$\displaystyle=\Delta_{H}^{-1/2}(\Delta_{H}^{-1/2}-\Delta_{H}^{1/2})^{-1}h-(\Delta_{H}^{-1/2}-\Delta_{H}^{1/2})^{-1}J_{H}h$
$\displaystyle=\Delta_{H}^{-1/2}(\Delta_{H}^{-1/2}-\Delta_{H}^{1/2})^{-1}h-(\Delta_{H}^{-1/2}-\Delta_{H}^{1/2})^{-1}\Delta_{H}^{1/2}h$
$\displaystyle=h$

and:

$\displaystyle\Delta_{H}^{-1/2}(\Delta_{H}^{-1/2}$
$\displaystyle-\Delta_{H}^{1/2})^{-1}h^{\prime}+J_{H}(\Delta_{H}^{-1/2}-\Delta_{H}^{1/2})^{-1}h^{\prime}$
$\displaystyle=\Delta_{H}^{-1/2}(\Delta_{H}^{-1/2}-\Delta_{H}^{1/2})^{-1}h^{\prime}-(\Delta_{H}^{-1/2}-\Delta_{H}^{1/2})^{-1}J_{H}h^{\prime}$
$\displaystyle=\Delta_{H}^{-1/2}(\Delta_{H}^{-1/2}-\Delta_{H}^{1/2})^{-1}h^{\prime}-(\Delta_{H}^{-1/2}-\Delta_{H}^{1/2})^{-1}\Delta_{H}^{-1/2}h$
$\displaystyle=0.$

This proves that the sum of these two operators is $P_{H}$.
In the general case, we consider the orthogonal decomposition
$\mathcal{H}=\mathcal{H}_{\epsilon}\oplus\mathcal{K}_{\epsilon}$, where
$\mathcal{K}_{\epsilon}=\mathcal{H}_{\epsilon}^{\bot}$ is the complementary
spectral subspace of $\Delta_{H}$. We also have the corresponding
decomposition $H=H_{\epsilon}\oplus K_{\epsilon}$. As
$0,1\not\in\mathrm{sp}(\Delta_{H}|_{\mathcal{H}_{\epsilon}})$ (because of our
construction of $\mathcal{H}_{\epsilon}$ with $\epsilon>0$ which does not
include $0,1$ in the spectrum), we conclude that $\mathcal{D}_{0}\subset
H+H^{\prime}$ and thus Eq. $\eqref{eq: projection}$ holds true with
$\mathcal{D}_{0}$ instead of $\mathcal{D}$ by following the same proof that we
used above.
Let $k\in\mathcal{D}$ and call
$k_{\epsilon}=E_{\epsilon}k\in\mathcal{H}_{\epsilon}$. As $\epsilon\to 0$ we
have $k_{\epsilon}\to k$ and:

$\displaystyle a(\Delta_{H})k_{\epsilon}$ $\displaystyle\to a(\Delta_{H})k$
$\displaystyle b(\Delta_{H})k_{\epsilon}$ $\displaystyle\to b(\Delta_{H})k.$

If we apply Eq. $\eqref{eq: projection}$ to $k_{\epsilon}$ we have that the
limiting element $P_{H}k_{\epsilon}\to P_{H}k$ must be well defined as a
consequence of $P_{H}$ being closed. So,
$k\in\mathrm{Dom}(P_{H})=H+H^{\prime}$ and:

$P_{H}k=a(\Delta_{H})k+J_{H}b(\Delta_{H})k,$

which shows Eq. (2.4) holds on $H+H^{\prime}$.
Let now $k\in H+H^{\prime}$. As $P_{H}$ commutes with $E_{\epsilon}$ as well,
we have that $k_{\epsilon}\to k$ and
$P_{H}k_{\epsilon}=P_{H}E_{\epsilon}k=E_{\epsilon}P_{H}k\to P_{H}k$. This
shows that $\mathcal{D}_{0}=\mathcal{K}_{\epsilon}$ is a core for $P_{H}$ as,
by taking the closure, we include the limiting elements $k$ obtaining $P_{H}$:
$\overline{P_{H}|_{\mathcal{D}_{0}}}=P_{H}$. ∎

The functions $a(\lambda),b(\lambda)$ are continuous functions on
$(0,+\infty)$ with a singularity at $\lambda=1$ as they are defined in terms
of $(\lambda^{-1/2}-\lambda^{1/2})^{-1}$, namely
$a(\lambda),b(\lambda)=o(\lambda^{-1})$ as $\lambda\to 1$.

We can now define the notion of entropy for a vector:

######  Definition 2.2.7.

Let $H$ be a standard subspace of $\mathcal{H}$. If $h\in H$, the entropy
$S_{h}$ of $h$ w.r.t. $H$ is defined as:

$S_{h}=-(h,\log\Delta_{H}h).$

We now want to generalize this to arbitrary vectors of $\mathcal{H}$. In order
to do so, let us first represent the modular flow in terms of its spectral
decomposition:

$\Delta_{H}=\int_{0}^{\infty}\lambda dE(\lambda).$

Then, we can generalize the above to:

######  Definition 2.2.8.

If $k\in\mathcal{H}$, we define the entropy $S_{k}$ of the vector $k$ with
respect to $H$ by:

$S_{k}=\imaginary((k,P_{H}Ak)\big{)}$

For $A=i\log\Delta_{H}$.

Indeed, by introducing the spectral decomposition of the modular flow and the
formula for $P_{H}$ we see that this is well defined for all
$k\in\mathcal{H}$:

$(k,P_{H}Ak)=i\int_{0}^{\infty}a(\lambda)\log\lambda
d(k,E(\lambda)k)-i\int_{0}^{\infty}b(\lambda)\log\lambda
d(k,J_{H}E(\lambda)k).$

Notice that $b(\lambda)\log\lambda$ is always a well defined bounded operator
on $(0,+\infty)$ as the singularity in $\lambda\to 1$ is removed:

$\displaystyle\lim_{\lambda\to
1}\frac{\log\lambda}{(\lambda^{-1/2}-\lambda^{1/2})}$
$\displaystyle=\lim_{\lambda\to
1}\frac{\log(1+(\lambda-1))}{((1+(\lambda-1))^{-1/2}-(1+(\lambda-1))^{1/2})}$
$\displaystyle\simeq\frac{\lambda-1}{1-\frac{\lambda-1}{2}-1-\frac{\lambda-1}{2}}$
$\displaystyle\simeq-1.$

Secondly, we know that $a(\lambda)\log\lambda$ is bounded in $(1,+\infty)$ as
well and positive in $(0,1)$. Therefore, the entropy is finite if
$-\int_{0}^{1}a(\lambda)\log\lambda d(k,E(\lambda)k)<\infty$, otherwise
$S_{k}=+\infty$. Indeed:

######  Proposition 2.2.5.

Let $k\in\mathcal{H}$. We have:

$S_{k}<+\infty\Leftrightarrow-\int_{0}^{1}\log\lambda\,\,d(k,E(\lambda)k)<+\infty,$
(2.5)

iff $k\in\mathrm{Dom}(\sqrt{|\log\Delta_{H}|}E_{-})$, with $E_{-}$ the
negative spectral projection for $\log\Delta_{H}$.
In particular, all vectors in $\mathrm{Dom}(\log\Delta_{H})$ have finite
entropy.

###### Proof.

Notice that:

$\lim_{\lambda\to 1^{-}}a(\lambda)\log\lambda\sim-1-\frac{\lambda-1}{2}$

with the same limit being finite for $\log\lambda$ as well. Moreover,
$a(\lambda)\to 1$ as $\lambda\to 0^{+}$. ∎

Finally, we prove the main properties of the entropy for vectors:

######  Proposition 2.2.6.

Let $H$ be a factorial standard subspace of $\mathcal{H}$ and
$k\in\mathcal{H}$. The following holds:

  * i)

If $k=h+h^{\prime}$, with $h\in H$, $h^{\prime}\in H^{\prime}$, then
$S_{k}=-(h,\log\Delta_{H}h)$

  * ii)

$S_{k}=(k,iP_{H}i\log\Delta_{H}k)=-(k,P^{*}_{H}\log\Delta_{H}k)$

  * iii)

$S_{k}\geq 0$ and $S_{k}=0$ iff $k\in H^{\prime}$

  * iv)

If $\mathcal{H}=\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ with $H=H_{1}\oplus
H_{2}$, and $k=k_{1}\oplus k_{2}$, then
$S^{H}_{k}=S^{H_{1}}_{k_{1}}+S^{H_{2}}_{k_{2}}$, in particular for $k=k_{1}$:
$S^{H}_{k_{1}}=S^{H_{1}}_{k_{1}}$

  * v)

$S_{k}=\lim_{\epsilon\to 0^{+}}S_{k_{\epsilon}}$; where
$S_{k_{\epsilon}}<+\infty$ and $S_{k_{\epsilon}}$ is non-decreasing as
$\epsilon\to 0^{+}$

  * vi)

If $k_{n}\to k$ in the graph norm of $\sqrt{|\log\Delta_{H}|}E_{-}$, then
$S_{k_{n}}\to S_{k}$

###### Proof.

Let us start proving $i)$. If $k\in\mathrm{Dom}(\log\Delta_{H})$ or
$k\in\mathrm{Dom}(P_{H})$, we have $k=h+h^{\prime}$, with $h\in H$ and
$h^{\prime}\in H^{\prime}$ and both
$h,h^{\prime}\in\mathrm{Dom}(\log\Delta_{H})$. Then:

$\displaystyle S_{k}$ $\displaystyle=\imaginary((k,P_{H}Ak)\big{)}$
$\displaystyle=\imaginary((h+h^{\prime},P_{H}A(h+h^{\prime}))\big{)}$
$\displaystyle=\imaginary((h+h^{\prime},Ah)\big{)}$
$\displaystyle=\imaginary((h,Ah)\big{)}$ $\displaystyle=-(h,\log\Delta_{H}h).$

We then have:

$S_{k}\geq 0.$

Indeed, by assuming $\|h\|=1$, we find:

$(h,\log\Delta_{H}\,\,h)=\int\log(\lambda)d(h,E(\lambda)h)\leq-\log\bigg(\int\lambda
d(h,E(\lambda)h)\bigg{missing})=\log\|h\|^{2}=0,$

where we have used Jensen’s inequality.
Statement $iv)$, follows from the very definition of entropy.
Result $v)$ is proven noticing that $S_{k_{\epsilon}}$ is finite, as the
restriction of the cutting projection and of $\log\Delta_{H}$ on
$\mathcal{H}_{\epsilon}$ with them being bounded operators, and thus by the
spectral theorem $S_{k_{\epsilon}}\to S_{k}$ as $\epsilon\to 0^{+}$. Moreover,
we have proven above that $S_{k_{\epsilon}}\geq 0$. Thus, by statement $iv)$,
we see that $S_{k_{\epsilon}}$ increases, in converging to $S_{k}$, as
$\mathcal{H}=\mathcal{H}_{\epsilon}\oplus\mathcal{K}_{\epsilon}$ and
$k_{\epsilon}\in\mathcal{H}_{\epsilon}$, which becomes larger by reducing
$\epsilon$
Let us now look at $iii)$. By $v)$, and the positivity of each
$S_{k_{\epsilon}}$, we also have $S_{k}\geq 0$. If, on the other hand, we
suppose $S_{k}=0$, by the monotone increase proven in $v)$, we must also have
$S_{k_{\epsilon}}=0$. But now, by $i)$ and the strict positivity in Def.
2.2.7, we have that $k_{\epsilon}\in H^{\prime}$ and therefore $k\in
H^{\prime}$.
The general case in $i)$ subsequently follows, as we have
$S_{k}=\lim_{\epsilon\to 0}S_{k_{\epsilon}}$, and using that
$\mathrm{Dom}(P_{H})$ is a core for $P_{H}$.
For $ii)$, start noticing that the first equality simply represents the
statement that $(k,iP_{H}i\log\Delta_{H}k)$ is real if finite. Now,
remembering that $A=i\log\Delta_{H}$, we have that
$(k_{\epsilon},iP_{H}i\log\Delta_{H}k_{\epsilon})$ is real from the identity
we have proven at the beginning:

$\imaginary((k_{\epsilon},P_{H}Ak_{\epsilon})\big{)}=\imaginary((k_{\epsilon},P_{H}i\log\Delta_{H}k_{\epsilon})\big{)}=-(h_{\epsilon},\log\Delta_{H}h_{\epsilon}),$

and from the simple result:

$\real((k_{\epsilon},iP_{H}i\log\Delta_{H}k_{\epsilon})\big{)}=\imaginary((k_{\epsilon},P_{H}i\log\Delta_{H}k_{\epsilon})\big{)},$

in conjunction with statement $v)$, we conclude that also $ii)$ holds.
Moreover, we have seen that $P_{H}^{*}=-iP_{H}i$, so the result follows.
∎

We now have everything that is needed to introduce second quantization for
bosonic scalar free fields and the notion of coherent excitations.
In particular, if $\mathcal{H}$ is a complex Hilbert space, we can construct
(namely making it the one-particle Hilbert space) the Fock space on it as:

$\mathfrak{S}(\mathcal{H})=\bigoplus_{n=0}^{\infty}\mathcal{H}_{s}^{\otimes^{n}},$

which for this reson is also called the exponential of $\mathcal{H}$, often
denoted as $e^{\mathcal{H}}$. Moreover, $\mathcal{H}_{0}=\mathbb{C}\Omega$ is
the one-dimensional Hilbert space for the vacuum vector $\Omega$ with
$\mathcal{H}_{s}^{\otimes n}$ being the symmetrized $n$-fold tensor product of
$\mathcal{H}$. Now, if we have $h\in\mathcal{H}$, we can define the coherent
vector $e^{h}$ as:

$e^{h}:=\bigoplus_{n=0}^{\infty}\frac{1}{\sqrt{n!}}(h^{\otimes^{n}})_{s},$

where the zeroth component of $e^{h}$ is given by $e^{0}=\Omega$. We can prove
that:

$\displaystyle(e^{h},e^{k})$
$\displaystyle=\bigg{(}\bigoplus_{n=0}^{\infty}\frac{1}{\sqrt{n!}}(h^{\otimes^{n}})_{s},\bigoplus_{n=0}^{\infty}\frac{1}{\sqrt{n!}}(k^{\otimes^{n}})_{s}\bigg{)}$
$\displaystyle=\bigoplus_{n=0}^{\infty}\frac{1}{n!}\big{(}(h^{\otimes^{n}})_{s},(k^{\otimes^{n}})_{s}\big{)}$
$\displaystyle=\bigoplus_{n=0}^{\infty}\frac{1}{\sqrt{n}!}\big{(}h^{\otimes^{n}},k^{\otimes^{n}}\big{)}_{s}$
$\displaystyle=e^{(h,k)}$

and $\\{e^{h},h\in\mathcal{H}\\}$ is a dense subset of $\Gamma(\mathcal{H})$.
For $h\in\mathcal{H}$, we can identify a corresponding Weyl algebra (see
1.1.1.1) generated by $V(h)$:

$V(h)\frac{e^{k}}{\|e^{k}\|}=\frac{e^{h+k}}{\|e^{h+k}\|},$

satisfying the Weyl commutation relations:

$V(h+k)=e^{i\imaginary((h,k)\big{)}}V(h)V(k).$

In particular:

$\displaystyle V(h)\ket{\Omega}$ $\displaystyle=V(h)e^{0}$
$\displaystyle=\frac{e^{h}}{\|e^{h}\|},$

and since:

$\|e^{h}\|=\sqrt{(e^{h},e^{h})}=\sqrt{e^{(h,h)}}=e^{\frac{1}{2}(h,h)},$

we have:

$V(h)\ket{\Omega}=e^{-\frac{1}{2}(h,h)}e^{h}.$

Therefore:

$(V(k)\Omega,V(h)\Omega)=e^{-\frac{1}{2}(\|h\|^{2}+\|k\|^{2})}(e^{k},e^{h})=e^{-\frac{1}{2}(\|h\|^{2}+\|k\|^{2})}e^{(k,h)}$

and if we denote by $\omega:=(\Omega,\cdot\Omega)$ the vacuum state, we
obtain:

$\omega(V(h))=e^{-\frac{1}{2}\|h\|^{2}}.$

Let $H\subset\mathcal{H}$ be a real linear subspace. Then we can define the
von Neumann algebra on $\Gamma(\mathcal{H})$:

$R(H):=\\{V(h):h\in H\\}^{\prime\prime}.$

If we further assume that $H$ is a standard subspace, we have:

######  Proposition 2.2.7.

Let $H$ be a standard, real, linear subspace, then:

  * •

If $K\subset H$ is dense, then $R(K)=R(H)$

  * •

$\Omega$ is cyclic and separating for $R(H)$

  * •

$R(H^{\prime})=R(H)^{\prime}$

  * •

The modular flow and conjugation associated with $(R(H),\Omega)$ are given by:

$\Delta^{it}_{R(H)}=\mathfrak{S}(\Delta^{it}_{H})\hskip
20.0ptJ_{R(H)}=\mathfrak{S}(J_{H})$

Where $\mathfrak{S}(\cdot)$ denotes the second quantization of the operator
acting on the one-particle Hilbert space.

###### Proof.

Let us start from the first. As $K\subset H$ we must have $R(K)\subset R(H)$.
Now, as $K$ is dense in $H$, for each $h\in H$ we can find a sequence
$(h_{n})_{n\in\mathbb{N}}\in K$ such that $h_{n}\to h$ in $H$. I claim, that
this implies $V(h_{n})\to V(h)$ in the weak topology. For that purpose,
consider the action of $V(h_{n})\in R(K)$ on a general element
$\Psi\in\mathfrak{S}(\mathcal{H})$ that, w.l.o.g., is assumed to be
normalized. We know that the set of normalized independent vectors
$e^{k}/\|e^{k}\|$ for $k\in\mathcal{H}$ is dense, so we can always find
$e^{k_{l}}/\|e^{k_{l}}\|\to\Psi$. Then:

$\displaystyle\lim_{n\to\infty}V(h_{n})\Psi$
$\displaystyle=\lim_{n\to\infty}V(h_{n})\lim_{l\to\infty}\frac{e^{k_{l}}}{\|e^{k_{l}}\|}$
$\displaystyle=\lim_{n\to\infty}\lim_{l\to\infty}\frac{e^{h_{n}+k_{l}}}{\|e^{h_{n}+k_{l}}\|}$
$\displaystyle=\lim_{l\to\infty}\frac{e^{h+k_{l}}}{\|e^{h+k_{l}}\|}$
$\displaystyle=V(h)\lim_{l\to\infty}\frac{e^{k_{l}}}{\|e^{k_{l}}\|}$
$\displaystyle=V(h)\Psi,$

proving the convergence in the weak topology of $V(h_{n})\in R(K)$ to $V(h)\in
R(H)$. But now, from Theorem 1.1.1 it follows that $\\{V(h):h\in
H\\}^{\prime\prime}=\overline{\\{V(h):h\in H\\}}$, with closure taken with
respect to the weak topology, that is $R(K)=R(H)$.
The second statement is simply the Reeh-Schlieder theorem
For the third statement, notice that as $\Omega$ is cyclic and separating for
$R(H)$ and by Reeh-Schlieder theorem it must be for $R(H)^{\prime}$ as well:

$\overline{R(H^{\prime})\ket{\Omega}}=\mathfrak{S}(\mathcal{H}^{\prime})=\mathfrak{S}(J_{H})\mathfrak{S}(\mathcal{H})=J_{R(H)}\overline{R(H)\ket{\Omega}}=\overline{J_{R(H)}R(H)J_{R(H)}\ket{\Omega}}=\overline{R(H)^{\prime}\ket{\Omega}},$

where we have used the fourth statement $J_{R(H)}=\mathfrak{S}(J_{H})$ and the
one for factorial standard subspaces, $J_{H}=J_{H^{\prime}}$, see Lemma 2.2.3.
The last statement follows from the very definition of second quantization of
a unitary operator on $\mathcal{H}$. Namely, if $U$ is a unitary operator on
$\mathcal{H}$, its second quantization $\mathfrak{S}(U)$ is defined as the
direct sum of:

$\mathfrak{S}(U)|_{\mathcal{H}_{s}^{\otimes^{n}}}=U\otimes
U\otimes\dots\otimes U.$

From this definition, it follows that $\mathfrak{S}(U)e^{h}=e^{Uh}$. Finally,
since $J_{H}$ and $\Delta_{H}^{it}$ are unitary and since for
$h\in\mathcal{H}$, we have:

$e^{h}=\|e^{h}\|V(h)\ket{\Omega},$

from which follows the claim. ∎

Let us now prove the analogy between relative entropy for coherent excitations
of the vacuum, of a free real scalar QFT, and the entropy of a vector of the
corresponding one particle Hilbert space. We will first prove the result for
$h\in H$ and subsequently generalize it for arbitrary $h\in\mathcal{H}$.

#####  2.2.5.1 Case $h\in H$

We saw that by taking $h\in H$ to be a real, linear, standard subspace of
$\mathcal{H}$, the entropy of $h$, relative to $H$ is:

$S_{h}=-(h,\log\Delta_{H}h).$

For what concerns the relative entropy on the Fock space, we start by
specifying a coherent state on $R(H)$: $\omega_{h}=(V(h)\Omega,\cdot
V(h)\Omega)$, for some $h\in\mathcal{H}$. We notice that:

$\omega_{h}=\omega\cdot\mathrm{Ad}V(h)^{*}|_{R(H)},$

where, for $A\in R(H)$:

$\mathrm{Ad}V(h)|_{R(H)}[A]:=V(h)AV(h)^{*}.$

We first focus on the case in which $h,k\in H$ and we study
$S(\omega_{h}\|\omega_{k})$.
Let us start proving the following important general results, that will turn
out to also be crucial for later purposes:

######  Proposition 2.2.8.

Let $\mathcal{A}$ be a von Neumann algebra and denote by
$\mathcal{A}^{\prime}$ its commutant. Let $\Omega$ be a cyclic separating
vector for $\mathcal{A}$ and $U\in\mathcal{A}$,
$U^{\prime}\in\mathcal{A}^{\prime}$ unitary operators. Then, the vector
$\Phi=U^{\prime}U\Omega$ is cyclic and separating and we further have:

$S_{\Omega,\Phi}=US_{\Omega}U^{\prime*}.$

Additionally, by polar decomposition, we find:

$\displaystyle J_{\Omega,\Phi}$ $\displaystyle=UJ_{\Omega}U^{\prime*}$
$\displaystyle\Delta_{\Omega,\Phi}$
$\displaystyle=U^{\prime}\Delta_{\Omega}U^{\prime*}.$

###### Proof.

We have:

$\displaystyle\overline{\mathcal{A}\ket{\Phi}}$
$\displaystyle=\overline{\mathcal{A}U^{\prime}U\ket{\Omega}}$
$\displaystyle=U^{\prime}\overline{\mathcal{A}U\ket{\Omega}}$
$\displaystyle=U^{\prime}\overline{\mathcal{A}\ket{\Omega}}$
$\displaystyle=\overline{\mathcal{A}\ket{\Omega}}=\mathcal{H},$

where in the third step we have used the fact that $U\in\mathcal{A}$. One can
prove the same for $\mathcal{A}^{\prime}$, obtaining that as $\ket{\Phi}$ is
cyclic also for the commutant algebra, it must be separating for the algebra
itself. This was proven in Prop. 2.2.1.
For any $A\in\mathcal{A}$ we have:

$\displaystyle(US_{\Omega}U^{\prime*})A\ket{\Phi}$
$\displaystyle=(US_{\Omega}U^{\prime*})AU^{\prime}U\ket{\Omega}$
$\displaystyle=US_{\Omega}U^{\prime*}AU^{\prime}U\ket{\Omega}$
$\displaystyle=US_{\Omega}AU\ket{\Omega}$
$\displaystyle=UU^{*}A^{*}\ket{\Omega}$ $\displaystyle=A^{*}\ket{\Omega}$
$\displaystyle=S_{\Omega,\Phi}A\ket{\Phi},$

where we have used $U^{\prime}\in\mathcal{A}^{\prime}$. From this, it follows
by polar decomposition that:

$J_{\Omega,\Phi}\Delta^{1/2}_{\Omega,\Phi}=UJ_{\Omega}U^{\prime*}U^{\prime}\Delta^{1/2}_{\Omega}U^{\prime*},$

from which we have:

$\displaystyle J_{\Omega,\Phi}$ $\displaystyle=UJ_{\Omega}U^{\prime*}$
$\displaystyle\Delta_{\Omega,\Phi}$
$\displaystyle=U^{\prime}\Delta_{\Omega}U^{\prime*}.$

∎

######  Remark 2.2.5.

Observe that we can take $U^{\prime}=\mathbb{1}$ and in this way obtain that
any $U\ket{\Omega}$ is cyclic and separating whenever $\ket{\Omega}$ is.

From this proposition, in the particular case in which we consider $U=V(h)\in
R(H)$ and $U^{\prime}=\mathrm{Ad}J_{\Omega}[V(h)]$, we have that:

$J_{\Omega,\Phi}=V(h)J_{\Omega}J_{\Omega}V^{*}(h)J_{\Omega}=V(h)V(-h)J_{\Omega}=J_{\Omega}.$

For what concerns the relative entropy between coherent states, we may cover
all possibilities, by computing the relative entropy between a single coherent
excitation and the vacuum vector:

######  Proposition 2.2.9.

The relative entropy between two coherent states can be computed to be:

$S(\omega_{h}\|\omega_{k})=S(\omega_{k-h}\|\omega),$

where $\omega$ denotes the vacuum state.

###### Proof.

Let us consider $\ket{\Omega},V(h)\ket{\Omega},V(k)\ket{\Omega}$ and denote by
$S_{k,h}$ the associated Tomita operator (well defined as $V(h)\ket{\Omega}$
is cyclic and separating for all $h\in H$ as long as $\ket{\Omega}$ is). Now,
for $U\in R(H)$ unitary and for any $A\in R(H)$:

$\displaystyle(US_{\Omega,\Phi}U^{*})AU\ket{\Phi}$
$\displaystyle=US_{\Omega,\Phi}(U^{*}AU)\ket{\Phi}$
$\displaystyle=U(U^{*}AU)^{*}\ket{\Omega}$
$\displaystyle=UU^{*}A^{*}U\ket{\Omega}$ $\displaystyle=A^{*}U\ket{\Omega},$

from which follows:

$US_{\Omega,\Phi}U^{*}=S_{U\Omega,U\Phi}\Rightarrow\Delta_{U\Omega,U\Phi}=U\Delta_{\Omega,\Phi}U^{*}.$
(2.6)

Then, however, if we take as cyclic and separating states $\ket{\Omega}$ and:

$\ket{\Phi}:=V(k)^{*}V(h)\ket{\Omega},$

and we look at the relative Tomita operator
$S_{V(k)\Omega,V(k)\Phi}=S_{V(k)\Omega,V(h)\Omega}$, we have:

$S_{V(k)\Omega,V(k)\Phi}=V(k)S_{\Omega,\Phi}V(k)^{*}\Rightarrow\Delta_{V(k)\Omega,V(k)\Phi}=V(k)\Delta_{\Omega,\Phi}V(k)^{*}.$

But since $V(k)^{*}V(h)=V(h-k)$, we have that the state functional associated
to $\ket{\Phi}$ is $\omega_{h-k}$. Therefore, we find:

$\displaystyle S(\omega_{k}\|\omega_{h})$
$\displaystyle=-\innerproduct{V(h)\Omega}{\log\Delta_{V(k)\Omega,V(h)\Omega}V(h)\Omega}$
$\displaystyle=-\innerproduct{V(h)\Omega}{V(k)\log\Delta_{\Omega,\Phi}V(k)^{*}V(h)\Omega}$
$\displaystyle=-\innerproduct{V(k)^{*}V(h)\Omega}{\log\Delta_{\Omega,\Phi}V(k)^{*}V(h)\Omega}$
$\displaystyle=-\innerproduct{V(h-k)\Omega}{\log\Delta_{\Omega,V(h-k)\Omega}V(h-k)\Omega}=S(\omega_{h-k}\|\omega).$

∎

Using the above results, we can rewrite the general relative entropy:

$S(\omega_{h}\|\omega)=-\innerproduct{\Omega}{\log\Delta_{V(h)\Omega,\Omega}\Omega}$

in terms of the modular operator $\Delta_{\Omega}$ only, noticing that
$V(h)\in R(H)$ is unitary for $h\in H$:

$\displaystyle\Delta_{V(h)\Omega,\Omega}$
$\displaystyle=V(h)\Delta_{V^{*}(h)V(h)\Omega,V^{*}(h)\Omega}V^{*}(h)$
$\displaystyle=V(h)\Delta_{\Omega,V^{*}(h)\Omega}V^{*}(h)$
$\displaystyle=V(h)\Delta_{\Omega}V^{*}(h),$

where in the first step we have used Eq. (2.6) and Prop. 2.2.8 in the third.
Therefore, it follows that:

$S(\omega_{h}\|\omega)=-\innerproduct{V^{*}(h)\Omega}{\log\Delta_{\Omega}V^{*}(h)\Omega}.$
(2.7)

We are able to prove the main theorem:

######  Theorem 2.2.4.

Let $h\in H$. If $h\in\mathrm{Dom}(\log\Delta_{H})$, the relative entropy on
$R(H)$ between $\omega$ and $\omega_{h}$ is given by:

$S(\omega_{h}\|\omega)=S_{h}=-(h,\log\Delta_{H}h).$

###### Proof.

The first thing to notice is that, from Eq. (2.7), we have for
$t\in\mathbb{R}$:

$\displaystyle S(\omega_{h}\|\omega)$
$\displaystyle=\innerproduct{V^{*}(h)\Omega}{K_{\Omega}V^{*}(h)\Omega}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{V^{*}(h)\Omega}{e^{iK_{\Omega}t}V^{*}(h)\Omega}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{V^{*}(h)\Omega}{\Delta^{it}_{\Omega}V^{*}(h)\Omega},$

where I have introduced the modular Hamiltonian, defined as:

$K_{\Omega}=-\log\Delta_{\Omega},$

since $\Delta_{\Omega}=\Delta_{R(H)}$ is the modular operator of $R(H)$ with
respect to the state $\omega_{\Omega}$ defined on it. By the second quantized
equivalent of the modular flow on the one particle Hilbert space, we have:
$\Delta_{R(H)}^{it}=\mathfrak{S}(\Delta_{H}^{it})$. Therefore:

$\displaystyle S(\omega_{h}\|\omega)$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{V^{*}(h)\Omega}{\mathfrak{S}(\Delta^{it}_{H})V^{*}(h)\Omega}$
$\displaystyle=ie^{-\|h\|^{2}}\frac{d}{dt}\bigg{|}_{t=0}e^{(h,\Delta_{H}^{it}h)}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}(h,\Delta_{H}^{it}h)$
$\displaystyle=-(h,\log\Delta_{H}h).$

∎

We note as well that, since $S(\omega_{h}||\omega)$ is real, we have:

$S(\omega_{h}||\omega)=-(h,\log\Delta_{H}h)=i(h,i\log\Delta_{H}h)=\imaginary(h,i\log\Delta_{H}h),$

where $\imaginary(\cdot,\cdot)$ defines a symplectic form. We have proven that
the relative entropy for coherent excitations for $h\in H$ is equivalent to
the entropy of the vector in the real and standard subset of the one particle
Hilbert space. We now seek to generalize this to the case in which the vector,
with respect to which we take the coherent excitations, is a general vector in
the one-particle Hilbert space.

#####  2.2.5.2 Case $h\in\mathcal{H}$

Let us now consider the general case in which $k\in\mathcal{H}$ and consider
$\omega_{k}$ and $\omega_{k}^{\prime}$ to be states on $R(H)$ and
$R(H)^{\prime}$, respectively.

######  Lemma 2.2.4.

Let $k\in\mathcal{H}_{\epsilon}$. Then $S(\omega_{k}\|\omega)=S_{k}$.

###### Proof.

We saw that a general element in $\mathcal{H}_{\epsilon}$ can be written as
$k=h+h^{\prime}$, where $h\in E_{\epsilon}H$, $h^{\prime}\in
E_{\epsilon}H^{\prime}$ and where $E_{\epsilon}$ is the spectral projection of
$\Delta_{H}$. So, starting from the general definition of relative entropy of
$k\in\mathcal{H}_{\epsilon}$ with respect to $H$, one has:

$\displaystyle S_{k}$
$\displaystyle=\imaginary((h+h^{\prime},P_{H}i\log\Delta_{H}(h+h^{\prime}))\big{)}$
$\displaystyle=\imaginary((h+h^{\prime},i\log\Delta_{H}h)\big{)}$
$\displaystyle=\imaginary((h,i\log\Delta_{H}h)\big{)}=S_{h},$

where we have used in the second step the fact that $P_{H}$ commutes with
$\Delta_{H}$ and in the third the fact that $\Delta_{H}$ is an automorphism of
$H$.
However, on the other hand, we have that $V(k)=V(h)V(h^{\prime})$ which gives:

$\displaystyle S(\omega_{k}\|\omega)$
$\displaystyle=-\innerproduct{V(h)V(h^{\prime})\Omega}{\log\Delta_{V(k)\Omega,\Omega}V(h)V(h^{\prime})\Omega}$
$\displaystyle=-\innerproduct{V(h)V(h^{\prime})\Omega}{V(h^{\prime})\log\Delta_{\Omega}V^{*}(h^{\prime})V(h)V(h^{\prime})\Omega}$
$\displaystyle=-\innerproduct{V(h)\Omega}{\log\Delta_{\Omega}V(h)\Omega},$

where we used $R(H^{\prime})=R(H)^{\prime}$ and again Prop. 2.2.8 to write:

$\Delta_{V(h)V(h^{\prime})\Omega,\Omega}=V(h^{\prime})\Delta_{\Omega}V(h^{\prime})^{*}.$

Consequently, we have shown that $S_{k}=S_{h}$ and
$S(\omega_{k}\|\omega)=S(\omega_{h}\|\omega)$. Therefore, from the results of
the preceding section, we have that:

$S(\omega_{k}\|\omega)=S_{k}.$

∎

Before presenting the main result, we need another technical lemma:

######  Lemma 2.2.5.

If $\mathcal{H}=\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ with $H=H_{1}\oplus
H_{2}$, and $k=k_{1}\oplus k_{2}$, then:

$S(\omega_{k}\|\omega)=S(\omega_{k_{1}}\|\omega_{k_{2}}),$

where $\omega_{k_{i}}$ is the coherent state associated with $k_{i}$ on
$\mathfrak{S}(\mathcal{H}_{i})$.

###### Proof.

First, let us prove that, by passing to second quantization,
$\mathfrak{S}(\mathcal{H})=\mathfrak{S}(\mathcal{H}_{1})\otimes\mathfrak{S}(\mathcal{H}_{2})$.
Let us look at the finite particle level:

$\displaystyle\mathcal{H}\otimes\mathcal{H}$
$\displaystyle=(\mathcal{H}_{1}\oplus\mathcal{H}_{2})\otimes(\mathcal{H}_{1}\oplus\mathcal{H}_{2})$
$\displaystyle=(\mathcal{H}_{1}\otimes\mathcal{H}_{1})\oplus(\mathcal{H}_{1}\otimes\mathcal{H}_{2})\oplus(\mathcal{H}_{2}\otimes\mathcal{H}_{1})\oplus(\mathcal{H}_{2}\otimes\mathcal{H}_{2}),$

where we notice that each of the factors of the direct sum is orthogonal to
the others. Let us now take the symmetrized tensor product:

$(\mathcal{H}\otimes\mathcal{H})_{s}=(\mathcal{H}_{1}\otimes\mathcal{H}_{1})\oplus(\mathcal{H}_{1}\otimes\mathcal{H}_{2})\oplus(\mathcal{H}_{2}\otimes\mathcal{H}_{2}),$

from which follows that at every order $n$, we have:

$\displaystyle\mathfrak{S}(\mathcal{H})$
$\displaystyle=\bigoplus_{n=0}^{\infty}(\mathcal{H}_{1}\oplus\mathcal{H}_{2})^{\otimes
n}_{s}$
$\displaystyle=\bigg{(}\bigoplus_{n=0}^{\infty}(\mathcal{H}_{1})^{\otimes
n}_{s}\bigg{)}\otimes\bigg{(}\bigoplus_{n=0}^{\infty}(\mathcal{H}_{2})^{\otimes
n}_{s}\bigg{)}=\mathfrak{S}(\mathcal{H}_{1})\otimes\mathfrak{S}(\mathcal{H}_{2}).$

For what concerns the states, we split them as $\omega=\omega\otimes\omega$
and $\omega_{k}=\omega_{k_{1}}\otimes\omega_{k_{2}}$. Now, using the
additivity of relative entropy under tensor products (see Eq. $(5.22)$ in
[54]), we have the claimed result. ∎

We can now state and prove the main theorem:

######  Theorem 2.2.5.

Let $k\in\mathcal{H}$, then:

$S(\omega_{k}\|\omega)=S_{k}=\imaginary((k,iP_{H}\log\Delta_{H}k)\big{)}$

###### Proof.

Take $k\in\mathcal{H}$ and decompose
$\mathcal{H}=\mathcal{H}_{\epsilon}\oplus\mathcal{K}_{\epsilon}$.
Correspondingly, we have decompositions for the $J_{H},\Delta_{H},P_{H}$
operators. Then, the above lemma and the positivity of relative entropy give:

$S(\omega_{k}\|\omega)=S(\omega_{k_{\epsilon}}\|\omega)+S(\omega_{(1-E_{\epsilon})k}\|\omega)\geq
S(\omega_{k_{\epsilon}}\|\omega).$

Now, in the limit $\epsilon\to 0^{+}$, we have that $k_{\epsilon}\to k$ and,
by lower semicontinuity of relative entropy, we have:

$\liminf_{\epsilon\to 0^{+}}S(\omega_{k_{\epsilon}}\|\omega)\geq
S(\omega_{k}\|\omega).$

Thus, taking both inequalities into account, we find:

$\lim_{\epsilon\to
0^{+}}S(\omega_{k_{\epsilon}},\omega)=S(\omega_{k}\|\omega).$

But since $S(k_{\epsilon}\|k)=S_{k_{\epsilon}}$, we must have:

$S(\omega_{k}\|\omega)=\lim_{\epsilon\to 0^{+}}S_{k_{\epsilon}}=S_{k},$

where, the last equality, follows from Prop. 2.2.6. ∎

The convenience of working with coherent excitations of the vacuum is manifest
: for such, the relative entropy between two different configurations of the
underlying QFT is computed, using first quantization techniques only, namely
just in terms of the inner product of vectors in the one-particle Hilbert
space.
To establish some analogy with the upcoming results in the next chapter, let
us reformulate the final result. In the case of a free scalar Quantum Field
Theory we have, as abstract algebra $\mathfrak{A}_{(\mathbf{P},\sigma)}$,
where $(\mathbf{P},\sigma)$ is the symplectic space of solutions of the
classical field equation. Then, by choosing a quasi-free state on
$\mathfrak{A}_{(\mathbf{P},\sigma)}$, we quoted in Theorem 1.1.5 that one can
obtain a Fock space representation. In particular, we mentioned that the inner
product over the one-particle Hilbert space $\mathcal{H}$ is related to the
symplectic structure by:

$\sigma(f,g)=2\imaginary(\innerproduct{f}{g}_{\mathcal{H}})$

, for $f,g\in\mathcal{H}$. Therefore, the above result for the relative
entropy for a coherent excitation of the vacuum, can be rewritten for
$\mathcal{H}\ni f=f_{H}+f_{H}^{\prime}$as:

$\displaystyle S(\omega_{f}\|\omega)$
$\displaystyle=\imaginary((f,iP_{H}\log\Delta_{H}f)\big{)}$
$\displaystyle=-(f_{H},\log\Delta_{H}f_{H})$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega}{V(f_{H})V^{*}(\Delta_{H}^{it}f_{H})}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega}{e^{i\imaginary((f_{H},\Delta_{H}^{it}f_{H})\big{)}}V(f_{H}-\Delta^{it}_{H}f_{H})}$
$\displaystyle=-\frac{d}{dt}\bigg{|}_{t=0}\imaginary((f_{H},\Delta_{H}^{it}f_{H})\big{)}$
$\displaystyle=-\frac{1}{2}\frac{d}{dt}\bigg{|}_{t=0}\sigma(f_{H},\Delta_{H}^{it}f_{H})$
$\displaystyle=\frac{1}{2}\frac{d}{dt}\bigg{|}_{t=0}\sigma(\Delta_{H}^{it}f_{H},f_{H}).$

Finally, if we want to compute the relative entropy in a concrete situation,
we need to explicitly determine the $t$-dependence of the last expression. In
fact, in absence of a time symmetry on the underlying spacetime on which the
test functions are defined, this is in general unknown. However, in the
specific case in which the fields are assumed to be localized in wedge-like
regions, the Bisognano-Wichmann theorem (see Appendix A.3) provides a
geometric action of the modular automorphism with the corresponding explicit
form of the $t$-dependence.

##  Chapter 3 Relative entropy for fermionic fields

In the last chapter, we have presented how the relative entropy for a free
scalar QFT, between a quasifree state and a coherent excitation of it is
computed, using Tomita-Takesaki modular theory and the Araki formula. As
already mentioned, the final explicit formula, allows to calculate relative
entropies in contexts like semiclassical gravity. The calculations are
simpler, as a quantity in second quantization, Araki’s entropy is defined in
terms of an operator acting on a vector in the Fock space, is computed just at
the one particle level. At the same time, however, the result holds just for
the specific type of coherent excitations and its generalization remains an
open problem.
Another open task, is the search for similar results for a Dirac/Majorana
Quantum Field Theory. As far as we know, a completely satisfactory result, in
the same spirit as that for the free scalar, seems to be missing in
literature.
In this chapter, that corresponds to the core of this thesis, we derive an
explicit formula for the relative entropy for a Fermionic QFT. The result,
allows the computation of the relative entropy between a quasifree state and a
specific type of unitary excitation of it, in terms of the inner product of
two vectors in the Hilbert space on which the self-dual CAR algebra is
defined. Later on, we attempt at giving a first generalization of this formula
for different types of excitations. We conclude the chapter presenting
concrete examples in finite and infinite dimension, in which the local
algebras are Type $I$ factors, where we will show the equivalence between the
relative entropy computed using the von Neumann formula and the one obtained
starting from out result.

###  3.1 Single Unitary Fermionic Excitation of the Vacuum

Following the work of Araki [13], and what we have discussed in Section
1.1.1.2, we start presenting the result in the most general case and just
later specify to a fermionic QFT.
Therefore, the starting object is a self-dual CAR algebra
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$, that from what we discussed in
Section 1.1.1.2 we know is a $C^{*}$-algebra, where $\mathcal{H}$ is an
Hilbert space, finite or infinite dimensional, on which we have an involution
$\Gamma$ with properties reported in Def. 1.1.9. Moreover, we assume that we
have a (strongly continuous) one parameter group of unitaries $V_{t}$
describing a dynamics over $\mathcal{H}$ with $t\in\mathbb{R}$. By Stone’s
theorem, this one parameter family, is related to a self-adjoint, positve
operator $\mathbf{h}$:

$V_{t}=e^{-it\mathbf{h}}.$

To be precise, if we want this to define a dynamics (think of the Heisenberg
picture), we need to make sure that it lifts to an automorphism over
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. In particular, as
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ is a $C^{*}$-algebra, we need to
demand it to be a $*$-automorphism. Therefore, if we define it as:

$\alpha_{t}(B(g)):=B(V_{-t}g)\hskip 20.0pt\forall g\in\mathcal{H}$

we see that:

$\displaystyle\alpha_{t}(B^{*}(g))$ $\displaystyle=\alpha_{t}(B(\Gamma
g))=B(V_{-t}\Gamma g)$ $\displaystyle\alpha_{t}(B(g))^{*}$
$\displaystyle=B^{*}(V_{-t}g)=B(\Gamma V_{-t}g).$

and, in order $V_{t}$ to define a dynamics, we need for each $t\in\mathbb{R}$:

$[V_{t},\Gamma]=0.$

Now that we have fixed the assumptions, we go back to the elements in the
algebra and identify a specific class of them, that squares to the identity
and is invariant under the $*$-operation. To obtain such a class, we start
considering those elements of the Hilbert space $f\in\mathcal{H}$ for which:

$\Gamma f=f$

Such elements always exist, as we can always construct one starting from a
random element in $\mathcal{H}$:

$\displaystyle f$ $\displaystyle:=(1+\Gamma)h\hskip 20.0pt\forall
h\in\mathcal{H}$ $\displaystyle:=i(1-\Gamma)h\hskip 20.0pt\forall
h\in\mathcal{H}$

If we now consider the corresponding elements, in the self-dual CAR algebra,
associated to these ”involution invariant” vectors of the Hilbert space, we
get elements that are invariant under the abstract $*$-operation:
$B^{*}(f)=B(f)$. Then, from the anticommutation relations:

$B(f)B(f)=\frac{(f,f)}{2}\mathbb{1}$

That, by a proper chooice of the function $h\in\mathcal{H}$ in the definition
of $f$, in order to have $(f,f)=2$, gives the idempotence:

$B(f)B^{*}(f)=B^{*}(f)B(f)=B(f)B(f)=\mathbb{1}$

######  Remark 3.1.1.

If we take two different $f,g\in\mathcal{H}$ such that $\Gamma f=f$ (same for
$g$), we have that for the corresponding
$B(f),B(g)\in\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$:

$[B(f),B(g)]_{+}=(f,g)_{\mathcal{H}}\mathbb{1}$

but at the same time:

$[B(f),B(g)]_{+}=[B(g),B(f)]_{+}=(g,f)_{\mathcal{H}}\mathbb{1}$

But this implies:

$(f,g)_{\mathcal{H}}=(g,f)_{\mathcal{H}}=\overline{(f,g)_{\mathcal{H}}}$

That gives $(f,g)_{\mathcal{H}}\in\mathbb{R}$.

These $B(f)$, for $f=\Gamma f$, are the elements of the algebra that we will
consider to excite a quasifree state and thus to compute the relative entropy.
Now, in order to be able to define the Tomita operator and a notion of
relative entropy between states using Araki’s formula, we need to represent
the abstract algebra as a von Neumann algebra over a Hilbert space. For this
purpose, we start by presenting the two following results (See [18] Lemma
$3.2$ and $3.3$):

######  Lemma 3.1.1.

For any state $\varphi$ over $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$, there
exists a bounded operator $S$ (called basis polarization) on $\mathcal{H}$,
satisfying:

$\displaystyle\varphi(B^{*}(f)B(g))$ $\displaystyle=(f,Sg)_{\mathcal{H}}$
(3.1) $\displaystyle\mathbb{1}\geq S^{*}$ $\displaystyle=S\geq 0$ (3.2)
$\displaystyle S+\Gamma S\Gamma$ $\displaystyle=\mathbb{1}$ (3.3)

###### Proof.

Being $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ a $C^{*}$-algebra, we have:

$B^{*}(f)B(f)\leq B^{*}(f)B(f)+B(f)B^{*}(f)=\|f\|^{2}$

It follows:

$\|B(f)\|=\|B^{*}(f)B(f)\|^{1/2}\leq\|f\|$

Hence, the relation (3.1) defines an operator $S$ that is bounded and linear.
The positivity of $\varphi$ gives:

$(f,Sf)_{\mathcal{H}}=(S^{*}f,f)_{\mathcal{H}}\geq 0$

but the inequality implies
$(S^{*}f,f)_{\mathcal{H}}=(f,S^{*}f)_{\mathcal{H}}$. Follows that $S^{*}=S\geq
0$.
Finally, from the anticommutation relations over the self-dual CAR algebra, we
have:

$\displaystyle\varphi(B^{*}(f)B(g))$
$\displaystyle=(f,g)_{\mathcal{H}}-\varphi(B(g)B^{*}(f))$
$\displaystyle=(f,g)_{\mathcal{H}}-\varphi(B^{*}(\Gamma g)B(\Gamma f))$
$\displaystyle=(f,g)_{\mathcal{H}}-(\Gamma g,S\Gamma f)_{\mathcal{H}}$

From the definition of the involution $\Gamma$, we have $(h,\Gamma
f)_{\mathcal{H}}=(f,\Gamma h)_{\mathcal{H}}$ that gives:

$(\Gamma g,S\Gamma f)_{\mathcal{H}}=(S\Gamma g,\Gamma
f)_{\mathcal{H}}=(f,\Gamma S\Gamma g)_{\mathcal{H}}$ (3.4)

Hence:

$(f,Sg)_{\mathcal{H}}=(f,g)_{\mathcal{H}}-(f,\Gamma S\Gamma g)_{\mathcal{H}}$

From which it follows Eq. (3.3). Morever, combining the positivity of $S$ with
Eq. (3.4), we get $\Gamma S\Gamma=\mathbb{1}-S\geq 0$. ∎

######  Lemma 3.1.2.

For any $S$ as above, there exist a unique quasifree state satisfying Eq.
(3.1).

###### Proof.

As a quasifree state is uniquely determined by its two point function and $S$
determines it, follows the uniqueness. The existence follows by Lemma $4.6$ in
[18] ∎

We denote such quasifree states by $\varphi_{S}$. But, any quasifree state
over the self-dual algebra has, by Lemma 3.1.1, an associated $S$ with the
properties Eq. (3.2) and (3.3) that determine it uniquely. As a consequence,
any quasifree state over the self-dual CAR algebra is of the form
$\varphi_{S}$.
In order to formulate Tomita-Takesaki modular theory, we need the state over
the self-dual CAR algebra to be faithful. Moreover, for later purposes, we
want it also to be quasifree. Therefore, we investigate the existence and the
properites of quasifree, faithful states over a self-dual CAR algebra. The
faithfulness condition lead us to closely look at Eq. (3.1), our state must
be:

$\varphi_{S}(B^{*}(f)B(f))=0\Longleftrightarrow B(f)=0$

However, by the following lemma (Lemma $4.3.$ in [18]) and by Theorem 1.1.5,
we see that if a quasifree state has an associated basis polarization $S$ that
is also a projection, i.e. $S^{2}=S$ and thus called basis projection, then it
cannot be faithful:

######  Lemma 3.1.3.

Let $P$ be a basis projection. If a state $\varphi$ of
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ satisfies:

$\varphi(B(f)B^{*}(f))=0\hskip 20.0pt\forall f\in P\mathcal{H}$

Then $\varphi$ is a quasifree state with $S=P$ i.e. $\varphi=\varphi_{P}$. The
representation $\pi_{P}$ is irreducible.

Therefore, we need to work with states that do not lead directly to
irreducible representations, when we perform the associated GNS construction,
if we want the GNS vector to be separable.
Anyway, for each quasifree state, the corresponding GNS construction leads to
a Fock space. In fact, for each $\varphi_{S}$, we have:

$\varphi_{S}(B^{*}(f)B(g))=(f,Sg)_{\mathcal{H}}=(S^{1/2}f,S^{1/2}g)_{\mathcal{H}}$

We define the one-particle Hilbert space as (see [25] Section II.$2$):

$\mathfrak{h}_{S}=\\{S^{1/2}f|f\in\mathcal{H}\\}$

and correspondingly, by taking the direct sum of the antisymmetrized tensor
product of it, the Fermionic Fock space:

$\mathcal{K}_{S}:=\mathbb{C}\oplus\bigg{(}\bigoplus_{n=1}^{\infty}(\mathfrak{h}_{S})^{\wedge
n}\bigg{)}$

With $\wedge$ denoting the antisymmetric tensor product. The corresponding
vacuum vector $\Omega_{S}$ is the vector spanning the $\mathbb{C}$ component.
The representation is:

$\pi_{S}(B(f))=a^{*}(S^{1/2}f)+a(S^{1/2}\Gamma f)$

Where $a^{*}(p)$ and $a(p)$ are creation and annihilation operators over the
Fock space $\mathcal{K}_{S}$. To check that this defines a representation, one
can check that this map defines a $*$-isomorphism of
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ into the algebra of creation and
annihilation operators over $\mathfrak{h}_{S}$. Furthermore, if $S$ happens to
be a projection, we can replace, in the latter construction of the Fock space,
$S^{1/2}$ with $S$. In particular, given
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$, a representation with respect to a
basis projection always exists (Lemma $3.3.$ in [13]):

######  Lemma 3.1.4.

If $\dim\mathcal{H}=\mathrm{even}$ or $+\infty$, there exists a basis
projection $P$, that generates a $*$-representation $\pi_{P}$ to the Fock
space $\mathcal{K}_{P}$:

$\displaystyle\pi_{P}(B^{*}(f))$ $\displaystyle=a^{*}(Pf)+a(P\Gamma f)$
$\displaystyle\pi_{P}(B(f))$ $\displaystyle=a(Pf)+a^{*}(P\Gamma f)$
$\displaystyle\pi_{P}(\mathbb{1})$ $\displaystyle=\mathbb{1}$

###### Proof.

From what we said above, we just need to prove existence. Consider a $\Gamma$
invariant basis of $\mathcal{H}$ denoted $\\{f_{i}\\}_{i=1,\dots}$. Since
$\mathcal{H}$ is even dimensional, we can pair $f_{2n}$ with $f_{2n-1}$ for
$n=1,2,\dots$ to define $P$ as the projector onto the subspace spanned by:

$\frac{1}{2^{1/2}}(f_{2n}+if_{2n-1})$

In this way $\Gamma P\Gamma=\mathbb{1}-P$. The existence of a $\Gamma$
invariant basis for $\mathcal{H}$ follows from the fact that, for a general
$h\in\mathcal{H}$ non zero, either $(1+\Gamma)h$ or $i(1-\Gamma)h$ are non
zero. Therefore, starting from an arbitrary non-zero element of $\mathcal{H}$
we form the corresponding $\Gamma$ invariant vector and then perform a Gram-
Schmidt procedure, as the orthogonal of the space spanned by the $\Gamma$
invariant vectors in $\mathcal{H}$, is itself $\Gamma$-invariant. ∎

We will need the existence of such a basis projection, for any
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ later on.

However, for the current purposes, let $\omega$ be just a faithful, quasifree
state over the self-dual CAR algebra, with associated basis polarization $S$.
Performing the corresponding GNS construction, we get the triple
$(\mathcal{K}_{\omega},\pi_{\omega},\Omega_{\omega})$ of a Fock
representation.
Let us call
$\mathcal{A}=\pi_{\omega}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))^{\prime\prime}$
the associated von Neumann algebra. Now that we have a von Neumann algebra, we
can define a relative Tomita operator between two cyclic and separating
states. For that purpose, we start by easing the notation, denoting from now
on $F:=\pi_{\omega}(B(f))$, for $f\in\mathcal{H}$ such that $\Gamma f=f$.
Then, by the properties of the corresponding $B(f)$ and the $*$-representation
$\pi_{\omega}$, the operator $F$ must be unitary. As a consequence, the vector
state $F\ket{\Omega_{\omega}}$, with associated vector functional $\omega_{F}$
over $\mathcal{A}$, must be cyclic and separating as well:

######  Lemma 3.1.5.

Let $\ket{\Omega_{\omega}}$ be a cyclic and separating vector for the algebra
$\mathcal{A}$. Then, for $F\in\mathcal{A}$ unitary, the vector
$F\ket{\Omega_{\omega}}$ is cyclic and separating.

###### Proof.

As $F\in\mathcal{A}$ is unitary we have $\mathcal{A}F=\mathcal{A}$. Then,
using the fact that $\Omega_{\omega}$ is cyclic:

$\displaystyle\overline{\mathcal{A}F\ket{\Omega_{\omega}}}$
$\displaystyle=\overline{\mathcal{A}\ket{\Omega_{\omega}}}$
$\displaystyle=\mathcal{K}_{\omega}$

Since $F\in\mathcal{A}$.
For what concerns the separating property, consider:

$AF\ket{\Omega_{\omega}}=0\Longrightarrow F^{*}AF\ket{\Omega_{\omega}}=0$

But, as $\ket{\Omega_{\omega}}$ is separating, we have $F^{*}AF=0$. But
$F^{*}AF\in\mathcal{A}$ as well, so by repeating the same argument:
$FF^{*}AFF^{*}=0$ that gives $A=0$. ∎

Now that we have two cyclic and separating vectors $\ket{\Omega_{\omega}}$ and
$F\ket{\Omega_{\omega}}$, for the von Neumann algebra $\mathcal{A}$, we can
compute the Araki’s relative entropy:

$S(\omega_{F}||\omega)=-\innerproduct{\Omega_{\omega}}{\log\Delta_{F\Omega_{\omega},\Omega_{\omega}}\Omega_{\omega}}$

Where $\Delta_{F\Omega_{\omega},\Omega_{\omega}}$ is the modular operator for
the relative Tomita operator $S_{F\Omega_{\omega},\Omega_{\omega}}$.
To simplify this expression, keeping in mind that $F\in\mathcal{A}$ is
unitary, we use the result of Proposition 2.2.8 and Eq. (2.6) to express the
relative modular operator, in terms of the modular operator of the vector
state $\omega$:

$\displaystyle\Delta_{F\Omega_{\omega},\Omega_{\omega}}$
$\displaystyle=F\Delta_{F^{*}F\Omega_{\omega},F^{*}\Omega_{\omega}}F^{*}$
$\displaystyle=F\Delta_{\Omega_{\omega},F^{*}\Omega_{\omega}}F^{*}$
$\displaystyle=F\Delta_{\Omega_{\omega}}F^{*}$

Therefore, the relative entropy becomes:

$S(\omega_{F}||\omega)=-\innerproduct{\Omega_{\omega}}{\log\big(F\Delta_{\Omega_{\omega}}F^{*}\big{missing})\Omega_{\omega}}=-\innerproduct{\Omega_{\omega}}{F\log\big(\Delta_{\Omega_{\omega}}\big{missing})F^{*}\Omega_{\omega}}$

Where the second equality is a consequence of the unitarity of $F$.

Now, as we later want the modular automorphism to act on the vector
$f\in\mathcal{H}$, we present the following lemma (Lemma $4.2.$ together with
Theorem $3$ in [18]):

######  Lemma 3.1.6.

Let $\omega$ be a faithful quasifree state with basis polarization $S$, over
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. If $S$ commutes with the one
parameter family of unitaries $V_{t}$ on $\mathcal{H}$, then we can choose the
modular flow such that for all
$B(f)\in\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$:

$\Delta_{\Omega_{\omega}}^{it}\pi_{\omega}(B(f))\Delta_{\Omega_{\omega}}^{-it}=\pi_{\omega}(B(V_{t}f))$

###### Proof.

If $[V_{t},S]=0$, then the state $\omega$ is in particular stationary. Hence,
we can define:

$U_{t}\sum_{i}c_{i}\pi_{\omega}(A_{i})\Omega_{\omega}=\sum_{i}c_{i}\pi_{\omega}(\alpha_{t}(A_{i}))\Omega_{\omega}$

and:

$U_{t}^{*}\sum_{i}c_{i}\pi_{\omega}(A_{i})\Omega_{\omega}=\sum_{i}c_{i}\pi_{\omega}(\alpha_{-t}(A_{i}))\Omega_{\omega}$

From the cyclicity of the GNS vector, these are isometric maps from a dense
subset of $\mathcal{K}_{\omega}$ into $\mathcal{K}_{\omega}$ satisfying:

$\displaystyle U_{t}U^{*}_{t}$ $\displaystyle=U_{t}^{*}U_{t}\subset\mathbb{1}$
$\displaystyle U_{t}$ $\displaystyle\subset(U_{t}^{*})^{*}$

Where as usual $A\subset B$ if $A=B$ on $\mathcal{D}(A)$, the domain of the
operator $A$, and $\mathcal{D}(A)\subset\mathcal{D}(B)$. Follows, that the
closure of $U_{t}$ is unitary and satisfies
$U_{t}\Omega_{\omega}=\Omega_{\omega}$ together with:

$U_{t}\pi_{\omega}(A)U^{*}_{t}=\pi_{\omega}(\alpha_{t}(A))$

Now, following the steps in the proof of Theorem $3$ in [18], one shows the
existence of an antiunitary operator $J$ over
$\pi_{\omega}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))^{\prime\prime}$ such
that for any
$A\in\pi_{\omega}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))^{\prime\prime}$:

$\displaystyle JU_{-i\beta/2}A\Omega_{\omega}$
$\displaystyle=A^{*}\Omega_{\omega}$ $\displaystyle J\Omega_{\omega}$
$\displaystyle=\Omega_{\omega}$ $\displaystyle
J\pi_{\omega}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))^{\prime\prime}J$
$\displaystyle=(\pi_{\omega}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))^{\prime\prime})^{\prime}$
$\displaystyle[J,U_{t}]$ $\displaystyle=0$

But then, from the faithfulness assumption, this defines the Tomita operator
with respect to $\Omega_{\omega}$ and, by the uniqeuness of the polar
decomposition, follows the statement. ∎

Therefore, from now on, we further assume that the state $\omega$ is such that
its associated basis polarization commutes with the one parameter family of
unitaries $V_{t}$ on $\mathcal{H}$ or at least that the modular automorphism
induces an action on $\mathcal{H}$ (for example if we are in a framework in
which we can apply the Bisognano-Wichmann theorem A.3.1).
Then, we further rewrite the above expression, introducing the Modular
Hamiltonian:

$K_{\Omega_{\omega}}=-\log\Delta_{\Omega_{\omega}}$

Then, implicitly using Stone’s theorem, we compute:

$\displaystyle K_{\Omega_{\omega}}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}e^{-iK_{\Omega_{\omega}}t}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}e^{it\log\Delta_{\Omega_{\omega}}}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\Delta_{\Omega_{\omega}}^{it}$

So we have obtained:

$-\log\Delta_{\Omega_{\omega}}=i\frac{d}{dt}\bigg{|}_{t=0}\Delta_{\Omega_{\omega}}^{it}$

that gives for the relative entropy:

$\displaystyle S(\omega_{F}||\omega)$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\omega}}{F\Delta_{\Omega_{\omega}}^{it}F^{*}\Delta_{\Omega_{\omega}}^{-it}\Delta_{\Omega_{\omega}}^{it}\Omega_{\omega}}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\omega}}{F\Delta_{\Omega_{\omega}}^{it}F^{*}\Delta_{\Omega_{\omega}}^{-it}\Omega_{\omega}}$

Where we have used
$\Delta_{\Omega_{\omega}}^{it}\Omega_{\omega}=\Omega_{\omega}$.

Finally, we let the modular automorphism act on the elements in $\mathcal{A}$:

$\Delta_{\Omega_{\omega}}^{it}\pi_{\omega}(B(f))\Delta_{\Omega_{\omega}}^{-it}=\pi_{\omega}(\alpha_{-t}(B(f)))=\pi_{\omega}(B(f_{t}))=:F_{t}$

and as a consequence, we get for the relative entropy:

$S(\omega_{F}||\omega)=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\omega}}{FF_{t}\Omega_{\omega}}=i\frac{d}{dt}\bigg{|}_{t=0}(f,Sf_{t})_{\mathcal{H}}$

Therefore, we have proven the following main result:

######  Proposition 3.1.1 (Araki’s formula for a Single Unitary Fermionic
Excitation of the Vacuum).

Let $\mathcal{A}$ be the corresponding von Neumann algebra of the abstract
self-dual CAR algebra obtained via the quasifree vector $\omega$ in the way
described above. Then, denoting by $F=\pi_{\omega}(B(f))$ for
$f\in\mathcal{H}$ such that $\Gamma f=f$, the relative entropy between the
state $\omega$ and the one obtained under an excitation by $F$, called
$\omega_{F}$, becomes:

$S(\omega_{F}\,||\,\omega)=i\frac{d}{dt}\bigg{|}_{t=0}(f,Sf_{t})_{\mathcal{H}}$
(3.5)

######  Remark 3.1.2.

Our result was derived for general self-dual CAR algebras, but we can of
course specialize it to the case of Dirac or Majorana fields since, as we have
outlined in Section 1.3.4, the field algebras in that case are self-dual CAR
algebras with $\mathcal{H}$ representing the space of solutions of the Dirac
equation on the globally hyperbolic spacetime $M$.

######  Remark 3.1.3.

The first thing we may notice, is that our result depends on the inner product
of the Hilbert space on which our self-dual CAR algebra is constructed. This
is in analogy with the coherent excitation in the bosonic case, where the
final result was depending just on the symplectic form. In this sense, both
relative entropies are computed just using the canonically defined structure
$s$ on the underlying space $(\mathbf{K},s)$ (see comment at the end of
Section 1.1.1.2). This result (together with the comment at the end of Section
3.3.2), suggests that the analogous of the coherent excitation in the bosonic
case is, in the fermionic case, the type of excitation that we are
considering. A summary of this, is reported in the following table:

Free scalar field (CCR) $(\mathbf{P,\sigma(\cdot,\cdot)})$ Coherent
excitation: $\omega_{k}$ $S(\omega_{k}\|\omega)\propto\sigma(f_{t},f)$ Free
Dirac/Majorana field (CAR) $(\mathcal{H},(\cdot,\cdot)_{\mathcal{H}})$ Unitary
field excitation: $\omega_{F}$
$S(\omega_{F}\|\omega)\propto(f,Sf_{t})_{\mathcal{H}}$

###  3.2 Multiple Unitary Fermion Excitation of the Vacuum

In this and in the following section, we aim at generalizing the result
obtained in Prop. 3.1.1, to more general types of excitations. The first
generalization is to the case of field polynomials, namely we consider a
product of such unitary field operators and we want to find the corresponding
expression for the relative entropy for this type of excitation.
Let us consider a set of vectors $\\{f^{(i)}\\}_{i=1,\cdots,N}\in\mathcal{H}$
such that $\Gamma f^{(i)}=f^{(i)}$ and, for simplicity, assuming that the
bounded operator associated to $\omega$ is $S>0$:

$(f^{(i)},Sf^{(j)})=0\hskip 20.0pt\mathrm{for}j\neq i$

Taking the same faithful and quasifree state, with the properties that we
discussed above, we get, in the GNS representation, the multiply excited
state:

$F_{1}\cdot\cdots\cdot
F_{N}\ket{\Omega_{\omega}}\doteq\ket{\Psi_{1,\cdots,N}}$

where once again:

$F_{i}\doteq\pi_{\omega}(B(f^{(i)}))$

Since $\ket{\Psi_{1,\cdots,N}}$ is again cyclic and separating for
$\mathcal{A}$, we can use Araki’s formula to determine the relative entropy:

$\displaystyle S(\omega_{F_{1}\cdots F_{N}}||\omega)$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\omega}}{F_{1}\dots
F_{N}\,F^{\prime}_{N}\cdots
F^{\prime}_{1}\,\Omega_{\omega}}=i\frac{d}{dt}\bigg{|}_{t=0}\omega\big{(}B(f^{(1)})\cdots
B(f^{(N)})B(f^{(N)}_{t})\cdots B(f^{(1)}_{t})\big{)}$

where $F^{\prime}_{i}\doteq\pi_{\omega}[B(f^{(i)}_{t})]$.
We use now our assumption of the state $\omega$ to be quasi-free, to write the
above $2N$-point function as a sum of products of $N$ 2-point functions. In
particular, from our choice of the vectors
$\\{f^{(i)}\\}_{i=1,\cdots,N}\in\mathcal{H}$, the only non vanishing
contributions come from permutations that couple a non-primed and a primed
index. Precisely, introducing the following convention in order to simplify
notation:

$\displaystyle\omega_{ij}$
$\displaystyle=\omega\big{(}B(f^{(i)})B(f^{(j)})\big{)}$
$\displaystyle\omega_{ij^{\prime}}$
$\displaystyle=\omega\big{(}B(f^{(i)})B(f^{(j)}_{t})\big{)}=\omega\big{(}B(f^{(i)})B(f^{{}^{\prime}(j)})\big{)}$
$\displaystyle\omega_{i^{\prime}j^{\prime}}$
$\displaystyle=\omega\big{(}B(f^{(i)}_{t})B(f^{(j)}_{t})\big{)}=\omega\big{(}B(f^{{}^{\prime}(i)})B(f^{{}^{\prime}(j)}_{t})\big{)}$

we have that:

$\omega_{1\cdots N\,N^{\prime}\cdots 1^{\prime}}=\sum_{\pi\in
P_{N}}(-1)^{\mathrm{sign}(\pi)}\prod_{i=1}^{N}\omega_{i\pi(i^{\prime})}$

which can be computed using the result of the previous section.

######  Remark 3.2.1.

Actually we would need to consider the sign of the permutation that realizes:

$\\{1,\cdots,N,N^{\prime},\cdots,1^{\prime}\\}\mapsto\\{1,1^{\prime},2,2^{\prime},\cdots,N,N^{\prime}\\}$

However, this is attained with a number of steps:

$\sum_{l=1}^{N}2(N-l)=2N^{2}-N(N+1)=N(N-1)$

which is always even.

As a consequence, for the relative entropy, we have:

$S(\omega_{F_{1}\cdots F_{N}}||\omega)=i\frac{d}{dt}\bigg{|}_{t=0}\sum_{\pi\in
P_{N}}(-1)^{\mathrm{sign}(\pi)}\prod_{i=1}^{N}\omega_{i\pi(i^{\prime})}$

Where we have kept the above notation. Moreover, from the above orthogonality
condition, we compute:

$\displaystyle S(\omega_{F_{1}\cdots F_{N}}||\omega)$
$\displaystyle=i\sum_{i=1}^{N}\frac{d}{dt}\bigg{|}_{t=0}\omega(B(f^{(i)})B(f_{t}^{(i)}))$
$\displaystyle=\sum_{i=1}^{N}S(\omega_{F_{i}}\|\omega)$

In particular, a general polynomial excitation of $\omega$, of this kind,
gives a relative entropy that is still computable just in terms of the inner
product that we have on $\mathcal{H}$ and in particular just in terms of the
result for a single unitary field excitation.

###  3.3 Extension to more general types of excitations

The result in Prop. 3.1.1 can be generalized to another type of excitation,
with respect to a more general type of test functions. In this section we
present two of these examples.

####  3.3.1 Standard subspaces for a fermionic QFT

Let $(\mathcal{H},\Gamma)$ be the Hilbert space of solutions of the Dirac
equation giving rise, for simplicity, to a Majorana field algebra. Therefore,
the condition on the spinor test function $\Gamma f=f$, makes the element
$B(f)\in\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ a unitary fermionic field.
However, if we choose a particular representation of the Dirac algebra, we
obtain:

$\Gamma f=\overline{f}$

Therefore, the condition for $B(f)$ to be unitary, is that $f\in\mathcal{H}$
should be real. Let us call $H$ the real closed (see Remark 3.1.1) linear
subspace of $\mathcal{H}$ containing the real spinor test functions. Then,
from the discussion outlined at the beginning of Section 2.2.5, allows us to
assume $H$ to be a closed, real standard subspace of $\mathcal{H}$.
Then, by considering elements $g\in H+iH$ such that:

$g=f+if\hskip 20.0pt\mathrm{for}\,\,f\in H$

we have in particular:

$B(g)=(1+i)B(f)$

That we can check to be unitary, after a proper choice of the normalization:

$B(g)B^{*}(g)=2B^{2}(f)=(f,f)_{\mathcal{H}}\mathbb{1}$

namely, if we pick $f$ such that $(f,f)_{\mathcal{H}}=1$. For this reason, we
may as well compute the relative entropy between a state $\omega$ (with the
properties discussed above) over $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ and
another state $\omega_{\tilde{F}}$ obtained by exciting the corresponding
vacuum state using such a $B(g)$. In particular, in the GNS representation of
$\omega$, let us call $\tilde{F}:=\pi_{\omega}(B(g))$. Then, repeating the
steps performed in the first section:

$\displaystyle S(\omega_{\tilde{F}}\|\omega)$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\omega}}{\tilde{F}\Delta_{\Omega_{\omega}}^{it}\tilde{F}^{*}\Omega_{\omega}}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\omega}}{(1+i)F(1-i)F_{t}\Omega_{\omega}}$
$\displaystyle=2i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\omega}}{FF_{t}\Omega_{\omega}}$
$\displaystyle=2S(\omega_{F}\|\omega)$

####  3.3.2 Unitary exponential excitation

Consider in the abstrac self-dual CAR algebra, not necessarily a
Dirac/Majorana algebra, for $f\in\mathcal{H}$ such that $\Gamma f=f$, the
following element:

$e^{iB(f)}=\sum_{k=0}^{+\infty}\frac{(iB(f))^{k}}{k!}$

The first thing we may notice is that, from the $*$-invariance of the
corresponding $B(f)$, we have:

$\big{(}e^{iB(f)}\big{)}^{*}=e^{-iB(f)}$

As a consequence, using the Baker-Campbell-Hausdorff formula, we can compute:

$\displaystyle\big{(}e^{iB(f)}\big{)}^{*}e^{iB(f)}$ $\displaystyle=\mathbb{1}$

Therefore, the excitation induced by $e^{iB(f)}$ (that from now on we call
unitary exponential excitation) will be represented by a unitary operator,
once we represent the algebra as bounded operators over a Hilbert space.
Moreover, using the properties of $B(f)$, we rewrite the unitary exponential
excitation as:

$\displaystyle e^{iB(f)}$
$\displaystyle=\mathbb{1}+iB(f)+\frac{(i)^{2}}{2!}B(f)^{2}+\frac{(i)^{3}}{3!}B(f)^{3}+\dots$
$\displaystyle=\mathbb{1}+iB(f)+\frac{(i)^{2}}{2!}\mathbb{1}+\frac{(i)^{3}}{3!}B(f)+\dots$
$\displaystyle=\bigg{(}1-\frac{1}{2!}+\frac{1}{4!}+\dots\bigg{)}\mathbb{1}+i\bigg{(}1-\frac{1}{3!}+\frac{1}{5!}+\dots\bigg{)}B(f)$
$\displaystyle=\cos(1)\mathbb{1}+i\sin(1)B(f)$

If we call $\omega_{C}$ the state obtained from $\omega$ by the unitary
exponential excitation, we can compute:

$\displaystyle S(\omega_{C}\|\omega)$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\omega}}{e^{iB(f)}e^{-iB(f_{t})}\Omega_{\omega}}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\omega}}{\big{(}\cos(1)\mathbb{1}+i\sin(1)B(f)\big{)}\big{(}\cos(1)\mathbb{1}-i\sin(1)B(f_{t})\big{)}\Omega_{\omega}}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\sin^{2}(1)\innerproduct{\Omega_{\omega}}{B(f)B(f_{t})\Omega_{\omega}}$
$\displaystyle=i\sin^{2}(1)\frac{d}{dt}\bigg{|}_{t=0}(f,Sf_{t})_{\mathcal{H}}$
$\displaystyle=\sin^{2}(1)S(\omega_{F}\|\omega)$

Therefore, in the case of a unitary exponential excitation, the relative
entropy can again be computed just in terms of the single unitary field
excitation.
In this sense, despite the analogy of this case with the bosonic coherent
excitation, it seems more fundamental to consider the single fermionic unitary
excitation as the fermionic analogue of the bosonic case.

###  3.4 Comparison with Von Neumann relative entropy

To check our result, we shall compute the relative entropy using Araki’s
formula, for a self dual CAR algebra constructed over a finite dimensional
Hilbert space. In this context, we know it should coincide with the relative
entropy computed using the usual relative entropy formula due to von Neumann.
Let $(\mathcal{H},\Gamma)$ be such that $\dim\mathcal{H}<+\infty$ and even.
Construct over it the self dual CAR algebra
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. Morever, assume that we have a
(strongly continuous) one parameter group of unitaries $V_{t}$, describing a
dynamics over $\mathcal{H}$. As explained at the beginning of Section 3.1,
this raises to a $*$-automorphism $\alpha_{t}$ over
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. The existence of such an
automorphism, allows to define notions like ground and KMS states over the
abstract self dual CAR algebra.

Therefore, let us pick a state $\omega$ over
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ assumed to be a ground state that is
also quasifree leading to a Fock representation (see Lemma 3.1.3). Let us call
$P$ the associated basis projection over $(\mathcal{H},\Gamma)$, and perform
the corresponding GNS construction over
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. This leads to a triple of Fock-
Hilbert space, Fock-representation map and a corresponding vacuum vector, all
denoted as $(\mathcal{K}_{\omega},\pi_{\omega},\Omega_{\omega})$. A ground
state always exists, as the assumption of $\mathcal{H}$ having a dynamics
$V_{t}$ generated by a self-adjoint operator $\mathbf{h}$ that anticommutes
with $\Gamma$, allows us to take $P$ to be the projection onto the positive
part of the spectrum of $\mathbf{h}$. Then, the corresponding quasifree state
associated to $P$, is the ground state:

$\displaystyle-i\partial_{t}\omega(B^{*}(g)\alpha_{t}(B(g)))|_{t=0}$
$\displaystyle=-i\partial_{t}|_{t=0}(g,PV_{-t}g)_{\mathcal{H}}$
$\displaystyle=-i\partial_{t}|_{t=0}(g,Pe^{it\mathbf{h}}g)_{\mathcal{H}}$
$\displaystyle=-i\partial_{t}|_{t=0}(g,\sum_{n}e^{itE^{+}_{n}}b_{n}^{+}\Psi_{n}^{+})_{\mathcal{H}}$
$\displaystyle=-i\partial_{t}|_{t=0}(g,\sum_{n}e^{itE^{+}_{n}}b_{n}^{+}\Psi_{n}^{+})_{\mathcal{H}}$
$\displaystyle=\sum_{n}\sum_{m}\overline{b}_{m}^{+}b_{n}^{+}\delta_{nm}E_{n}^{+}\geq
0$

Where we have expanded a general $g\in\mathcal{H}$ on an eigenbasis of
$\mathbf{h}$:

$g=\sum_{n}b_{n}^{-}\Psi_{n}^{-}+b_{n}^{+}\Psi_{n}^{+}+b_{n}^{0}\Psi_{n}^{0}$

where the sum is finite and $b_{n}^{\pm}\in\mathbb{C}$. Moreover, we have
denoted by $E_{n}^{\pm}$ the positive/negative eigenvalues of $\mathbf{h}$.
As a ground state, $\omega$ is also stationary. This means that
$\alpha^{*}_{t}\omega=\omega$, where we have raised the action of the
automorphism $\alpha_{t}$ to state functionals:

$\alpha^{*}_{t}\omega(A):=\omega(\alpha_{t}A)\hskip 20.0pt\forall
A\in\mathfrak{A}_{SDC}(\mathcal{H},\Gamma),$

The stationarity implies, from the uniqueness of the GNS construction up to
unitary equivalence, that $\alpha_{t}$ is implementable in
$\mathcal{K}_{\omega}$ by a one-parameter family of unitaries
$U_{t}=e^{-itH}$. In particular, $H$ is the second quantization of the one-
particle Hamiltonian $\mathbf{h}$, as $\mathcal{K}_{\omega}$ is the Fock space
consructed over the one-particle Hilbert space $P\mathcal{H}$.
Moreover, as $\mathcal{H}$ is finite dimensional and we are dealing with
fermions: $\dim\mathcal{K}_{\omega}<\infty$. Then, we can consider a general
density matrix, associated to a Gibbs state of inverse temperature $\beta>0$,
on $\mathcal{K}_{\omega}$:

$\rho_{\beta}=\frac{e^{-\beta H}}{\Tr(e^{-\beta H})}$

and this is well defined, i.e. is of trace class, as
$\dim\mathcal{K}_{\omega}<\infty$.

Having a density matrix of a Gibbs state in this finite dimensional setting,
allows us to go back from the representation to the abstract algebra:

$\rho_{\beta}\longrightarrow\omega_{\beta}$

defining $\omega_{\beta}$ as a KMS state over
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. The argument behind this, is based
on the fact that the KMS state $\omega_{\beta}$ is defined over a finitely
generated algebra $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$.

The finite dimension of the algebra, gives that $\omega_{\beta}$ is
quasiequivalent to $\omega$. This follows from the fact that the GNS
constructions with respect to them, leads to finite dimensional vector spaces
that, as such, are all isomorphic to some $\mathbb{C}^{n}$. Therefore, we can
define an isomorphism (denoting the GNS triple associated to $\omega_{\beta}$
as $(\mathcal{K}_{\beta},\pi_{\beta},\Omega_{\beta})$):

$B:\mathcal{K}_{\beta}\to\mathcal{K}_{\omega}$

But then, whenever we take a normal state $\phi$ in the representation
$\pi_{\beta}$, denoted as
$\phi\in\mathfrak{S}^{(\pi_{\beta})}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))$,
we have:

$\displaystyle\phi(A)$
$\displaystyle=\Tr_{\beta}\big{(}\rho_{\phi}\pi_{\beta}(A)\big{)}$
$\displaystyle=\Tr_{\omega}\big{(}(B^{-1})^{*}\rho_{\phi}\pi_{\beta}(A)B^{-1}\big{)}$
$\displaystyle=\Tr_{\omega}\big{(}(B^{-1})^{*}\rho_{\phi}B^{-1}\pi_{\omega}(A)BB^{-1}\big{)}$
$\displaystyle=\Tr_{\omega}\big{(}(B^{-1})^{*}\rho_{\phi}B^{-1}\pi_{\omega}(A)\big{)}$

for any $A\in\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. Therefore,
$\phi\in\mathfrak{S}^{(\pi_{\omega})}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))$.
In this way, showing with the analogous argument also the opposite inclusion,
we have
$\mathfrak{S}^{(\pi_{\beta})}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))=\mathfrak{S}^{(\pi_{\omega})}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))$
proving the quasiequivalence. The immediate consequence, is that
$\omega_{\beta}$ has an associated density matrix in $\mathcal{K}_{\omega}$,
as it is normal with respect to its representation by considering in
$\mathcal{K}_{\beta}$ simply $\ket{\Omega_{\beta}}\bra{\Omega_{\beta}}$.

Let us assume, that also the thus constructed state $\omega_{\beta}$ is
quasifree and that its associated basis polarization is $S$. Therefore, by
considering $\omega_{\beta}$ over $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ we
can perform another GNS construction, leading to:
$(\mathcal{K}_{\beta},\pi_{\beta},\Omega_{\beta})$.

After this necessary introduction of the setup, let us go back to our elements
of the self dual CAR algebra $B(f)$, for $f\in\mathcal{H}$ such that $\Gamma
f=f$, and introduce the following convenient notation:

$\displaystyle F_{g}$ $\displaystyle:=\pi_{\omega}(B(f))$ $\displaystyle
F_{\beta}$ $\displaystyle:=\pi_{\beta}(B(f)).$

We aim at computing the relative entropy between the KMS state
$\omega_{\beta}$ and the one obtained by acting on it with $B(f)$ that we will
denote as:

$\omega_{F_{\beta}}(A):=\omega_{\beta}(B(f)AB(f))\hskip 20.0pt\forall
A\in\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$

We start by computing the von Neumann relative entropy. For that purpose, we
need to derive the form of the density matrix in the Hilbert space
$\mathcal{K}_{\omega}$ associated to our considered unitary excitation of the
KMS state. To derive it, notice that the quasiequivalence discussed above,
gives:

$\displaystyle\omega_{F_{\beta}}(\cdot)$
$\displaystyle=\omega_{\beta}(B(f)\cdot B(f))$
$\displaystyle=\Tr_{\omega}\big{(}\rho_{\beta}F_{g}\pi_{\omega}(\cdot)F_{g}\big{)}$
$\displaystyle=\Tr_{\omega}\big{(}F_{g}\rho_{\beta}F_{g}\pi_{\omega}(\cdot)\big{)}$
$\displaystyle=\Tr_{\omega}\big{(}\sigma_{\beta}\pi_{\omega}(\cdot)\big{)}$

where we have denoted by $\sigma_{\beta}$ the density matrix associated with
the new state in $\mathcal{K}_{\omega}$.
Therefore, we compute the corresponding von Neumann relative entropy in
$\mathcal{K}_{\omega}$:

$\displaystyle S_{vN}(\rho_{\beta},\sigma_{\beta}):=S_{vN}$
$\displaystyle=\Tr\bigg(\rho_{\beta}(\log\rho_{\beta}-\log\sigma_{\beta})\bigg{missing})$
$\displaystyle=\Tr\bigg(\frac{e^{-\beta H}}{\Tr(e^{-\beta
H})}\bigg{(}\log(e^{-\beta H})-\log(\Tr(e^{-\beta
H}))-\log\big(F_{g}\rho_{\beta}F_{g}\big{missing})\bigg{)}\bigg{missing}).$

Now, as the $F_{g}$ is a unitary operator, we can drag it out of the
logarithm:

$\displaystyle\log(F_{g}\rho_{\beta}F_{g})$
$\displaystyle=\log(1+(F_{g}\rho_{\beta}F_{g}-1))$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{n!}(F_{g}\rho_{\beta}F_{g}-\mathbb{1})^{n}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{n!}(F_{g}(\rho_{\beta}-\mathbb{1})F_{g})^{n}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{n!}F_{g}(\rho_{\beta}-\mathbb{1})^{n}F_{g}$
$\displaystyle=F_{g}\bigg{(}\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{n!}(\rho_{\beta}-\mathbb{1})^{n}\bigg{)}F_{g}$
$\displaystyle=F_{g}\log(\rho_{\beta})F_{g}.$

The series expansion is well defined as:

$\displaystyle\|F_{g}\rho_{\beta}F_{g}-\mathbb{1}\|_{op}^{2}$
$\displaystyle=\|F_{g}(\rho_{\beta}-\mathbb{1})F_{g}\|_{op}^{2}$
$\displaystyle=\sup_{\|v\|_{\mathcal{K_{\omega}}}=1}\|F_{g}(\rho_{\beta}-\mathbb{1})F_{g}v\|_{\mathcal{K}_{\omega}}^{2}$
$\displaystyle=\sup_{\|v\|_{\mathcal{K_{\omega}}}=1}\|(\rho_{\beta}-\mathbb{1})F_{g}v\|_{\mathcal{K}_{\omega}}^{2}$
$\displaystyle=\sup_{\|v\|_{\mathcal{K_{\omega}}}=1}\|\rho_{\beta}v-v\|_{\mathcal{K}_{\omega}}^{2}=\|\rho_{\beta}-\mathbb{1}\|_{op}^{2}$
$\displaystyle=\sup_{\|v\|_{\mathcal{K_{\omega}}}=1}\langle(\rho_{\beta}v-v),(\rho_{\beta}v-v)\rangle$
$\displaystyle=\sup_{\|v\|_{\mathcal{K_{\omega}}}=1}(\langle\rho_{\beta}v,\rho_{\beta}v\rangle+\langle
v,v\rangle-2\langle v,\rho_{\beta}v\rangle)$
$\displaystyle=\sup_{\|v\|_{\mathcal{K_{\omega}}}=1}(\langle
v,\rho_{\beta}^{2}v\rangle+\langle v,v\rangle-2\langle
v,\rho_{\beta}v\rangle)$
$\displaystyle\leq\sup_{\|v\|_{\mathcal{K_{\omega}}}=1}(1-\langle
v,\rho_{\beta}v\rangle)$ $\displaystyle<1.$

Where we have used that a density matrix is positive definite, self-adjoint
and has unit trace so $\rho_{\beta}^{2}\leq\rho_{\beta}$. Moreover, at the
third step we have used that $F_{g}$ is unitary, at the fourth that still by
unitarity we have $\|F_{g}v\|_{\mathcal{K}_{\omega}}=1$ allowing us to include
everything in the supremum and at the last step the continuity of the
expectation value together with the compacteness of the unit ball in finite
dimension.
Going back to the computation of the von Neumann relative entropy:

$\displaystyle S_{vN}$ $\displaystyle=\Tr\bigg(\frac{e^{-\beta
H}}{\Tr(e^{-\beta H})}\bigg{(}\log(e^{-\beta H})-\log(\Tr(e^{-\beta
H}))-F_{g}\log(\rho_{\beta})F_{g}\bigg{)}\bigg{missing})$
$\displaystyle=\Tr\bigg(\frac{e^{-\beta H}}{\Tr(e^{-\beta
H})}\bigg{(}\log(e^{-\beta H})-\log(\Tr(e^{-\beta
H}))-F_{g}\big{(}\log(e^{-\beta H})-\log(\Tr(e^{-\beta
H}))\big{)}F_{g}\bigg{)}\bigg{missing})$
$\displaystyle=\Tr\bigg(\frac{e^{-\beta H}}{\Tr(e^{-\beta
H})}\bigg{(}\log(e^{-\beta H})-F_{g}\log(e^{-\beta
H})F_{g}\bigg{)}\bigg{missing})$ $\displaystyle=\Tr\bigg(\frac{e^{-\beta
H}}{\Tr(e^{-\beta H})}\big{(}-\beta H+F_{g}(\beta
H)F_{g}\big{)}\bigg{missing})$ $\displaystyle=\beta\Tr\bigg(\frac{e^{-\beta
H}}{\Tr(e^{-\beta H})}\big{(}F_{g}[H,F_{g}]\big{)}\bigg{missing}).$

On the other hand, let us now compute the relative entropy using our result
starting from Araki’s formula:

$\displaystyle S_{A}$ $\displaystyle:=S(\omega_{F_{\beta}}||\omega_{\beta})$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\innerproduct{\Omega_{\beta}}{F_{\beta}\Delta_{\beta}^{it}F_{\beta}\Omega_{\beta}}.$

In the derivation of our result, we were assuming that the modular flow was
the raising of the dynamics on $\mathcal{H}$. Therefore, the abovely discussed
dynamics $V_{t}=e^{-it\mathbf{h}}$ is related to the modular flow in the GNS
construction of $\omega_{\beta}$ as follows:

$\displaystyle\Delta_{\Omega_{\beta}}^{it}B(f)\Delta_{\Omega_{\beta}}^{-it}=B(e^{-it\beta\mathbf{h}}f)$

Therefore, the quasiequivalence between $\omega_{\beta}$ and $\omega$, gives:

$\displaystyle S_{A}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\omega_{\beta}(B(f)B(e^{-it\beta\mathbf{h}}f))$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\Tr\big(\rho_{\beta}F_{g}F_{g}^{t}\big{missing})$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\Tr\big(\rho_{\beta}F_{g}e^{-it\beta
H}F_{g}e^{it\beta H}\big{missing})$
$\displaystyle=i\beta\Tr\big(\rho_{\beta}F_{g}(-iHF_{g}+iF_{g}H)\big{missing})$
$\displaystyle=\beta\Tr\big(\rho_{\beta}F_{g}[H,F_{g}]\big{missing})$
$\displaystyle=\beta\Tr\bigg(\frac{e^{-\beta H}}{\Tr(e^{-\beta
H})}\big{(}F_{g}[H,F_{g}]\big{)}\bigg{missing})$

where now $H$ is the second quantization of $\mathbf{h}$ with respect to the
ground state.
The abstract steps performed in the proof for the Araki entropy, are outlined
in the following diagram:

In this way, we have proven the equivalence between our result and the von
Neumann relative entropy, providing a consistency check for it.
The result just presented, gives even a simpler way to compute relative
entropies in the finite dimensional context, we just need a specific dynamics
over $\mathcal{H}$. Once that is known, the relative entropy becomes just a
derivative, with respect to the parameter of the dynamics, of the inner
product over $\mathcal{H}$. In this sense, this result shows once more the
power of working with abstract algebras, where we can perhaps move from one
representation to a more convenient one to compute the same quantity.

###  3.5 Relative entropy for Majorana fields on ultrastatic spacetimes

In this section we aim at applying our result for the relative entropy between
a KMS state and the state obtained from it by the unitary excitation $B(f)$.
These, are taken, for simplicity, as states over the abstract algebra of
Majorana fields. However, as the explicit form of the modular automorphism on
the corresponding von Neumann algebra is generally unknown, we will consider a
configuration in which the underlying Hilbert space of solutions of Dirac’s
equation has a canonically implemented unitary dynamical evolution. This is
achieved by assuming the spacetime $(M,g)$, on which the elements of
$\mathcal{H}$ are defined, to be ultrastatic.

An ultrastatic spacetime $(M,g)$ is:

$M=\mathbb{R}\times\mathcal{C}\hskip 20.0ptg=dt^{2}-h(\mathbf{x})$

where $h(\mathbf{x})$ is a Riemannian metric over $\mathcal{C}$, that is
assumed to be a compact Cauchy surface, and $\mathbf{x}$ denotes the
coordinates on it. In order to define the notion of spinors, we further assume
$\mathcal{C}$ to be parallelizable. We put this as an assumption as, in
general, we do not fix the dimension of $\mathcal{C}$ to $3$ where this will
be the case by Stiefel’s theorem [36].
Under such assumptions, we can write the Dirac’s equation
$(-i\cancel{\nabla}+m)\psi=0$ as:

$\displaystyle 0$
$\displaystyle=(i\gamma^{0}\nabla_{0}+i\gamma^{j}\nabla_{j}+m)\psi$
$\displaystyle i\gamma^{0}\partial_{t}\psi$
$\displaystyle=-(-i\sum_{j=1}^{D}\gamma_{j}\nabla_{j}+m)\psi$ $\displaystyle
i\partial_{t}\psi$
$\displaystyle=-(-i\sum_{j=1}^{D}\gamma_{0}\gamma_{j}\nabla_{j}+\gamma_{0}m)\psi$
$\displaystyle:=\mathbf{h}\psi$

where we have denoted by $D=\dim M$.
To find the solutions to this equation, we do the ansatz:

$\psi(t,\mathbf{x})=e^{-i\epsilon t}\Psi(\mathbf{x}).$

We remark that such an ansatz is possible, just because of the ultrastatic
assumption of the spacetime. Therefore, we can find the stationary solutions
by solving the eigenvalue problem:

$\epsilon\Psi(\mathbf{x})=\mathbf{h}\Psi(\mathbf{x}).$

Let us postpone the task of finding explicit solutions to this equation for
later when we will study a specific model for the spacetime manifold. For the
moment, we just call the corresponding Hilbert space of solutions
$\mathcal{H}$. In this case, as there’re no bounds on the energy, we will have
$\dim\mathcal{H}=+\infty$.
As mentioned at the beginning, we will study Majorana fields. Therefore, the
involution over $\mathcal{H}$ becomes:

$\Gamma v=C\overline{v}$

where $C$ is the charge conjugation operator that, by definition, satisfies:

$C\gamma_{a}C^{-1}=-\overline{\gamma_{a}}$

for $\gamma_{a}=\pi(l_{a})$ a chosen representation of the Dirac algebra (see
Appendix A.4). As a consequence, if we compute:

$\displaystyle\Gamma^{-1}\mathbf{h}\Gamma$
$\displaystyle=-i\Gamma^{-1}\gamma_{0}\Gamma\Gamma^{-1}\gamma_{i}\Gamma\nabla_{i}-\Gamma^{-1}\gamma_{0}\Gamma
m$ $\displaystyle=-i\gamma_{0}\gamma_{i}\nabla_{i}+\gamma_{0}m=-\mathbf{h}.$

Therefore, the involution $\Gamma$ maps eigenspinors with positive energy, to
eigenspinors with negative energy. From this, we can already conclude that the
spectrum must be symmetric with respect to $0$ and, from the form of
$\mathbf{h}$, has a finite gap $(-m,m)$. Furthermore, from
$[\Gamma,\mathbf{h}]_{+}=0$, follows that the associated dynamical evolution
$V_{t}$, commutes with $\Gamma$. Therefore, we will be allowed to raise this
map as a $*$-automorphism over the algebra constructed on $\mathcal{H}$.
Finally, the spatial compacteness, implies the discretization of the allowed
values for the momentum $p$ and consequently also of the energy spectrum.
Now, from the definition of $\Gamma$ and the form of $\mathbf{h}$, we define a
projection $P$, over the space of solutions, corresponding to the projection
into the positive energy eigenspinors of $\mathbf{h}$. Such a projection will,
by construction, satisfy:

$\Gamma P\Gamma=\mathbb{1}-P.$

Therefore, picking the quasifree state $\omega$ over the abstract self dual
CAR algebra, that one can construct over $(\mathcal{H},\Gamma)$, associated
with the above projection $P$, by performing a GNS construction, we obtain a
Fock representation of the fields on the one-particle Hilbert space
$\mathcal{H}_{1}=P\mathcal{H}$.
We now specify this problem for a specific model of the spacetime, where we
will solve the spatial Dirac equation.

####  3.5.1 Majorana fields in 1+1 dimensions

The goal of this subsection is to explicitly compute the relative entropy for
a simple spacetime model. Namely, we will assume the spacetime to be $1+1$
dimensional.
First, we need to find stationary solutions to the Dirac equation, that
corresponds to solve a $1$ dimensional differential equation with spinorial
solutions in $\mathbb{C}^{2}$, as the fields are taken to be Majorana.
Moreover, assuming the spatial sections to be compact intervals
$\mathcal{C}\subset\mathbb{R}$, the equation becomes:

$\epsilon\Psi(x)=\bigg{(}i\gamma_{0}\gamma_{i}\frac{d}{dx}-\gamma_{0}m\bigg{)}\Psi(x).$

By the compacteness assumption, we can approach the problem as that of a
particle in an infinite well potential, imposing vanishing boundary conditions
(by conservation of probability).

To simplify the computations, we choose as representation of the Dirac
algebra:

$\displaystyle C$ $\displaystyle=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\hskip
20.0ptA=\begin{pmatrix}0&i\\\ -i&0\end{pmatrix}$ $\displaystyle\gamma_{0}$
$\displaystyle=A\hskip 20.0pt\gamma_{1}=\begin{pmatrix}0&i\\\
i&0\end{pmatrix}$

that one can check to be a consistent choice. The first consequence is that
$\Gamma v=\overline{v}$ for any solution $v$ of the Dirac equation. Finally,
due to these choices, we make the space of solutions an Hibert space, by
completing it with respect to the inner product:

$\innerproduct{f_{1}}{f_{2}}_{\mathcal{H}}:=\int_{\mathcal{C}}f_{1}^{\dagger}f_{2}(x)dx$

with $dx$ the Lebesgue measure. Namely, our Hilbert space of solutions is
$\mathcal{H}=L^{2}(I,dx;\mathbb{C}^{2})$ and we define on it the self dual CAR
algebra $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. Such an Hilbert space is of
the kind that was constructed in Section 1.3.4.3.

To solve the stationary Dirac equation, we start by finding the eigenvectors
of the one particle hamiltonian:

$\displaystyle\mathbf{h}$ $\displaystyle=i\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\frac{d}{dx}-\begin{pmatrix}0&im\\\ -im&0\end{pmatrix}$
$\displaystyle=\begin{pmatrix}p&im\\\ -im&-p\end{pmatrix}$

where we have introduced the momentum operator $p:=-id/dx$. In this simple
form, we can diagonalize the one-particle Hamiltonian to get the energy
eigenvalues:

$\epsilon_{p}^{\pm}=\pm\sqrt{p^{2}+m^{2}}$

for corresponding eigenvectors:

$v^{\pm}=\frac{1}{\sqrt{2\big{(}(\epsilon_{p}^{\pm})^{2}-p\epsilon_{p}^{\pm}\big{)}}}\begin{pmatrix}-im\\\
p-\epsilon_{p}^{\pm}\end{pmatrix}.$

One can even get an explicit form for the projection $P$:

$P=\frac{1}{2((\epsilon_{p}^{+})^{2}-p\epsilon_{p}^{+})}\begin{pmatrix}m^{2}&-im(p-\epsilon_{p}^{+})\\\
im(p-\epsilon_{p}^{+})&(p-\epsilon_{p}^{+})^{2}\end{pmatrix}$

that, from our choice of $\Gamma$, can be checked to be compatible with
$\Gamma P\Gamma=\mathbb{1}-P$.
For what concerns the $x$ dependence of the solutions, we need to solve the
differential equation, in position space, for the infinite well potential.
Assume for simplicity, without loss of generality, $\mathcal{C}=[0,a]$ for
some $a\in\mathbb{R}$. Then, a general solution is:

$\Psi_{p}^{\pm}(x)=Av_{p}^{\pm}e^{ipx}+Bv_{p}^{\pm}e^{-ipx}.$

Imposing the vanishing boundary conditions:

$\Psi^{\pm}_{n}(x)=Av_{n}^{\pm}\sin\bigg(\frac{\pi n}{a}x\bigg{missing})$

where we have already substituted:

$p=\frac{\pi n}{a}\hskip 20.0ptn\in\mathbb{N}.$

Therefore, by the compacteness of $\mathcal{C}$, we have proven that the
momenta are discretized, by the index $n$, and thus also the energy
eigenvalues are:

$\epsilon_{n}^{\pm}=\pm\sqrt{\bigg{(}\frac{\pi n}{a}\bigg{)}^{2}+m^{2}}.$

Moreover, by computing the inner product between two such eigenspinors:

$\int_{\mathcal{C}}(\Psi^{s}_{n}(x))^{\dagger}\Psi^{s^{\prime}}_{n^{\prime}}(x)dx=\delta_{n,n^{\prime}}\delta_{s,s^{\prime}}|A|^{2}\frac{a}{2}$

for $s,s^{\prime}=\\{+,-\\}$. From the above orthogonality, fix the
normalization to get:

$\Psi^{\pm}_{n}(x)=\sqrt{\frac{2}{a}}v_{p}^{\pm}\sin\bigg(\frac{\pi
n}{a}x\bigg{missing}).$

In this way $\\{\Psi_{n}^{+}(x),\Psi_{n}^{-}(x)\\}_{n\in\mathbb{N}}$ provides
a basis for $\mathcal{H}$. For this reason, our $f\in\mathcal{H}$ such that
$\Gamma f=f$, can be decomposed:

$f(x)=\sum_{n\in\mathbb{N}}(1+\Gamma)(a_{n}^{+}\Psi_{n}^{+}(x)+a_{n}^{-}\Psi_{n}^{-}(x)).$

assuming, without loss of generality, that this is nonzero. Moreover, from
$[\mathbf{h},\Gamma]_{+}=0$ we also see that:

$\displaystyle\mathbf{h}\Gamma\Psi_{n}^{+}$
$\displaystyle=-\Gamma\mathbf{h}\Psi_{n}^{+}$
$\displaystyle=-\epsilon_{n}^{+}\Gamma\Psi_{n}^{+}$
$\displaystyle=\epsilon_{n}^{-}\Gamma\Psi_{n}^{+}.$

That gives $\Gamma\Psi_{n}^{+}=\chi_{n}\Psi_{n}^{-}$ for some
$\chi_{n}\in\mathbb{C}$ that, by the orthonormality of the basis
$\\{\Psi_{n}^{+},\Psi_{n}^{-}\\}_{n\in\mathbb{N}}$, is such that:
$|\chi_{n}|^{2}=1$. Finally, from our choice of $\Gamma$ and the abovely
computed eigenvectors, we conclude: $\chi_{n}=-1$. Therefore, we can rewrite:

$f(x)=\sum_{n\in\mathbb{N}}(a_{n}^{+}-\overline{a}_{n}^{-})\Psi_{n}^{+}(x)+(a_{n}^{-}-\overline{a}_{n}^{+})\Psi_{n}^{-}(x)$

On $(\mathcal{H},\Gamma)$, we construct the the self-dual CAR algebra
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ on which we compute now the relative
entropy. In particular, that will be done between a general quasifree $KMS$
state $\omega_{\beta}$, for the Majorana QFT over the $1+1$ dimensional
ultrastatic spacetime, with associated basis polarization $S$ and the
corresponding state $\omega_{F_{\beta}}$ (where we are adopting the same
notation as last section).
From our result in Prop. 3.1.1, we know we need to compute:

$S(\omega_{\beta}\|\omega_{F_{\beta}})=i\frac{d}{dt}\bigg{|}_{t=0}(f,Sf_{\beta
t})_{\mathcal{H}}$

for $f=\Gamma f$ and the time evolution is the one at the one-particle level,
coming from that on the Fock space of the KMS state (that is why we have also
denoted the $\beta$). In order to explicitly compute this expression, we need
to know the explicit form of $S$. For that purpose, we first need to show that
given a basis polarization, by doubling the Hilbert space, we can construct a
basis projection. Therefore, let us define
$\hat{\mathcal{H}}=\mathcal{H}\oplus\mathcal{H}$ and on it the involution
$\hat{\Gamma}=\Gamma\oplus(-\Gamma)$. With respect to such a doubled Hilbert
space, construct a self-dual CAR algebra
$\mathfrak{A}_{SDC}(\hat{\mathcal{H}},\hat{\Gamma})$ and notice that:

$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)\subset\mathfrak{A}_{SDC}(\hat{\mathcal{H}},\hat{\Gamma})$

As we can see $\mathcal{H}$ as a subset of $\hat{\mathcal{H}}$, identifying it
with $\mathcal{H}\oplus 0$.
Now, let us consider over $\hat{\mathcal{H}}$ the following operator:

$P_{S}=\begin{pmatrix}S&S^{1/2}(1-S)^{1/2}\\\
S^{1/2}(1-S)^{1/2}&(1-S)\end{pmatrix}$

Where $S$ is the basis polarization associated to the state $\omega$. But
then, by direct computation using that $S^{*}=S$ and that $\Gamma
S\Gamma=\mathbb{1}-S$:

$\displaystyle P_{S}^{2}$
$\displaystyle=\begin{pmatrix}S&S^{1/2}(1-S)^{1/2}\\\
S^{1/2}(1-S)^{1/2}&(1-S)\end{pmatrix}=P_{S}$ $\displaystyle P_{S}$
$\displaystyle=P_{S}^{*}$ $\displaystyle\hat{\Gamma}P_{S}\hat{\Gamma}$
$\displaystyle=1-P_{S}$

Where for the last equality, from $\Gamma S\Gamma=\mathbb{1}-S$, we get using
the fact that $\Gamma$ is involutive:

$(\Gamma S^{1/2}\Gamma)(\Gamma
S^{1/2}\Gamma)=(\mathbb{1}-S)^{1/2}(\mathbb{1}-S)^{1/2}$

That gives $\Gamma S^{1/2}\Gamma=(\mathbb{1}-S)^{1/2}$.
Since $P_{S}$ is a basis projection over $\hat{K}$, we take the associated
quasifree state $\omega_{P_{S}}$ over
$\mathfrak{A}_{SDC}(\hat{\mathcal{H}},\hat{\Gamma})$ and notice that this is
related to the initially chosen state $\omega$ by the following result of
Araki (see Lemma $4.6$ in [18]):

######  Lemma 3.5.1.

Let $S$, $P_{S}$, $\hat{\mathcal{H}}$, $\hat{\Gamma}$ as introduced above.
Then, the restriction of the Fock state $\omega_{P_{S}}$ of
$\mathfrak{A}_{SDC}(\hat{\mathcal{H}},\hat{\Gamma})$ to
$\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ is the quasifree state $\omega$.

###### Proof.

Since $\omega_{P_{S}}$ is quasifree, also its restriction will of course
remain such. Now, if we take elements in $\mathcal{H}$ as $f=\hat{f}\oplus 0$
and $g=\hat{g}\oplus 0$, we can compute:

$\omega_{P_{S}}(B^{*}(f)B(g))=(f,P_{S}g)=(\hat{f},S\hat{g})$

That follows from the explicit form of $P_{S}$ given above. ∎

Let us denote by $(\mathcal{K}_{P_{S}},\pi_{P_{S}},\Omega_{P_{S}})$ the GNS
triple assocaited to $\omega_{P_{S}}$. Then, taking advantage of this doubling
of the Hilbert space, we can define the von Neumann algebra associated to
$\mathcal{A}_{SDC}(\mathcal{H},\Gamma)$ with respect to the representation
$\pi_{P_{S}}$ for:

$R_{S}:=\pi_{P_{S}}(\mathfrak{A}_{SDC}(\mathcal{H},\Gamma))^{\prime\prime}$

Then, provided that such an algebra is a factor, by Theorem $3$ in [18] we
have that the basis polarization associated to our starting KMS state with
inverse temperature $\beta$ is:

$S=(1+e^{-\beta\mathbf{h}})^{-1}$

Then, we compute:

$\displaystyle S_{A}$ $\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}(f,Sf_{\beta
t})_{\mathcal{H}}$
$\displaystyle=i\frac{d}{dt}\bigg{|}_{t=0}\sum_{n,m\in\mathbb{N}}\bigg{[}\overline{(a_{n}^{+}-\overline{a}_{n}^{-})}(a_{n}^{+}-\overline{a}_{n}^{-})\frac{e^{-i\beta
t\epsilon_{n}^{+}}}{1+e^{-\beta\epsilon_{n}^{+}}}(\psi_{m}^{+},\psi_{n}^{+})_{\mathcal{H}}+\overline{(a_{n}^{+}-\overline{a}_{n}^{-})}(a_{n}^{+}-\overline{a}_{n}^{-})\frac{e^{i\beta
t\epsilon_{n}^{+}}}{1+e^{\beta\epsilon_{n}^{+}}}(\psi_{m}^{-},\psi_{n}^{-})_{\mathcal{H}}\bigg{]}$
$\displaystyle=\sum_{n\in\mathbb{N}}\beta\epsilon_{n}^{+}\overline{(a_{n}^{+}-\overline{a}_{n}^{-})}(a_{n}^{+}-\overline{a}_{n}^{-})\bigg{[}\frac{1}{1+e^{-\beta\epsilon_{n}^{+}}}-\frac{1}{1+e^{\beta\epsilon_{n}^{+}}}\bigg{]}$
$\displaystyle=\beta\sum_{n\in\mathbb{N}}\epsilon_{n}^{+}\overline{(a_{n}^{+}-\overline{a}_{n}^{-})}(a_{n}^{+}-\overline{a}_{n}^{-})\tanh\bigg(\frac{\beta\epsilon_{n}^{+}}{2}\bigg{missing})$

That, defining the energy content in each mode of $f$ as:

$\tilde{E}_{n}^{+}:=\epsilon_{n}^{+}\overline{(a_{n}^{+}-\overline{a}_{n}^{-})}(a_{n}^{+}-\overline{a}_{n}^{-})$

Becomes:

$S_{A}=\beta\sum_{n\in\mathbb{N}}\tilde{E}_{n}^{+}\tanh\bigg(\frac{\beta\epsilon_{n}^{+}}{2}\bigg{missing})$

That gives us the relative entropy between the two configurations, of the
Majorana field, given by the KMS state $\omega_{\beta}$ and its single unitary
excitation $\omega_{F_{\beta}}$

The analysis carried out here, is easily generalizable to other ultrastatic
spacetime models. For example, we may consider $M=\mathbb{R}\times S$, for $S$
a sphere with radius $r\in[0,a]$. In this case, the problem reduces to solve
the spherically symmetric spatial Dirac equation. However, the form of the
final result will remain the same: the inverse temperature of the considered
KMS state times the energy content of the considered excitation damped by the
hyperbolic tangent term.

###  3.6 Conclusion

We started off introducing the Algebraic approach to Quantum Theory and
provided arguments supporting the considerable advantages of working with
abstract algebras, in particular in the last chapter the strengths of working
with abstract algebras. Subsequently, we introduced fermionic fields, in the
algebraic spirit, over a globally hyperbolic spacetime, putting special
emphasis on the fact that the corresponding algebra falls under the general
category of self-dual CAR algebras.
Moreover, thanks to the algebraic perspective, we were able to better
understand the issues encountered in defining relative entropy for a theory
with an uncountable number of degrees of freedom and that is defined by a type
$III$ von Neumann algebra, such as QFT. On the other hand, the algebraic point
of view simultaneously allowed us to define a generalization of relative
entropy as a measure of distinguishability for states defined over the
abstract von Neumann algebra.
Finally, in order to deal with such an abstract definition, we presented the
case of a coherent excitation of the vacuum for a free scalar Quantum Field
Theory, in which the relative entropy is computed at the one-particle level
([1], [2]). Inspired by this result, we were able to derive an expression for
the relative entropy for a unitary excitation of a quasifree faithful state
given by:

$S(\omega_{F}\,||\,\omega)=i\frac{d}{dt}\bigg{|}_{t=0}(f,Sf_{t})_{\mathcal{H}},$
(3.6)

presented in this work. As became apparent in the last chapter, the
computation of relative entropy then reduces to the knowledge of the form of
the basis polarization $S$.
We argued, presenting a series of examples, how this result can be generalized
to different kinds of excitations and how in those cases, the relative entropy
is expressible in terms of the one of the unitary excitation. Based on our
current understanding, we believe that this result can be generalized further,
for instance, in order to obtain a lower bound for the relative entropy. In
fact, from the assumption of the $\Gamma$-invariant subspace $H$ to be of the
standard type, any $g\in H+iH$ can be written as:

$g=f_{1}+if_{2}\Longrightarrow B(g)=B(f_{1})-iB(f_{2}),$

for $f_{1},f_{2}\in H$. In this case, the associated $B(g)$ does not square to
the identity. This implies that the corresponding vector state representative
cannot be cyclic and separating. Nevertheless, as discussed in [39], we may
define a notion of relative entropy even if only one of the states is
faithful. The expression that one should study involves
$\Delta_{\pi_{\omega}(B(g))\Omega_{\omega},\Omega_{\omega}}$, where the state:

$\pi_{\omega}(B(g))\Omega_{\omega}=(\pi_{\omega}(B(f_{1}))-i\pi_{\omega}(B(f_{2})))\Omega_{\omega},$

is the sum of two cyclic and separating vectors. We believe it to be
worthwhile to investigate this case further in the future as it might lead to
some additional and more general result.

##  Appendix A Appendix

###  A.1 Elements of Lorentzian geometry

This section aims to recall the most important definitions and results of
Lorentzian geometry, which are used in the thesis. The purpose is, by no
means, to give a complete review of Lorentzian geometry, for a complete
treatement, I instead refer to [27], to which our notation will be affine and
I further refer to [55] for a more mathematically rigorous discussion.

Firstly, remember the convention for the spacetime, to be a Lorentzian
Manifold $M$ with metric $g$ of signature $(+---)$. Let me quote a set of
standard definitions:

######  Definition A.1.1.

For a closed subset $C\subset M$, one defines the causal future/past as:

$\displaystyle J^{\pm}(C):=\\{p\in M|$
$\displaystyle\exists\,\,\mathrm{future}/\mathrm{past}\,\,\mathrm{directed}\,\,\mathrm{causal}\,\,\mathrm{curve}\,\gamma(\tau)$
$\displaystyle\mathrm{and}\,\tau_{1}\geq\tau_{0}\,\,\mathrm{s.t.}\,\,\gamma(\tau_{0})\in
C,\gamma(\tau_{1})=p\\}$

######  Definition A.1.2.

Let $S\subset M$ be a closed achronal set. We define the future/past domain of
dependence or future/past causal developement of $S$ as:

$D^{\pm}(S):=\\{p\in
M|\mathrm{Every}\,\,\mathrm{past/future}\,\,\mathrm{inextendible}\,\,\mathrm{causal}\,\,\mathrm{curve}\,\,\mathrm{through}\,\,p\,\,\mathrm{intersects}\,\,S\\}.$

The domain of dependence of $S$ is then defined as: $D(S)=D^{+}(S)\cup
D^{-}(S)$

######  Definition A.1.3.

Given $\mathcal{O}\subset M$, we define its spacelike complement
$\mathcal{O}^{\prime}$ as the set of all points that are spacelike separated
with all points of $\mathcal{O}$:

$\mathcal{O}^{\prime}:=\mathrm{int}\\{p\in M|p\not\in J^{\pm}(q)\,\,\forall
q\in\mathcal{O}\\}$

and its causal completion as $\mathcal{O}^{\prime\prime}$. It is always true
that $(\mathcal{O}^{\prime\prime})^{\prime}=\mathcal{O}^{\prime}$. We say that
$\mathcal{O}$ is causally complete if
$\mathcal{O}=\mathcal{O}^{\prime\prime}$. Finally, let us denote the set of
all causally complete regions on $M$ as $\mathcal{K}$.

######  Definition A.1.4.

A curve $\gamma:I\to M$ has a future (past) endpoint $p\in M$ if for any open
neighborhood $U$ containing $p$, there exist some value $\tau_{0}\in I$ for
the affine parameter such that $\gamma(\tau)\in U$ for $\tau>\tau_{0}$ (resp.
$\tau<\tau_{0}$)

######  Definition A.1.5.

A curve that does not have future or past endpoints is called future/past
inextendible.

######  Definition A.1.6.

A Cauchy surface $\Sigma$ is a hypersurface of $(M,g)$, such that any
inextendible causal curve intersects $\Sigma$ exactly once111One can show this
to be equivalent to the definition of a Cauchy surface as the hypersurface
whose causal developement is the entire spacetime manifold $M$.

######  Definition A.1.7.

A spacetime $(M,g)$ is said to be globally hyperbolic, if it possesses a
Cauchy surface.

For a globally hyperbolic spacetime, we have the following known result:

######  Proposition A.1.1.

Let $(M,g)$ be a globally hyperbolic spacetime. For any compact
$K,K^{\prime}\subset M$, $J^{\pm}(K)$ is closed, $J^{+}(K)\cap
J^{-}(K^{\prime})$ is compact and, for any Cauchy surface $\Sigma$, the
intersection $\Sigma\cap J^{\pm}(K)$ is compact.

###  A.2 The Reeh-Schlieder theorem

We will present the theorem in the case of a free scalar $QFT$ on Minkowski
spacetime but it can be easily generalized also to the case of an arbitrary
$QFT$. I refer to [56] for the original proof, while here we present a
revisited version of it using the Schwarz reflection principle.

######  Theorem A.2.1 (Reeh-Schlieder).

Consider a free scalar Quantum Field Theory on Minkowski spacetime
$M_{D}:=(\mathbb{R}^{D},\eta)$. Then, given an open subset $\mathcal{V}$ of a
Cauchy surface $\Sigma$ with neighbourhood $U_{\mathcal{V}}\subset M_{D}$ and
an arbitrary number of test functions
$f_{1},\cdots,f_{n}\in\mathcal{C}^{\infty}_{0}(U_{\mathcal{V}},\mathbb{C})$
the set of vectors:

$\phi_{f_{1}}\cdots\phi_{f_{n}}\ket{\Omega}$

Defines a dense subset of the Hilbert space $\mathcal{H}$ of the quantum field
theory and $\ket{\Omega}$ is the cyclic and separating vector representing the
vacuum of the theory. The above notation means:

$\phi_{f}=\int_{M_{D}}d^{D}x\phi(x)f(x)$

###### Proof.

The idea of the proof is the following: we take for all
$f_{1},\cdots,f_{n}\in\mathcal{C}^{\infty}_{0}(U_{\mathcal{V}},\mathbb{C})$
and all $n\in\mathbb{N}$ any vector $\ket{\chi}\in\mathcal{H}$ such that:

$0=\bra{\chi}\phi_{f_{1}}\cdots\phi_{f_{n}}\ket{\Omega}\iff
0=\bra{\chi}\phi(x_{1})\cdots\phi(x_{n})\ket{\Omega}$

i.e. that is in the orthogonal complement in $\mathcal{H}$ of the space
generated by the field polynomial for $x_{1},\dots,x_{n}\in U_{\mathcal{V}}$,
then we want to show that such $\ket{\chi}$ is trivial, proving that the
orthogonal complement contains just the trivial vector.
Let us start by considering a future directed timelike vector $\mathbf{t}$ and
let $u\in\mathbb{R}$. Consider the following timelike translation of the
$n$-th coordinate: $x_{n}\mapsto x_{n}+u\mathbf{t}$. Consider then the
function:

$g(u):=\bra{\chi}\phi(x_{1})\cdots\phi(x_{n}+u\textbf{t})\ket{\Omega}$

But, from the axioms of an AQFT that we have listed before, we know that the
fields must transform covariantly:

$\phi(x_{n}+u\textbf{t})=\exp\big(iHu\big{missing})\phi(x_{n})\exp\big(-iHu\big{missing})$

Where $\exp\big(iHu\big{missing})$ is a time translation operator and $H$ is
the self adjoint operator that is positive semidefinite such that
$H\ket{\Omega}=0$. It follows that:

$g(u)=\bra{\chi}\phi(x_{1})\cdots\exp\big(iHu\big{missing})\phi(x_{n})\ket{\Omega}$

Since $0=\bra{\chi}\phi(x_{1})\cdots\phi(x_{n})\ket{\Omega}$ as long as
$x_{1},\dots,x_{n}\in U_{\mathcal{V}}$, we have that $g(u)=0$ as long as $u$
is small. So $g(u)=0$ for $u\in I=[-\varepsilon,\varepsilon]$.
If we now displace $u$ in the complex plane, as long as the displacement is in
the upper half plane, the function $g(u)$ is holomorphic. This can be seen
from the $u$-dependence of $g(u)$ that is just in the exponential factor that,
by the spectral theorem, can be decomposed as:

$\exp\big(iHu\big{missing})=\int_{\mathbb{R}^{+}}\exp\big(i\lambda
u\big{missing})dE_{\lambda}$

But the function $\exp\big(i\lambda u\big{missing})$ is holomorphic for
$\lambda>0$ and $\imaginary(u)>0$ since:

$\exp\big(i\lambda
u\big{missing})=\exp\big(i\lambda\real(u)\big{missing})\exp\big(-\lambda\imaginary(u)\big{missing})$

and if we study the holomorphicity of this function in the upper half plane,
we see that there’re no pole. So, from Cauchy integral theorem, whenever we
integrate on a closed curve in the upper half plane the function
$\exp\big(i\lambda u\big{missing})$, we get zero. But then, by Morera’s
theorem, this implies that $g(u)$ is holomorphic in the upper half plane.

Figure A.1: The choice of a circle is random, we should have chosen any other
region symmetric w.r.t. the real line, also a square or a ellipse

We can now use a Lemma of complex analysis:

######  Lemma A.2.1 (Schwarz reflection principle).

Let $U$ be a region symmetric about the real axis. If $f(z)$ is a holomorphic
function in the part of $U$ in the upper half plane and is such that:

$\lim_{Im(z)\to 0+}f(z)=0$

Then $f(z)$ extends to a holomorphic function on $U$

###### Proof.

See Section $11.10$ of [57]. ∎

This lemma allows us to say that the function $g(u)$ is not just holomorphic
in the upper half plane, but has domain of analiticity that extends through
the real axis also to a piece of the lower half plane, provided that the
symmetric region is taken with respect to $I$ or a subset of it. As a
consequence, $g(u)$ for $u\in I$ is a holomorphic function that vanishes
identically on the closed interval $I$. A holomorphic function that vanishes
identically on a closed interval of the real line must be identically zero for
all $u\in\mathbb{R}$, so $g(u)=0$. This means that, no matter how much we move
in the timelike direction $\mathbf{t}$ the coordinate $x_{n}$, the inner
product $g(u)$ is identically zero.
This was just a displacement in a chosen timelike direction of $x_{n}$, if we
want to extend this for any $x_{n}\in M_{D}$ we can compose timelike
translations (either future or past directed) to reach any point in $M_{D}$

Where in the case in which the timelike translation is past directed we will
perform the holomorphicity argument starting from the lower half plane. In
this way we have shown that, independently from the choice of $x_{n}\in
M_{D}$: $0=\bra{\chi}\phi(x_{1})\cdots\phi(x_{n})\ket{\Omega}$.
We can now repeat the argument for all the other coordinates in an iterative
way. Namely, if we consider now the coordinate $x_{n-1}$ and another timelike
vector $\mathbf{t}^{\prime}$ we perform the following timelike translation:

$x_{n-1}\mapsto x_{n-1}+u\mathbf{t}^{\prime}\hskip 20.0ptx_{n}\mapsto
x_{n}+u\mathbf{t}^{\prime}$

Being aware that for any $x_{n}$ we have
$0=\bra{\chi}\phi(x_{1})\cdots\phi(x_{n})\ket{\Omega}$. Then, repeating the
same above arguments, we prove that also the choice of $x_{n-1}\in M_{D}$ does
not affect the above quantity. Repeating the argument for all $n\in\mathbb{N}$
we have:

$0=\bra{\chi}\phi(x_{1})\cdots\phi(x_{n})\ket{\Omega}\,\,\,\,\,\,\forall
x_{1},\dots,x_{n}\in M_{D}$

Showing in this way, from axiom A4 of an AQFT, that
$\phi_{f_{1}}\cdots\phi_{f_{n}}\ket{\Omega}$ for
$f_{1},\cdots,f_{n}\in\mathcal{C}^{\infty}_{0}(U_{\mathcal{V}},\mathbb{C})$
generates a dense subset of $\mathcal{H}$ ∎

Generalizations of the theorem on curved spacetimes were given by Verch in
[58] for the case of a quasifree ground-state of a free scalar massive field
theory on an ultrastatic spacetime, by Strohmaier [59] for general local
Quantum field theories on stationary spacetimes and by [60] in the case of
spacetimes diffeomorphic to spacetimes with the Reeh-Schlieder property.

###  A.3 The Bisognano-Wichmann theorem

The Bisognano-Wichmann theorem is a statement regarding the algebras of a
general QFT localized in wedge-like regions. In this extent is one of the few
known cases in which the modular flow of the theory is said to be geometric
as, due to this theorem, is expressed in terms of a geometric transformation
on the wedge. This result was first proven to hold by the authors for the free
scalar case in [23] and generalized later for arbitrary QFT by the same
authors in [24]. In both cases, the algebras are localized on Minkowski
spacetime, a generlization of this result to more general curved backgrounds
admitting wedge-like regions was obtained in [61].
In this section I will just present the original version of the theorem for
arbitrary Quantum field theories on Mikowski spacetimes and refer to the cited
litterature for its generalization to curved backgrounds.

Let us start recalling the definition of Rindler wedges on Minkowski spacetime
$M_{D}$:

$W_{R}=\\{x\in M_{D}:x_{1}>|x_{0}|\\}$

called the right wedge, and:

$W_{L}=\\{x\in M_{D}:x_{1}<-|x_{0}|\\}$

called the left wedge. We will denote as usual the local algebras of operators
as $\mathfrak{A}(\mathcal{O})$ where $\mathcal{O}\subset M_{D}$. These are
assumed to be $C^{*}$-algebras, thus, when represented, give rise to von
Neumann algebras still denoted as $\mathfrak{A}(\mathcal{O})$ for simplicity
(see Theorem 1.1.2).
Moreover, if the local algebras have a $\mathbb{Z}_{2}$-grading we can define
the twisted algebra:

$\mathfrak{A}(\mathcal{O})^{t^{\prime}}=\\{ZXZ^{-1}|X\in\mathfrak{A}(\mathcal{O})\\}$

Where $Z$ is the twisting operator we defined in Definition 1.2.4. In
particular, from the discussion we had after the Definition 1.2.4, we see that
the only difference occurs if the net of local algebrs
$\mathfrak{A}(\mathcal{O})$ is homogeneous Fermi because in the Bose case:
$\mathfrak{A}(\mathcal{O})^{t^{\prime}}=\mathfrak{A}(\mathcal{O})$.
Considering the von Neumann algebras $\mathfrak{A}(W_{R})$ and
$\mathfrak{A}(W_{L})^{t^{\prime}}$, recalling that from Tomita-Takesaki
modular theory on von Neumann algebras we have the existence of modular
conjugation and flow, the statement of the Bisognano-Wichmann theorem is:

######  Theorem A.3.1 (Bisognano-Wichmann).

If $J_{W_{R}}$ and $\Delta_{W_{R}}$ denote the modular operators for the pair
$(\mathfrak{A}(W_{R}),\Omega)$, then:

$J_{W_{R}}=ZU(R_{23}(\pi),0)\Theta\hskip
20.0pt\Delta_{W_{R}}^{it}=U(\Lambda_{W_{R}}(t),0)$

Where: $U(R_{23}(\pi),0)$ is the unitary representation over the Hilbert space
of a rotation of $\pi$ in the plane of the free coordinates,
$U(\Lambda_{W_{R}}(t),0)$ is the unitary representation of a boost of
parameter $t$ in the direction $x_{1}$ and $\Theta$ is a CPT transformation
such that:

$\displaystyle\Theta^{2}$ $\displaystyle=U(-\mathbb{1},0)$
$\displaystyle\Theta\Omega$ $\displaystyle=\Omega$ $\displaystyle\Theta
U(g,x)\Theta^{-1}$ $\displaystyle=U(g,-x)$

for $g\in Spin^{0}_{1,3}$ and $x\in M_{D}$.
Finally, it holds
$J_{W_{R}}\mathfrak{A}(W_{R})J_{W_{R}}=\mathfrak{A}(W_{L})^{t^{\prime}}$

###### Proof.

For a proof of the theorem I refer to Lemma $7$ and Theorem $1$ in the
original work [24]. Notice that, in the original work, the authors deal with
field algebras that are not von Neumann algebras. For this reason, their proof
starts by showing the existence of the $J_{W_{R}}$ operator of the above form.
With that, they prove that for any $X$ (possibly unbounded):

$J_{W_{R}}U(\Lambda_{W_{R}}(i\pi),0)X\Omega=X^{*}\Omega$

That, in the case in which the algebras are turned into von Neumann, in
analogy with Tomita-Takesaki modular theory, gives the claimed result. ∎

The result for the modular conjugation $J_{W_{R}}$, modulo a twisting factor
$Z$, shows that is a $CRT$ symmetry. This is the transformation in coordinate:
$t\to-t$, $x_{1}\to-x_{1}$ and $x_{i}\to x_{i}$ for all $i>2$. In this way
becomes more intuitive, from a geometrical point of view , that
$J_{W_{R}}\mathfrak{A}(W_{R})J_{W_{R}}=\mathfrak{A}(W_{L})^{t^{\prime}}$ as
such a transformation of coordinates maps exactly $W_{R}\to W_{L}$. For this
reason, and the form of the modular flow, that this theorem is often mentioned
as giving a geometric action of the modular theory.

###  A.4 Clifford algebra

This section is devoted to a short survey of Clifford algebras that will allow
us to study the universal covering group of the Lorentz group, used to define
and relate Spin and Frame bundles. Let $\mathbb{R^{r,s}}$ be the real vector
space of dimension $n=r+s$ and equip it with a non-degenerate bilinear form
$\Omega_{ab}$ that has $r$ many positive and $s$ many negative eigenvalues. A
special example of this is Minkowski spacetime $M_{D}=\mathbb{R}^{1,3}$, where
the bilinear form is the usual Minkowski metric tensor that in the orthonormal
basis $\\{l_{a}\\}_{a=0,\dots,3}$ is: $\eta=\mathrm{diag}(1,-1,-1,-1)$.

######  Definition A.4.1.

The Clifford algebra $Cl_{r,s}$ of $\mathbb{R}^{r,s}$ is defined as the
associative unital algebra generated by an orthonormal basis $e_{a}$ of
$\mathbb{R}^{r,s}$ subject to the Clifford relations222One can show that the
definition of the Clifford relations is independent on the choice of the basis
$e_{a}$ see [15] Prop. 1.1:

$e_{a}e_{b}+e_{b}e_{a}=2\Omega_{ab}\mathbb{1}$

We can identify the subspaces of $Cl_{r,s}$ spanned by monomials of even or
odd degree in the basis vectors, and call them $Cl_{r,s}^{0}$ resp.
$Cl_{r,s}^{1}$.
We define the Dirac algebra as the Clifford algebra of Minkowski:
$D:=Cl_{1,3}$. Therefore the Clifford condition becomes in this case:

$l_{a}l_{b}+l_{b}l_{a}=2\eta_{ab}\mathbb{1}$

Where now $\\{l_{a}\\}_{a=0,1,2,3}$ is an orthonormal basis of Minkowski
spacetime.

Recognize, that the real vector space itself is a subspace of the algebra,
namely $\mathbb{R}^{r,s}\subset Cl_{r,s}$ generated by the monomials of order
$1$ in the basis $e_{a}$.

######  Remark A.4.1.

The even subspace $Cl_{r,s}^{0}$ is a subalgebra as it is closed under product
of its elements, while for the odd case the product of two elements will give
an even monomial. Despite this, both subspaces are well defined, as the
Clifford relations are purely even and thus, independently from the order of
the monomials, we have a sum of even monomials.

######  Remark A.4.2.

The dimension of the Clifford algebra is $2^{r+s}$. To see it consider as
basis of $\mathbb{R}^{r+s}$ an orthogonal basis $e_{a}$, then the Clifford
relations give:

$\displaystyle e_{i}e_{j}$ $\displaystyle=-e_{j}e_{i}$ $\displaystyle
e_{i}^{2}$ $\displaystyle=\Omega_{ii}\mathbb{1}$

but this implies that, in searching for all the independent monomials, we can
define a unique independent order and we cannot have repetitions of $e_{j}$.
Then:

$\dim Cl_{r,s}=\sum_{k=0}^{r+s}\begin{pmatrix}r+s\\\ k\end{pmatrix}=2^{r+s}$

As it is customary in introducing the Dirac fields on Minkowski spacetime, and
from the relations with Dirac $\gamma$ matrices that we will introduce later,
we shall call the volume element $l_{5}:=l_{0}l_{1}l_{2}l_{3}$.
The real Dirac algebra $D$ can be represented as complex matrices via a
complex representation: $\pi:D\to M(n,\mathbb{C})$ for some $n\in\mathbb{N}$.
We look for such a representation because, in treating Dirac fields, we want
the Dirac algebra to act on spinor fields that are vectors in a
$\mathbb{C}^{n}$. For this purpose, we quote a theorem due to Pauli (see [62])
about the representation theory of the Dirac algebra:

######  Theorem A.4.1 (Fundamental Theorem).

The Dirac abstract algebra $D$ is simple333A simple abstract algebra
$\mathfrak{A}$ is an algebra for which each homomorphism that has as domain
the entire algebra $\mathfrak{A}$ is injective and has a unique irreducible
complex representation up to equivalence. This representation is denoted as:

$\displaystyle\pi_{0}:D$ $\displaystyle\to M(4,\mathbb{C})$ $\displaystyle
l_{a}$ $\displaystyle\mapsto\pi_{0}(l_{a})=:\gamma_{a}$

that gives the famous Dirac gamma matrices $\gamma_{a}$:

$\gamma_{0}:=\begin{pmatrix}0&\mathbb{1}_{2\times 2}\\\ \mathbb{1}_{2\times
2}&0\end{pmatrix},\hskip 15.0pt\gamma_{i}:=\begin{pmatrix}0&-\sigma_{i}\\\
\sigma_{i}&0\end{pmatrix}$

Where $\sigma_{i}$ are the Pauli matrices.
The equivalence with another complex irreducible representation $\pi$ of $D$
is impllemented by $\pi(S)=L\pi_{0}(S)L^{-1}$ for all $S\in D$, where $L\in
GL(4,\mathbb{C})$ unique up to a non-zero complex factor. Define, for
notational convenience, also $\gamma_{5}:=\pi_{0}(l_{5})$

This theorem, defines and gives an explicit form to the famous Dirac $\gamma$
matrices, that were used in defining the Dirac derivative in Section 1.3.
Furthermore, in that Section, the notions of adjoint and charge conjugation of
Dirac spinors were used starting from matrices $A,C\in GL(4,\mathbb{C})$. Now,
we define them and list their properties:

######  Definition A.4.2.

Let $\pi$ be an irreducible complex representation of the Dirac algebra. We
define the matrices $A,C\in GL(4,\mathbb{C})$ via the conditions:

$\displaystyle A$ $\displaystyle=A^{*},\hskip
15.0pt\pi(l_{a})^{*}=A\pi(l_{a})A^{-1},\hskip
15.0ptA\pi(n)=An^{a}\gamma_{a}>0$ $\displaystyle\overline{C}C$
$\displaystyle=\mathbb{1},\hskip
15.0pt-\overline{\pi(l_{a})}=C\pi(l_{a})C^{-1}$

for all future pointing timelike vectors $n$ and the Hermitean conjugation and
adjoint are the standard ones on $\mathbb{C}^{4}$.

######  Remark A.4.3.

As both sets of gamma matrices $\pi(l_{a})^{*}$ and $-\overline{\pi(l_{a})}$
satisfy the Clifford relations, from the above theorem, the matrices $A,C$ are
uniquely determined up to a multiplicative constant.

In fact, one can prove that such matrices always exist given a complex
irreducible representation:

######  Theorem A.4.2.

For any complex irreducible representation $\pi$ of $D$, there are $A,C\in
GL(4,\mathbb{C})$ which satisfy properties of Definition A.4.2 with respect to
$\pi$. We also have $A=-C^{*}\overline{A}C$.
Moreover, $A$ is uniquely determined up to a positive factor while $C$ up to a
phase factor.
Let $A_{i},C_{i}\in GL(4,\mathbb{C})$ for $i=1,2$ satisfying Definition A.4.2
with respect to irreducible complex representations $\pi_{i}$ of $D$. Then
there exist $L\in GL(4,\mathbb{C})$, unique up to a sign, such that
$L^{*}A_{1}L=A_{2}$, $\overline{L}^{-1}C_{1}L=C_{2}$ and
$\pi_{2}=L^{-1}\pi_{1}L$ on $D$

###### Proof.

We start proving the existence for the representation $\pi_{0}$. For that,
take $A_{0}:=\gamma_{0}$ and $C_{0}:=\gamma_{2}$. Then, we need to check the
properties in Definition A.4.2 to be fulfilled. First of all, as
$\pi_{0}(l_{a})=\gamma_{a}$, for what concerns $A_{0}$, we have:

$\displaystyle\gamma_{0}$ $\displaystyle=(\gamma_{0})^{*}$
$\displaystyle\gamma_{a}^{*}$
$\displaystyle=-\gamma_{a}=\gamma_{0}\gamma_{a}\gamma_{0}$
$\displaystyle\gamma_{0}n^{a}\gamma_{a}$
$\displaystyle=\begin{pmatrix}n^{0}\mathbb{1}+n^{i}\sigma_{i}&0\\\
0&n^{0}\mathbb{1}-n^{i}\sigma_{i}\end{pmatrix}>0$

Where the last inequality follows from $\det(n^{0}\mathbb{1}\pm
n^{i}\sigma_{i})=n_{0}^{2}-|\mathbf{n}|=1$ and also $\Tr(n^{0}\mathbb{1}\pm
n^{i}\sigma_{i})=2n^{0}>0$. For what concerns $C_{0}$:

$\overline{\gamma_{2}}\gamma_{2}=\begin{pmatrix}0&\sigma_{2}\\\
-\sigma_{2}&0\end{pmatrix}\begin{pmatrix}0&-\sigma_{2}\\\
\sigma_{2}&0\end{pmatrix}=\mathbb{1}$

and also:

$\displaystyle\gamma_{2}\gamma_{a}\gamma_{2}^{-1}$
$\displaystyle=(-\gamma_{a}\gamma_{2}+2\eta_{a2})\gamma_{2}^{-1}$
$\displaystyle=-\gamma_{a}+2\eta_{a2}\gamma_{2}^{-1}$

that, for $a\neq 2$, is precisely $-\overline{\gamma_{a}}$ from the reality of
gamma matrices and as the second term vanishes. For $a=2$, we have that
$-\overline{\gamma_{2}}=\gamma_{2}$ and since $\gamma_{2}^{-1}=-\gamma_{2}$:

$-\gamma_{2}-2\gamma_{2}^{-1}=-\gamma_{a}+2\gamma_{2}=-\overline{\gamma_{2}}$

In this same representation, we can also compute the relation between the $A$
and $C$ matrices, claimed in the statement of the theorem, to hold:

$-C_{0}^{*}\overline{A_{0}}C_{0}=\gamma_{2}\gamma_{0}\gamma_{2}=-\gamma_{0}\gamma_{2}^{2}=\gamma_{0}=A_{0}$

In the case of a general complex representation $\pi$, we use the fundamental
theorem to write $\pi(l_{a})=K^{-1}\gamma_{a}K$ for $K\in GL(4,\mathbb{C})$.
But then, relating $A=K^{*}A_{0}K$ and $C=\overline{K}^{-1}C_{0}K$ and using
the proved properties for $A_{0},C_{0}$, we have that the thus defined $A$ and
$C$ still satisfy Definition A.4.2:

$\displaystyle(K^{*}A_{0}K)^{*}$ $\displaystyle=K^{*}A_{0}K$ $\displaystyle
A\pi(l_{a})A^{-1}$
$\displaystyle=K^{*}A_{0}KK^{-1}\gamma_{a}KK^{-1}A_{0}^{-1}(K^{-1})^{*}$
$\displaystyle=K^{*}\gamma_{a}^{*}(K^{-1})^{*}=\pi(l_{a})^{*}$ $\displaystyle
A\pi(n)$ $\displaystyle=K^{*}A_{0}\pi_{0}(n^{a}l_{a})K>0$
$\displaystyle\overline{(\overline{K}^{-1}C_{0}K)}\overline{K}^{-1}C_{0}K$
$\displaystyle=K^{-1}\overline{C}_{0}C_{0}K$
$\displaystyle=K^{-1}K=\mathbb{1}$ $\displaystyle C\pi(l_{a})C^{-1}$
$\displaystyle=\overline{K}^{-1}C_{0}KK^{-1}\gamma_{a}KK^{-1}C_{0}^{-1}\overline{K}$
$\displaystyle=\overline{K}^{-1}C_{0}\gamma_{a}C_{0}^{-1}\overline{K}$
$\displaystyle=-\overline{K}^{-1}\overline{\gamma_{a}}\overline{K}=-\overline{\pi(l_{a})}$

and they also satisfy the general relation:

$\displaystyle-C^{*}\overline{A}C$
$\displaystyle=-K^{*}C_{0}^{*}(\overline{K}^{-1})^{*}\overline{K}^{*}\overline{A}_{0}\overline{K}\overline{K}^{-1}C_{0}K$
$\displaystyle=-K^{*}C_{0}^{*}\overline{A}_{0}C_{0}K$
$\displaystyle=K^{*}A_{0}K=A$

For the statement regarding the uniqueness, we start noticing that the
matrices $A$ and $C$ are determined uniquely up to non-zero complex factors as
noticed in the remark A.4.3 that we may call $a,c$. Because $A=A^{*}$ we must
have $a\in\mathbb{R}$ and because of $\overline{C}C=1$ we have $|c|=1$ proving
already that $C$ is determined up to a phase factor. Moreover, as we also have
$A\pi(n)>0$ we must have $a>0$.
For the last part, fix a $K\in GL(4,\mathbb{C})$ such that
$\pi_{1}=K\pi_{2}K^{-1}$ by the fundamental theorem. Set
$A^{\prime}_{2}:=K^{*}A_{1}K$ and $C^{\prime}_{2}:=\overline{K}^{-1}C_{1}K$.
Then, with respect to $\pi_{2}$:

$\displaystyle A^{\prime}_{2}\pi_{2}(l_{a})A^{\prime}_{2}{}^{-1}$
$\displaystyle=K^{*}A_{1}\pi_{1}(l_{a})A_{1}^{-1}(K^{*})^{-1}$
$\displaystyle=K^{*}\pi_{1}(l_{a})^{*}(K^{*})^{-1}$
$\displaystyle=(K^{-1}\pi_{1}(l_{a})K)^{*}=\pi_{2}(l_{a})^{*}$ $\displaystyle
A^{\prime}_{2}\pi_{2}(n)$ $\displaystyle=K^{*}A_{1}\pi_{1}(n)K>0$
$\displaystyle C^{\prime}_{2}\pi_{2}(l_{a})C^{\prime}_{2}{}^{-1}$
$\displaystyle=\overline{K}^{-1}C_{1}\pi_{1}(l_{a})C_{1}^{-1}\overline{K}$
$\displaystyle=\overline{K}^{-1}(-\overline{\pi_{1}(l_{a})})\overline{K}=-\overline{\pi_{2}(l_{a})}$

So, as also $A^{\prime}_{2}$ and $C^{\prime}_{2}$ satisfy A.4.2 with respect
to $\pi_{2}$ as well as $A_{2}$ and $C_{2}$, by the above uniqueness:

$A^{\prime}_{2}=aA_{2}\hskip 20.0ptC^{\prime}_{2}=cC_{2}$

for $a>0$ and $|c|=1$. Then, the desired matrix $L\in GL(4,\mathbb{C})$ must
be $L=zK$ for some $z\neq 0$ still by the fundamental theorem. In particular
to have the right intertwining relations:

$\displaystyle A_{2}$
$\displaystyle=L^{*}A_{1}L=|z|^{2}K^{*}A_{1}K=|z|^{2}A^{\prime}_{2}$
$\displaystyle C_{2}$
$\displaystyle=\overline{L}^{-1}C_{1}L=\overline{z}^{-1}z\overline{K}^{-1}C_{1}K=\overline{z}^{-1}zC^{\prime}_{2}$

we must have $|z|^{2}=a$ and $z=c\overline{z}$ that fixes $z$ up to a sign. ∎

We are now in the position of introducing the universal covering group of the
Lorentz group, using the elegant formalism just introduced:

######  Definition A.4.3.

The Pin and Spin groups of $Cl_{r,s}$ are defined as:

$\displaystyle Pin_{r,s}$ $\displaystyle:=\\{S\in Cl_{r,s}|S=u_{1}\cdots
u_{k},\hskip 15.0ptk\in\mathbb{N},\hskip 15.0ptu_{i}\in\mathbb{R}^{r,s},\hskip
15.0ptu_{i}^{2}=\pm\mathbb{1}\\}$ $\displaystyle Spin_{r,s}$
$\displaystyle:=Pin_{r,s}\cap Cl^{0}_{r,s}$

In fact, Pin is a group: it certainly contains the inverse,as
$u_{i}^{2}=\pm\mathbb{1}$, is closed under composition, the product operation
is by definition associative and the neutral element is given by
$S=\mathbb{1}$. This statement, can be seen also from the following equivalent
characterization444The notion of determinant and trace for elements in $D$,
are defined via the trace and determinant of the corresponding images under
the unique, up to equivalence, complex representation of the Dirac algebra:

######  Proposition A.4.1.

$Pin_{1,3}=\\{S\in D|\det X=1;\,\,\forall v\in M_{D}\,\,\,\,SvS^{-1}\in
M_{D}\\}$

In order to prove this proposition, we first need the following lemma:

######  Lemma A.4.1.

We have $l_{5}^{2}=-\mathbb{1}$ and:

$l_{5}vl_{5}^{-1}=-vl_{5}l_{5}^{-1}=-v,\hskip 15.0ptv\in M_{D}$ (A.1)

Moreover, if $u\in M_{D}$ has $u^{2}=\|u\|^{2}\mathbb{1}\neq 0$, with norm in
$M_{D}$, then $u^{-1}=\frac{1}{\|u\|^{2}}u$ and $v\mapsto-uvu^{-1}$ defines a
reflection of $M_{D}$ in the hyperplane perpendicular to $u$

###### Proof of lemma.

From the Clifford relations, we have $l_{5}l_{a}=-l_{a}l_{5}$ for each $a$, as
one always needs to perform $3$ switches. This implies that Eq. (A.1) holds,
and by direct computation one sees $l_{5}^{2}=-\mathbb{1}$.
Finally, compute:

$-uvu^{-1}=v-(uv+vu)u^{-1}=v-\frac{2(u,v)}{\|u\|^{2}}u$

Where we have used the Clifford relations and denoted with $(\cdot,\cdot)$ the
inner product on $M_{D}$. ∎

###### Proof of proposition.

Whenever we take an $S\in Pin_{1,3}$ the map $v\mapsto SvS^{-1}$ on $M_{D}$ is
just a product of reflections from the above lemma, so $SvS^{-1}\in M_{D}$ for
all $M_{D}$. Now, from the identity $u^{2}=\|u\|^{2}$ for all $u\in M_{D}$:

$\displaystyle\det u^{2}$
$\displaystyle=\det\big(\|u\|^{2}\mathbb{1}\big{missing})$ $\displaystyle(\det
u)^{2}$ $\displaystyle=\|u\|^{8}\det\big(\mathbb{1}\big{missing})$
$\displaystyle\det u$ $\displaystyle=\|u\|^{4}$

Then as $\|u_{i}\|=1$ for the $u_{i}$ in the definition of the $Pin$ group, we
have: $\det S=1$.
For the converse suppose that $S\in D$ is such that $\det S=1$ and
$SvS^{-1}\in M_{0}$ for all $v\in M_{0}$. If we look at:

$vu+uv=2v^{a}u_{a}\mathbb{1}$

we notice that the right hand side of this remains invariant under the adjoint
action of $S$ on the vectors. As a consequence $S$ preserves the Minkowski
inner product, hence the adjoint action of $S$ determines a Lorentz
transformation $\Lambda$. In general, a Lorentz transformation can always be
written as a finite product of reflections in non-null hyperplanes (see
Theorem $3.20$ of [63]), so we can take them be $v\mapsto-u_{i}vu_{i}$ as this
is a reflection from previous lemma. Define then:

$T:=\left\\{\begin{aligned} u_{k}\cdots u_{1}\hskip
15.0pt\mathrm{if}\,\,k\,\,\mathrm{is\,\,even}\\\ u_{k}\cdots u_{1}l_{5}\hskip
15.0pt\mathrm{if}\,\,k\,\,\mathrm{is\,\,odd}\\\ \end{aligned}\right.$

By definition we have $T\in Pin_{1,3}$ and from the above lemma:

$T^{-1}:=\left\\{\begin{aligned} u_{1}^{-1}\cdots u_{k}^{-1}\hskip
15.0pt\mathrm{if}\,\,k\,\,\mathrm{is\,\,even}\\\ -l_{5}u_{1}^{-1}\cdots
u_{k}^{-1}\hskip 15.0pt\mathrm{if}\,\,k\,\,\mathrm{is\,\,odd}\\\
\end{aligned}\right.$

So in both cases, still from the above lemma:

$\displaystyle TvT^{-1}$ $\displaystyle=\left\\{\begin{aligned} &u_{k}\cdots
u_{1}vu_{1}^{-1}\cdots u_{k}\\\ &-u_{k}\cdots u_{1}l_{5}vl_{5}u_{1}^{-1}\cdots
u_{k}^{-1}=u_{k}\cdots u_{1}vu_{1}^{-1}\cdots u_{k}^{-1}\end{aligned}\right.$
$\displaystyle=\Lambda(v)=SvS^{-1}$

Where we used the fact that $v\mapsto-u_{i}vu_{i}$ is a reflection and
$u_{i}^{-1}=\pm u_{i}$. In particular, it follows that
$T^{-1}Sl_{a}S^{-1}T=l_{a}$, so, by Theorem A.4.1 we must have
$T^{-1}S=c\mathbb{1}$ and $S=cT$ for some nonzero $c\in\mathbb{C}$.
Furthermore, one can prove that
$M(4,\mathbb{C})\simeq\mathbb{C}\otimes_{\mathbb{R}}D$ (see [15] Section $4$
together with Theorem $3.7$). Therefore, as $S,T\in D$ we must have that they
differ just by a real constant, i.e. $c\in\mathbb{R}$. Moreover, as $\det S=1$
and one can compute from its definition that $\det T=1$, it follows $T=\pm S$.
Finally, as also $-T=(l_{5})^{2}T\in Pin_{1,3}$, we must have also $S\in
Pin_{1,3}$ ∎

From this proposition, that characterizes the Pin group, one can prove that
$Pin_{1,3}$ and $Spin_{1,3}$ are indeed Lie groups. This can be seen from the
identification we mentioned in the above proof:

$M(4,\mathbb{C})\simeq\mathbb{C}\otimes_{\mathbb{R}}D$ (A.2)

That implies that $D$ is the subset of square matrices with unit determinant.
As a Lie group, let us denote by $Spin^{0}_{1,3}$ the connected component of
$Spin_{1,3}$ containing the identity.
We are now ready to relate the Lie group $Pin_{1,3}$ with the Lorentz group,
by defining a map:

$\displaystyle\Lambda:Pin_{1,3}$ $\displaystyle\to\mathcal{L}$ $\displaystyle
S$ $\displaystyle\mapsto\Lambda^{a}_{\,\,b}(S)$

such that: $Sl_{b}S^{-1}=l_{a}\Lambda^{a}_{\,\,b}(S)$. The matrix
$\Lambda^{a}_{\,\,b}(S)$ exists and is a Lorentz transformation by the
previous proposition. At this point, one can show (see Theorem $I.2.10$ in
[15]):

######  Proposition A.4.2.

The map $\Lambda$, is a surjective double covering homomorphism of Lie groups,
which restricts to a double covering homomorphism
$Spin_{1,3}^{0}\to\mathcal{L}^{\uparrow}_{+}$. We have:

$\displaystyle\Lambda^{a}_{\,\,b}(S)$
$\displaystyle=\frac{1}{4}\eta^{ac}\Tr(l_{c}Sl_{b}S^{-1})$
$\displaystyle\Lambda^{a}_{\,\,b}(S^{-1})$
$\displaystyle=\eta^{ac}\eta_{bd}\Lambda^{d}_{\,\,c}(S)$
$\displaystyle(d\Lambda)^{-1}(\lambda^{b}_{\,\,a})$
$\displaystyle=\frac{1}{4}\lambda^{b}_{\,\,a}\eta^{ac}l_{b}l_{c}$

Where
$d\Lambda:\mathfrak{lie}(Spin_{1,3}^{0})\to\mathfrak{lie}(\mathcal{L}^{\uparrow}_{+})$.

Respectively, one can see that $Pin_{1,3}$, $Spin_{1,3}$ and $Spin_{1,3}^{0}$
are the universal coverings of $\mathcal{L}$, $\mathcal{L}_{+}$ and
$\mathcal{L}_{+}^{\uparrow}$.
Another result, that was needed in introducing the Dirac bundles is the
following:

######  Lemma A.4.2.

Let $\pi$ be a complex irreducible representation of $D$ and let $A,C\in
GL(4,\mathbb{C})$ be as in Definiiton A.4.2. Then for all $S\in
Spin_{1,3}^{0}$ we have:

$\pi(S)^{*}A\pi(S)=A,\hskip 15.0pt\pi(S^{-1})C^{-1}\overline{\pi(S)}=C^{-1}$

###### Proof.

Consider a unit vector $u=u^{a}l_{a}$, then we have $u^{2}=\pm\mathbb{1}$.
Hence:

$\displaystyle\pi(u)^{*}A\pi(u)$
$\displaystyle=u^{a}u^{b}\pi(l_{a})^{*}A\pi(l_{b})$
$\displaystyle=u^{a}u^{b}A\pi(l_{a}l_{b})$ $\displaystyle=A\pi(u^{2})=\pm A$

But then, from the definition of $S\in Pin_{1,3}$ in terms of vectors $u_{i}$
with these same properties we must have: $\pi(S)^{*}A\pi(S)=\pm A$. Of course,
if we take $S=\mathbb{1}$ we will get a plus sign. But, as $Spin^{0}_{1,3}$ is
the connected component of the Lie group containing the identity, by
continuity we must have that also for all $S\in Spin^{0}_{1,3}$ we must keep
having the plus sign.
For what concerns the charge conjugation, notice that for $u\in M_{D}$:

$\pi(u^{-1})C^{-1}\overline{\pi(u)}=-\pi(u)^{-1}\pi(u)C^{-1}=-C^{-1}$

Now, as $S\in Spin_{1,3}$ is a product of an even number of such $u$’s we must
have: $\pi(S^{-1})C^{-1}\overline{\pi(S)}=C^{-1}$. ∎

## References

  * [1] R. Longo, Entropy of Coherent Excitations, Lett. Math. Phys. 109, 2587 (2019).
  * [2] H. Casini, S. Grillo, and D. Pontello, Relative entropy for coherent states from Araki formula, Physical Review D 99 (2019).
  * [3] De Broglie, Louis, Recherches sur la théorie des Quanta, Ann. Phys. 10, 22 (1925).
  * [4] P. A. M. Dirac, The Quantum Theory of the Emission and Absorption of Radiation, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 114, 243 (1927).
  * [5] R. Haag, On quantum field theories, Kong. Dan. Vid. Sel. Mat. Fys. Med. 29N12, 1 (1955).
  * [6] R. Haag and D. Kastler, An Algebraic approach to quantum field theory, J. Math. Phys. 5, 848 (1964).
  * [7] S. W. Hawking, Black hole explosions, Nature 248, 30 (1974).
  * [8] H. Araki, Relative Entropy of States of Von Neumann Algebras, Publ. Res. Inst. Math. Sci. Kyoto 1976, 809 (1976).
  * [9] A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Communications in Mathematical Physics 54, 21 (1977).
  * [10] F. Kurpicz, N. Pinamonti, and R. Verch, Temperature and entropy–area relation of quantum matter near spherically symmetric outer trapping horizons, Letters in Mathematical Physics 111 (2021).
  * [11] E. D’Angelo, Entropy for spherically symmetric, dynamical black holes from the relative entropy between coherent states of a scalar quantum field, Class. Quant. Grav. 38, 175001 (2021).
  * B [12] V. Moretti, Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, Springer, 2 edition (2013).
  * [13] H. Araki, On the Diagonalization of a Bilinear Hamiltonian by a Bogoliubov Transformation, Publications of the Research Institute for Mathematical Sciences, Kyoto University. Ser. A 4, 387 (1968).
  * B [14] J. Dereziński and C. Gérard, Mathematics of Quantization and Quantum Fields, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2013).
  * B [15] H. B. Lawson and M.-L. Michelsohn, Spin Geometry, Princeton University Press (1989).
  * B [16] R. Haag, Local quantum physics, Springer Berlin, Heidelberg (1992).
  * [17] J. M. G. Fell, The Dual Spaces of C*-Algebras, Transactions of the American Mathematical Society 94, 365 (1960).
  * [18] H. Araki, On Quasifree States of CAR and Bogoliubov automorphisms, Publications of The Research Institute for Mathematical Sciences 6, 385 (1970).
  * [19] B. S. Kay and R. M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon, Physics Reports 207, 49 (1991).
  * B [20] O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics $II$, Springer Berlin, Heidelberg (1996).
  * [21] A. S. Wightman and L. Garding, Fields as operator-valued distributions in relativistic Quantum theory, Arkiv Fys. Vol: 28 (1965).
  * B [22] S. Hollands and K. Sanders, Entanglement measures and their properties in quantum field theory, Springer (2018).
  * [23] J. J. Bisognano and E. H. Wichmann, On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys. 16, 985 (1975).
  * [24] J. J. Bisognano and E. H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17, 303 (1976).
  * [25] C. D'Antoni and S. Hollands, Nuclearity, Local Quasiequivalence and Split Property for Dirac Quantum Fields in Curved Spacetime, Communications in Mathematical Physics 261, 133 (2005).
  * B [26] R. M. Wald, Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics, Chicago Lectures in Physics, University of Chicago Press, Chicago, IL (1995).
  * B [27] R. M. Wald, General Relativity, Chicago Univ. Pr., Chicago, USA (1984).
  * B [28] C. Bär, Spin Geometry, Lecture notes (2018).
  * [29] J. Dimock, Dirac quantum fields on a manifold, Transactions of the American Mathematical Society 269, 133 (1982).
  * [30] C. J. Fewster and R. Verch, A Quantum Weak Energy Inequality for Dirac Fields in Curved Spacetime, Communications in Mathematical Physics 225, 331 (2002).
  * [31] K. Sanders, Aspects of locally covariant quantum field theory (2008).
  * [32] E. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, Annals of Mathematics 40, 149 (1939).