Clocks, Algebras and Cosmology

Cesar<EMAIL_ADDRESS>
a Instituto de Física Teórica UAM-CSIC
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

Gauge invariant local observables describing primordial scalar quantum
fluctuations in Inflationary Cosmology are identified as elements of a type
$II$ de Sitter crossed product algebra. This algebra is defined, after adding
a reference frame clock, as the algebra of clock dressed local operators.
Clock dressing sets, in the weak gravity limit, the Schrodinger equation for
gauge invariant quantum fluctuations. Instead of using a slow roll inflaton
potential to define the clock Hamiltonian and the clock state we suggest a
natural double de Sitter clock making the whole algebra associated with the
planar patch a type $II$ factor. The corresponding clock states are EPR
squeezed states. Using this clock to define the needed clock dressing leads to
concrete model independent predictions of the inflationary parameters. Some
speculative remarks on potential ways to define a type $I$ upgrading are
briefly discussed.

###### Contents

  1. 1 Introduction
  2. 2 The algebraic picture of de Sitter
  3. 3 Clock Algebra
    1. 3.1 The quantum mechanical picture
    2. 3.2 Clock dressing
    3. 3.3 A comment on gravitational limits to clock dressing
  4. 4 dS maximal entropy state
  5. 5 Gauge invariant observables in primordial Inflationary Cosmology
  6. 6 The de Sitter double squeezed clock
  7. 7 Some speculative conjectural final remarks
  8. 8 Acknowledgments

##  1 Introduction

The problem of time in Quantum Gravity is an old and extensively discussed
problem along the years. In a nutshell the problem is that in QG physical
local observables should commute with the Hamiltonian i.e. $[H,{\cal{O}}]=0$
and therefore they are stationary. This, as originally pointed out by D.Page
and K.Wootters [1], is similar to the problem with superselection charges
where physical local observables commute with the charge generator implying
that quantum superpositions of states with different charge are unobservable.
A natural way to account for the non trivial time dependence we observe in the
world is by either adding an external reference frame/ observer/clock, in the
sense originally suggested by Aharonov and Susskind [2]. Recently this
approach has acquired a new impulse thanks to the work of E.Witten and
collaborators [3, 4, 5, 6, 7] on the von Neumann algebras of local observables
for local quantum field theory in a fixed background with horizons 222The
notion of reference frames to define superselection charges appears in quantum
information theory [8]. The interpretation of clocks as the reference frame
needed to define gauge invariant observables in QG was discussed in [9] and
more recently in [10]..

The problem of how to define a clock, or more precisely an hermitian operator
telling time, goes back to the early days of quantum mechanics. More
specifically to the discussions on how to interpret the uncertainty relation

$\Delta E\Delta t\geq\hbar$ (1)

The simplest interpretation of (1) is to consider a quantum wave packet of
width $\Delta(E)$ and to interpret $\Delta t$ as the time during which the
wave packet does not change significantly. In this interpretation (1) follows
from the standard Fourier analysis and simply tells us about a typical life
time of the system. As originally observed by Mandelstam and Tamm [11] in this
approach we use as clock some observable $A$ of the system that varies with
time i.e.

$[H,A]=\hbar\dot{A}$ (2)

for $H$ the Hamiltonian of the system and we select a quantum state on which
the average $\langle\dot{A}\rangle$ is not changing significantly and such
that the uncertainty in $\dot{A}$ satisfies
$\frac{\Delta\dot{A}}{\langle\dot{A}\rangle}<<1$. In this case we easily get
on this state that

$\Delta t\Delta H\geq\hbar$ (3)

for $\Delta t\equiv\frac{\Delta A}{\langle\dot{A}\rangle}$. This
interpretation is, however, not making any reference to any underlying
measurement process.

The other popular interpretation of (1) is as implying that energy
measurements cannot be carried out in arbitrary short periods of time. In
order to substantiate this interpretation we need to have some physical
procedure to identify the time at which the measurement is performed. It is in
order to define this measurement time that we will need an external clock
[12].

This will be also the case whenever we intend to implement, in local quantum
field theory, invariance of physical amplitudes under coordinate time
reparametrizations. In such a case we need to define the reference clock time
at which the local operators involved in the amplitude are acting. This will
means that local observables should depend not only on the coordinate time $t$
but also on the clock coordinate used to define the way the clock, as an
independent quantum system, tells us the measurement time. These clock dressed
operators are part of an extended algebra and they act on the extended Hilbert
space representing the observed system and the external clock. Now gauge
invariance under coordinate time reparametrizations can be implemented
requiring these dressed operators to commute with $H+H_{c}$ with $H$
representing the observed system Hamiltonian and with $H_{c}$ representing the
clock Hamiltonian. This algebraic structure defines, in certain cases, a
crossed product algebra [3, 4, 5] 333Note that both $H$ and $H_{c}$ define
automorphisms i.e. coordinate time $t$ translations..

The key observation in [3, 4] is that for systems characterized by a type
$III_{1}$ factor while the algebra of gauge invariant local observables is
trivial ( only the identity ) once you include an external clock you can
transform this type $III$ factor into a type $II$ factor and to identify a non
trivial set of gauge invariant local observables. Moreover in the Hilbert
space of the so defined extended system you have finite density matrix mixed
states with well defined von Neumann entropy that can be interpreted as
Bekenstein generalized entropy [13]. The deep ingredient of such construction
does not lies in defining an operative way to tell time using the clock but on
how adding the clock you manage to define a non trivial set of gauge invariant
local observables.

As stressed in [9] the need of an external clock to define non trivial gauge
invariant local observables is potentially crucial in understanding primordial
Inflationary Cosmology. In this case gauge invariant primordial quantum
fluctuations become naturally part of the crossed product of the de Sitter
type $III$ factor and a clock algebra with the inflaton playing the role of
clock coordinate. We will see how the gauge invariant Mukhanov-Sasaki
variables [14] are a canonical example of clock dressed operators. In this
approach the crucial cosmological question of Inflation is how to define the
right clock Hamiltonian. We will suggest to use as clock the analog of a free
”particle” moving in the planar patch. The clock Hamiltonian generating, for
the so defined clock, translations in the conformal time coordinate, is given
by the pure de Sitter Bogolyubov transformation for conformal time
translations on the planar patch. This Bogolyubov transformation defines a
natural automorphism on the algebra describing a clock in dS. However since,
when doing Cosmology we are working on the whole planar patch, containing the
static patch as well as the region beyond the cosmological horizon, we
naturally expect a ”two mode” clock algebra
${\cal{A}}_{c}\times\tilde{{\cal{A}}_{c}}$ with each clock algebra making the
algebra of gauge invariant observables, as well as its commutant, a type $II$
factor. The Bogolyubov transformation defines a natural automorphism of this
”two mode” clock algebra. Moreover the natural clock states are squeezed EPR
entangled two modes states. In this frame the non trivial and observable
features of Inflationary Cosmology will appear when we consider amplitudes of
operators clock dressed with this two mode dS-clock 444The need in Inflation
of two clocks is physically quite natural. We need to define, using clock
dressing, the local gauge invariant quantum fluctuations describing the CMB
postinflationary correlators as well as the gauge invariant primordial quantum
fluctuations during inflation. A basic assumption in Inflation is that
postinflationary correlators can be expressed in terms of primordial
correlators at horizon exit.

##  2 The algebraic picture of de Sitter

In this first section we will review some aspects of the algebraic approach to
weakly coupled QFT in a fixed de Sitter background [4]. Let us denote
${\cal{H}}$ the QFT Hilbert space. Now let us consider an observer with a
static patch as her causal domain and let us define ${\cal{A}}_{dS}$ the
algebra of local observables that this observer can measure performing local
experiments 555See [6] for a recent discussion and precise definitions.. The
first key assumption is that it exists a representation of ${\cal{A}}_{dS}$ in
the algebra ${\cal{B}}({\cal{H}})$ of bounded operators such that the so
defined sub algebra of ${\cal{B}}({\cal{H}})$ is a von Neumann factor. This
means: i) that ${\cal{A}}_{dS}$ = ${\cal{A}}^{{}^{\prime\prime}}_{dS}$ for
${\cal{A}}^{\prime}_{dS}$ the commutant of ${\cal{A}}_{dS}$ 666This is defined
as the set of bounded operators $x$ in ${\cal{B}}({\cal{H}})$ such that they
commute with all elements in ${\cal{A}}_{dS}$. and ii) that the center of
${\cal{A}}_{dS}$ is trivial. The GNS representation of ${\cal{A}}_{dS}$
defines a Hilbert space ${\cal{H}}_{GNS}$ with states $|a\rangle$ associated
to the elements in ${\cal{A}}_{dS}$ with scalar product defined by a linear
form $f:{\cal{A}}_{dS}\rightarrow R$ with $\langle
a|b\rangle=f(a^{+}b)$777More precisely each element in the GNS Hilbert space
is associated with an equivalence class defined relative to ideal kernel of
the linear form $f$.. This representation is characterized by a ”vacuum”
defined as the state in ${\cal{H}}_{GNS}$ associated with the identity. We
will denote this state $\Psi_{dS}$. Formally identifying ${\cal{H}}_{GNS}$
with the QFT Hilbert space ${\cal{H}}$ we can associate $\Psi_{dS}$ with the
globally de Sitter invariant Bunch-Davis vacuum [15] in ${\cal{H}}$.

General arguments imply that ${\cal{A}}_{dS}$ is a type $III_{1}$ factor. This
in particular means that it exists a state dependent modular Hamiltonian
$\hat{h}_{\Psi_{dS}}$ generating outer automorphisms of ${\cal{A}}_{dS}$ and
such that $\hat{h}_{\Psi_{dS}}|\Psi_{dS}\rangle=0$. Geometrically
$\hat{h}_{\Psi_{dS}}$ is associated with a global Killing of full de Sitter,
inducing inverse time translations on opposite static patches. In that sense
the action of $\hat{h}_{\Psi_{dS}}$ on the observer static patch can be
identified with the generator of time translations on that static patch.
However this action cannot be identified with the action of the naive static
patch Hamiltonian $H$ associated with time translation is this static patch
and no action on the opposite static patch. The crucial property of
${\cal{A}}_{dS}$ as a type $III_{1}$ factor is that this ”static patch
Hamiltonian” $H$ is not defined in ${\cal{H}}_{GNS}$. This means that acting
with $H$ on a generic state $|\psi\rangle$ in ${\cal{H}}_{GNS}$ we move out of
the space or equivalently that the norm of $H|\psi\rangle$ is infinity. This
is the direct manifestation of the divergent entanglement associated with type
$III_{1}$ factors.

The QG ”time problem” in this algebraic picture becomes specially neat [4].
Indeed the set of physical observables for the static patch observer should be
defined by requiring invariance under time reparametrizations. Defining those
by the action of $\hat{h}_{\Psi_{dS}}$ on the static patch we get as invariant
algebra ${\cal{A}}_{dS}^{\hat{h}}$ which unfortunately is trivial. Note that
we cannot use $H$ defined above since this operator is not defined due to the
type $III_{1}$ divergent entanglement.

Intuitively as discussed in [9] and previously in [1] this problem is similar
to the one we found with superselection charges. Indeed if we declare the
algebra of physical observables to be defined by $[Q,a]=0$ for some SS charge
$Q$ we cannot find, in the so defined algebra, local physical operators
creating any non vanishing charge and consequently superpositions of states
with different charges are unobservable. In the case of using $\hat{h}$
instead of the SS charge $Q$ we cannot find quantum superpositions of states
with different ”modular energy”. The natural way to address this problem in
the case of SS charges is, following [2], the introduction of an external
reference frame system [8] that in the case of time reparametrizations means
adding an external clock.

Thus the suggestion in [4] is to define the algebra of physical local
observables i.e. invariant under time reparametrizations of the observer
static patch, as

${\cal{A}}^{cr}=({\cal{A}}_{dS}\otimes{\cal{A}}_{clock})^{(\hat{h}+\hat{H}_{c})}$
(4)

provided we can define a clock algebra ${\cal{A}}_{clock}$ and a clock
Hamiltonian $\hat{H}_{c}$. Before discussing the nature of this crossed
product algebra let us pause to discuss how to define a physical clock in
quantum mechanics.

##  3 Clock Algebra

Following [12] we will define a generic quantum mechanical clock as follows.
Let us first define the clock as an independent quantum system. In the
simplest case we can define this system as having just one degree of freedom.
Thus quantum mechanically this clock is characterised by position $\hat{z}$
and momentum $\hat{p}_{z}$ operators satisfying

$[\hat{p}_{z},\hat{z}]=i\hbar$ (5)

To use this system as a clock i.e. to use the value of the coordinate $z$ to
tell us the clock time, we need to define its Hamiltonian $\hat{H}_{c}$ and to
define $\dot{\hat{z}}$ by the corresponding Heisenberg equation of motion
$i\hbar\dot{\hat{z}}=[\hat{H}_{c},\hat{z}]$. In the simplest illustrative case
we can use the free Hamiltonian $\hat{H}_{c}=\frac{\hat{p}_{z}^{2}}{2M}$ and
to use as a clock a free particle that will tell us the time by its position.
In this discussion we consider the clock, not as a classical external
apparatus, but as an external purely quantum system. Obviously to use this
quantum mechanical system as a clock we need to consider a clock quantum state
$|\Phi_{clock}\rangle$ for which the variance of $\dot{\hat{z}}$ defined as
$\frac{\Delta\dot{\hat{z}}}{|\langle\dot{\hat{z}}\rangle|}$ is negligible. Let
us now ask ourselves under what conditions we can define, associated with this
quantum system, a clock algebra formally defined by two self adjoint operators
$\hat{H}_{c}$ and $\hat{t}$ satisfying

$[\hat{H}_{c},\hat{t}]=i\hbar$ (6)

For the simplest case of $\hat{H}_{c}$ the free Hamiltonian we can define the
hermitian time operator $\hat{t}$ as [12]

$\hat{t}=\frac{1}{2}M(\hat{z}\frac{1}{\hat{p}_{z}}+\frac{1}{\hat{p}_{z}}\hat{z})$
(7)

that indeed formally leads to

$[\hat{H}_{c},\hat{t}]=i\hbar$ (8)

and to the standard Heisenberg uncertainty relation
$\Delta(\hat{t})\Delta(\hat{H}_{c})\geq\hbar$.

Obviously the previous definition of $\hat{t}$ should be wrong. Indeed the
problem of defining a self adjoint time operator $\hat{t}$ canonically
conjugated to some Hamiltonian is that in such a case both $\hat{t}$ and
$\hat{H}_{c}$ should have as spectrum the whole real line. This cannot be the
case if the clock Hamiltonian is positive. In the previous simple example,
where we used a free particle Hamiltonian, this problem is revealed by the
fact that the operator $\hat{t}$ defined by (7) is not well defined for
$p_{z}=0$.

The Hilbert space representation of (5), describing the physical system used
as clock, is $L^{2}(R)$, however the clock time operator $\hat{t}$ given by
(7) is, in principle, only well defined in the subspace of states satisfying
$|\langle\psi|\frac{p_{z}}{M}|\psi\rangle|>0$ i.e. only on clock states with
some non vanishing $\dot{z}$.

In order to make a preliminary contact with the discussion in [4] notice that
we have, in principle, two alternative ways to define the external clock. In
the case presented above you start with a quantum system characterised by the
algebra (5) and for a given positive Hamiltonian $\hat{H}_{c}$ you try to
define a clock time operator $\hat{t}$ satisfying (8). This as discussed is
not possible for $\hat{H}_{c}$ bounded. Hence you need to define $\hat{t}$ on
an appropriated subspace characterised, in the case of using as clock a free
particle, by the constraint $|\langle\dot{\hat{z}}\rangle|>0$. However, you
can also start with the algebra (8) that implies the full real line as
spectrum of $\hat{H}_{c}$ and try to implement the positivity constraint
projecting on states with positive eigenvalue of $\hat{H}_{c}$. Thus, in one
version, in which you keep track of the physical degrees of freedom defining
the external clock, you need to introduce the constraint
$|\langle\dot{\hat{z}}\rangle|>0$ while in the case you directly start with
the algebra (8) you need to introduce the positivity constraint on
$\hat{H}_{c}$. This is the constraint that in the case of de Sitter leads to a
type $II_{1}$ crossed product by contrast to a type $II_{\infty}$ factor [4].

More specifically if we define the algebra ${\cal{A}}_{clock}$ by (8) then, in
the case of de Sitter, you need to implement the positivity constraint on
$\hat{H}_{c}$ at the algebraic level as

$\tilde{\cal{A}}=\Pi{\cal{A}}^{cr}\Pi$ (9)

with $\Pi$ the positivity projector and ${\cal{A}}^{cr}$ the crossed product
(4).

The previous problem can be presented, in simpler terms, as the impossibility
to define, in quantum mechanics, an ideal clock with positive Hamiltonian and
vanishing probability amplitude to move backwards in time [16] 888The essence
of the problem is that for a bounded clock Hamiltonian if we impose that the
clock goes always forward or backwards in time then general analyticity
arguments based on the positivity of the clock Hamiltonian will imply that the
clock is not working at all. .

###  3.1 The quantum mechanical picture

Let us momentarily ignore this problem and consider first generic quantum
states of the clock. These, for a free particle clock, are characterised by
wave packets

$\Phi(z,t)=\int
dp_{z}f(p_{z})e^{\frac{i}{\hbar}(p_{z}z-\frac{p_{z}^{2}t}{2M})}$ (10)

To define a reasonable clock it is natural to use for $\Phi$ a narrow wave
packet in coordinate $z$ and to use a large value of $M$ to be sure that the
wave packet spreads slowly. Let us denote this wave packet $\Phi_{0}$.

If we intend to use this clock to tell at what clock time the observed system
has a certain property ${\cal{O}}$ we need to associate with the measurement
of ${\cal{O}}$ a measurement time and to read such measurement time in our
clock using the clock coordinate $z$. In other words, when the observer
measures the observable ${\cal{O}}$ she needs to identify at what clock time
as indicated by the value of $z$ this measurement was performed. In standard
quantum mechanics this means that the wave function of the observed system on
which you measure ${\cal{O}}$ will depend not only on the degrees of freedom
$x$ of the observed system and on the particular way the observable
${\cal{O}}$ is acting but also on the degrees of freedom $z$ of the clock used
to define the measurement time. Thus if we consider a clock characterised by a
wave packet in $z$ coordinates that spreads slowly the wave function
describing the observed system after acting with the observable ${\cal{O}}$ at
the clock time indicated by $z$ will be a product state of the type

$\Psi_{{\cal{O}}}=\Phi_{0}(z,t)\psi_{{\cal{O}}}(x,z,t)$ (11)

with $\Phi_{0}$ the clock wave packet and with $\psi_{{\cal{O}}}(x,z,t)$
representing the wave function of the observed system after acting with the
observable ${\cal{O}}$ at clock time characterised by the clock coordinate
$z$.

The Schrodinger equation for $\Psi$ is

$i\hbar\frac{d}{dt}\Psi=(\hat{H}_{c}+\hat{H})\Psi$ (12)

where $\hat{H}$ is the Hamiltonian of the observed system. Assuming (11) and
slow spread of the clock wave packet i.e.

$\Phi_{0}(z,t)=f(z-\dot{z}t)e^{\frac{i}{\hbar}(\bar{p}_{z}z-\frac{\bar{p}_{z}^{2}t}{2M})}$
(13)

with $\bar{p}_{z}$ the average value and $\dot{z}=\frac{\bar{p}_{z}}{M}$, we
get for $\psi_{{\cal{O}}}$ the Schrodinger equation

$i\hbar\frac{d}{dt}\psi_{{\cal{O}}}=(\hat{H}-i\hbar\dot{z}\frac{d}{dz})\psi_{{\cal{O}}}$
(14)

###  3.2 Clock dressing

Formally for any local operator ${\cal{O}}$ and any initial product state
$\Psi=\Phi_{0}\psi_{0}$ in the extended Hilbert space describing the observed
system and the physical clock, we can define the action of ${\cal{O}}$ on the
extended Hilbert space as

$\hat{\cal{O}}\Psi=\Psi_{{\cal{O}}}=\Phi_{0}(z,t)\psi_{\cal{O}}(x,z,t)$ (15)

for $\psi_{{\cal{O}}}$ satisfying (14). We can think of the so defined
$\hat{\cal{O}}$ as the clock dressed operator.

Intuitively for an observer equipped with an algebra ${\cal{A}}$ of local
observables once we want to tell at what clock time the observed system has
the properties described by measuring the observables in ${\cal{A}}$ she needs
to extend the algebra by adding the clock algebra and to dress the operators
${\cal{O}}$ in ${\cal{A}}$ appropriately, for instance defining the former
$\hat{\cal{O}}$. Note that in case we impose invariance under time coordinate
reparametrizations the clock frame and the corresponding dressing is
compulsory in order to define physical amplitudes.

The simplest way to think about clock dressing is to imagine the observable
${\cal{O}}$ as a local operator and to identify the representation of
measuring ${\cal{O}}$ at certain coordinate time $t_{0}$ as the smearing of
${\cal{O}}$ with some test function with support in coordinate time in a small
region around $t_{0}$. However, invariance under reparametrizations of the
time coordinate will force that operator to commute with the Hamiltonian
$\hat{H}$ of the system we are observing and eventually, as it happens in the
case of de Sitter where this Hamiltonian is the modular Hamiltonian of a type
$III_{1}$ factor, to be just the identity operator. Nevertheless if we use a
quantum clock to tell at what clock time the measurement of ${\cal{O}}$
occurred, we could do it dressing the operator and making it to depend on the
clock coordinate ($z$ in the previous example ). This dressed local operator,
let us say $\hat{\cal{O}}(t,z)$, depends on coordinate time $t$ and on clock
time $z$. In case we now impose invariance under coordinate time
reparametrizations we need to require

$[\hat{H}+\hat{H}_{c},\hat{\cal{O}}]=0$ (16)

The dressed operator that automatically satisfies this condition is

$\hat{\cal{O}}=e^{i\hat{H}\hat{t}}{\cal{O}}e^{-i\hat{H}\hat{t}}$ (17)

for $\hat{H}$ the Hamiltonian of the observed system and $\hat{t}$ the clock
time operator satisfying $[\hat{H}_{c},\hat{t}]=i\hbar$.

Intuitively the operator (17) acting on the extended Hilbert space of the
observed system and the external clock sets the measurement of ${\cal{O}}$ at
the clock time measured by $\hat{t}$.

Note that the time operator $\hat{t}$, conjugated to the clock Hamiltonian
$\hat{H}_{c}$, naturally appears when we try to solve the equation (16)
defining the set of gauge invariant physical observables. The dressing defined
by (17) is the typical dressing defining the de Sitter crossed product algebra
in case we use for ${\cal{O}}$ operators in ${\cal{A}}_{dS}$. The dressing
defined in (15) agrees with that defined in (17) if

$i\hbar\frac{d}{dt}\Psi_{{\cal{O}}}=0$ (18)

This exemplifies that the clock dressing agrees with the crossed product
dressing once we impose invariance under reparametrizations of the coordinate
time 999 Recently in [10] the relation between clock dressing and time
reversal is discussed. In essence the problem is that generic correlators
$\langle A(t_{1})A(t_{2})\rangle$ only make sense, if we impose invariance
under time reparametrizations, for the corresponding clock dressed operators.
As discussed these dressed operators can be formally thought as depending on
clock time $z$. For a clock state $\Phi_{0}$ the way the clock move forward or
backwards in clock time depends on the sign of
$\dot{z}=\frac{\bar{p}_{z}}{M}$. The observation in [10] is that once we
choose a clock state $\Phi_{0}$ with a definite sign of $\dot{z}$ we fix time
reversal gauge invariance. The interesting aspect for us is that, as stressed
above, $\dot{z}=0$ should be projected out in order to have a well defined
$\hat{t}$ operator.. The important lesson to be extracted from the previous
exercise is that in order to define gauge invariant observables with respect
to time coordinate reparametrizations we need i) to add a quantum system
working as a clock and characterised by a clock Hamiltonian ii) to define the
operator $\hat{t}$ conjugated to the clock Hamiltonian and iii) to implement
the positivity constraint on the spectrum of the clock Hamiltonian.

###  3.3 A comment on gravitational limits to clock dressing

In the previous simple discussion of clock dressing in quantum mechanics we
were using as the clock needed to define the measurement time  a simple
system, namely a free moving particle that tells us time by its position.
Implicitly we were assuming [12] that the mass $M$ of this external clock was
large enough. Indeed equation (14) was written in the limit where we ignore
corrections of order $\frac{1}{M}$. Physically this condition on the clock
mass $M$ can be easily understood once we use the clock to define dressed
local operators. If we want to give meaning to the measurement time, as
defined by the clock, associated to acting with a local observable
${\cal{O}}(x)$ we need to assume that the action of the operator ${\cal{O}}$
on the observed system is taking place in a region of space smaller than the
typical wave length of the clock. Only in that case we can define a dressed
operator that inform us about the value of the observable ${\cal{O}}$ at some
concrete clock time. In more concrete terms this means that the support of the
smearing function used to define the local observable should be larger than
the clock wave length. Once this condition is satisfied we can, in particular,
ignore $\frac{1}{M}$ corrections to the Schrodinger equation for the clock
dressed operator. In case we work in the limit of $G_{N}=0$ this condition can
be easily satisfied since we don’t have a priori any limitation on the clock
mass $M$. However the situation changes when gravity is turned on i.e. when we
work with $G_{N}$ non vanishing.

In this case the clock mass cannot be larger than $M_{P}$ if we don’t want to
have as clock a black hole. This makes clear that the $\frac{1}{M}$
corrections to the Schrodinger equation for the clock dressed operators
become, once gravity is turned on, quantum gravity corrections. Hence the
picture we get is the following. Once we impose general invariance under time
coordinate reparametrizations we need to dress our operators which in practice
means to make them dependent on the clock coordinate. The effect of the clock
Hamiltonian on the amplitudes for clock dressed operators defines, depending
on the particular clock state, a power series in the clock mass. Moreover the
clock mass is restricted by the condition that the clock should not become a
black hole. Obviously this gravitational limit on the clock mass implies also
a limit on the clock precision, in particular on the clock wave packet. In
particular the clock dressed operators defining the gauge invariant
observables i.e. the ones invariant under time coordinate reparametrizations,
should be defined using smearing functions with support larger than the clock
wave length. In what follows we will work in the weak gravity limit, however
we will make some general comments on the effects of clock dressing when
$G_{N}$ is turned on in the last section.

##  4 dS maximal entropy state

Assuming that the algebra (9) for de Sitter is a type $II$ factor implies the
existence of a trace $tr$ on $\tilde{\cal{A}}$ as well as the existence, for
any state $|\hat{\Phi}\rangle$ in the extended Hilbert space representation of
$\tilde{\cal{A}}$, of a density matrix operator $\rho_{\hat{\Phi}}$ such that
for any $\hat{a}\in\tilde{\cal{A}}$ we have

$\langle\hat{\Phi}|\hat{a}|\hat{\Phi}\rangle=tr(\rho_{\hat{\Phi}}\hat{a})$
(19)

Let us consider the state $|\hat{\Psi}_{dS}\rangle$ representing the QFT
vacuum in fixed pure de Sitter and a clock in thermal equilibrium at
temperature $\beta_{dS}=r_{dS}$101010Recall that thermal states in dS are
formally defined by Euclidean continuation where the Killing vector associated
to the modular Hamiltonian $\hat{h}_{dS}$ defines translation on a circle.
Before adding the clock and extending the algebra the Bunch Davis vacuum
$|\Psi_{dS}\rangle$ is not associated to any density matrix. By a clock in
thermal equilibrium we will mean a clock state defined by a simple
superposition of eigenstates of the clock Hamiltonian defined by a formal
thermal probability distribution $f(\epsilon)=e^{-\beta_{dS}\epsilon}$. The
meaning of this state will become clear once we define a trace form on the
algebra [3]. . Let us now assume that this state is in the extended Hilbert
space representing $\tilde{\cal{A}}$ and therefore that we can associate with
it a density matrix operator $\rho_{\hat{\Psi}_{dS}}$. We expect that the von
Neumann entropy
$s_{vN}(\rho_{\hat{\Psi}_{dS}})=tr(\rho_{\hat{\Psi}_{dS}}\ln\rho_{\hat{\Psi}_{dS}})$
defined for this density matrix will be the Gibbons Hawking (GH) entropy [17]
of pure de Sitter since the external added clock is in thermal equilibrium.
The novelty, relative to Gibbons and Hawking, is that now we are using the
extended algebra $\tilde{\cal{A}}$, including the clock, in order to have a
well defined density matrix and a well defined von Neumann entropy. In GR the
GH entropy is given by $\frac{A}{4G_{N}}$ for $A$ the area of the cosmological
horizon. Thus we need to discover the relation between this GH entropy and the
vN entropy $s_{vN}(\rho_{\hat{\Psi}_{dS}})$. In the QFT weak gravity $G_{N}=0$
limit where we are defining $\tilde{\cal{A}}$ and the corresponding vN
entropy, this GR entropy is infinity. For $\tilde{\cal{A}}$ a type $II_{1}$
factor we can define a finite vN entropy by formally substracting this
infinity. Indeed we can identify $\rho_{\hat{\Psi}_{dS}}$ with the identity in
$\tilde{\cal{A}}$ and to normalize $tr1=1$. If we do that we get from (19)
that for any $\hat{a}$ in $\tilde{\cal{A}}$ we have

$tr\hat{a}=tr(\hat{a}\rho_{\hat{\Psi}_{dS}})=\langle\hat{\Psi}_{dS}|\hat{a}|\hat{\Psi}_{dS}\rangle$
(20)

Assuming $\hat{\Psi}_{dS}$ is a product state $\Psi_{dS}\Phi_{clock}$ with
$\Psi_{dS}$ the Bunch Davis vacuum and $\Phi_{clock}$ representing the clock
in thermal equilibrium i.e. defined by a thermal distribution
$e^{-\beta\epsilon}$ on the spectrum $\epsilon$ of $\hat{H}_{c}$ we get
$\hat{\Psi}_{dS}=\int
d{\epsilon}e^{-\frac{\beta\epsilon}{2}}\Psi_{dS}|\epsilon\rangle$. The key
consistency check with the assumption that $\rho_{\hat{\Psi}_{dS}}$ for this
product state is the identity i.e. flat entanglement, requires to show that
the trace defined by (20) satisfies indeed the trace property. This, non
trivial check, was done in [3] using the KMS property. The state
$\hat{\Psi}_{dS}$ is, in the weak gravity limit, the maximal entropy state
[18, 19, 20, 21]. What is here special is not that we can renormalize the
$G_{N}=0$ limit of Gibbons Hawking entropy, what is special is that we can fix
the substracted value of the entropy as the finite trace of the identity
$tr(1)$ in $\tilde{\cal{A}}$ in the case $\tilde{\cal{A}}$ is a type $II_{1}$
factor. Recall that the type $II_{1}$ property and consequently the existence
of a maximal entropy state is a consequence of imposing positivity for the
clock Hamiltonian 111111Note that although the von Neumann entropy for
$\rho_{\hat{\Psi}_{dS}}$ is normalized to zero with $\rho_{\hat{\Psi}_{dS}}$
identified with a flat entanglement density matrix we need to keep a finite
$\beta_{dS}$ in the definition (20) of the trace. In other words what
represents flat entanglement is not the non existent density matrix describing
pure dS but the density matrix describing the extended system including a
clock with a simple clock state characterised by the GH finite $\beta$..

The clock defined in ”thermal equilibrium” is a very bad clock. Indeed the
time uncertainty in $\hat{t}$ is order $\beta$ since the typical energy
uncertainty of the clock is given by the GH temperature. What about using a
better clock ? Such a clock must have average energy larger than GH
temperature and therefore we should expect that it will modify the de Sitter
metric. To see how this is working let us define the clock quantum state as a
wave packet with average value $E$ of the clock energy and such that it
spreads slowly. We can characterise this wave packet as $\Phi_{clock}=\int
d\epsilon f(\epsilon)|\epsilon\rangle$ for an appropriated $f$. We can now
assume that the state in the extended Hilbert space representing just the
presence of this clock is a product state $\hat{\Phi}=\Phi\Phi_{clock}$ with
$\Phi$ some state in the QFT Hilbert space ${\cal{H}}$ different from the BD
vacuum. The state $\hat{\Phi}$ will account for the back reaction of the clock
energy on the background metric. The first hint about this state is to expect
that the corresponding vN entropy will differ from the vN entropy of the
maximal entropy state, that is normalised to zero, by an amount of the order
$E\beta$. This is in particular the expected deviation of GH entropy if we
assume that the metric in the presence of a localised clock of energy $E$ is a
Schwarzschild-de Sitter metric. Thus we expect the vN entropy of
$\rho_{\hat{P}hi}$ to contain a piece of the order $\langle\hat{\Phi}|\beta
H_{c}|\hat{\Phi}\rangle$ for $H_{c}$ the clock Hamiltonian. As shown in [4]
this is indeed a piece of the vN entropy associated to the density matrix
$\rho_{\hat{\Phi}}$. This vN entropy is a Bekenstein generalized entropy [13]
and it contains two pieces. The other piece will depend on what is, in this
case, the QFT state $\Phi$ and in particular how different is this state,
representing now the analog of the QFT Bunch Davis vacuum but for the
Schwarzschild-de Sitter background, from $\Psi_{dS}$. Assuming the clock state
to spread slowly i.e. to have small time uncertainty, we can use Araki’s
relative operator $\Delta(\Psi_{dS}|\Phi)$ to measure the distinguishability
of both states relative to the de Sitter algebra. However this general
argument does not fix the state $\Phi$.

Naively we should expect that

$i\hbar\frac{d}{dt}|\hat{\Phi}\rangle=(\hat{h}_{\Psi_{dS}}+\beta\hat{H}_{c})|\hat{\Phi}\rangle=0$
(21)

where $\hat{h}$ is the modular Hamiltonian. This is a naiv description of the
physical Hilbert space as those states of the combined system – that includes
the external clock– that are invariant under time reparametrizations. The
important subtlety with (21) is that the state $|\hat{\Phi}\rangle$ describes
the full space i.e. not only the region associated with $\tilde{\cal{A}}$ but
also the region associated with its commutant $\tilde{\cal{A}}^{{}^{\prime}}$
and therefore it generically represents two entangled clocks 121212Ignoring
one of the clocks will lead to some extra entropy deficit [4]..

##  5 Gauge invariant observables in primordial Inflationary Cosmology

The first ingredient needed to define the weakly coupled QFT description of
quantum fluctuations during the inflationary period is to identify those
scalar quantum fluctuations invariant under time reparametrizations. According
to our previous discussion if the inflationary period is defined using pure de
Sitter, the potential set of local gauge invariant fluctuations reduces to the
invariant subalgebra ${\cal{A}}_{dS}^{\hat{h}}$ for $\hat{h}$ the modular
Hamiltonian defining time translations on the observer static patch. As
discussed this invariant subalgebra is, for pure de Sitter, trivial and only
contains the identity operator. This means that in pure de Sitter the observer
cannot identify neither any gauge invariant quantum fluctuation created in her
static patch nor its potential horizon exit. Following our previous discussion
the most reasonable way to get a non trivial set of gauge invariant local
fluctuations describing the inflationary primordial period will be to add an
external clock algebra and to define the gauge invariant quantum fluctuations
using the non trivial crossed product algebra
$({\cal{A}}_{dS}\times{\cal{A}}_{clock})^{(\hat{h}+\hat{H}_{c})}$ or its
constrained version if we include the positivity constraint on the clock
energy spectrum.

Hence we expect that the algebra of gauge invariant fluctuations describing
the primordial inflationary phase should be a crossed product algebra for some
external added clock. This is in essence the approach discussed in [9, 22].

In order to identify the clock let us first recall the standard definition of
gauge invariant scalar fluctuations in Inflationary Cosmology. These gauge
invariant scalar fluctuations are known as Mukhanov-Sasaki variables [14].

The most general parametrisation of linear scalar fluctuations is given by the
metric

$ds^{2}=a^{2}((1+2\phi)d\eta^{2}-2B_{i}dx^{i}d\eta-((1-2\Psi)\delta_{i,j}+2E_{i,j})dx^{i}dx^{j})$
(22)

with $\phi,E,B,\Psi$ arbitrary functions of space and time and $\eta$ the
conformal time related to the physical time $t$ by $t=\int ad\eta$. We are
interested in identifying gauge invariant scalar fluctuations i.e. those
fluctuations that are not due to pure reparametrizations of time. From (22) it
follows that both $\phi$ and $\Psi$ change under time reparametrizations thus
we need ”to dress” the corresponding QFT operators, describing these
fluctuations, to define gauge invariant observables. In order to define this
gauge invariant dressing we could try to use a clock dressing, in the sense
defined above. Mukhanov’s variable [14] achieves naturally this goal.

To see how that is working let us formally define the clock coordinate as the
homogeneous inflaton vev $\varphi_{0}$. To define the clock algebra we also
consider the canonically conjugated momentum operator $p_{0}$ with
$[\varphi_{0},p_{0}]=i\hbar$ and a clock Hamiltonian $\hat{H}_{c}$ such that
$i\hbar\dot{\varphi}_{0}=[\varphi_{0},\hat{H}_{c}]$. Thus, at this level, the
external clock is defined by just one degree of freedom that we identify with
the homogeneous added inflaton field and by a clock Hamiltonian $\hat{H}_{c}$.
As discussed above once we have defined this quantum mechanical system we can
define, using (7) and the appropriated projection, a clock time hermitian
operator $\hat{t}$ formally satisfying $[\hat{t},\hat{H}_{c}]=i\hbar$. From
that it follows Heisenberg relation $\Delta t\Delta H_{c}\geq\hbar$. The
typical uncertainty $\Delta t$ for the clock time operator $\hat{t}$ will
depend on the particular quantum state representing the clock as
$\frac{\Delta\varphi_{0}}{|\langle\dot{\varphi}_{0}\rangle|}$.

Let us now think of the metric fluctuation defined by $\Psi$ as an element $a$
in ${\cal{A}}_{dS}$ and let us look for the corresponding crossed product
(clock) dressing $\hat{a}=e^{i\hat{h}\hat{t}}ae^{-i\hat{h}\hat{t}}$ for
$\hat{t}$ the clock time operator. For the inflaton clock just defined we have
$\hat{t}\sim\frac{\varphi_{0}}{\dot{\varphi}_{0}}$ and consequently the clock
dressing of $\Psi$ becomes Mukhanov-Sasaki variable

$\chi=\Psi+\frac{\delta\varphi_{0}H}{\dot{\varphi}_{0}}$ (23)

for $H$ the Hubble parameter. Thus we conclude that the gauge invariant scalar
fluctuations defined by MS variables are in the crossed product algebra
defined for an external clock with clock coordinate $\varphi_{0}$ and some
clock Hamiltonian $\hat{H}_{c}$ ( not yet specified ). It is important to
stress that before adding the clock we don’t have any non trivial gauge
invariant way to describe scalar fluctuations in pure de Sitter. Naively we
could think of using the Hamiltonian $H$, generating time translations in the
static patch of the observer and doing nothing on the complementary static
patch, to define invariance under time reparametrizations. However, and that
is a crucial consequence of the type $III_{1}$ nature of the de Sitter algebra
${\cal{A}}_{dS}$, this Hamiltonian is not acting on the GNS Hilbert space and
consequently is not acting on the QFT Hilbert space describing the weakly
coupled quantum fluctuations. Moreover using $H$ leads to undesired
divergences due to the fact that $H|\psi\rangle$ for generic $|\psi\rangle$ is
not normalizable.

This simple comment explains why when we stop the clock i.e. when we impose
$\dot{\varphi}_{0}=0$ or equivalently pure de Sitter without clock, the
physical amplitudes of gauge invariant observables become divergent and ill
defined. This is specially clear for the power spectrum of scalar curvature
fluctuations that becomes divergent in the no clock limit
$\dot{\varphi}_{0}=0$.

Another important comment concerns the way the clock is introduced using the
homogeneous inflaton field $\varphi_{0}$. In this case we don’t jump into
adding an abstract clock algebra defined by $[\hat{t},\hat{H}_{c}]=i\hbar$ but
we insist on defining a quantum mechanical system with coordinate
$\varphi_{0}$ and the corresponding conjugated momentum $p_{0}$ and with clock
Hamiltonian $\hat{H}_{c}(\varphi_{0},p_{0})$ defining $\dot{\varphi}_{0}$. It
is on the basis of this clock data that the clock time $\hat{t}$ is formally
related with $\frac{\varphi_{0}}{\dot{\varphi}_{0}}$.

In order to proceed as we did in the previous sections we will need to
consider product states in the extended Hilbert space representing the weakly
coupled QFT on fixed de Sitter background and the external clock defined using
the inflaton field $\varphi_{0}$. As we did in section 3 let us consider the
state $\Psi_{QFT;\chi}\Phi_{clock}$ obtained by acting on the extended Hilbert
space with the dressed and gauge invariant MS observable $\chi$131313Note that
this dressed operator as defined in (23) belongs to the crossed product
algebra.. We can think of $\Phi_{clock}$ as some localised wave packet
$\Phi(\varphi_{0},t)$ but for the time being it will not be important to
identify this clock state. What is crucial is to impose the gauge invariant
constraint

$i\hbar\frac{d}{dt}\Psi_{QFT;\chi}\Phi_{clock}=(\hat{H}+\hat{H}_{c})\Psi_{QFT;\chi}\Phi_{clock}=0$
(24)

for $\hat{H}$ representing the pure de Sitter evolution. This leads, for a
clock wave packet that spreads slowly, to a simple equation for
$\Psi_{QFT;\chi}$ namely

$(\hat{H}+\langle\hat{H}_{c}\rangle)\Psi_{QFT;\chi}=0$ (25)

for $\langle\hat{H}_{c}\rangle$ defined relative to the clock state
$\Phi_{clock}$.

What is the meaning of (25) ? It is the Schrodinger equation for the gauge
invariant quantum fluctuations $\chi$ defined by a clock dressing with clock
Hamiltonian $\hat{H}_{c}$ and clock state $\Phi_{clock}$. This equation for
$\Psi_{QFT;\chi}$ representing scalar curvature fluctuations is the well known
Chibisov-Mukhanov equation [23].

Before describing the connection between (25) and Chibisov Mukhanov equation
let us make some comments to link the present discussion with the discussion
above on the effect of a clock artefact defined with a localised energy. In
that case the state that now we are calling generically $\Psi_{QFT}$ was the
QFT state that accounts for the back reaction on the background metric of the
energy of the external localised clock. This back reacted metric can be
Schwarzschild de Sitter. In such a case $\Psi_{QFT}$ was the analog of Bunch
Davis vacuum but for the Schwarzschild de Sitter metric describing the clock
artefact with some localised energy $E$. Now we are using as clock the
inflaton vev $\varphi_{0}$ so the corresponding back reaction, to the presence
of the inflaton clock, should be some quasi de Sitter metric. What particular
quasi de Sitter metric will depend on what particular clock Hamiltonian we
decide to use. Recall that to add a clock algebra is not optional if we want
to have a non trivial set of gauge invariant observables. What in principle
could be optional is what particular clock we decide to use and how that
decision depends on some underlying physics argument.

A second relevant comment concerns the tensor modes. Those are by construction
gauge invariant so they don’t need to be dressed by any external clock.
Algebraically we can think of these tensor modes as central with respect to
the crossed product algebra.

Once that is clarified let us recall the form of Chibisov-Mukhanov equation.
You define $v=z\chi$ with $z=a\frac{\dot{\varphi}}{H}\equiv a\sqrt{\epsilon}$.
The Schrodinger equation for the Fourier mode of comoving momentum $k=0$ is

$v^{{}^{\prime\prime}}-\frac{z^{{}^{\prime\prime}}}{z}v=0$ (26)

where $v^{\prime}$ represents derivative with respect to conformal time
$\eta$. Let us rewrite
$\frac{z^{{}^{\prime\prime}}}{z}=\frac{2}{\eta^{2}}(1+\Delta)$. To make
contact with (25) we set the pure de Sitter part defined by $\hat{H}$ as
$\hat{H}v=v^{{}^{\prime\prime}}-\frac{2}{\eta^{2}}v$ and
$\frac{2\Delta}{\eta^{2}}$ as $\langle\hat{H}_{c}\rangle$. In other words we
map the deviation $\frac{\Delta}{\eta^{2}}$ from pure de Sitter of Chibisov
Mukhanov equation into the contribution of the average clock energy defined by
the clock Hamiltonian on some particular clock state $\Phi_{clock}$. This
identification assumes a non vanishing but constant $\dot{\varphi}$ 141414It
is in this approximation that we can write
$\frac{z^{{}^{\prime\prime}}}{z}=\frac{2}{\eta^{2}}(1+\Delta)$ with $\Delta$
constant and time independent..

In this simplest approximation we have

$\frac{z^{{}^{\prime\prime}}}{z}=2H^{2}a^{2}+\epsilon H^{2}a^{2}$ (27)

The clock interpretation of this equation links $\epsilon H^{2}a^{2}$ with
$\langle\hat{H}_{c}\rangle$. Thus the quasi de Sitter slow roll parameter
$\epsilon$ is fully determined by the dynamics of the added clock.

In words a non vanishing cosmological constant forces us to add a clock and to
work on a Hilbert space representing the crossed product algebra. When we
imagine this clock as an artefact we move, by imposing gauge invariance, into
some Schwarzschild de Sitter metric while when we add the clock defined by the
inflaton dynamics we naturally move into a quasi de Sitter metric.

Let us now consider equation (27). The clock Hamiltonian that leads to
$\epsilon H^{2}a^{2}$ corresponding to constant $\epsilon$ is like a clock
defined by a free particle with Hamiltonian $a^{2}(\dot{\varphi})^{2}$ and
$\dot{\varphi}=\sqrt{\epsilon}H$. Thus $\epsilon$ non vanishing and constant
corresponds to a ”free particle clock” with non vanishing $\dot{\varphi}$ and
vanishing ”acceleration”.

Once we include the space dependence of the inflaton field we can consider the
different Fourier modes $\chi_{k}$ for $k$ the comoving momentum. Note that
for each mode $k$ we are formally defining a $k$ clock with clock coordinate
$\varphi_{k}$ and clock Hamiltonian $\hat{H}_{c,k}$. The Schrodinger equation
for these modes becomes

$v_{k}^{{}^{\prime\prime}}+(k^{2}-\frac{z^{{}^{\prime\prime}}}{z})v_{k}=0$
(28)

that we can rewrite formally as

$v_{k}^{{}^{\prime\prime}}+(k^{2}-\frac{2}{\eta^{2}}-\langle\hat{H}_{c,k}\rangle)v_{k}=0$
(29)

In the short wave regime where $k^{2}$ is much larger than
$\frac{z^{{}^{\prime\prime}}}{z}$ the effect of the clock is effectively
suppressed. After our algebraic discussion this is dangerous since once the
clock is suppressed we move back into the type $III$ ill defined regime. This
is a different way to present the so called transplanckian problem in
Cosmology. In the long wave regime the solution to the Schrodinger equation
goes as $z$ and therefore it strongly depends on the clock contribution.

Let us now consider equation (29) in the long wave limit $k\sim 0$ or more
precisely $k<<\frac{z^{{}^{\prime\prime}}}{z}$ and let us parametrise
$\langle\hat{H}_{c}\rangle=\frac{\Delta}{\eta^{2}}$ with $\Delta$ independent
on time. This, as said, corresponds to a constant speed clock. In this case
the solution can be written as

$v=f(k)\frac{(\tilde{k}\eta)^{\delta}}{\eta}$ (30)

with $\tilde{k}$ an arbitrary ( in the $k\sim 0$ regime ) pivot scale,
$\delta$ determined by the clock contribution $\Delta$ i.e. $\delta(\Delta)$
and $f$ defining the normalization. This normalization can be fixed by
matching at horizon exit $k\sim aH$ with the free wave solution in the short
wave limit $k^{2}>>\frac{z^{{}^{\prime\prime}}}{z}$.

Using this normalization the power spectrum of scalar curvature
fluctuations151515Defined as $k^{3}|\chi_{k}(\eta)|^{2}$. at horizon exit for
the mode $k$ goes like

${\cal{P}}\sim(\frac{\tilde{k}}{k})^{2\delta}\frac{H^{2}}{\epsilon}$ (31)

with $2\delta=(1-n_{s})$ where recall $\dot{\varphi}=\sqrt{\epsilon}H$. This
provides the first lesson. Namely in the very long wave regime the spectral
index $(1-n_{s})$ [24, 25] is determined by the clock energy
$\langle\hat{H}_{c}\rangle$ once we define the clock with a constant but non
vanishing $\dot{\varphi}$. Obviously this clock corresponding to non vanishing
but constant $\epsilon$ is by no means the only possible clock neither the
best candidate for the right clock. Using a more general clock makes the
equation for the dressed clock gauge invariant variable (29) more complicated
than the Bessel equation we have been considering above.

Now we can use the time delay interpretation of ${\cal{P}}$ as
$(H\Delta(t))^{2}$ [26]. In the clock picture this should be done using for
$\Delta(t)$ the clock uncertainty that is determined by $\hat{H}_{c}$ by the
standard Heisenberg relation $\Delta t\Delta H_{c}\geq\hbar$. Thus in the
approximation where we assume the clock $\dot{\varphi}$ non vanishing but
constant we get $\Delta H_{c}\sim\frac{\dot{\varphi}\hbar}{\Delta\varphi}$ and
consequently ${\cal{P}}\sim\frac{H^{2}(\Delta\varphi)^{2}}{\dot{\varphi}^{2}}$
that leads to (31) once we use $\Delta\varphi\sim\frac{v}{a}$ for $v$ the
solution (30). In summary the former simple discussion can be seem as an
argument in two steps. First of all we use the clock Hamiltonian and the clock
state to identify the Schrodinger equation for clock dressed gauge invariant
scalar fluctuations and we formally solve that equation. In the simplest case
we are considering we do that in the limit of small comoving momentum $k$. If
we don’t impose this restriction we can try to solve the equation for
arbitrary $k$ provided, in our scheme, we know the corresponding clock
Hamiltonian for those modes. The solution to this dressed Schrodinger equation
gives us information about the value of $\Delta\varphi$. In the second step we
use an approximate time delay representation of the power spectrum as
$(H\Delta(t))^{2}$ but for $\Delta(t)$ defined by the clock time as determined
by $\varphi$. The consistency of both steps requires that the $\Delta\varphi$
determined by the dressed Schrodinger equation is precisely the one that leads
to the correct time delay representation of the power spectrum as
$(H\Delta(t))^{2}$. Note that this consistency requirement is far from trivial
and as we will discuss in the next section it requires to identify a concrete
clock with comoving momentum mode $k$ and with a time uncertainty that
accounts for the anomalous scale dependence $(\frac{k}{\tilde{k}})^{2\delta}$.
Above we were presenting the consistency condition for this two step argument
in the simpler case where we assume a clock with $\dot{\varphi}$ non vanishing
but not changing in time. However the previous formal argument was done
without using any concrete clock Hamiltonian.

Finally let us make a comment on the potentially dangerous transplanckian
regime. As already pointed out this regime acquires a different meaning in the
clock dressing approach. In such a regime it looks superficially that we
suppress the effect of the clock and that consequently we move back into the
type $III$ regime. However we can also address this problem from the point of
view of our previous discussion on clock precision. In the weak gravity limit
where $G_{N}=0$ we do’t need to be worried about increasing the ”clock energy”
and consequently the ”clock precision”. However this changes whenever we turn
on a non vanishing value of $G_{N}$. In this case the clock ”mass” should be
smaller than $M_{P}$ which in this algebraic approach means that local and
gauge invariant scalar fluctuations should be reduced to be subplanckian.
Otherwise the needed clock dressing will make the corresponding gauge
invariant operators non local. This simple algebraic argument likely leads to
the transplanckian bound on the maximal number of e-foldings [27].

The previous discussion should not come as a surprise. It simply reflects the
fact that gauge invariant cosmological observables describing the quantum
fluctuations of a primordial exponentially expanding phase should be thought
as elements of a dS crossed product algebra i.e. as clock dressed operators.
In this picture the inflaton dynamics is just defining the external clock,
namely the clock coordinate as well as the clock Hamiltonian. The interesting
problem is what can we do to identify a natural clock without getting lost
into a landscape of clocks161616A landscape of potential inflationary
potentials.. In the next section we will make a suggestion on how to identify
a natural and model independent clock and how the so defined clock leads
naturally to a non vanishing value for the spectral index $(1-n_{s})$.

##  6 The de Sitter double squeezed clock

At this point of the discussion our problem is to identify a quantum system
that can be used to define a physical clock in de Sitter and to use it to
define gauge invariant quantum fluctuations through the corresponding clock
dressing. Physical amplitudes for these clock dressed operators are determined
by the condition of invariance under time coordinate reparametrizations. Thus
the Schrodinger equation governing these gauge invariant quantum fluctuations
depend, at the linear level where we are working, on the initial reference
pure de Sitter background on the clock Hamiltonian and on the clock state. In
principle the clock data, namely the clock Hamiltonian and the clock state can
be thought as an abstract way, in case we use the inflaton vev as clock
coordinate, to define a particular inflationary model potential. This will
lead to an undesired richness of potential clocks to use in order to define a
non trivial set of gauge invariant clock dressed observables. Thus the
question we want to address in this section is if we could figure out a
natural and model independent clock.

In order to answer this question is convenient to use the planar patch of de
Sitter to describe the Cosmology during the primordial inflationary period. In
the planar patch we have two regions. One is the static patch of the observer
and the other piece, representing the planar patch complement of the static
patch, describes the part of space-time that is beyond the observer
cosmological horizon. The Killing vector $\frac{\partial}{\partial t}$ in
static coordinates defines opposite time translations in the two static
patches, let us say $L$ and $R$, as well as opposite time translations on the
two planar patch complements (the upper $U$ and lower $L$ triangles of the
Penrose diagram ). Time translations in the $L$ or $R$ static patch define
automorphisms of ${\cal{A}}_{dS}^{L,R}$ with the corresponding modular
hamiltonian defining the automorphism on
${\cal{A}}_{dS}^{L}\times{\cal{A}}_{dS}^{R}$. Let us now define the algebras
${\cal{A}}_{dS}^{U,L}$ as the algebras of local observables with support on
the region beyond the static patch cosmological horizon ( for instance the
upper $U$ or lower $L$ triangle ). We could expect to define an automorphism
of these algebras and the corresponding modular hamiltonian using the
corresponding action of $\frac{\partial}{\partial t}$ on these regions. In
doing Cosmology we are effectively interested on the planar patch alebra
${\cal{A}}_{dS}^{L}\times{\cal{A}}_{dS}^{U}$171717Or the one defined by $R$
and $L$. Note also that the action of time automorphisms on
${\cal{A}}_{dS}^{L}$ is not leading ( for any finite time) to elements of
${\cal{A}}_{dS}^{U}$. . Let us now assume that both algebras
${\cal{A}}_{dS}^{L}$ and ${\cal{A}}_{dS}^{U}$ are both type $III_{1}$ factors
and let us define two type $I$ clock algebras let us say ${\cal{A}}_{c}^{L,U}$
to define the corresponding type $II$ factors. Let us formally define these
two clock algebras by two clock coordinates, let us say $z$ and $\tilde{z}$,
with the corresponding momentum operators $p_{z}$ and $p_{\tilde{z}}$ as well
as two clock Hamiltonians $H_{c}$ and $\tilde{H_{c}}$.

A priori we should expect the corresponding double clock state should
represent an entangled state of both ”clocks”. Since the planar patch can be
foliated in terms of equal conformal time $\eta$ hypersurfaces the two-clock
state will depend on $\eta$ as $\Phi_{clock}(\eta)$. The role of the reference
pure de Sitter background we are using is to define a non trivial time
evolution of the clock state reflecting the fact that the conformal time is
not globally defined on the planar patch. Moreover this pure de Sitter time
evolution of $\Phi_{clock}(\eta)$ is determined by the well known Bogolyubov
transformation that naturally defines an automorphism on the clock product
algebra ${\cal{A}}_{c}^{L}\times{\cal{A}}_{c}^{U}$. The clock state in the
Hilbert space representation of ${\cal{A}}_{c}^{L}\times{\cal{A}}_{c}^{U}$
will be an entangled state, actually an EPR like entangled squeezed state
181818Note that we are using pure de Sitter input to define the double clock
that should account for the natural clock dressing on the full planar patch..

The two clock Heisenberg algebras defined by $z,p_{z}$ and
$\tilde{z},\tilde{p}_{z}$ leads to a squeezed state only if $[z,p_{z}]=i\hbar$
and $[\tilde{z},\tilde{p}_{z}]=-i\hbar$. Once we include clock modes with
commoving momentum $k$ and we use the standard creation annihilation operator
basis both algebras correspond to $a_{k},a^{\dagger}_{k}$ and
$a_{-k},a^{\dagger}_{-k}$. As said the squeezed state is generated by the
Bogolyubov time transformation and it represent the vacuum for modes $k$ at
conformal time $\eta$. This state $|k,\eta\rangle$ is given by (see [30])

$|k,\eta\rangle=\sum_{n}c((k\eta),n)e^{in\Phi((k\eta))}|n_{k},n_{-k}\rangle$
(32)

where we sum over all integers and where $c((k\eta),n)=C\tanh({r(k\eta)})^{n}$
with $C$ a normalization constant, $r(k\eta)=-\sinh^{-1}(\frac{1}{2k\eta})$
and with the phase given by
$\Phi(k\eta)=-\frac{\pi}{4}-\frac{1}{2}\tan^{-1}(\frac{1}{2k\eta})$. We will
suggest to use this entangled and squeezed state as the natural two clock
entangled state to define gauge invariant quantum scalar fluctuations on the
planar patch.

Now we need to identify for this clock the analog of the $\hat{t}$ operator
that, as discussed we need to use to define the gauge invariant clock dressed
operators. As discussed formally $\hat{t}$ is the conjugated to the clock
Hamiltonian. For the former description of the two clock state as a squeezed
state and by analogy with what is familiar for quantum coherent states we can
think of $\hat{t}$ as conjugated to the operator number defining the two clock
state energies. Thus we can map $\hat{t}$ with the phase $\Phi$ conjugated to
the operator number and to identify the clock algebra as $[N,\Phi]=-i\hbar$
with $\Phi$ the phase in (32).

This last comment provides the hint we need to guess how to extract from the
squeezed state $|k,\eta\rangle$ the corresponding mode dependent clock time
uncertainty $\Delta(\hat{t}_{k,\eta})$. Since all functions defining the
squeezed clock state depend on the combination $k\eta$ we can extract the
information about the clock time uncertainty in terms of the variance of
$\frac{\partial\Phi}{\partial(k\eta)}$, that we will denote
${\cal{F}}(k\eta)$, evaluated on the squeezed state (32):

${\cal{F}}(k\eta)=(\sum_{n}c(k\eta,n)^{2}(\frac{\partial\Phi(k\eta,n)}{\partial(k\eta)})^{2}-(\sum_{n}c(k\eta,n)^{2}(\frac{\partial\Phi(k\eta,n)}{\partial(k\eta)}))^{2})$
(33)

Defining $\hat{\cal{F}}(k\eta)=(k\eta)^{6}{\cal{F}}(k\eta)$ we can now try to
extract direct information on the power spectrum
$(\frac{k}{\tilde{k}})^{(1-n_{s})}\frac{1}{\epsilon}$ in terms of the squeezed
clock time uncertainty set by ${\hat{\cal{F}}(k\eta)}$. The numerical
evaluation of $\hat{\cal{F}}(k\eta)$ leads [28, 29] to

$\hat{\cal{F}}(k\eta)\sim\frac{(k\eta)^{\alpha_{F}(k\eta)}}{(k\eta)^{2}}$ (34)

for a ”quantum tilt” $\alpha_{F}(k\eta)$ [28]191919This quantum tilt evaluated
in [28] is a well defined function of $k\eta$ that depends numerically on how
we cut the sum in (33) i.e. on the numerical bound on the number of entangled
pairs used to evaluate ${\cal{F}}(k\eta)$. The numerical sensibility of the
result on this cut was discussed in [28].. Since we want to use
${\cal{F}}(k\eta)$ to identify the power spectrum at horizon exit we should
think of the argument $k\eta$ as determined by the threshold condition
$k^{2}\sim\frac{z^{{}^{\prime\prime}}}{z}$ but reduced to the clock correction
i.e. $k\delta(k)\sim\frac{\Delta}{\eta^{2}}$. This leads to two conditions,
namely the one that relates $\Delta$ to some $k\eta$ as $k\eta\sim 2\Delta$
and $(1-n_{s})$ to the value of $\alpha_{F}$ at this particular value of
$k\eta$ i.e. $\alpha_{F}(2\Delta)=2\delta(\Delta)$ with $\delta(\Delta)$
defined by (30). Thus, taking as input the numerical value of the function
$\alpha_{F}(k\eta)$, we can extract the concrete numerical values of the
inflationary slow roll parameters202020As well as the pivot scale defining the
CMB regime..

Since our target in this note is simply to show the existence of a predictive
recipe we will not insist neither on the concrete numerical values nor on the
potential phenomenology delivered by this scheme already discussed in [28,
29]212121In the approximation used in [28] we get a nice value of
$(1-n_{s})=0.0328$.. The key message can be summarized saying that the
squeezed two mode clock state $|k,\eta\rangle$ naturally delivers a clock time
uncertainty depending on $(k\eta)$ and with a non trivial scale dependence on
$k\eta$ encoded in $\alpha_{F}$. Matching this clock uncertainty with the
power spectrum for scalar perturbations at horizon exit it sets the values of
the inflationary slow roll parameters as well as the typical threshold scale
i.e the range of wave lengths at which
$k^{2}\sim\frac{z^{{}^{\prime\prime}}}{z}$.

The key lesson we would like to extract from this exercise is that we can use
pure de Sitter information to define a two modes natural planar clock and to
learn about the observable power spectrum, using, to define the needed clock
dressing of cosmological observables, this de Sitter double squeezed clock.

##  7 Some speculative conjectural final remarks

The main target of this note has been to define amplitudes for local and gauge
invariant cosmological observables using a type $II$ clock dressing. In
addition we have suggested the existence of a natural dS clock with clock
Hamiltonian defined by the planar dS Bogolyubov transformations and with a
double EPR squeezed clock state.

This construction motivates some natural questions. The first and probably the
most fundamental one is to unveil how to use quantum gravity effects in dS (
recall we have been working in the weak gravity limit $G_{N}=0$ ) to define a
potential upgrading to a type $I$ factor. Adding a reference frame clock in
the $G_{N}=0$ limit makes the type $III$ QFT factor into a more friendly type
$II$ factor. Thus what do we need ”to add” to get a type $I$ factor ? The type
$II$ appears because invariance under time reparametrizations forces us to add
the reference frame clock. As briefly discussed this reference frame clock can
have in the weak gravity limit $G_{N}=0$ arbitrary precision, however that is
not the case once we turn on gravity. At the level of amplitudes for dressed
operators this means that the root of the intrinsic locality constraints, once
we include quantum gravity $G_{N}$ non vanishing effects, lies in the needed
clock dressing [31]. Hence, pushing into the UV gauge invariant dressed
observables, comes with an unavoidable lost of clock precision i.e. the clock
wave length is pushed into the IR. A conjectural way to understand this UV-IR
bouncing for dressed operators could be in terms of what we can call clock
classicalization [32, 33]. In other words once gravity is turned on the needed
added clock flows with energy into the IR. The question then is if this clock
classicalization implies effective type $I$ dynamics.

A second natural remark concerns the natural limits on the number of
e-foldings that inflation lasts. A possibility briefly discussed in [9] could
be to link this number of e-foldings to the von Neumann-Murray dimension for
the entropy deficit for the two needed clocks. Namely one for the primordial
period and the other to account for clock dressing at the CMB scales.

Finally we can ask ourselves in the holographic AdS context about the meaning
of the analog ”clock dressing” at the dual holographic boundary. As pointed
out in [34] we expect that for Yang Mills at large $N$ and in finite volume we
will have a type $III$ description of the high temperature Hagedorn phase.
What is the meaning of the analog clock dressing in this non gravitational
holographically dual set up ? In [9] we have suggested to use the dynamics of
the gapped and confined matrix model eigenvalues in the Hagedorn phase as the
natural candidate to define the Yang Mills analog of the type $II$ added
reference frame.

##  8 Acknowledgments

This work was supported by grants SEV-2016-0597, FPA2015-65480-P and
PGC2018-095976-B-C21. Parts of this work were presented in the Cambridge
workshop Quantum de Sitter Universe. I thank the organisers for the
opportunity and for the great and insightful meetting.

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