# A Modular Operator Approach to Entanglement of Causally Closed Regions

Cosmo Gallaro  Rupak Chatterjee  corresponding author:
<EMAIL_ADDRESS>Center for Quantum Science and Engineering
Department of Physics, Stevens Institute of Technology, Castle Point on the
Hudson, Hoboken, NJ 07030

###### Abstract

Quantum entanglement is shown for causally separated regions along the radial
direction by using a conformal quantum mechanical correspondence with
conformal radial Killing fields of causal diamonds in Minkowski space. In
particular, the theory of local von Neumann algebras and Tomita Takesaki
modular operators is applied in the entanglement structure of causal diamonds
in conformal quantum mechanics. The entanglement of local states in their
respective causal regions is shown through the measures of concurrence and
entanglement entropy using the Tomita Takesaki modular conjugation operator. A
holographic entropy formula is derived for the conformal quantum mechanics
causal diamond correspondence. A new connection is made between the thermal
time flow defined by the modular group of automorphisms to the physical time
flow in a causal diamond via the aforementioned correspondence. The thermal
interpretation of these results via two-point thermal Green’s functions and
modular group flow supports the idea of a possible emergent theory of
spacetime.

##  I Introduction

Conformal symmetry is a local scale symmetry that keeps the metric tensor of
spacetime invariant up to a spacetime dependent scale,
$ds^{2}=\Omega(x)g_{\mu\nu}dx^{\mu}dx^{\nu}$. Physically, this means that a
system is $locally$ invariant under the change of observation scale while
leaving spacetime angles invariant. Indeed it is an extension of the Poincare
group possessing 15 degrees of freedom, the extra five being four special
conformal transformations and one dilation. This class of symmetry has been
previously used to study critical phenomena such as phase transitions where
the diverging correlation lengths indicate scale invariance. However, the
symmetry need not only be applied to systems of infinite degrees of freedom.
Conformal invariance in a quantum mechanical system was first investigated in
Alfaro1976  and subsequently, with the interest in conformal field theories,
seen as formally equivalent to a conformal quantum field theory in $(0+1)$
dimensions Boozer2007 . The conformal invariance analogy in a quantum
mechanical system is made with respect to a conformal algebra rather than a
metric tensor as in classical relativity.

A $causal$ $diamond$ is some finite region of spacetime that encompasses the
past, present and future of a particles spacetime world-line, whose causal
geometric structure emerges from the collective emanation of light from the
said region. Such a region, and the energy within it, could be interpreted as
an observer with a finite lifetime Unruh1989 . In Arzano2020 , Arzano made a
connection between the conformal radial symmetries of causal diamonds in
Minkowski space-time and time evolution in conformal quantum mechanics.
Specifically, the two point function of conformal quantum mechanics was
related to a two point function for an observer in Minkowski space, as well as
the size of a causal diamond to a temperature parameter. A deep relationship
between the generators of the radial conformal Killing vector field in
Minkowski space, which describes the conformal symmetry of the radial
dimension of space, and the generators of the conformal symmetry in quantum
mechanics was found.

Here, we investigate the entanglement properties within this correspondence by
considering causal diamonds which are separated by a space-like distance in
Minkowski space. In particular, we construct a Tomita-Takesaki modular
operator formalism within a von Neumann algebra structure for two causal
diamonds. This allows us to discuss the relation between conformal radial
Killing fields of two causal diamonds in Minkowski space and their
entanglement. Specifically, we use the modular conjugation operator to express
Wooter’s concurrence for arbitrary states in two causally separated regions.
The construction of concurrence in terms of the modular conjugation operator
was first discussed using a super-symmetric quantum mechanical system in
Chatterjee2021 . It is known in holography theory that bulk causality can
emerge from a boundary conformal field theory via a definition of modular flow
and its entanglement between distant regions Maldacena1999 ; Maldacena2003 .
Conformal quantum mechanics is an excellent model to better understand the
emergence of bulk causality from a (0+1) conformal field theory and arising
entanglement due to separation of causal regions through use of von Neumann
algebras and Tomita-Takesaki modular theory. An entropy formula is derived
which is found to be proportional to the length of the causal diamond with a
corrective logarithmic term, similar to that found in Carlip2000 . The $(0+1)$
conformal field theory is mapped to a $(1+1)$ causal physical system, leading
to the concept of a "line density" of entanglement entropy, similar to the
holographic area density of Bekenstein and Hawking. The framework developed in
this paper may further elucidate the holographic emergence of bulk causality.
The problem that we solve is constructing local von Neumann algebras for
causal diamonds using the correspondence with conformal quantum mechanics.
Furthermore, this allows us to define Tomita-Takesaki modular operators and
provide holographic entanglement entropy for a cyclic vector state of the von
Neumann algebra representing the radial dimension of a causal diamond.
Finally, we then comment on the interpretation of the model and its possible
origination in thermality.

##  II The Correspondence between Classical and Quantum Symmetry

de Alfaro et al. Alfaro1976  were the first to deduce from group theoretic
principles the algebra of quantum mechanical operators that correspond to the
algebra of infinitesimal generators of conformal transformations. They do this
by examining projective Möbius transformations for a time variable $t$ in a
$(0+1)$ conformal field theory. Using Heisenberg’s equations of motion, they
are able to match the conformal generators to their unitary time evolution
operators for each transformation generator. It it thus interpreted that the
theory is performing projective conformal transformations over time. The
conformal algebra is infinite in two dimensions making any problem in 2D
exactly solvable. Though we have $(0+1)$ for this model, the theory is still
exactly solvable since the generators of conformal transformations are
constants of motion. The Lagrangian for a quantum mechanical theory can be
written as

$L=\dfrac{1}{2}\left(\partial_{t}Q-\dfrac{g}{Q^{2}}\right)$ (1)

for some Hermitian quantum field $Q(t)$ that only depends on time, hence a
(0+1) field theory. It is the action, $A=\int L\;dt$, that is invariant under
conformal transformations. The transformations of the conformal group can be
found via an infinitesimal analysis Alfaro1976 . These transformations include
translations generated by $H=\dfrac{d}{dt}$, dilations generated by
$D=-t\dfrac{d}{dt}$, and special conformal transformations generated by
$K=-t^{2}\dfrac{d}{dt}$. The generators of time evolution in conformal quantum
mechanics thus form the $\mathfrak{s}l(2,\mathbb{R})$ algebra Alfaro1976 ,

$[H,P]=iH,\;[K,D]=-iK,\;[H,K]=2iD$ (2)

Furthermore, a combination of these generators yields the observable quantity,

$G=uH+vD+wK.$ (3)

This generator leaves the action of the quantum field invariant and is a
constant of motion. This means that one could choose any one of these
conformal generators to study time evolution, though they have different
properties.

Consider now (3+1) dimensional Minkowski space associated with some
symmetries. A conformal radial Killing field is a radial vector field such
that the Lie derivative acting on the metric is proportional to the metric
itself, $\mathcal{L}_{\chi}\eta_{\mu\nu}$ $\alpha$ $\eta_{\mu\nu}$,
demonstrating conformal invariance of the spacetime. Such a Killing field,
$\chi$, can be written as a combination of three conformal generators:

$\begin{array}[]{c}\chi=\left\\{a(t^{2}+r^{2})+bt+c\right\\}\partial_{t}+r(2at+b)\partial_{r}\\\
=aP_{0}+bD_{0}+cK_{0}\end{array}$ (4)

Arzano Arzano2020  obtains the relationship between generators of conformal
quantum mechanics and the conformal radial killing field as:

$\begin{array}[]{c}G=i\chi\\\ \\\ H=iP_{0},\;D=iD_{0},\;K=iK_{0}\\\ \\\
a=u,\;b=v,\;c=w\end{array}$ (5)

That is, the generators of conformal symmetries in quantum mechanics, $H$,
$D$, and $K$ are directly related to the conformal radial Killing vector field
that describes the symmetry of the radial dimension of a causal diamond in
Minkowski space. The generators of conformal radial symmetries of a causal
diamond in Minkowski space are analytically continued into the complex plane
and are expressed with only quantum mechanical time evolution. Therefore, the
classical conformal symmetry of the radial and temporal coordinate in
Minkowski space, $(r,t)$, appear to emerge from generators of conformal
symmetry in quantum mechanics Chamon2011 . In other words, a (1+1) conformal
spacetime can be described with a (0+1) conformal quantum theory. The first
theory of this kind was the AdS/CFT correspondence of Maldacena Maldacena1999
.

There are classes of Hamiltonians that are characterized the determinant of
the generator $G$, as it is invariant under general transformations. The
Hamiltonian corresponding to a negative determinant of $G$ is a rotational
Hamiltonian that generates elliptical compact rotations. For now, we recognize
that this operator is convenient to use for calculations as its basis is
discrete and complete, and we thus proceed with the model in this rotational
basis. The operator $R$ is given as,

$\begin{array}[]{cc}R=\dfrac{1}{2}\left(\alpha
H+\dfrac{K}{\alpha}\right)\end{array}$ (6)

and generates elliptical rotations. The other classes of Hamiltonians are $H$
and $K$ generating parabolic transformations around the light-like axis when
the determinant is zero, while $D$ and $S=\dfrac{1}{2}\left(\alpha
H-\dfrac{K}{\alpha}\right)$ generate hyperbolic transformations (Lorentz
boosts) when the determinant is positive. Note that the number $\alpha$ has
units of time Alfaro1976 ; Arzano2020 .

The connection between the quantum conformal generators and the Killing field
is illustrated through the fact that their classification depends upon the
determinant of the Casimir operator of the conformal group as explained in
Arzano2020 . This connection highlights how the three classes of
’Hamiltonians’ generate different types of time evolution. Indeed, one can
express the three ’Hamiltonians’ $H$, $R$, and $S$ in terms of three different
time coordinates,

$H=i\partial_{t},\;R=i\partial_{T},\;S=i\partial_{\tau},$ (7)

where the times are related to each other through

$t=\alpha tan(T/2)=\alpha tanh(\tau/2)$ (8)

Upon quantization, the rotation $operator$ $R$ has eigenstates and a
corresponding rotational vacuum $\ket{n=0}$ such that Alfaro1976

$\begin{array}[]{c}R\ket{n}=r_{n}\ket{n}\\\ \\\
L_{\pm}\ket{n}=\sqrt{r_{n}(r_{N}\pm 1)-r_{0}(r_{0}-1)}\ket{n\pm 1}\\\ \\\
R\ket{n=0}=r_{0}\ket{n=0}\\\ \\\ r_{n}=r_{0}+n,\;n=0,1,2,...\end{array}$ (9)

with orthonormal eigenstates. One can introduce creation and annihilation
operators for the rotational eigenstates leading to

$\begin{array}[]{c}R=\frac{1}{2}(a^{\dagger}a+\frac{1}{2})\rightarrow
r_{n}=\frac{1}{2}(n+\frac{1}{2})\\\ \\\ L_{+}=\frac{1}{2}a^{\dagger 2}\\\ \\\
L_{-}=\frac{1}{2}a^{2}\end{array}$ (10)

where we identify $L_{+}$ and $L_{-}$ as raising and lowering operators.

Furthermore, one can write an expression for raising and lowering operators of
rotations,

$L_{\pm}=\frac{1}{2}\left(\alpha H-\frac{K}{\alpha}\right)\pm iD$ (11)

to realize the algebra in the Cartan sub-basis,

$[R,L_{\pm}]=\pm L_{\pm},\;[L_{-},L_{+}]=2R$ (12)

thus allowing us to define a scaling dimension and spin for the generators of
dilations and rotations in conformal quantum mechanics by observing the
conformal weights of the two point functions Henkel1999 . We can identify a
relationship between the conformal radial Killing field in Minkowski space and
the quantum rotational generator $R$ by noticing that the generator
$R=i\partial_{T}$ can also be written as

$R=\dfrac{1}{2\alpha}(\alpha^{2}H+K)=\dfrac{i}{2\alpha}\left[(t^{2}+r^{2}+\alpha^{2})\partial_{t}++2\;t\;r\partial_{r}\right]=i\chi$
(13)

with $a=\dfrac{1}{2\alpha}$, $b=0$ and $c=\dfrac{\alpha}{2}$ using the
relations $H=iP_{0}$ and $K=iK_{0}$. The parameter $\Delta=b^{2}-4ac$
determines the causal character of the radial conformal Killing field. For our
case, $\Delta=-1$ and thus the causal character is time-like everywhere. It is
also interesting to view the Killing field in terms of the $(0+1)$ quantum
field $Q(t)$. Using the expressions for the generators in terms of the field
$Q(t)$ found in Alfaro1976 , we obtain the relationship between $R$ and
$Q(t)$,

$R=\frac{1}{2}\left(\alpha(\partial_{t}Q^{2}+g/Q^{2})+\frac{1}{\alpha}\left[t^{2}/2(\partial_{t}Q^{2}+g/Q^{2})-\frac{1}{2}t(Q\partial_{t}Q+\partial_{t}QQ)+\frac{1}{2}Q^{2}\right]\right)$
(14)

and we can express the conformal radial Killing field for a causal diamond in
Minkowski space in terms of the quantum field $Q(t)$,

$\begin{array}[]{c}\chi=\dfrac{-i}{2}\left(\alpha(\partial_{t}Q^{2}+\dfrac{g}{Q^{2}})+\dfrac{1}{\alpha}\left[t^{2}/2(\partial_{t}Q^{2}+\dfrac{g}{Q^{2}})-\dfrac{1}{2}t(Q\;\partial_{t}Q+\partial_{t}Q\;Q)+\dfrac{1}{2}Q^{2}\right]\right)\end{array}$
(15)

providing a relationship between the classical radial conformal Killing field
of a causal diamond and the $(0+1)$ dimensional quantum field. From the
relation $-iR=\chi=\partial_{T}$ one can adopt diamond coordinates as

$t=\alpha\frac{sin(T)}{cos(x)+cos(T)},\;\;r=\alpha\frac{sin(x)}{cos(x)+cos(T)}$
(16)

and transform the typical Minkowski line element
$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\Omega^{2}=-dt^{2}+dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ into

$ds^{2}=\frac{\alpha^{2}}{(cos(T)+cos(x))^{2}}(-dT^{2}+dx^{2}+sin^{2}xd\Omega^{2})$
(17)

sufficing as a $conformal$ transformation in the radial and time dimensions
only. That invites one to think of this as a holographic theory from which
$(1+1)$ spacetime can be described as a $(0+1)$ conformal field theory.

Using the integral curves of conformal radial Killing fields found in
Herrero1999 , we set $a=\dfrac{1}{2\alpha}$, $b=0$, and $c=\dfrac{\alpha}{2}$
for the case of $-iR=\chi$ to obtain the one-parameter family of world-lines,

$\begin{array}[]{c}\dfrac{1}{2\alpha}(t^{2}-r^{2})-\omega
r+\dfrac{\alpha}{2}=0\\\ \\\
t^{2}-(r-\alpha\omega)^{2}=-\alpha^{2}(1+\omega^{2})\end{array}$ (18)

for some parameter $\omega$ giving a family of hyperbola in the $(t,r)$
Minkowski space. From these curves, we note the corresponding acceleration of
an observer in these coordinates as Herrero1999 ,

$a=\dfrac{1}{\alpha\sqrt{\omega^{2}+1}}=\dfrac{sin(x)}{\alpha}$ (19)

for the parameter $\omega=\dfrac{1}{tan(x)}$. This corresponds to hyperbolic
relativistic motion. The acceleration is zero when passing through $x=0$ where
we can also notice that

$t=\alpha\dfrac{sin(T)}{1+cos(T)}=\alpha tan(T/2)$ (20)

relating the diamond time $t$ to the time variable of the conformal quantum
rotation operator $T$. As is well known, there is a deep connection between
temperature and acceleration, as illustrated through the Unruh/Hawking effect
Unruh1976 ; Hawking1975 . There is also a direct relationship between the
acceleration of a massless scalar field along the world-line of a static
observer in a causal diamond and the temperature of the primary conformal two
point function of conformal quantum mechanics Arzano2020 ; Anninos2012 ;
Nakayama2012  with the appropriate conformal weight. The length of this causal
diamond is $2\alpha$ Unruh1989  as seen from the line element (17), and is
inversely proportional to the temperature found in the thermal Greens function
as $T=\dfrac{1}{\pi\alpha}$. It was in fact mentioned in Unruh1989 ;
Arzano2020 ; Anninos2012  that $2\alpha$ is the lifetime of a static observer
in Minkowski space. The construction in terms of the operator $R$ was built in
order to highlight the connections between two separate causal diamonds via
the correspondence reviewed in this section. Indeed, it is the operator $R$
whose eigenbasis we use to define the cyclic vector for Tomita-Takesaki
modular operators in the next section.

##  III Causality in Conformal Quantum Mechanics: Modular Operators

We begin by imagining that there are two causal diamonds in Minkowski space
that are separated by some spacelike distance such that there is no causal
relationship between observations in each region. The system is illustrated in
Figure 1. Each diamond has associated with their conformal radial symmetries a
set of conformal quantum mechanical operators through the correspondence
reviewed in the previous section. In this section, we show that a holomorphic
representation of these quantum mechanical operators may be used to form von
Neumann algebras of observables such that the second diamond’s algebra of
observables is shown to be the commutant of the first diamonds algebra. The
conformal symmetries of each diamond is interpreted to emerge from the quantum
generators $R$ of the kets $\ket{n}$ as reviewed in the previous section.

The relationship between these two systems of algebras, shown below, may be
generated by Tomita-Takesaki modular operators from a standard form of von
Neumann algebras (see the appendix and Haag1996 ). Proceeding with the GNS
construction (see appendix), let $\mathcal{A}$ be a representation of a von
Neumann algebra of bounded linear operators on a Hilbert space $\mathcal{H}$,
$\mathcal{A}\subset\mathcal{B(H)}$. $\mathcal{A}$ is a $C^{*}$-algebra with a
commutant $\mathcal{A}^{\prime}$ (the set of elements in $\mathcal{B(H)}$
commuting with $\mathcal{A}$) such that
$\mathcal{A}^{\prime\prime}=\mathcal{A}$ (the double commutant returns one to
$\mathcal{A}$ as defined in the appendix). Let $|\Omega\rangle\in\mathcal{H}$
be a separating and cyclic vector for the $C^{*}$-algebra $\mathcal{A}$ (the
elements in $\mathcal{A}$ can create a space dense in $\mathcal{H}$ by acting
on $|\Omega\rangle\in\mathcal{H}$). The Reeh-Schlieder theorem states that
mathematically, the vacuum state of some quantum field theory is cylic
$\mathcal{A}(\mathcal{O})\ket{\Omega}$ is dense in the whole Hilbert space
$\mathcal{H}$ for any open region $\mathcal{O}$ in Minkowski space. Further,
$\mathcal{A}(\mathcal{O}^{*})\ket{\Omega}$ is also dense in $\mathcal{H}$,
where $\mathcal{O}^{*}$ is the causal compliment to $\mathcal{O}$. Physically,
this suggests that energy can be created or annihilated anywhere in the
universe from anywhere in the universe, though there are of course caveats to
this statement. See Witten’s review Witten2018  for an excellent discussion
and proof. In the current theory, however, the vacuum is not cyclic as there
is not an invariant vacuum state for $\mathfrak{s}l(2,\mathbb{R})$ Chamon2011
and so the Reeh-Schlieder theorem does not immediately follow. It is of great
interest to us, however, to understand how the Reeh-Schlieder theorem can be
analogously defined in a conformal quantum mechanics model.

Tomita-Takesaki theory states that there exists an anti-linear map
$S:\mathcal{H}\rightarrow\mathcal{H},$ such that
$SA|\Omega\rangle=A^{*}|\Omega\rangle,\forall A\in\mathcal{A}$. $S$ has a
polar decomposition given by
$S=J\Delta^{1/2}=\Delta^{-1/2}J,\,\,\Delta=S^{*}S$ where the modular
conjugation operator $J$ has the following properties (see appendix),

$J\Delta^{\frac{1}{2}}J=\Delta^{-\frac{1}{2}},\;\;\;J^{2}=I,\,\,\,J^{*}=J,\;\;\;J|\Omega\rangle=|\Omega\rangle,\;\;\;J\mathcal{A}J=\mathcal{A}^{\prime}.$
(21)

One of the key feature for our purposes is that of the modular conjugation
operator $J$ that takes an algebra $\mathcal{A}$ into its commutant
$\mathcal{A}^{\prime}$.

Two causal diamonds separated by some spacelike distance will have local
algebras of bounded operators acting on their respective local Hilbert spaces
as define through the conformal quantum mechanical correspondence. To define
some notion of causality, we demonstrate that the Hermitian operators
corresponding to observables commute with the operators in the other causal
diamond using Tomita-Takesaki modular theory. This defines the commutant of
the algebra, and allows us to define local von Neumann Algebras in the two
distant causal diamonds Haag1996 . While this is known in local algebraic
quantum field theory, it is through the conformal quantum mechanical
correspondence that we construct the commutant structure of local von Neumann
algebras for the radial direction of causal diamonds.

In the first causal diamond, we have local rotational quanta $\ket{n}$
described by a set of operators $a$ and $a^{\dagger}$. In the second diamond,
we have local rotational quanta described by a different set of operators $b$
and $b^{\dagger}$ that will be shown to be related via involution. The
algebraic structure of the double diamond system is given by,

$\begin{array}[]{cc}\mathcal{H}=\mathcal{H}_{a}\otimes\mathcal{H}_{b}\\\ \\\
\mathcal{A}_{a}=\set{\mathcal{O}_{a}}\subset\mathcal{B}(\mathcal{H}_{a})\\\
\\\
\mathcal{A}_{b}=\set{\mathcal{O}_{b}}\subset\mathcal{B}(\mathcal{H}_{b})\\\
\\\ \mathcal{A}=\mathcal{A}_{a}\otimes\mathcal{A}_{b}\end{array}$ (22)

Figure 1: Double Causal Diamond Structure

We place the operators $a$ and $a^{\dagger}$ in a holomorphic representation
as in Chatterjee2021 ,

$a=\dfrac{1}{\sqrt{2}}\left(z+\partial_{\bar{z}}\right),\;a^{\dagger}=\dfrac{1}{\sqrt{2}}\left(\bar{z}-\partial_{z}\right)$
(23)

such that $[a,a^{\dagger}]=1$. We define the involution of the structure as
transforming $a$ to $b$ such that

$\begin{array}[]{c}a^{*}=\dfrac{1}{\sqrt{2}}\left(\bar{z}+\partial_{z}\right)=b\\\
\\\
a^{\dagger*}=\dfrac{1}{\sqrt{2}}\left(z-\partial_{\bar{z}}\right)=b^{\dagger}\end{array}$
(24)

thus $[b,b^{\dagger}]=1$ and therefore

$[a,b]=[a^{\dagger},b^{\dagger}]=0$ (25)

Then, $R_{b}=\dfrac{1}{2}\left(b^{\dagger}b+\dfrac{1}{2}\right)$. The modular
conjugation operator $J$ will transform states from one diamond to another.

The Tomita-Takesaki modular conjugation operator for this theory fulfills the
identity for an arbitrary state,

$\begin{array}[]{c}J(a\otimes I)J\ket{n}\otimes\ket{m}=J(a\otimes
I)\ket{m}\otimes\ket{n}\\\ \\\
=J\sqrt{m}\ket{m-1}\otimes\ket{n}=\sqrt{m}\ket{n}\otimes\ket{m-1}\\\ \\\
=(I\otimes b)\ket{n}\otimes\ket{m}\end{array}$ (26)

for the modular operator

$\Delta=e^{-\beta(N_{a}\otimes I-I\otimes N_{b})}$ (27)

and the physical state

$\begin{array}[]{cc}\ket{\Omega}=N(\beta)\sum_{n}e^{-\beta
n/2}\ket{n}\otimes\ket{n}\\\ \\\ =N(\beta)\sum_{n}\dfrac{e^{-\beta
n/2}}{n!}(a^{\dagger})^{n}\ket{0}\otimes(b^{\dagger})^{n}\ket{0}\end{array}$
(28)

where $N(\beta)=(1-e^{-\beta})^{1/2}$, which is indeed cyclic and separating
as there are no vanishing Schmidt coefficients Lashkari2019 . Further
relationships are fulfilled,

$\begin{array}[]{c}J\Delta^{1/2}(a\otimes
I)\ket{\Omega}=Je^{-\frac{\beta}{2}(N_{a}\otimes I-I\otimes N_{b})}(a\otimes
I)\ket{\Omega}\\\ \\\ =Je^{-\frac{\beta}{2}(N_{a}\otimes I-I\otimes
N_{b})}(a\otimes I)N(\beta)\sum_{n}e^{-\beta n/2}\ket{n}\otimes\ket{n}\\\ \\\
=Je^{-\frac{\beta}{2}(N_{a}\otimes I-I\otimes N_{b})}N(\beta)\sum_{n}e^{-\beta
n/2}\sqrt{n}\ket{n-1}\otimes\ket{n}\\\ \\\ =JN(\beta)\sum_{n}e^{-\beta
n/2}e^{-\frac{\beta}{2}(2n-1-2n)}\sqrt{n}\ket{n-1}\otimes\ket{n}\\\ \\\
=N(\beta)\sum_{n}e^{-\beta
n/2}e^{\frac{\beta}{2}}\sqrt{n}\ket{n}\otimes\ket{n-1}\\\ \\\
=N(\beta)\sum_{n}e^{-\beta n/2}e^{\frac{\beta}{2}}(I\otimes
b)\ket{n}\otimes\ket{n}=a^{*}\ket{\Omega}\end{array}$ (29)

where the vector has been renormalized. The commutant structure ensures
Einstein causality as the commutators vanish for operators in the two algebras
and therefore generate the conformal radial symmetry of causally disconnected
causal diamonds. Using these relations we can identify the commutant,

$\begin{array}[]{c}JR_{a}J=R_{b}\end{array}$ (30)

The physical interpretation is made a little more clear by recalling the
representation of $R$ in terms of some time coordinate $T$,

$R_{a}=i\partial_{T}$ (31)

such that

$R_{b}=-i\partial_{T}$ (32)

which may hint at opposite flows of complex time evolution for space-like
separated causal diamonds. Furthermore, since we also know that
$R_{a}=i\chi_{a}$, we have $R_{b}=i\chi_{b}=-i\chi_{a}$. Clearly,

$[R_{a},R_{b}]=[i\chi_{a},i\chi_{b}]=0$ (33)

since two space-like separated casual diamonds will not have any causal
influence on each others measurements or observations, making their respective
symmetries commutative. Though there is non local interaction between these
two diamonds, as we will see through the non-zero entanglement of the two
diamonds which will be expressed below through the conformal quantum
mechanical correspondence.

Within one causal diamond, there exists a bipartite structure as discussed in
Arzano2021  of which this formalism may be applied. However, it is
illuminating to consider one diamond and its causal compliment, of which a
second diamond is a subset, as it is a common relationship to study in
algebraic quantum field theory. Indeed, the identified commutant algebra is
physically a subset of the causal compliment of the first algebra representing
a causal diamond itself. In particular, the operators composing the commutant
of conformal quantum mechanics also can be mapped to the radial conformal
Killing field of another diamond.

The exponential argument $N_{a}\otimes I-I\otimes N_{b}$ of the modular
operator is serving as the modular Hamiltonian in this model generating
rotations, as the quanta generated by the operator $R$ are indeed some type of
abstract angular momentum, analogous to our familiar notion of quantum "spin"
as being some angular momentum. The modular operator allows us to define a one
parameter group of automorphisms on the algebra, $\mathcal{A}$, that generates
the local time flow for the theory (see appendix),

$\alpha_{t}A=\Delta^{it}A\Delta^{-it}=e^{it\beta(N_{a}\otimes I-I\otimes
N_{b})}Ae^{-it\beta(N_{a}\otimes I-I\otimes N_{b})}$ (34)

experienced by observables on the algebras such that they will remain in such
algebras. Furthermore, due to the fact that our system contains finite degrees
of freedom, we can define a suitable density matrix,
$\rho=Z^{-1}e^{-\beta(N_{a}\otimes I-I\otimes N_{b})}$ where
$Z=Tr(e^{\beta(N_{a}\otimes I-I\otimes N_{b})})$ is the partition function. An
algebraic state can thus be written for some operator $A\in\mathcal{A}$,

$\omega(A)=\mathrm{Tr}(\rho A)$ (35)

which indeed satisfies the KMS condition for the correlation function,

$F(A,B,t+i\beta)=\omega((\alpha_{t}B)A)=\omega(A\alpha_{t}B)=F(A,B,t)$ (36)

via cyclicity of the trace operation, meaning that this state is analytic on
the boundary of the strip $0<Im(t)<\beta$. This constitutes the cyclic vector
$\ket{\Omega}$ as a thermal equilibrium state. Jacobson has emphasized
Jacobson2016  that the vacuum in any quantum field theory can be expressed as
a thermal density matrix when restricted to a causal diamond. This may suggest
that our cyclic vector may be related to a vacuum of some type, a point we
will return to in a later section.

Due to the fact that the time flow emerges from the modular group of a
specific KMS equilibrium state, $\omega$, Connes and Rovelli Connes1994  have
hypothesized that physical time flow has thermodynamic origin in the sense
that the physical time flow, $\alpha_{t}$, depends upon the thermal state of
the system, $\omega$. This is known as the thermal time hypothesis. They were
unable to connect such thermal modular time flow to a suitable metric
coordinate for time as in relativity theory. It is within the current model
that this connection can be made, as the (modular) thermal time evolution in
conformal quantum mechanics is deeply connected to the conformal radial
Killing symmetries of a causal diamond in Minkowski space as known from the
relation $R=i\chi$. The fact that a conformal radial Killing field is related
to quantum $operators$ as opposed to quantum states is a reflection of the
fact that it is really the operators, and the algebraic relations they obey,
that define a theory as is the philosophy of algebraic quantum field theory
Hollands2015 . This is a step toward bridging the gap between thermal modular
flow of time and dynamic geometric time flow of a causal diamond.

Entropy of entanglement can be used as a measure of entanglement for a
bipartite quantum system such as the vector $\ket{\Omega}$ by computing the
von Neumann entropy for the reduce density matrix of the subsystems. Observing
that the entanglement entropy is non zero results in a finite von Neumann
entropy, meaning that the reduced density is a mixed state thus indicating
entanglement for the bipartite state. We can compute the entanglement entropy
of the cyclic vector $\ket{\Omega}$ as,

$\begin{array}[]{c}S_{\Omega}=-Tr_{a}(\rho_{\Omega}^{a}\;ln\rho_{\Omega}^{a})=-Tr_{b}(\rho_{\Omega}^{b}\;ln\rho_{\Omega}^{b})\\\
\\\ =-N^{2}(\beta)\sum_{n=0}^{\infty}e^{-\beta n}(-\beta
n+ln(N^{2}(\beta))=\dfrac{N^{2}(\beta)\beta
e^{-\beta}}{(1-e^{-\beta})^{2}}-2\dfrac{N^{2}(\beta)}{1-e^{-\beta}}ln(N(\beta))\end{array}$
(37)

indicating entanglement for the vector $\ket{\Omega}$. Plugging in the
normalization $N(\beta)=(1-e^{-\beta})^{1/2}$ and using the relation
$\beta=\dfrac{1}{T}=\pi\alpha$, we obtain

$S_{\Omega}=\dfrac{\pi\alpha}{(e^{\pi\alpha}-1)}-ln(1-e^{-\pi\alpha})$ (38)

which is an expression for the entanglement entropy of the cyclic vector
representing two causal diamonds in terms of the length, $\alpha$ of one
diamond, minus a logarithmic corrective term. This is similar to a well known
logarithmic correction in holography theory Carlip2000 . This is expected
since the quantum theory represents only $(1+1)$ components of the full
$(3+1)$ causal diamond.

In summary, to generate the conformal radial symmetries of two causally
disconnected diamonds, one need define two sets of commuting conformal quantum
operators. The non-locality of the entanglement in algebraic quantum field
theory is natural in a conformal model where the causal diamonds themselves
are seen to emerge from an algebraic structure of conformal quantum operators
that compose von Neumann algebras. We will see in the next section how the
entanglement of the model is quantified using the modular conjugation operator
$J$.

##  IV A Concurrence Measurement for Entanglement using Modular Operators

A well known way to measure entanglement is Wooter’s Concurrence Wooters1997 .
The entanglement of formation of a generic bi-partite state $\rho$ (quantum
density operator) is related to the concurrence of that state $C(\rho)$ as
follows:

$E_{Formation}(\rho)=H_{bin}\left(\dfrac{1+\sqrt{1+C(\rho)}}{2}\right)$

where $H_{bin}$ is the Shannon binary entropy. One can define the concurrence
for a bi-partite state using the anti-unitary modular conjugation operator as
Chatterjee2021

$C(\ket{\psi})=|\bra{\psi}J\ket{\psi}|$ (39)

The concurrence relation (39) has given a physical meaning to the modular
conjugation operator as a quantitative measure of entanglement for a bi-
partite state. It is known that one needs an anti-linear, anti-unitary
operator as the key driving force behind the entanglement of formation of
bipartite systems and it is now understood in this context as the modular
conjugation operator $J$.

The most convenient basis to use for an entanglement calculation here is that
of quantized rotational states, for these are discrete, normalized states that
complete the Hilbert space. For an entangled rotational state
$\ket{\Psi}=\alpha\ket{n}\otimes\ket{m}+\beta\ket{m}\otimes\ket{n}$, and for
$n\neq m$, we have a concurrence given by the modular conjugation operator $J$
as

$C(\ket{\Psi})=|\bra{\Psi}J\ket{\Psi}|=2|\alpha\beta|.$ (40)

Clearly when $\alpha=\beta=\dfrac{1}{\sqrt{2}}$ we have a maximally entangled
state of $C=1$. The nonzero entanglement again is interpreted to arise from
the active separation of causal compliments, $\mathcal{A}$ and
$\mathcal{A}^{{}^{\prime}}$. This gives a measure of the entanglement between
two spacelike separated causal diamonds in Minkowski space as was achieved by
representing the radial and temporal dimensions of each diamond as a set of
conformal quantum mechanical operators that generate rotational states in a
(0+1) conformal field theory.

##  V Thermal Structure of Conformal Quantum Mechanics

It was mentioned above that the thermal time flow can be derived from the
modular automorphism group for the current theory,
$\alpha_{t}A=\Delta^{it}A\Delta^{-it}=e^{it\beta(N_{a}\otimes I-I\otimes
N_{b})}Ae^{-it\beta(N_{a}\otimes I-I\otimes N_{b})}$. The modular flow induced
by the thermality of the system can be clearly viewed in conformal quantum
mechanics. This is due to the fact that the quantization of time was hinted at
in the original observation of conformal symmetry in a $(0+1)$ field theory
Alfaro1976 . This view leads to the quantization of states $\ket{t}$ labeled
by a time variable $t$. These states are in fact the Fourier transform of the
energy eigenkets of the Hamiltonian,

$\ket{E}=2^{r_{0}}E^{\frac{1}{2}-r_{0}}\int_{-\infty}^{\infty}dte^{-iEt}\ket{t}$
(41)

that form a continuous, orthonormal basis. Indeed, one can construct quantized
time states from the rotational vacuum through a complex time translation
Alfaro1976 ,

$\ket{t}=e^{iHt}\ket{t=0}=\frac{\Gamma^{\frac{1}{2}}(2r_{0})}{2^{2r_{0}}}e^{(\alpha+it)H}\ket{n=0}$
(42)

where $\alpha$ is half the length of the causal diamond. One may interpret
this to mean that quantized time emerges from the thermal evolution of the
vacuum. The quantization of time was thoroughly discussed in Caldirola1956 ;
Farias2010 ; Caldirola1980 ; Albanese2004 ; Olkhovsky2007 ; Yang1947  among
many other works, though it is currently unclear whether the time states
$\ket{t}$ corresponds to states of the hypothetical "chronon", which is a
discrete unit of time of $6.266*10^{-24}s$. Furthermore, it has been shown
that these are thermal states by observing the two point function for "$\tau$"
time states of the quantum operator $S$ Arzano2021 ,

$G(\tau_{1},\tau_{2})=\braket{\tau_{1}}{\tau_{2}}=\frac{\Gamma(2r_{0})\alpha^{2r_{0}}}{(4isinh(\frac{\tau_{1}-\tau_{2}}{2}))^{2r_{0}}}$
(43)

at a temperature of $\dfrac{1}{2\pi}$ and conformal weight $\Delta=r_{0}$. It
was mentioned in Arzano2020 ; Gupta2017  that there is a direct connection
between this thermal two-point function of conformal quantum mechanics and the
one associated to scalar field along the world line of a static diamond
observer in spacetime. Note that our thermal cyclic vector (28) is related to
the temporal vacuum of Arzano2021 ,
$\ket{\tau=0}=\sum_{n}(-1)^{n}\ket{n}\otimes\ket{n}$, coinciding with
arguments that the vacuum is thermal when restricted to a causal diamond
Jacobson2016 . Other thermal effects of conformal quantum mechanics and
thermal time flow were investigated in Martinetti2003 ; Martinetti2009 . This
link also provides a possible path to the Reeh-Schlieder theorem for conformal
quantum mechanics, which will be made more precise in future work. Further,
this thermal structure supports the interpretation that spacetime and
causality may emerge from thermality. The physical interpretation can be made
more clear by noticing that the creation operator for time states,
$\mathcal{O}(t)$, is related to the creation operator for rotational states,
$L_{+}$, as in Jackiw2012

$\begin{array}[]{cc}\mathcal{O}(t)=N(t)e^{\omega(t)L_{+}}\\\ \\\
N(t)=\left(\Gamma(2r_{0})\right)^{1/2}\left(\dfrac{\omega(t)+1}{2}\right)^{2r_{0}},\\\
\\\ \omega(t)=\dfrac{\alpha+it}{\alpha-
it}=e^{i\theta},\;t=\alpha\;tan\theta/2\end{array}$ (44)

suggesting that the construction of a quantum time variable labeled by the ket
$\ket{t}$ may be interpreted as a form of abstract angular momentum scaled by
some temperature.

Using the current construction of von Neumann algebras and modular operators
of conformal quantum mechanics from section III, we can determine the modular
structure of time states created from the commutant algebra for a second
causal diamond. Let $O_{a}(t)=e^{(\alpha+it)H_{a}}$ such that

$\ket{t}=\ket{t_{a}}\otimes\ket{t_{b}}=\biggl{(}O_{a}(t)\otimes
O_{b}(t)\biggr{)}\ket{0}\otimes\ket{0}$ (45)

Our task is to determine the form of the operator $O_{b}(t)$ such that it
composes the commutant of $\mathcal{A}_{a}$. Using the above relations, we
find that,

$\begin{array}[]{cc}J(O_{a}\otimes I)J\ket{t}=J(e^{(\alpha+it)H_{a}}\otimes
I)J(\ket{t_{a}}\otimes\ket{t_{b}})\\\ \\\ =J(e^{(\alpha+it)H_{a}}\otimes
I)(\ket{t_{b}}\otimes\ket{t_{a}})=J(e^{(\alpha+it)H_{a}}\otimes
I)(O_{b}(t)\otimes O_{a}(t)\ket{0}\otimes\ket{0})\\\ \\\ =(I\otimes
e^{(\alpha-it)H_{b}})(O_{a}(t)\otimes O_{b}(t)\ket{0}\otimes\ket{0})=(I\otimes
O_{b}(t))\ket{t}.\end{array}$ (46)

with $O_{b}(t)=e^{(\alpha-it)H_{b}}$. Indeed, in a similar fashion to
equations (31, 32) we have the commutant structure $J(O_{a}(t)\otimes
I)J=(I\otimes O_{b}(t))$. Therefore we find that the state $\ket{t_{b}}$ is
created from the vacuum at an inverse temperature $\dfrac{1}{\alpha-it}$ with
opposite complex time $-it$ as opposed to $\ket{t_{a}}$ which is created from
the vacuum at the inverse temperature $\dfrac{1}{\alpha+it}$ and complex time
$it$. This therefore suggests that such a quantum time generated from those
states is flowing in opposite complex directions. If one associates the
quantum time states as representing the time in a causal diamond, then this
would suggest opposite temperatures and thus complex flows of time for
causally disconnected regions. The concept of negative temperature in the
context of causal diamonds was discussed thoroughly in Jacobson2019 ;
Visser2019 . However, it is currently unclear how to relate the time kets,
$\ket{t}$, and their eigenvalues Chamon2011 , $-r_{0}\omega$, to physical
time, $t$.

The modular group flow, $\alpha_{t}A=\Delta^{it}A\Delta^{-it}$, allows us to
define a notion of thermal time flow in the current model. We therefore find
that the ordinary time evolution of a quantum observable,
$\gamma_{t}A=e^{itH/\hbar}Ae^{-itH/\hbar}$, can be equated with the modular
time flow, $\alpha_{t}A=\Delta^{it}A\Delta^{-it}$, when
$H=\hbar\beta(N_{a}\otimes I-I\otimes N_{b})$ thus giving
$\gamma_{t}A=\alpha_{t}A$ Connes1994 . The thermal time hypothesis, namely
that the time flow is determined by the thermal state, is then supported for a
theory including thermal states of quantized time. It is a topic of future
work to further investigate the application of modular operators to time
states $\ket{t}$.

Indeed, one can approximately describe the dynamics of an eternal black hole
in a maximally extended AdS by considering two entangled conformal field
theories on each boundary, which is the theory of thermo-field dynamics
Maldacena2003 ; Israel1976 . In that framework, a thermal density operator in
the conformal field theory corresponds to a black hole geometry. The formalism
developed in this work can be understood to be associated with the formalism
of thermo-field dynamics, as our cyclic vector is the thermo-field double for
two conformal field theories in $(0+1)$ dimensions.

##  VI Discussion

We have elucidated how the entanglement of two causally closed regions emerges
from modular operator structure developed for conformal quantum field theory
in (0+1) dimensions. This was achieved by constructing local von Neumann
algebras for conformal quantum mechanics, which is known to be holographically
dual to a causal diamond spacetime structure. The representation of two von
Neumann algebras is then studied using Tomita-Takesaki theory, and connections
between two causal diamonds are made by representing each of them as local
conformal quantum mechanical algebras. Entanglement between the quantum states
representing each diamond was quantified using Tomita-Takesaki modular
operators through entropy and Wooter’s Concurrence as defined in
Chatterjee2021 . The entanglement entropy of the cyclic vector of conformal
quantum mechanics was found to be proportional to the length of the causal
diamond with a logarithmic correction, which is expected for a holographic
conformal theory. A physical interpretation is provided by seeing the radial
conformal Killing field in terms of the quantum field as in equation (15).
This provides us with a simplified model for understanding how spacetime and
causality may emerge from the entanglement entropy of causal diamonds,
temperature, and quantum states.

It was also mentioned how to understand the thermal time hypothesis in terms
of the quantized states found in conformal quantum mechanics. By examining the
algebraic structure of the quantized time states in conformal quantum
mechanics, two causally disconnected regions were found to be at opposite
temperatures and thus opposite flows of complex time. The states were shown to
be KMS states and thus generated the modular automorphisms from which local
time flow emerges. It is of interest to us to understand better the relation
between quantum time in conformal quantum mechanics and physical time flow in
a causal diamond. Perhaps at the scale when quantum gravitational effects
become relevant, the current model suggests at such a scale that space is not
a fundamental construct, but rather temperature is the fundamental concept
from which classical spacetime, causality and gravitation emerge. We plan to
investigate further the bulk geometric spacetime structure that emerges from
the correspondence in a future work.

##  VII Appendix: Von Neumann Algebras and Tomita-Takesaki Modular Operators

The general references for this section are KR1983 ; Tak1979 ; Bratteli1987 .

Normed Algebra: Consider an algebra $A\in\mathcal{A}$ over $\mathbb{C}$. A
normed algebra has a Norm Map: $A\rightarrow||A||\in\mathbb{R}^{+}$ such that

$||A||>0$

$||A||=0\;\text{iff}\;A=0$

$\alpha\in\mathbb{C},||\alpha A||=|\alpha|||A||$

$||A+B||\leq||A||+||B||$

$||AB||\leq||A||\;||B||$

Banach Algebra: Let $\mathcal{A}$ be an algebra. A normed algebra has a norm
map:
$\mathcal{A}\rightarrow\mathbb{R}^{+},\,\,\,A\rightarrow||A||\in\mathbb{R}^{+},\forall
A\in\mathcal{A}$ such that

$||A||\geq 0,$

$||A||=0\iff A=0,$

$\alpha\in\mathbb{C},||\alpha A||=|\alpha|\,||A||,$

$||A+B||\leq||A||+||B||,$

$||AB||\leq||A||\,||B||.$

A Banach algebra is a complete normed algebra (complete in the norm map):
Consider a Cauchy sequence $A_{1},A_{2},...\in\mathcal{A}$. For any
$\epsilon>0,\exists$ an integer $N$ such that $\forall$ natural numbers
$m,n>N$, $||A_{m}-A_{n}||<\epsilon$ (a sequence whose elements become
arbitrarily close to each other as the sequence progresses). By introducing a
"distance" metric $d(A,B)=||A-B||$, once induces a topology on $\mathcal{A}$
where a neighborhood $U$ in $\mathcal{A}$ is given by
$U(A,\epsilon)=\\{B;\;B\in\mathcal{A},\;d(A,B)<\epsilon,\;\epsilon>0\\}$. A
metric space $(\mathcal{A},d)$ in which every Cauchy sequence converges to an
element in $\mathcal{A}$ is called complete in the standard norm.
Counterexample: Rational numbers $(p/q)\in\mathbb{Q}$ are not complete. The
Cauchy sequence $x+0=1,x_{n+1}=\dfrac{x_{n}+2/x_{n}}{2}$ converges to the
irrational number $\sqrt{2}$.

Furthermore, for example, let $\mathcal{A}=C^{0}(X,\mathbb{C})\equiv$ complex
valued continuous functions $f$ on a compact space $X$ (a bounded region):
$f:X\rightarrow\mathbb{C}$.

Let $x\in X,\;\;\ f,g\in\mathcal{A}$

$(f+g)(x)=f(x)+g(x)$ (Addition)

$(\alpha f)(x)=\alpha f(x)$ (scalar multiplication)

$\Big{(}fg\Big{)}(x)=f(x)g(x)$ (Algebra product)

$||f||=\displaystyle\sup_{x\in X}|f(x)|$ (norm)

$\bm{C^{*}}$-Algebra: A $\bm{C^{*}}$-algebra $\mathcal{A}$ is a Banach algebra
with an involutive map
${}^{*}:\mathcal{A}\rightarrow\mathcal{A},\,\,\,A\rightarrow
A^{*}\,\,\,\forall A\in\mathcal{A},\lambda\in\mathbb{C}$ such that

$(A^{*})^{*}=A$

$(AB)^{*}=B^{*}A^{*}$

$(\lambda A)^{*}=\bar{\lambda}A^{*}$

$(\lambda A+\alpha B)^{*}=\bar{\lambda}A^{*}+\bar{\alpha}B^{*}$ (anti-linear)
(so far, we have a $*$-algebra)

$||A^{*}||=||A||$ (norm condition for involutive Banach algebra),

$||AA^{*}||=||A||\,||A^{*}||=||A^{*}A||=||A^{*}||\,||A||=||A||^{2}$ ($C^{*}$
condition for $\bm{C^{*}}$-Algebra).

If $A^{*}=A$, this element is called self-adjoint. If $A^{*}A=AA^{*}=I$ this
element is called unitary.

For example, let $\mathcal{A}=C^{0}(X,\mathbb{C})\equiv$ complex valued
continuous functions $f$ on a compact space $X$ (a bounded region). From
above, we know that this is a Banach algebra. We know introduce an involution
map through complex conjugation:

$f^{*}:X\rightarrow\mathbb{C},\;\;\;f^{*}\in\mathcal{A}$

$f^{*}(x):=\overline{f(x)},\;\forall x\in X$ (Involution)

$||f^{*}||=\displaystyle\sup_{x\in X}|f^{*}(x)|=\sup_{x\in
X}|\overline{f(x)}|=\sup_{x\in X}|f(x)|=||f||$ (norm condition)

Furthermore, consider complex $n\times n$ matrices $A$. Here, we have

$A^{*}:=A^{\dagger}$ (involution is the adjoint operation)

$||A||:=\sqrt{Tr(AA^{\dagger})}$ (norm is the square root of trace)

The algebra is given by commutative matrix addition $A+B$ and non-commutative
matrix multiplication $AB\neq BA$.

Bounded Linear Operators $\mathcal{B}(\mathcal{H})$ on a Hilbert Space: Let
$\mathcal{H}$ be a Hilbert space. A bounded linear operator
$A\in\mathcal{B}(\mathcal{H})$ acts on the Hilbert space
$A:\mathcal{H}\rightarrow\mathcal{H}$ such that

$A(\psi+\phi)=A(\psi)+A(\phi),\;\;\forall\psi,\phi\in\mathcal{H}$ (linearity)

$\exists$ a positive real number $c<\infty$ such that $||A(\psi)||\leq
c||\psi||,\;\;\forall\psi\in\mathcal{H}$ (bounded)

$||A||=\displaystyle\sup_{\psi\in\mathcal{H},||\psi||=1}\\{||A(\psi)||\\}$
(norm of an operator)

For all $A\in\mathcal{B}(\mathcal{H}),\;\exists$ a unique element called the
adjoint operator $A^{\dagger}\in\mathcal{B}(\mathcal{H})$ such that (using the
Hilbert space inner product $\langle\cdot,\cdot\rangle$

$\langle A^{\dagger}(\psi),\phi\rangle=\langle\psi,A(\phi)\rangle$

$(\alpha A+\lambda
B)^{\dagger}=\overline{\alpha}A^{\dagger}+\overline{\lambda}B^{\dagger}$

$(AB)^{\dagger}=B^{\dagger}A^{\dagger}$

$(A^{\dagger})^{\dagger}=A$

$||A^{\dagger}A||=||AA^{\dagger}||=||A||^{2}$

$||A^{\dagger}||=||A||$

The $\bm{C^{*}}$-Algebra features can be given to
$\bm{\mathcal{B}(\mathcal{H})}$ via the following

Banach Norm $\longrightarrow$ Operator Norm

Involution $*$-operator $\longrightarrow$ Adjoint operation †

$||A^{*}||=||A^{\dagger}||=||A||$ (Norm condition)

$||AA^{*}||=||AA^{\dagger}||=||A||\;||A^{\dagger}||=||A||^{2}$ ($C^{*}$
condition for $\bm{C^{*}}$-Algebra).

∗-Homomorphism $\bm{\xi}$: Let $\mathcal{A}$ and $\mathcal{B}$ be
$\bm{C^{*}}$-algebras. A ∗-homomorphism $\xi$ is a mapping
$\xi:\mathcal{A}\rightarrow\mathcal{B}$ that preserves the algebraic and ∗
structures of $\mathcal{A}$. That is, $\forall A,A_{1},A_{2}\in\mathcal{A}$

$\xi(A_{1}+\ A_{2})=\xi(A_{1})+\xi(A_{2})$ (linearity)

$\xi(A_{1}A_{2})=\xi(A_{1})\xi(A_{2})$ (homomorphism)

$\xi(A^{*})=\xi(A)^{*}$ (∗-preserving)

In general, $\bm{\xi}$ is norm decreasing, i.e. $||\xi(A)||\leq||A||$. If
$\bm{\xi}$ is an ∗-isomorphism (one-to-one, onto), then it is norm preserving,
$||\xi(A)||=||A||$. A ∗-automorphism is a ∗-isomorphism from a
$\bm{C^{*}}$-algebra to itself, i.e. $\xi:\mathcal{A}\rightarrow\mathcal{A}$.

Representation $\bm{\pi}$ of a $\bm{C^{*}}$-Algebra: A representation of a
$\bm{C^{*}}$-algebra $\mathcal{A}$ is a pair $(\mathcal{H},\pi)$ where
$\mathcal{H}$ is a Hilbert space and $\pi$ is a ∗-Homomorphism from
$\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$,
$\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{H})$. If $\pi$ is an
∗-isomorphism, it is called faithful such that

$\text{ker}\;\pi=\\{0\\}$ (faithful)

$||\pi(A)||=||A||,\forall A\in\mathcal{B}(\mathcal{H})$ (norm preserving)

$||\pi(A)||>0,\forall A>0$ (positive)

A subset $S\subset\mathcal{H}$ is called invariant under $\mathcal{A}$ if
$\pi(\mathcal{A})S:=\\{\mathcal{A}\psi|A\in\mathcal{A},\psi\in\mathcal{H}\\}\subset
S$

A representation $\pi$ is called irreducible if the only subspace of
$\mathcal{H}$ invariant under $\pi(\mathcal{A})$ are $\\{0\\}$ and
$\mathcal{H}$.

An operator ${A\in\mathcal{B}(\mathcal{H})}$ is called dense if its range
exists only in the dense subspace of the Hilbert space
$\mathcal{D}\in\mathcal{H}$.

Cyclic Vector $\;\Omega\in\mathcal{H}$: A vector $\;\Omega\in\mathcal{H}$ is
called a cyclic vector for a set of bounded operators
$\mathcal{B}(\mathcal{H})$ if $\\{A\;\Omega|A\in\mathcal{B}(\mathcal{H})\\}$
is dense in the whole $\mathcal{H}$.

Cyclic Representation $\bm{\pi}$ of a $\bm{C^{*}}$-Algebra: A cyclic
representation of a $\bm{C^{*}}$-algebra $\mathcal{A}$ is a triple
$(\mathcal{H},\pi,\Omega)$ where $(\mathcal{H},\pi)$ is a representation of
$\mathcal{A}$ and $\Omega\in\mathcal{H}$ is cyclic in the representation
$\pi$.

Linear Functionals (and States): Given an algebra $\mathcal{A}$, a linear
functional (or state) on $\mathcal{A}$ (a scalar valued ’function’ on
$\mathcal{A}$) is a map $\omega:\mathcal{A}\rightarrow\mathbb{C}$ such that
$\omega(\lambda A+\beta B)=\lambda\omega(A)+\beta\omega(B)$. Furthermore,

If $\mathcal{A}$ is a ∗-algebra, $\omega$ is called a positive functional if
$\omega(A^{*}A)\geq 0,\forall A\in\mathcal{A}$

A state $\omega$ is a positive functional with
$\omega(A^{*})=\overline{\omega(A)}$.

If $\mathcal{A}$ is a ∗-algebra, and $\omega_{i}$ are positive functionals
then for $\lambda_{i}\in\mathbb{R}^{+},\sum_{i}\lambda_{i}=1$, one may
construct a positive convex linear functional as
$\omega=\sum_{i}\lambda_{i}\omega_{i}$.

If $\mathcal{A}^{sa}$ is self-adjoint subspace, i.e. $A^{*}=A$ and $\omega$ is
a positive functional, then $\omega:\mathcal{A}^{sa}\rightarrow\mathbb{R}$
(real scalars)

If $\mathcal{A}$ is a $\bm{C^{*}}$-algebra, every positive functional $\omega$
is continuous.

Let $\mathcal{A}$ be a $\bm{C^{*}}$-algebra. $\forall A\in\mathcal{A},\exists$
a positive functional $\omega_{A}$ such that, $||\omega_{A}||=1$ and
$\omega_{A}(A^{*}A)=||A||^{2}$.

GNS Construction-Representation from a State: Let $\mathcal{A}$ be a
$\bm{C^{*}}$-algebra (of observables). Let $\omega$ be a functional (state) on
$\mathcal{A}$. A representation of $\mathcal{A}$ constructed from $\omega$ is
a pair $(\mathcal{H},\pi_{\omega})$ where given the inner product on
$\mathcal{H}$,
$\left\langle\;\;|\;\;\right\rangle:\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{C}$
and a cyclic vector $\Omega_{\omega}$ with respect to $\pi_{\omega}$
$\bigl{(}\pi_{\omega}(\mathcal{A})|\left.\Omega_{\omega}\right\rangle$ is
dense in $\mathcal{H}\bigr{)}$, one has the action of operators on
$\mathcal{H}$ such that

$\omega(A):=\dfrac{\left\langle\Omega_{\omega}|\pi_{\omega}(A)\Omega_{\omega}\right\rangle}{\left\langle\Omega_{\omega}|\Omega_{\omega}\right\rangle}$

Remark: This method requires one to choose a preferred representation
$\pi_{\omega}$. In quantum mechanics, the Stone von Neumann theorem states
that all representations of the Weyl algebra are unitarily equivalent to the
Schrödinger representation so this is not a problem. In Minkowski QFT,
Poincaré invariance chooses an appropriate cyclic vector (vaccuum state). The
choice of a representation becomes an issue in QFT on curved spacetime
manifolds.

Von Neumann Algebra: Consider a $C^{*}$-algebra $\mathcal{B(H)}=\\{A\\}$ of
bounded linear operators on a Hilbert space,
$A:\mathcal{H}\rightarrow\mathcal{H}$. Let $\mathcal{C}$ be a subset of
$\mathcal{B(H)}$. An operator $A\in\mathcal{B(H)}$ belongs to the commutant
$\mathcal{C}^{\prime}$ of the set $\mathcal{C}$ $\iff$ $AC=CA,\,\,\,\forall
C\in\mathcal{C}$. A von Neumann algebra $\mathcal{A}$ is a unital
$C^{*}$-subalgebra of $\mathcal{B(H)}$ such that
$\mathcal{A}^{\prime\prime}=\mathcal{A}$. Consider a von Neumann algebra
$\mathcal{A}\subset\mathcal{B(H)}$. A von Neumann algebra in standard form is
one where there exists an element $|\Omega\rangle\in\mathcal{H}$ which is both
cyclic (operating on $|\Omega\rangle$ with elements in $\mathcal{A}$ can
generate a space dense in $\mathcal{H}$) and separating (if
$A|\Omega\rangle=0$, then $A=0$).

Tomita Takesaki Modular Operators: Consider a von Neumann algebra
$\mathcal{A}\subset\mathcal{B(H)}$ in standard form with a cyclic and
separating vector $|\Omega\rangle\in\mathcal{H}$. Let
$S:\mathcal{H}\rightarrow\mathcal{H}$ be a anti-unitary operator defined by
$SA|\Omega\rangle=A^{*}|\Omega\rangle$. Let the closure of $S$ have a polar
decomposition given by $S=J\Delta^{\frac{1}{2}}=\Delta^{-\frac{1}{2}}J$, where
$J$ is called the modular conjugation operator and $\Delta$ is called the
modular operator. $J$ is anti-linear and anti-unitary whereas $\Delta$ is
self-adjoint and positive. Furthermore, the following relations hold:

1\. $J\Delta^{\frac{1}{2}}J=\Delta^{-\frac{1}{2}}$

2\. $J^{2}=I,\,\,\,J^{*}=J$

3\. $J|\Omega\rangle=|\Omega\rangle$

4\. $J\mathcal{A}J=\mathcal{A}^{\prime}$

5\. $\Delta=S^{*}S$

6\. $\Delta|\Omega\rangle=|\Omega\rangle$

7\. $\Delta^{it}\mathcal{A}\Delta^{-it}=\mathcal{A}$ (one parameter-$t$ group
of automorphisms of $\mathcal{A})$

8\. If $\omega(A)=\langle\Omega|A\Omega\rangle$, $\forall A\in\mathcal{A}$,
then $\omega$ is a KMS (Kubo-Martin-Schwinger) functional (state) on
$\mathcal{A}$ with respect to the automorphism of 7.

9.$|\Omega\rangle$ is cyclic for $\mathcal{A}$ if and only if $|\Omega\rangle$
is separating for $\mathcal{A}^{\prime}$

##  VIII Acknowledgements

We would like to thank Michele Arzano for helpful discussions.

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