau_{0})\widehat{b}_{\sf
p}\widehat{b}_{{\sf p}^{\prime}})$ (5.120a)
$\displaystyle\langle\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf
p}^{\prime}}^{\dagger}\rangle=$
$\displaystyle\mathrm{tr}(\widehat{\rho}\,\widehat{b}_{\sf
p}^{\dagger}\widehat{b}_{{\sf
p}^{\prime}}^{\dagger})=\mathrm{tr}(\widehat{\rho}_{\rm
LE}(\tau_{0})\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf
p}^{\prime}}^{\dagger})$ (5.120b) $\displaystyle\langle\widehat{b}_{\sf
p}\widehat{b}_{{\sf p}^{\prime}}^{\dagger}\rangle=$
$\displaystyle\mathrm{tr}(\widehat{\rho}\,\widehat{b}_{\sf p}\widehat{b}_{{\sf
p}^{\prime}}^{\dagger})=\mathrm{tr}(\widehat{\rho}_{\rm
LE}(\tau_{0})\widehat{b}_{\sf p}\widehat{b}_{{\sf p}^{\prime}}^{\dagger})$
(5.120c) $\displaystyle\langle\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf
p}^{\prime}}\rangle=$
$\displaystyle\mathrm{tr}(\widehat{\rho}\,\widehat{b}_{\sf
p}^{\dagger}\widehat{b}_{{\sf p}^{\prime}})=\mathrm{tr}(\widehat{\rho}_{\rm
LE}(\tau_{0})\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf p}^{\prime}}).$
(5.120d)

These are most readily given by evaluating (5.66a)–(5.66d) at $\tau_{0}$

$\displaystyle\langle\widehat{b}_{\sf p}\widehat{b}_{{\sf
p}^{\prime}}\rangle=$
$\displaystyle-\frac{1}{2}\sinh\left(2\Theta(\tau_{0})\right){\rm
e}^{-i\chi(\tau_{0})}\left(2n_{\rm B}(\tau_{0})+1\right)\delta^{2}({\bf
p}_{\rm T}-{\bf p}_{\rm T}^{\prime})\,\delta(\mu-\mu^{\prime})$ (5.121a)
$\displaystyle\langle\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf
p}^{\prime}}^{\dagger}\rangle=$
$\displaystyle-\frac{1}{2}\sinh\left(2\Theta(\tau_{0})\right){\rm
e}^{i\chi(\tau_{0})}\left(2n_{\rm B}(\tau_{0})+1\right)\delta^{2}({\bf p}_{\rm
T}-{\bf p}_{\rm T}^{\prime})\,\delta(\mu-\mu^{\prime})$ (5.121b)
$\displaystyle\langle\widehat{b}_{\sf p}\widehat{b}_{{\sf
p}^{\prime}}^{\dagger}\rangle=$
$\displaystyle\left[\cosh\left(2\Theta(\tau_{0})\right)n_{\rm
B}(\tau_{0})+\cosh^{2}\Theta(\tau_{0})\right]\delta^{2}({\bf p}_{\rm T}-{\bf
p}_{\rm T}^{\prime})\,\delta(\mu-\mu^{\prime})$ (5.121c)
$\displaystyle\langle\widehat{b}_{\sf p}^{\dagger}\widehat{b}_{{\sf
p}^{\prime}}\rangle=$
$\displaystyle\left[\cosh\left(2\Theta(\tau_{0})\right)n_{\rm
B}(\tau_{0})+\sinh^{2}\Theta(\tau_{0})\right]\delta^{2}({\bf p}_{\rm T}-{\bf
p}_{\rm T}^{\prime})\,\delta(\mu-\mu^{\prime}).$ (5.121d)

Taking once again the energy density as a prototype, its non-equilibrium
thermal expectation value is obtained from (5.70) by replacing (5.66a)–(5.66d)
with (5.121a)–(5.121d)

$\begin{split}\rho(\tau)=&\mathrm{tr}(\widehat{\rho}\widehat{T}^{\mu\nu})u_{\mu}u_{\nu}=\mathrm{tr}(\widehat{\rho}_{\rm
LE}(\tau_{0})\widehat{T}^{\mu\nu}(\tau))u_{\mu}u_{\nu}\\\ =&\int\frac{{\rm
d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)\left[K(\tau)K(\tau_{0})-{\rm
Re}\left(\Lambda(\tau)\Lambda^{*}(\tau_{0})\right)\right]\left(n_{\rm
B}(\tau_{0})+\frac{1}{2}\right).\end{split}$ (5.122)

As in the local thermodynamic equilibrium analysis, the pressures are worked
out following the same steps. Thus, the non-equilibrium analog of equation
(5.80) reads

$\Gamma^{\gamma}(\tau)=\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\omega^{2}(\tau)\left[K_{\gamma}(\tau)K(\tau_{0})-{\rm
Re}\left(\Lambda_{\gamma}(\tau)\Lambda^{*}(\tau_{0})\right)\right]\left(n_{\rm
B}(\tau_{0})+\frac{1}{2}\right).$ (5.123)

Note that at $\tau=\tau_{0}$, namely at local thermodynamic equilibrium, this
equation reduces to (5.80), and so the results of the local thermodynamic
equilibrium analysis are recovered as expected. However, at later times
$\tau>\tau_{0}$, because of the mixing of some functions at $\tau$ and others
at $\tau_{0}$, the Wronskian of the Hankel functions is, unfortunately, not
reconstructed. When written explicitly, the above equation is given by a non-
trivial combination of Hankel fuctions of the same order but with different
arguments which cannot be recast in any simple form. We can therefore tell
that for $\tau>\tau_{0}$ the thermal expectation value of the energy-momentum
tensor differs from its local thermodynamic equilibrium form, but we are not
able to perform the integration of this combination of Hankel functions in
general. Indeed, since we are dealing with a free Quantum Field Theory, one
expects to find the same expression as for the free-streaming solution of
Boltzmann equation in Milne coordinates, which is reported in Appendix A.
However, there will also be quantum corrections due to vacuum subtraction.

In the next Subsection, we will consider a suitable approximation in which the
calculation of the non-equilibrium thermal expectation value of the energy-
momentum tensor becomes feasible.

####  5.3.2 Asymptotic analysis

In [49], the behaviour of the stress-energy tensor and related quantities for
late times $\tau$ was studied. The starting point is the function $h(\tau)$
defined in (5.30), that is

$h(\tau)=-i{\rm e}^{\frac{\pi}{2}\mu}{\rm H}^{(2)}_{i\mu}(m_{\rm T}\tau).$
(5.124)

The asymptotic expansion of the Hankel functions for large arguments is [91]

${\rm H}^{(2)}_{\nu}(x)\sim\sqrt{\frac{2}{\pi x}}{\rm
e}^{-i\left(x-\frac{\pi}{2}\nu-\frac{\pi}{4}\right)}\sum_{n}\frac{1}{(2ix)^{n}}\frac{\Gamma(\nu+1/2+n)}{n!\Gamma(\nu+1/2-n)}$
(5.125)

which is valid for ${\rm Re}(\nu)>-1/2$ and $|{\rm arg}(x)|<\pi$. By making
use of the property $z\Gamma(z)=\Gamma(z+1)$, substituting $x=m_{\rm T}\tau$
and $\nu=i\mu$, and plugging into (5.124), we get

$h(\tau)\sim\sqrt{\frac{-2i}{\pi m_{\rm T}\tau}}{\rm e}^{-im_{\rm
T}\tau}\sum_{n}\frac{1}{(2im_{\rm
T}\tau)^{n}}\frac{\left(i\mu+\frac{1}{2}-n\right)^{(2n)}}{n!},$ (5.126)

valid for large $m_{\rm T}\tau$. Similarly, using the exact relation [91]

$z\partial_{z}{\rm H}^{(2)}_{\nu}(z)=\nu{\rm H}^{(2)}_{\nu}(z)-z{\rm
H}^{(2)}_{\nu+1}(z),$ (5.127)

along with the expansion (5.125), one obtains the expansion for the derivative
$\partial_{\tau}h$

$\begin{split}\partial_{\tau}h(\tau)\sim&-im_{\rm T}\sqrt{\frac{-2i}{\pi
m_{\rm T}\tau}}{\rm e}^{-im_{\rm T}\tau}\times\\\
&\times\left[1+\sum_{n>0}\frac{1}{(2im_{\rm
T}\tau)^{n}}\left(-2i\mu\frac{\left(i\mu+\frac{3}{2}-n\right)^{(2n-2)}}{(n-1)!}+\frac{\left(i\mu+\frac{3}{2}-n\right)^{(2n)}}{n!}\right)\right].\end{split}$
(5.128)

Particularly, retaining the terms up to first order (next to leading) in
$m_{\rm T}\tau$ we get

$\begin{split}h(\tau)\simeq&\sqrt{\frac{-2i}{\pi m_{\rm T}\tau}}{\rm
e}^{-im_{\rm T}\tau}\left[1-\frac{i}{2m_{\rm
T}\tau}\left(i\mu-\frac{1}{2}\right)\left(i\mu+\frac{1}{2}\right)\right]\\\
=&\sqrt{\frac{-2i}{\pi m_{\rm T}\tau}}{\rm e}^{-im_{\rm
T}\tau}\left[1+i\frac{1+4\mu^{2}}{8m_{\rm T}\tau}\right],\end{split}$ (5.129)
$\partial_{\tau}h(\tau)\simeq-im_{\rm T}\sqrt{\frac{-2i}{\pi m_{\rm
T}\tau}}e^{-im_{\rm T}\tau}\left[1-i\frac{3-4\mu^{2}}{8m_{\rm T}\tau}\right].$
(5.130)

Feeding the above expansions in the definitions (5.77) and (5.78), we obtain

$K_{\gamma}(\tau)\simeq\frac{1}{2m_{\rm T}\omega(\tau)}\left[m_{\rm
T}^{2}+\gamma(\tau)\right],$ (5.131)
$\Lambda_{\gamma}(\tau)\simeq\frac{1}{2m_{\rm T}\omega(\tau)}{\rm
e}^{-2im_{\rm T}\tau}\left[-m_{\rm T}^{2}\left(1-i\frac{3-4\mu^{2}}{4m_{\rm
T}\tau}\right)+\gamma(\tau)\left(1+i\frac{1+4\mu^{2}}{4m_{\rm
T}\tau}\right)\right].$ (5.132)

In the case of $K_{\gamma}(\tau)$, the remainder is of the order $1/[m_{\rm
T}\omega(\tau)(m_{\rm T}\tau)^{2}]$, while for $\Lambda_{\gamma}$ is of order
${\rm e}^{-2im_{\rm T}\tau}/[m_{\rm T}\omega(\tau)(m_{\rm T}\tau)^{2}]$.

The last formulae are indicative of the behaviour both at late times and for
large values of the transverse momentum ${\rm p_{T}}$, whence the transverse
mass $m_{\rm T}$, necessary for the convergence check of the renormalized
thermal expectation values. In both cases, $m_{T}\tau\gg 1$ and the previous
asymptotic expansion is appropriate. Concerning the large transverse momentum
behaviour, equation (5.131) tells us that $K(\tau)\to 1$, therefore the
leading order of $K(\tau)-1$ is zero, at least at first order. However, from
equation (5.132) and the exact relation (5.46), one can obtain the value of
the second order expansion. Indeed, for $\gamma(\tau)=\omega(\tau)$, in the
large $m_{\rm T}$ limit

$\Lambda(\tau)=\frac{1}{2m_{\rm T}\tau}{\rm e}^{-2im_{\rm T}\tau}+{\rm
ord}\left(\frac{{\rm e}^{-2im_{\rm T}\tau}}{m_{\rm T}^{2}}\right),$ (5.133)

therefore

$\frac{1}{2m_{\rm
T}\tau}\simeq|\Lambda(\tau)|=\sinh\Theta(\tau)\simeq\Theta(\tau),$ (5.134)

then, still in the limit of large $m_{\rm T}$,

$K(\tau)-1\simeq\frac{1}{2}\Theta^{2}(\tau)\simeq\frac{1}{8m_{\rm
T}^{2}\tau^{2}}.$ (5.135)

A very similar approach can be used for the late time behaviour
$\tau\to\infty$. Namely, using the asymptotic formulae (5.131) and (5.132),
making an expansion of $\omega(\tau)$ and $\gamma(\tau)$ for large $\tau$
keeping only the first orders, that is up to the order $1/\tau$ for
$K_{\gamma}(\tau)$ and to the order ${\rm e}^{-2im_{\rm T}\tau}/\tau$ for
$\Lambda_{\gamma}(\tau)$. This difference between the order of
$K_{\gamma}(\tau)$ and $\Lambda_{\gamma}(\tau)$ is because of the different
gauge functions in the two cases from the large argument expansion of the
Hankel functions. Therefore, one has

$K_{\gamma}(\tau)\simeq\frac{m_{\rm T}^{2}+\tilde{\gamma}}{2m_{\rm T}^{2}},$
(5.136) $\Lambda_{\gamma}(\tau)\simeq\frac{1}{2m_{\rm T}^{2}}{\rm
e}^{-2im_{\rm T}\tau}\left[\tilde{\gamma}-m_{\rm T}^{2}+i\frac{m_{\rm
T}^{2}(3-4\mu^{2})+\tilde{\gamma}(1+4\mu^{2})}{4m_{\rm T}\tau}\right],$
(5.137)

with $\tilde{\gamma}$ being

$\tilde{\gamma}\equiv\lim_{\tau\to\infty}\gamma(\tau)=\begin{cases}m_{\rm
T}^{2},&\mbox{for }\rho(\tau)\\\ -m^{2},&\mbox{for }p^{\rm T}(\tau)\\\ -m_{\rm
T}^{2},&\mbox{for }p^{\rm L}(\tau).\end{cases}$ (5.138)

It is important to note that, beside the case related to the energy density
and only at the leading order, $\Lambda_{\gamma}(\tau)$ has a rapidly
oscillating phase preventing a proper limit in the function domain. However,
it converges in the distribution domain, which is fine since it has to be
integrated. In fact the limits

$\lim_{\tau\to\infty}\sin(2m_{\rm
T}\tau),\qquad\lim_{\tau\to\infty}\cos(2m_{\rm T}\tau)$ (5.139)

are proportional to Dirac delta functions. Making use of the formula for the
delta families

$\delta(x)=\lim_{\epsilon\to
0}\frac{1}{\epsilon}f\left(\frac{x}{\epsilon}\right),$ (5.140)

for a generic function $f$ normalized to $1$, by indicating $\epsilon\equiv
1/\tau$ in the first case and $\epsilon\equiv 1/\sqrt{\tau}$ in the second
one, we have

$\displaystyle\lim_{\epsilon\to
0}\left(\frac{1}{\pi}\frac{\sin(x/\epsilon)}{x}\right)=\delta(x)\qquad$
$\displaystyle\Rightarrow\qquad\sin(2m_{\rm
T}\tau)\;\stackrel{{\scriptstyle\tau\to\infty}}{{\longrightarrow}}\;2\pi
m_{\rm T}\,\delta(2m_{\rm T}),$ (5.141) $\displaystyle\lim_{\epsilon\to
0}\left(\frac{1}{\epsilon\sqrt{\pi}}\cos\left(\frac{x^{2}}{\epsilon^{2}}\right)\right)=\delta(x)\qquad$
$\displaystyle\Rightarrow\qquad\cos(2m_{\rm
T}\tau)\;\stackrel{{\scriptstyle\tau\to\infty}}{{\longrightarrow}}\;\sqrt{\frac{\pi}{\tau}}\,\delta(\sqrt{2m_{\rm
T}}).$ (5.142)

In both cases, the Dirac delta function is outside of the domain of
integration, and all these integrals are vanishing in the long proper time
limit.

These expressions can now be used to calculate the asymptotic value of the
non-equilibrium thermal expectation value (5.123) at late times. In
particular, we want to focus on the energy density, in which case
$K_{\gamma}(\tau)$ and $\Lambda_{\gamma}(\tau)$ coincide with $K(\tau)$ and
$\Lambda(\tau)$ respectively. As $\tau$ grows to infinity, $K(\tau)$ tends to
the constant value $1$, whereas $\Lambda(\tau)$ vanishes, thus

$\begin{split}\rho(\tau)=&\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)\left[K(\tau)K(\tau_{0})-{\rm
Re}\left(\Lambda(\tau)\Lambda^{*}(\tau_{0})\right)\right]\left(n_{\rm
B}(\tau_{0})+\frac{1}{2}\right)\\\ \simeq&\int\frac{{\rm d}^{2}{\rm
p_{T}}\,{\rm d}\mu}{(2\pi)^{3}\tau}\omega(\tau)K(\tau_{0})\left(n_{\rm
B}(\tau_{0})+\frac{1}{2}\right)\\\ =&\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)\left(n_{\rm
B}(\tau_{0})+\frac{1}{2}\right)+\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)\left(K(\tau_{0})-1\right)\left(n_{\rm
B}(\tau_{0})+\frac{1}{2}\right)\\\ =&\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)\left(n_{\rm
B}(\tau_{0})+\frac{1}{2}\right)+\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)2\sinh^{2}\Theta(\tau_{0})\left(n_{\rm
B}(\tau_{0})+\frac{1}{2}\right)\end{split}$ (5.143)

where equation (5.46) was used in the last step. Renormalization is about to
be discussed in Subsection 5.3.3. As we shall shortly see there, once suitably
renormalized, the first integral will correspond to the classical free-
streaming solution in Milne coordinates, reported in Appendix A. On the other
hand, the second integral will be a pure quantum correction due to vacuum
effects, for it will vanish only for $\Theta(\tau_{0})=0$.

####  5.3.3 Renormalization

The non-equilibrium thermal expectation values (5.123) as well as the
asymptotic energy density (5.143) are divergent due to the $1/2$ terms
stemming from the commutation relations between $\widehat{\xi}_{\sf
p}(\tau_{0})$ and $\widehat{\xi}_{\sf p}^{\dagger}(\tau_{0})$ that must be
suitably renormalized. What we are facing here is essentially the same
ambiguity of which vacuum contribution should be subtracted as the one
discussed in Subsection 5.2.2 in the context of the local thermodynamic
equilibrium analysis. The choice there was between the two vacua
$|0_{\tau}\rangle$ and $|0_{\rm M}\rangle$, the former being the
$\tau$-dependent vacuum of the local thermodynamic equilibrium density
operator whereas the latter being the static Minkowski vacuum. In particular,
it was shown that by taking the standpoint of the inertial or the Milne
observer and renormalizing with respect to the Minkowski vacuum, we ended up
with thermal expectation values that were still divergent. This persuaded us
that the Minkowski vacuum renormalization scheme should be discarded. That
conclusion still holds in the present non-equilibrium analysis, but now we
have to take a closer look at what the subtraction of the contribution of the
$\tau$-dependent vacuum $|0_{\tau}\rangle$ implies.

The renormalized thermal expectation value of the energy-momentum tensor ought
to fulfill the hydrodynamic equations $\partial_{\mu}T^{\mu\nu}=0$. However,
in the $|0_{\tau}\rangle$ scheme, the renormalized tensor

$T^{\mu\nu}_{\tau}\equiv\mathrm{tr}(\widehat{\rho}\,\widehat{T}^{\mu\nu})-\langle
0_{\tau}|\widehat{T}^{\mu\nu}|0_{\tau}\rangle=\mathrm{tr}\left[(\widehat{\rho}-|0_{\tau}\rangle\langle
0_{\tau}|)\widehat{T}^{\mu\nu}\right]$ (5.144)

has a non-vanishing divergence due to the $\tau$-dependence of
$|0_{\tau}\rangle$

$\begin{split}\partial_{\mu}T^{\mu\nu}_{\tau}=&\partial_{\mu}\mathrm{tr}\left[(\widehat{\rho}-|0_{\tau}\rangle\langle
0_{\tau}|)\widehat{T}^{\mu\nu}\right]\\\
=&\mathrm{tr}\left[(\widehat{\rho}-|0_{\tau}\rangle\langle
0_{\tau}|)\partial_{\mu}\widehat{T}^{\mu\nu}\right]-\mathrm{tr}\left[(\partial_{\mu}|0_{\tau}\rangle\langle
0_{\tau}|)\widehat{T}^{\mu\nu}\right]\\\
=&-\mathrm{tr}\left[u_{\mu}\left(\partial_{\tau}|0_{\tau}\rangle\langle
0_{\tau}|\right)\widehat{T}^{\mu\nu}\right]\neq 0,\end{split}$ (5.145)

where the conservation law $\partial_{\mu}\widehat{T}^{\mu\nu}=0$ and the
stationarity of $\widehat{\rho}$ were used. Therefore, although the energy-
momentum tensor quantum operator is conserved, its renormalized thermal
expectation value does not fulfill the hydrodynamic equations in this scheme.
We conclude that, in order to have a thermal expectation which is both finite
and conserved, the vacuum to be subtracted must be stationary. Since the
Minkowski vacuum can neither be chosen, our most reasonable alternative is the
vacuum of the non-equilibrium density operator (5.115), namely the vacuum of
$\widehat{\Pi}(\tau_{0})$. That is but the state annihilated by the operators
$\widehat{\xi}_{\sf p}(\tau_{0})$, so, in analogy with (5.88), we indicate it
as $|0_{\tau_{0}}\rangle$

$\widehat{\xi}_{\sf p}(\tau_{0})|0_{\tau_{0}}\rangle\equiv 0.$ (5.146)

As mentioned already, the operators $\widehat{\xi}_{\sf p}(\tau_{0})$ are just
$\widehat{\xi}_{\sf p}(\tau)$ evaluated at a fixed $\tau_{0}$, hence
$|0_{\tau_{0}}\rangle$ is $|0_{\tau}\rangle$ at $\tau_{0}$, and as such it is
a stationary state suitable for a proper renormalization.

Much in the same way as $|0_{\tau}\rangle$ is obtained from
$\widehat{\rho}_{\rm LE}(\tau)$ in the $T(\tau)\to 0$ limit as in (5.93),
$|0_{\tau_{0}}\rangle$ is obtained from $\widehat{\rho}$ in the
$T(\tau_{0})\to 0$ limit

$\lim_{T(\tau_{0})\to 0}\widehat{\rho}=\lim_{T(\tau_{0})\to
0}\widehat{\rho}_{\rm LE}(\tau_{0})=\lim_{T(\tau_{0})\to 0}\frac{1}{Z_{\rm
LE}(\tau_{0})}\exp\left[-\frac{\widehat{\Pi}(\tau_{0})}{T(\tau_{0})}\right]=|0_{\tau_{0}}\rangle\langle
0_{\tau_{0}}|.$ (5.147)

Thus, renormalization with respect to $|0_{\tau_{0}}\rangle$ is but the
subtraction of the $T(\tau_{0})=0$ contribution. The non-equilibrium thermal
expectation value of the energy-momentum tensor thereby renormalized is

$T^{\mu\nu}_{\tau_{0}}\equiv\mathrm{tr}(\widehat{\rho}\widehat{T}^{\mu\nu})-\langle
0_{\tau_{0}}|\widehat{T}^{\mu\nu}|0_{\tau_{0}}\rangle=\mathrm{tr}(\widehat{\rho}_{\rm
LE}(\tau_{0})\widehat{T}^{\mu\nu})-\langle
0_{\tau_{0}}|\widehat{T}^{\mu\nu}|0_{\tau_{0}}\rangle,$ (5.148)

in particular, equation (5.123) becomes

$\Gamma^{\gamma}_{\tau_{0}}(\tau)=\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau\omega(\tau)}\omega^{2}(\tau)\left[K_{\gamma}(\tau)K(\tau_{0})-{\rm
Re}\left(\Lambda_{\gamma}(\tau)\Lambda^{*}(\tau_{0})\right)\right]n_{\rm
B}(\tau_{0}),$ (5.149)

whereas the asymptotic expression of the energy-density at late times (5.143)
reads

$\rho_{\tau_{0}}(\tau)=\int\frac{{\rm d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)n_{\rm B}(\tau_{0})+\int\frac{{\rm
d}^{2}{\rm p_{T}}\,{\rm
d}\mu}{(2\pi)^{3}\tau}\omega(\tau)2\sinh^{2}\Theta(\tau_{0})n_{\rm
B}(\tau_{0}).$ (5.150)

The first term corresponds to the classical free-streaming solution in Milne
coordinates, as shown in Appendix A. On the other hand, the second one is a
pure quantum correction due to vacuum effects, interpretation supported by the
fact that it vanishes only if $\Theta(\tau_{0})=0$. Interestingly enough, the
latter does not vanish at later times, in fact it can as well be comparable
with the former if the main argument of $\Theta(\tau_{0})$, namely $m_{\rm
T}\tau_{0}$, is of order ${\cal O}(1)$, that is for an early decoupling of the
system.

###  5.4 Summary and outlook

In this Chapter, we considered a relativistic quantum fluid with boost
invariance. This is of particular concern for the physics of the quark-gluon
plasma as boost invariance is approximately realized in the central-rapidity
region in heavy-ion collisions, although modern hydrodynamic calculations can
go beyond this model by including transverse expansion and, eventually, by
breaking boost invariance itself. A boost-invariant fluid is inherently out of
thermodynamic equilibrium, so the thermodynamic fields are in general unknown.
However, geometrical constraints descended from this particular symmetry that
provided us with enough information on the thermodynamic fields and other
quantities to at least attempt actual calculations.

After thoroughly presenting boost invariance and its properties, we considered
as a Quantum Field Theory underlying the hydrodynamic theory a free real
scalar field theory in the future light-cone. This had quite some differences
with respect to the case considered in Chapter 4 in the right Rindler wedge.
In the Rindler wedges, the quantum field was expanded in terms of Rindler
creation and annihilation operators, which were different from the usual
operators appearing in the plane waves expansion, and the density operator was
diagonal with respect to them. In particular, these two sets of operators were
related by a non-trivial Bogolyubov transformation, which, together with the
four-temperature having a Killing horizon, ultimately gave rise to the Unruh
effect. On the other hand, in the future light-cone the quantum field was
expanded in terms of creation and annihilation operators linearly related to
the plane waves ones, but the density operator was diagonalized by a new set
of operators related to the former by a non-trivial Bogolyubov transformation.
Moreover, the four-temperature had no Killing horizon for it was not a Killing
vector field. In fact, the future-light-cone is not a stationary spacetime,
therefore it does not possess any global timelike Killing vector field. In
view of these facts, there appeared to be no analog of the Unruh effect in the
future light-cone.

With a diagonal density operator, we were in a good position to calculate
thermal expectation values. First, we performed a local thermodynamic
equilibrium analysis. At local thermodynamic equilibrium, in fact, due to the
reconstruction of the Wronskian of the Hankel functions, thermal expectation
values of operators of physical interest, such as the energy-momentum tensor,
could be calculated in an exact analytical way. Somehow unexpectedly, the
ideal form of the energy-momentum tensor was obtained. Renormalization was
also discussed, pointing out peculiar vacuum effects: the subtraction of the
Minkowski vacuum contribution ended up in thermal expectation values which
were still infinite, at variance, for instance, with the case studied in
Chapter 4. The entropy current thereby obtained seemed not to contain any
apparent quantum correction, which was equally unexpected.

Finally, we performed a full non-equilibrium analysis. This time, however,
because of the mixing of terms calculated at different times, thermal
expectation values became non-trivial combinations of Hankel functions
evaluated at different arguments. Neither the Wronskian could be
reconstructed, nor we were able to carry out exact integration of such
combinations. In order to go on, we considered an asymptotic expansion which
simplified the expressions and eventually allowed us to obtain results.
Renormalization was also discussed in this limit, highlighting again the
interesting vacuum effects. The subtraction of the Minkowski vacuum
contribution still ended up in divergent thermal expectation values, therefore
we had to subtract the $|0_{\tau_{0}}\rangle$ contribution, with
$|0_{\tau_{0}}\rangle$ being the stationary vacuum of $\widehat{\xi}_{\sf
p}(\tau_{0})$ at fixed $\tau_{0}$. Interestingly, in this renormalization
scheme the energy density turned out to be the sum of two terms: the former
being the classical free-streaming solution in Milne coordinates, while the
second one being a pure quantum correction due to a vacuum term. No entropy
current or entropy production rate could be calculated since no formula is
currently known for such quantities in the case of a Quantum Field Theory
fully out of thermodynamic equilibrium.

## 6Conclusions

The leitmotif of this work is the determination of the entropy current of
relativistic quantum fluids at local thermodynamic equilibrium. The entropy
current is, in fact, a fundamental ingredient of relativistic hydrodynamics
which makes sure that the second law of thermodynamics is enforced in a
covariant language, and whose expression entails the constitutive equations of
the conserved currents in terms of the derivatives of the intensive
thermodynamic parameters. However, in the Israel-Stewart theory the structure
of the entropy current is essentially postulated rather than derived.

In this work, we proved that, from the point of view of the quantum
statistical foundations of relativistic hydrodynamics, a rigorous definition
of the entropy current can indeed be provided at local thermodynamic
equilibrium and a method to calculate it can be put forward. This was done by
using the Zubarev formalism of relativistic quantum statistical mechanics,
which fully embeds the quantum field theoretical nature of the system
underlying its hydrodynamic picture. As such, it is particularly suitable for
the description of strongly interacting fluids such as the quark-gluon plasma
produced in heavy-ion collisions, for which the relativistic kinetic
theoretical approach breaks down. It should be emphasized that our method
holds at local thermodynamic equilibrium at most, since no equation is
currently known for the entropy current or the entropy production of
relativistic quantum fluids fully out of thermodynamic equilibrium. In this
framework, we showed that a sufficient condition for the existence of the
entropy current at local thermodynamic equilibrium is that the quantum
operator appearing in the argument of the exponential function in the density
operator should be bounded from below with non-degenerate lowest eigenvalue.
In this case, the logarithm of the partition function was proven to be
extensive, meaning that it is expressed as the integral of a vector field,
called the thermodynamic potential current, on a 3-dimensional spacelike
hypersurface, a usually tacitly understood hypothesis. The expression of the
entropy current was thereby obtained in terms of the thermal expectation
values at local thermodynamic equilibrium of the conserved currents, namely
the energy-momentum tensor and some possible charged current quantum
operators.

The next step was the actual calculation of the entropy current for two
different kinds of fluids. This, however, should not be mistaken for a mere
test of our method, but rather it should be regarded as an opportunity for us
to study two systems interesting in their own right, especially in the context
of heavy-ion collisions, and use the framework of relativistic quantum
statistical mechanics to work out exact analytical results. Thus, this ought
to be considered as an equally important objective of this work.

To begin with, we explored the case of a relativistic quantum fluid at global
thermodynamic equilibrium with acceleration field of constant magnitude, a
non-trivial instance of global thermodynamic equilibrium in Minkowski
spacetime. We managed to work out the thermal expectation value of the energy-
momentum tensor of a free real scalar quantum field in the right Rindler wedge
in an exact analytical way, at least in the massless case. This revealed to
possess a longitudinal anisotropy in the pressure due to the acceleration
field, which was indeed a quantum term. The Unruh effect also emerged, not
surprisingly, thanks to the combination of two major points. The first one was
the density operator being diagonal in the same set of creation and
annihilation operators appearing in the quantum field expansion, namely the
Rindler ones, and these were related to the familiar plane waves set by a non-
trivial Bogolyubov transformation. The second one being the fact that the
four-temperature possessed a Killing horizon. Renormalization was then
discussed by pointing out the differences between the two natural schemes at
hand, namely the subtraction of either the Minkowski vacuum contribution or
the Rindler vacuum one. After that, we calculated the entropy current by
employing our method: the result revealed to be divergenceless, in agreement
with the general condition of global thermodynamic equilibrium, and also to
possess a quantum term proportional to the acceleration field. A non-vanishing
entropy current was found in the Minkowski vacuum, probably owing to having
traced out the field degrees of freedom in the left Rindler wedge. The entropy
thereby obtained by integration was in full agreement with previously known
literature and it turned out to coincide with the entanglement entropy of the
right Rindler wedge with the left Rindler wedge by virtue of the factorization
of the density operator. The calculation of the entanglement entropy in
Quantum Field Theory is a notoriously hard task to accomplish: only few exact
calculations are present in literature and most of them heavily rely on either
the 2-dimensional conformal symmetry or the AdS/CFT correspondence. Having
been able to perform it in an exact analytical way by using a method
completely different from the standard ones for a system of physical concern
adds all the more interest to our result. The entropy current certainly
depends on the specific form of the energy-momentum tensor quantum operator,
whereas the entropy should not. The above findings were worked out by using
the canonical tensor, corresponding to minimal coupling, but we also compared
them with those we would have obtained with an “improved” tensor, which is
traceless for a massless field and corresponding to conformal coupling. The
entropy current did indeed change, in particular the quantum term proportional
to the acceleration field disappeared. However, the entropy seemed to change
as well by a constant factor, which was somewhat unexpected. This will be
subject for further studies.

Finally, we examined the case of a relativistic quantum fluid with boost
invariance in the future light-cone, which is inherently out of thermodynamic
equilibrium. Very little information, if any, is usually known about the
thermodynamic fields for non-equilibrium systems, making them remarkably hard
to study. Nevertheless, we showed that boost invariance implied special
constraints providing us with enough information to at least attempt actual
calculations. After a thorough presentation of boost invariance, we considered
free real scalar field theory in the future light-cone. Interestingly, the
quantum field was expanded in a set of creation and annihilation operators
related to the familiar ones of the plane waves expansion by a linear
transformation, therefore their vacuum did coincide with the Minkowski vacuum.
On the other hand, the density operator was diagonalized in a different set of
creation and annihilation operators which were related to the previous ones by
a non-trivial and time-dependent Bogolyubov transformation, so its time-
dependent vacuum was different from the Minkowski vacuum. Because of these
differences with respect to the previously analyzed system and because of the
four-temperature not being a Killing vector field, whence not possessing any
Killing horizon, no analog of the Unruh effect emerged in the future light-
cone. With a diagonalized density operator, we worked out the thermal
expectation value of the canonical energy-momentum tensor at local
thermodynamic equilibrium in an exact analytical way. The result proved to
possess the well-known ideal form with the transverse and longitudinal
pressures being equal to each other, which was somewhat unexpected.
Renormalization was then discussed. Curiously, the subtraction of the
Minkowski vacuum contribution ended up giving thermal expectation values which
were still divergent, therefore it had to be discarded. The remaining natural
choice was the subtraction of the contribution corresponding to the time-
dependent vacuum of the density operator, which gave indeed finite quantities.
This gave us the opportunity to use our method to calculate the entropy
current at local thermodynamic equilibrium. Surprisingly, the result coincided
with the classical free-streaming one in Milne coordinates, without any
apparent quantum correction.Finally, we also performed an analysis fully out
of thermodynamic equilibrium. However, because of non-trivial combinations of
Hankel functions evaluated at different arguments, we were forced to introduce
some approximation to eventually carry out calculations. In particular, we
considered the asymptotic behaviour at late times. Renormalization was the
discussed in this limit highlighting interesting vacuum effects. The
subtraction of the Minkowski vacuum contribution still gave us divergent
quantities, so we subtracted the contribution corresponding to the vacuum of
the non-equilibrium density operator, which was stationary by definition.
Interestingly, the energy density thereby obtained was given by the sum of two
terms, the one being the classical free-streaming one in Milne coordinates,
while the other one being a pure quantum correction due to the vacuum. Neither
the entropy current nor the entropy production rate could be calculated since
no formula is currently known for a Quantum Field Theory fully out of
thermodynamic equilibrium.

##  Appendix A Free-streaming in Milne coordinates

The free-streaming equation in classical relativistic kinetic theory reads

$p^{\mu}\partial_{\mu}f(x,{\bf p})=0,$ (A.1)

where $p^{\mu}$ is the four-momentum and $f$ the distribution function. Its
explicit solution in Cartesian coordinates is

$f(x,{\bf p})=f_{0}\left({\bf x}-\frac{t-t_{0}}{\varepsilon}{\bf p},{\bf
p}\right),$ (A.2)

wherein $f_{0}({\bf x},{\bf p})=f(t_{0},{\bf x};{\bf p})$ is the initial
condition in a generic inertial reference frame, and
$\varepsilon=\sqrt{m^{2}+p^{2}}$ is the (on-shell) energy.

In the the longitudinal boost-invariant symmetry, the initial condition is
given at some Milne time $\tau_{0}$ rather than the time $t_{0}$ in Cartesian
coordinates. Nevertheless, there is a very simple solution in this case too.
Since the distribution function is a scalar, it must be invariant under the
symmetry transformations at stake, that are longitudinal boosts and rotations
and translations in the transverse plane. Hence, it depends only on the
independent scalars that may be formed with combinations of the space and time
coordinates and the momentum vector which are invariant under the group of
transformations ${\rm IO}(2)\otimes{\rm SO}(1,1)$. These scalars are

$\tau=\sqrt{t^{2}-z^{2}},\qquad{\rm p_{T}}=\sqrt{p_{x}^{2}+p_{y}^{2}},\qquad
w=z\varepsilon-tp_{z}.$ (A.3)

The last variable can be shown to be equivalent to the covariant component
$p_{\eta}$ of the the four-momentum vector in Milne coordinates. Indeed, there
is a fourth invariant scalar

$v=t\varepsilon-zp_{z}=\tau\sqrt{m^{2}+{\rm p}_{\rm
T}^{2}+\frac{w^{2}}{\tau^{2}}},$ (A.4)

but it is redundant because of the on-shell condition (and positivity) of the
energy and because $t>|z|$ in the future light cone. The invariance under
space reflections $(x,y,z)\mapsto(-x,-y,-z)$ makes $f$ depending on the square
of $w$ rather than $w$ itself. By plugging these arguments into equation
(A.1), we obtain

$\frac{v}{\tau}\frac{\partial}{\partial\tau}f(\tau,{\rm p_{T}},w^{2})=0,$
(A.5)

since the contribution in the partial derivatives with respect to $w$ cancels
out. The free-streaming solution is then very simple, a constant in $\tau$:

$f(\tau,{\rm p_{T}},w^{2})=f(\tau_{0},{\rm p_{T}},w)\equiv f_{0}({\rm
p_{T}},w^{2}).$ (A.6)

We are now in a position to calculate the free-streaming solution for the
energy-momentum tensor from its classical kinetic definition

$T^{\mu\nu}=\int\frac{{\rm d}^{3}{\rm
p}}{(2\pi)^{3}\varepsilon}\,p^{\mu}p^{\nu}\,f,$ (A.7)

where the energy density, the transverse pressure and the longitudinal
pressure read

$\displaystyle\rho=$ $\displaystyle T^{\mu\nu}u_{\mu}u_{\nu}=\int\frac{{\rm
d}^{3}{\rm p}}{(2\pi)^{3}\varepsilon}\frac{v^{2}}{\tau^{2}}f,$ (A.8a)
$\displaystyle p^{\rm T}=$ $\displaystyle
T^{\mu\nu}\frac{1}{2}\left(\hat{i}_{\mu}\hat{i}_{\nu}+\hat{j}_{\mu}\hat{j}_{\nu}\right)=\int\frac{{\rm
d}^{3}{\rm p}}{(2\pi)^{3}\varepsilon}\frac{{\rm p}_{\rm T}^{2}}{2}f,$ (A.8b)
$\displaystyle p^{\rm L}=$ $\displaystyle
T^{\mu\nu}\hat{\eta}_{\mu}\hat{\eta}_{\nu}=\int\frac{{\rm d}^{3}{\rm
p}}{(2\pi)^{3}\varepsilon}\frac{w^{2}}{\tau^{2}}f.$ (A.8c)

By changing the integration variable as

$w=z\varepsilon-tp_{z}\qquad\Rightarrow\qquad{\rm
d}w=\left|-\frac{v}{\varepsilon}\right|{\rm
d}p_{z}\qquad\Rightarrow\qquad\frac{{\rm d}p_{z}}{\varepsilon}=\frac{{\rm
d}w}{v},$ (A.9)

we obtain

$\rho=\int\frac{{\rm d}^{2}{\rm p}_{\rm T}}{(2\pi)^{3}}\frac{{\rm
d}w}{v}\,\frac{v^{2}}{\tau^{2}}\,f_{0}({\rm p_{T}},w^{2})=\int\frac{{\rm
d}^{2}{\rm p}_{\rm T}\,{\rm d}w}{(2\pi)^{3}\tau}\,\sqrt{m_{\rm
T}^{2}+\frac{w^{2}}{\tau^{2}}}\,f_{0}({\rm p_{T}},w^{2}),$ (A.10) $p^{\rm
T}=\int\frac{{\rm d}^{2}{\rm p}_{\rm T}}{(2\pi)^{3}}\frac{{\rm
d}w}{v}\,\frac{{\rm p}_{\rm T}^{2}}{2}\,f_{0}({\rm
p_{T}},w^{2})=\int\frac{{\rm d}^{2}{\rm p}_{\rm T}\,{\rm
d}w}{(2\pi)^{3}\tau\sqrt{m_{\rm T}^{2}+\frac{w^{2}}{\tau^{2}}}}\,\frac{{\rm
p}_{\rm T}^{2}}{2}\,f_{0}({\rm p_{T}},w^{2}),$ (A.11) $p^{\rm
L}=\int\frac{{\rm d}^{2}{\rm p}_{\rm T}}{(2\pi)^{3}}\frac{{\rm
d}w}{v}\,\frac{w^{2}}{\tau^{2}}\,f_{0}({\rm p_{T}},w^{2})=\int\frac{{\rm
d}^{2}{\rm p}_{\rm T}\,{\rm d}w}{(2\pi)^{3}\tau\sqrt{m_{\rm
T}^{2}+\frac{w^{2}}{\tau^{2}}}}\,\frac{w^{2}}{\tau^{2}}\,f_{0}({\rm
p_{T}},w^{2}).$ (A.12)

The change of variable introduces in the integral an explicit dependence on
the proper time. Equations (A.10)–(A.12) are the classical relativistic
expressions of the energy density and pressures of a free-streaming gas and
coincide with the leading terms obtained in Subsection 5.3.1 with the
substitution $w\mapsto\mu$ and with the initial distribution equal to the
Bose-Einstein local equilibrium distribution function $f_{0}=n_{\rm B}^{0}$.

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