# A sign of the times

João Magueijo<EMAIL_ADDRESS>Theoretical Physics Group, The
Blackett Laboratory, Imperial College, Prince Consort Rd., London, SW7 2BZ,
United Kingdom

###### Abstract

We examine in greater detail the recent proposal that time is the conjugate of
the constants of nature. Fundamentally distinct times are associated with
different constants and we should select the one related to the constant
dominating the dynamics in each region or epoch. We show in detail how in
regions dominated by a single constant the Hamiltonian constraint can be
reframed as a Schrodinger equation in the corresponding time, solved in the
connection representation by outgoing-only monochromatic plane waves moving in
a “space” that generalizes the Chern-Simons functional. We pay special
attention to the issues of unitarity and the measure employed for the inner
product. Normalizable superpositions can be built, including solitons, “light-
rays” and coherent/squeezed states saturating a Heisenberg uncertainty
relation between constants and their times. A healthy classical limit is
obtained for factorizable coherent states, with classical cosmology seen
through the prism of the connection (the comoving Hubble length) rather than
the more conventional expansion factor (metric). A brief discussion of the
arrow of time within this framework is included. In this multi-time setting we
show how to deal with transition regions, where one is passing on the baton
from one time to to another, and investigate the fate of the subdominant
clock. For this purpose minisuperspace is best seen as a dispersive medium,
with packets moving with a group speed distinct from the phase speed. We show
that the motion of the packets’ peaks reproduces the classical limit even
during the transition periods, and for subdominant clocks once the transition
is over. Strong deviations from the coherent/semi-classical limit are expected
in these cases, however. Could these be a “sign of the times”, accessible
notably in the transition period (from matter to Lambda domination) we live
in?

##  I Introduction

The problem of time in General Relativity Isham ; Kuch ; marina ; rovelli  and
the mystery of the origin and value of the constants of nature Barrowtip  are
two well-known nuisances embedded in the foundations of physics. In Myletter
we suggested that they could be inextricably intertwined. A priori this should
certainly be the case. Physical time concerns relational change. In contrast,
the constants of nature, if true to their name, are the hallmarks of
immutability. We could therefore expect that time and the constants are
conjugate dynamical variables, or at least complementary in a quantum sense.
Should we, therefore, promote the constants to observables, with their
complementaries providing physical definitions of time?

Naturally, the devil is in the detail, and a great many questions pour in.
Foremost, given the plethora of constants (some more fundamental than others
Duff ), we have to settle on whether we should contend with different physical
times, or if, instead, a select constant is the progenitor of a single time.
In Myletter  we suggested following the first route. A clock is crafted with
what is at hand. Depending on the constant(s) dominating the dynamics,
different phase space regions should employ different times. Thus, we are led
to times variously conjugate to the cosmological constant, $\Lambda$, the
gravitational constant, $G_{N}$, or even the speed of light, $c$. Within such
a pragmatic democracy we need to know how to pass on the baton from one clock
to another. This adjustment of clocks should be seen as a physical feature of
our world.

The roots of this proposal go back to well-known literature. In the context of
$\Lambda$ it resonates with findings for unimodular gravity unimod1 ; unimod ,
where the conjugate of $\Lambda$ is identified with a time variable (which
turns out to be Misner’s volume time misner ). A further point of contact is
the concept of Chern-Simons time Chopin , related to York time york . The
proposal in Myletter  amounts to reinterpreting these ideas and extending them
to constants other than $\Lambda$. The implications will be investigated
further here.

Schematically, in this paper we posit the following:

  * •

Time is the conjugate of the constants of nature and vice versa.

  * •

Given the multitude of options, the constant chosen for the progenitor of time
in a given region of phase space is the one that allows for a clear separation
of space and time in that region. This is a dynamical issue.

  * •

Dealing with multi-time situations, or settings where one must handover from
one time to another, is part of physics.

These precepts will emerge from an examination of minisuperspace quantum
gravity in the connection representation. As explained later (and in more
detail elsewhere bruno1 ) our constructions often generalize beyond
minisuperspace. When they do not (due to the choice of constant), we should
replace “time” by “cosmological time” and assume that locality and Lorentz
invariance have been broken.

The plan of this paper is as follows. A formal and aspirational general
prescription will first be proposed in Section II, in the form of 3 precepts.
These will be purposefully of great generality, because we do not wish to wed
our proposal to a particular theory (even though we will be using Einstein-
Cartan gravity for the rest of the paper). Putting the Hamiltonian constraint
in a given format, and extending the action and phase space, transforms the
Wheeler-DeWitt (WDW) equation into a Schrodinger equation, with the conjugate
of a constant playing the role of “time”. The WDW equation then appears as the
condition for defining a monochromatic (fixed constant) wave.

In Section III we apply this procedure to pure $\Lambda$, showing that in the
connection representation the monochromatic plane waves move in a time
conjugate to $1/\Lambda$ (seen as a “frequency”) and a space which is the
Chern-Simons (CS) functional (so that the spatial part of the wave is the real
CS state kodama ). Crucially, we can superpose the monochromatic plane waves
into normalizable solutions, as we show in Section IV, where we identify
solitons, light rays and coherent/squeezed states. In Section V we present a
straightforward generalization to Universes dominated by radiation and fluids
with generic equation of state. In each of these the constant of choice is
different, so that the chosen time is different.

Having a quantum time variable and the ability to find peaked wave-packets is
instrumental in finding the correct classical limit. We explain how this is
achieved by coherent states in Section VI, once the reader is taught how to do
cosmology in the connection representation and with constant-times.

The rest of the paper is spent formalizing what to do in multi-time situations
and how to change the “time zone”, as already sketched in Myletter  and
reviewed in Section VII. For this purpose, we need to generalize the
admittedly “aspirational” constructions in Section II, typically possible only
when one fluid dominates the Universe. This is done in Section VIII, where we
show how minisuperspace can be seen as a dispersive medium, with wave-numbers
that can be position and frequency dependent. By examining the associated
group speed we can then find the equations of motion of the peak of suitable
wave functions. Using this technique, and assuming that the wave function
remains peaked, in Section IX we prove that the correct semi-classical limit
is still obtained in cross-over regions. We also illuminate the fate of the
minority clock once the handover of clocks is completed. In Section X we
formalize the reasons why a clock should indeed be built with “what is at
hand”, exposing the limitations of minority clocks. In a concluding Section we
summarize our findings and speculate on their ultimate implications.

##  II Clocks and rods in an ideal minisuperworld

We first propose a number of aspirational principles for the definition of
clocks derived from constants. These clocks naturally lead to a measure in
connection space (the associated “rod”) which generalizes the CS invariant. We
stress that some of these principles are too restrictive once we go beyond the
ideal situation of minisuperspace and single fluids

###  II.1 Precept 1: constants as the generators of clocks

Let a generic gravity-plus-matter Hamiltonian depend on variables $q_{i}$ and
momenta $p_{i}$, as well as “constants” $\alpha_{j}$. The idea is to put the
standard Hamiltonian constraint in minisuperspace:

$H(q_{i},p_{i};\alpha_{j})=0$ (1)

into the form:

$H_{0}-\alpha=0$ (2)

where we have singled out one of the constants, $\alpha$. We then expand phase
space by promoting $\alpha$ to a d.o.f. and adding to the minisuperspace
action a new term:

$S\rightarrow S+\int dt\dot{\alpha}p_{\alpha},$ (3)

so that $\alpha$ has conjugate momentum $p_{\alpha}$, with:

$\\{\alpha,p_{\alpha}\\}=1.$ (4)

The momentum $p_{\alpha}$ does not appear anywhere else (in particular, in
$H$), and so one of Hamilton’s equations is:

$\dot{\alpha}=\\{\alpha,H\\}=0$ (5)

implying that $\alpha$ is now not a “constitutional” constant or parameter of
the system, but a constant of motion. Quantization therefore leads to a
Schrodinger equation:

$\left[H_{0}(p_{i},q_{i})-i\hbar\frac{\partial}{\partial
T_{\alpha}}\right]\psi=0$ (6)

where before there was the WDW equation. Its time variable is
$T_{\alpha}\equiv p_{\alpha}$. The conjugate of the no-longer-fundamental-
constant $\alpha$ is the quantum time variable we were missing.

The following comments are in order:

  * •

Not all constants are amenable to this treatment. Singling out $G_{N}$ or
$\Lambda$ is easier than targeting the electron charge $e$ or the elementary
particles’ masses. Still, setting $\alpha$ to $c$ or even $\hbar$ is not as
outlandish as it might appear, and raises important foundational questions
PlanckIsham .

  * •

Not in all models or regions of phase space can this prescription for a clock
be applied. For example, it may be hard to implement for a scalar field with a
potential, even if this is just a mass potential. In general it may be
cumbersome to implement the precept for multi-fluids exchanging energy even in
a stable scaling regime.

  * •

Even when, for a given constant and model, the prescription can be applied in
one or several regions, it may not be doable everywhere. An example is a
multi-fluid situation, where in regions where one fluid dominates a clock
emerges, but not in the transition regions, where more than one fluid
dominates.

All of this leads up to the view that clocks are at best effective and tied to
the circumstances. Maybe one can find many ideal clocks, one per region/epoch.
Perhaps in some regions one can even find more than one ideal clock bruno1 .
Or perhaps there are no ideal clocks at all in some other regions. All of this
is OK. It is better than nothing.

In addition the following structural points can be made:

  * •

Not in all cases does the stated prescription generalize beyond
minisuperspace. As we will describe in more detail elsewhere bruno1  we can
lift the equivalent procedure for unimodular gravity

$S_{0}\rightarrow S=S_{0}+\int d^{4}x\Lambda\partial_{\mu}T^{\mu}_{\Lambda}$
(7)

to achieve a covariant and local version of our proposal (for example,
targetting is $G_{N}$). However, this is by no means always possible (for
example if the target constant is $c$) or even necessary. In that case we may
still accept the minisuperspace model and state that we are breaking Lorentz
invariance: indeed we can regard such models as a class of Lorentz symmetry
breaking models.

  * •

Eq. (2) must be equivalent to (1), but off-shell the relation between $H$ and
$H_{0}-\alpha$ can be as complicated as wanted. Sometimes the two will be
proportional (and so the two conditions amount to nothing but a redefinition
of lapse function; see the case of radiation). Other times the relation can be
very complicated, so that the theories are expected to be very different off-
shell (as in the case of $\Lambda$ in Chern-Simons theory, where (2) is
obtained inverting identities implied by (1)).

###  II.2 Precept 2: “time” independence of the Hamiltonian $H_{0}$

Precept 1 typically leaves us with an embarrassment of riches. Furthermore,
numerous ambiguities remain: the choice of constant (and function of constant)
to be taken as $\alpha$, the choice of the proportionality constant for
$p_{\alpha}$, etc. Some of these ambiguities should be seen as a fact of life
(e.g. in bruno1  we will investigate the implications of concurrent multi-
times); others should be fixed. Hence (3) should be seen as schematic before
the next two precepts are introduced.

As a rule of thumb we should target the constant which is most “at hand” in a
given context (model and epoch). One makes a clock with what there is. The
next 2 precepts help us further select the appropriate time/constant. They
depend on the concrete dynamics.

First, we would like $H_{0}$ to NOT depend on $\alpha$. This may be a strong
criterium for selecting a constant in a given context (model and/or phase
space region), and for abandoning it in other, as we shall see. If $H_{0}$
depends on $\alpha$ we have a situation where the Hamiltonian depends on the
“energy” rather than being time-dependent. This will be unavoidable in some
situations (see Section IX), but when it can be avoided one should cherish the
chance. As we will see later in this paper, revisiting this precept will be
unavoidable where one sort of time passes over the baton to another.

###  II.3 Precept 3: rod-clock segregation

Furthermore, we would like to see a clear separation of space and time. What
good is a clock, if it is also a rod? The existence of monochromatic (plane,
with regards to some measure) waves in superspace is a criterium for such a
separation. This will resolve another ambiguity (besides the choice of
$\alpha$): the choice of the function of $\alpha$ to be taken as spanning
phase space. If $\alpha$ is a constant, then so is any function of $\alpha$,
but this affects the definition of the conjugate momentum. The introduction of
this precept removes this ambiguity111Strictly speaking this ambiguity amounts
to a canonical transformation in $(\alpha,p_{\alpha})$, leading to equivalent
classical theories, but different quantum theories (for example the measure is
different; coherent states in one variable will not be coherent in any other,
etc)..

The concrete implementation of this principle depends on the classical
dynamics and on the representation and ordering chosen for the quantum theory.
Let us illustrate it with the Einstein-Cartan action reduced to homogeneity
and isotropy CSHHV ; MZ :

$S_{g}=6\kappa V_{c}\int dt\bigg{(}a^{2}\dot{b}+Na(b^{2}+k)\bigg{)}.$ (8)

where $\kappa=1/(16\pi G_{N})$, $k=0,\pm 1$ is the normalized spatial
curvature, $a$ is the expansion factor (the only metric variable), and the
connection variable $b$ is the off-shell version of the Hubble parameter
(since $b=\dot{a}$ on-shell, if there is no torsion). The Lagrange multiplier
$N$ is the lapse function and $V_{c}=\int d^{3}x$ is the comoving volume of
the region under study (which could be the whole manifold, should this be
compact). To this action one should add a matter or Lambda action, as
appropriate. We may also need to define $\kappa$ with a fixed fiducial
$G_{N0}$ Myletter  or not bruno1 .

Hence, the Poisson bracket is:

$\\{b,a^{2}\\}=\frac{1}{6\kappa V_{c}},$ (9)

leading to commutator:

$\left[\hat{b},\hat{a^{2}}\right]=i\frac{l_{P}^{2}}{3V_{c}},$ (10)

where $l_{P}=\sqrt{8\pi G_{N}\hbar}$ is the reduced Planck length. In the $b$
representation this implies:

$\hat{a}^{2}=-i\frac{l_{P}^{2}}{3V_{c}}\frac{\partial}{\partial b}.$ (11)

What follows is tied to the connection representation and it would need to be
adapted for a metric representation counterpart.

In the light of Precept 3, we now show that it is desirable to put the
Hamiltonian constraint (2) in the specific form:

$h_{\alpha}(b)a^{2}-\alpha=0,$ (12)

that is replace whichever constant $\alpha$ we started from by a function
thereof, for the sake of obtaining $H_{0}=h_{\alpha}(b)a^{2}$, if
possible222Although $h$ and its related functions should not explicitly depend
on $\alpha$, and be only a function of the connection $b$, this function
depends on the $\alpha$ selected as the progenitor of time, as we shall see.
Hence we will keep $\alpha$ as a subscript.. This may require more than a
redefinition of lapse function. We will also normalize the momentum
$p_{\alpha}$ (to be identified with time $T_{\alpha}$) according to:

$S\rightarrow S+6\kappa V_{c}\int dt\dot{\alpha}p_{\alpha},$ (13)

implying the refined version of (4):

$\\{\alpha,p_{\alpha}\\}=\frac{1}{6\kappa V_{c}}.$ (14)

The factor of $V_{c}$ is there to avoid introducing a “time per unit
volume”333One may wonder, why not?, but that would be another paper. It is
always assumed that time is intensive, but it could be extensive towerturtles
in some regimes, or even assume any fractal dimension.. The factor of
$6\kappa$ is pure convenience (and amounts to setting to one the “speed of
light” in minisuperspace in “linearizing” variables DSR . Thus:

$\left[\alpha,p_{\alpha}\right]=i\frac{l_{P}^{2}}{3V_{c}},$ (15)

leading to a Schrodinger equation of the form (6) with “Planck’s
constant”444Throughout the paper we will use the shorthand $\mathfrak{h}$ or
not depending on convenience. Its interpretation as the actual Planck constant
BarrowPlanck  will not be relevant here.:

$\hbar\rightarrow\mathfrak{h}=\frac{l_{P}^{2}}{3V_{c}},$ (16)

and $T_{\alpha}\equiv p_{\alpha}$, that is:

$\left[H_{0}(b)-i\mathfrak{h}\frac{\partial}{\partial
T_{\alpha}}\right]\psi=0.$ (17)

Its monochromatic solutions are:

$\psi=\psi_{s}(b;\alpha)\exp{\left[-\frac{i}{\mathfrak{h}}\alpha
p_{\alpha}\right]}$ (18)

with the “spatial” $\psi_{s}$ satisfying

$(H_{0}-\alpha)\psi_{s}=0.$ (19)

The advantage of $H_{0}=h_{\alpha}(b)a^{2}$ as in (12) (and of its implied
ordering in the quantum theory) is now obvious. Given (11), it translates in
the $b$ representation into:

$\left[-i\frac{l_{P}^{2}}{3V_{c}}h_{\alpha}(b)\frac{\partial}{\partial
b}-\alpha\right]\psi_{s}(b;\alpha)=0.$ (20)

Setting:

$X_{\alpha}(b)=\int\frac{db}{h_{\alpha}(b)}$ (21)

this becomes a plane-wave equation in $X_{\alpha}$:

$\left(-i\frac{l_{P}^{2}}{3V_{c}}\frac{\partial}{\partial
X_{\alpha}}-\alpha\right)\psi_{s}=0,$ (22)

with solution:

$\psi_{s}(b;\alpha)={\cal N}\exp{\left[i\frac{3V_{c}}{l_{P}^{2}}\alpha
X_{\alpha}(b)\right]}.$ (23)

The monochromatic solutions are therefore plane-waves in $X_{\alpha}$ moving
at fixed speed (set to 1):

$\psi(b;\alpha)={\cal
N}\exp{\left[i\frac{3V_{c}}{l_{P}^{2}}\alpha(X_{\alpha}(b)-T_{\alpha})\right]},$
(24)

with the choice

${\cal N}=\frac{1}{\sqrt{2\pi\mathfrak{h}}}$ (25)

to be justified later. Notice that $X_{\alpha}(b)$ is like a linearizing
variable in DSR: the speed of propagation is variable if measured in terms of
the more physically available $b$, rather than $X_{\alpha}$.

In summary, under Precept 3 the WDW equation becomes a free Schrodinger
equation, which in fact is a wave equation accepting only retarded waves:

$\left(\frac{\partial}{\partial X_{\alpha}}+\frac{\partial}{\partial
T_{\alpha}}\right)\psi=0.$ (26)

Its associated conserved current is:

$j^{0}=j^{1}=|\psi|^{2}$ (27)

i.e. all the waves are outgoing (or retarded time) solutions. The general
solution takes the form:

$\psi(b)=F(T_{\alpha}-X_{\alpha}),$ (28)

where $F$ can be any function. This can be written as a superposition of
monochromatic solutions indexed by $\alpha$ as is:

$\psi(b)=\int\frac{d\alpha}{\sqrt{2\pi\mathfrak{h}}}{\cal
A}(\alpha)\exp{\left[\frac{i}{\mathfrak{h}}\alpha(X_{\alpha}(b)-T_{\alpha})\right]}.$
(29)

##  III Pure Lambda and a reinterpretation of Chern-Simons time

We first illustrate these principles with the cosmological constant $\Lambda$,
showing that the implications are a twist on both unimodular gravity unimod1 ;
unimod  (specifically the time variable defined in unimod ), and the concept
of Chern-Simons time Chopin . Indeed our proposal leads to a hybrid between
these works, with a significant reinterpretation of Chern-Simons “time”.

We start by reviewing some standard results. For a pure $\Lambda$ we have in
minisuperspace (ignoring torsion GB ; MZ ):

$\displaystyle H$ $\displaystyle=$ $\displaystyle 6\kappa
V_{c}Na\left(-(b^{2}+k)+\frac{\Lambda}{3}a^{2}\right).$ (30)

A direct solution to the quantum Hamiltonian constraint in the $b$
representation:

$\left[-(b^{2}+k)-i\frac{\Lambda l_{P}^{2}}{9V_{c}}\frac{\partial}{\partial
b}\right]\psi=0$ (31)

is given by the (real) Chern-Simons state CS ; jackiw ; kodama  reduced to
minisuperspace GB ; MZ ; CSHHV :

$\psi_{CS}={\cal N}\exp{\left[i\frac{9V_{c}}{\Lambda
l_{P}^{2}}\left(\frac{b^{3}}{3}+bk\right).\right]},$ (32)

As is well known, this is a pure phase, which is the product of a “frequency”
proportional to $1/\Lambda$, and the “Chern-Simons time” as proposed by Smolin
and Soo Chopin .

Something similar can be obtained from our prescription. We can put the
Hamiltonian constraint associated with (30) in the form:

${\cal H}=\frac{1}{b^{2}+k}a^{2}-\frac{3}{\Lambda}=0.$ (33)

By doing this the Hamiltonian constraint acquires the form (12) required by
Precept 3 :

${\cal H}=h_{\alpha}(b)a^{2}-\alpha=0$ (34)

in this case with:

$\displaystyle h_{\alpha}(b)$ $\displaystyle=$
$\displaystyle\frac{1}{b^{2}+k}$ (35) $\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\phi=\frac{3}{\Lambda}.$ (36)

A Schrodinger equation (17) follows, with a time variable $T_{\phi}$
identified with $p_{\phi}$, normalized such that:

$\left[\phi,p_{\phi}\right]=i\frac{l_{P}^{2}}{3V_{c}},$ (37)

(cf. (13) and (15)). Its monochromatic solutions are:

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\psi_{s}(b;\phi)\exp{\left[-i\frac{3V_{c}}{l_{P}^{2}}\phi
p_{\phi}\right]}$ (38) $\displaystyle=$
$\displaystyle\psi_{s}(b;\phi)\exp{\left[-i\frac{9V_{c}}{l_{P}^{2}\Lambda}T_{\phi}\right]}.$

with the “spatial” factor of the wave-function satisfying:

$\left[-i\frac{l_{P}^{2}}{3V_{c}}h_{\alpha}(b)\frac{\partial}{\partial
b}-\frac{3}{\Lambda}\right]\psi_{s}=0.$ (39)

This is the point of bringing the Hamiltonian to form (34) and choosing the
ordering we chose. As in (24), the $\psi_{s}$ are plane waves:

$\psi_{s}(b;\phi)={\cal N}\exp{\left[i\frac{3V_{c}}{l_{P}^{2}}\phi
X_{\phi}(b)\right]}.$ (40)

in “spatial” variable:

$X_{\phi}(b)=\int\frac{db}{h_{\alpha}(b)}=\frac{b^{3}}{3}+kb.$ (41)

This is nothing but the Chern-Simons functional (in minisuperspace) and (40)
is the Chern-Simons state (32).

However, our interpretation of “Chern-Simons time” is different from that of
Smolin and Soo. The full monochromatic solution is:

$\psi(b;\phi)={\cal
N}\exp{\left[i\frac{3V_{c}}{l_{P}^{2}}\phi(T_{\phi}-X_{\phi}(b))\right]}$ (42)

with $T_{\phi}\equiv p_{\phi}$. Hence the (unitary) time evolution happens in
terms of a time which is not the Chern-Simons functional, but the momentum
conjugate to $1/\Lambda$ (up to a conventional proportionality constant). Here
$X_{i}=\Im(Y_{CS})$ is not a time, but a spatial variable. Time, instead, is
the conjugate of $\alpha=3/\Lambda$. The waves, however, move at constant
speed (set to one by the conventional proportionality factors) in terms of
$X_{\phi}$ and $T_{\phi}$. This is true of the phase speed and also of the
group speed if we construct wave packets, as we shall do in the next Section.
Hence the spatial $X_{\phi}$ and the time $T_{\phi}$ can be loosely confused
if $\psi$ is peaked. Its peak moves along the outgoing “light-ray”
$T_{\phi}=X_{\phi}$, and hence confusing the two may be harmless for some
purposes.

##  IV More general states

By demoting $\Lambda$ to a circumstantial constant we gain more than a time
variable in the quantum theory: we enlarge the space of solutions. Instead of
being restricted to (42) we can now admit the most general superposition of
these monochromatic plane waves:

$\psi(b)=\int\frac{d\phi}{\sqrt{2\pi\mathfrak{h}}}{\cal
A}(\phi)\exp{\left[\frac{i}{\mathfrak{h}}\phi(X_{\phi}(b)-T_{\phi})\right]},$
(43)

with the probability for the cosmological constant given from:

$P(\phi)=|{\cal A}(\phi)|^{2}$ (44)

with measure $d\phi$ (fast-forward to the end of this Section for more
details; also see GBMSS  for alternatives).

Everything we state in this Section about general states for $\Lambda$,
parameterized by $\phi$, works for any other $\alpha$, with suitable
modifications (i.e. $\phi\rightarrow\alpha$ in all relevant quantities).

###  IV.1 Extreme cases

Two limiting cases are of interest. At one extreme we may have a completely
undetermined $\phi$:

${\cal A}(\phi)=\epsilon$ (45)

leading to:

$\psi=\sqrt{2\pi\mathfrak{h}}\epsilon\delta(T_{\phi}-X_{\phi}).$ (46)

This is the conformal constrain present in the parity-even branch of the
quasi-topological theories of GB ; MZ ; GBMSS , where $\Lambda$ is allowed to
vary by virtue of multiplying a Gauss-Bonnet topological term. In such a
theories $1/\Lambda$ has a conjugate momentum which is forced to equal the
Chern-Simons functional by a primary constraint. Here we see that this
constraint is interpreted as a “light-ray” in minisuperspace: of the many
waves generally acceptable only a delta function ray is possible in this
theory. Time $T_{\phi}$ is fully fixed by the position $X_{\phi}$ along this
ray. Whereas in standard Relativity any state (43) is a solution, in these
quasi-topological theories one is forced to have an infinitely sharp clock,
with total delocalization in “constant” $\phi$.

At the opposite extreme we may completely fix $\phi$:

${\cal A}(\phi)=\delta(\phi-\phi_{0})$ (47)

leading to:

$\psi={\cal
N}\exp{\left[-i\frac{3V_{c}\phi_{0}}{l_{P}^{2}}(T_{\phi}-X_{\phi}(b))\right]}.$
(48)

This is the Chern-Simons state in the usual Einstein-Cartan theory, where
Lambda is fully fixed. It implies a uniform distribution in $X_{\phi}$. The
crests of this infinite plane wave still move at the speed light, but its
“location” does not, because it is not localized. Hence time effectively
disappears, since the wave function is fully smeared in $X$ and $T$. This is
an example of a more general fact: infinitely sharp constants are failed
clocks. They imply complete delocalization in time. This clarifies and
reinterprets the discussion on a time-like tower of turtles in towerturtles .
Obviously these states are not strictly speaking normalizable.

###  IV.2 Coherent/squeezed states

In between these two extremes we can build coherent/squeezed states centred at
$\phi_{0}$:

${\cal A}(\phi)=\sqrt{{\bf
N}(\phi_{0},\sigma_{i})}=\frac{\exp{\left[-\frac{(\phi-\phi_{0})^{2}}{4\sigma^{2}}\right]}}{(2\pi\sigma^{2})^{1/4}},$
(49)

(where $\bf N$ denotes a normal distribution). Evaluating the integral we get:

$\displaystyle\psi(b,T)$ $\displaystyle=$ $\displaystyle{\cal
N}^{\prime}\exp\left[\frac{-\sigma^{2}(X_{\phi}-T_{\phi})^{2}+i\phi_{0}(X_{\phi}-T_{\phi})}{\mathfrak{h}^{2}}\right]$
(50) $\displaystyle=$ $\displaystyle{\cal
N}^{\prime}\psi(b,T_{\phi};\phi_{0})\exp\left[-\frac{\sigma^{2}(X_{\phi}-T_{\phi})^{2}}{\mathfrak{h}^{2}}\right].$

The last expression relates the infinite norm Chern-Simons state for
$\phi_{0}$ to the finite norm wave packet built around a fixed $\Lambda$. We
see that it is dressed by a Gaussian, which regularizes it. This is just a
Gaussian distribution in $X-T$, with variance:

$\sigma_{T}^{2}=\sigma_{X}^{2}=\frac{\mathfrak{h}^{2}}{4\sigma^{2}}.$ (51)

A Heisenberg uncertainty principle can therefore be established, with:
$\sigma_{X}=\sigma_{T}$ and

$\sigma_{T}\sigma_{\phi}\geq\frac{l_{P}^{2}}{6V_{c}}.$ (52)

For a coherent state:

$\sigma_{T}^{2}=\sigma_{\phi}^{2}=\frac{l_{P}^{2}}{6V_{c}}.$ (53)

###  IV.3 Solitons

Note also that we do not need to use monochromatic waves to build states. As
already explained, any function of the form:

$\psi=F(X_{\phi}-T_{\phi})$ (54)

would work, as we can see from [see before; this is repetition]. Namely $F$
can be just a Gaussian, without the plane wave factor, as for (50):

$\displaystyle\psi(b,T)$ $\displaystyle=$ $\displaystyle{\cal
N}^{\prime\prime}\exp\left[\frac{(X_{\phi}-T_{\phi})^{2}}{4\sigma^{2}}\right]$
(55)

Such “solitons” could be interesting. We stress that unlike coherent states,
such solitons are not well localized in $\phi$: for that we need the internal
beats of a plane-wave, for which this $F$ would be the envelope.

###  IV.4 Normalizability and measure

All of these solutions are normalizable with the “naive” inner product. No
longer do we need to blame the trivial inner product for the non-
normalizability of the monochromatic solutions. Of course monochromatic
solutions are strictly speaking non-normalizable by themselves; their
superpositions, on the other hand, are normalizable in the standard sense.

Specifically, mimicking the procedure in vilenkin  for the simpler current
(27), we can infer the inner product:

$\langle\psi_{1}|\psi_{2}\rangle=\int
dX_{\phi}\psi_{1}^{\star}(b,T_{\phi})\psi_{2}(b,T_{\phi})$ (56)

with unitarity:

$\frac{\partial}{\partial T_{\phi}}\langle\psi_{1}|\psi_{2}\rangle=0$ (57)

enforced by current conservation:

$\frac{\partial}{\partial T_{\phi}}\langle\psi_{1}|\psi_{2}\rangle=\int
dX_{\phi}\frac{\partial}{\partial
X_{\phi}}(\psi_{1}^{\star}(b,T_{\phi})\psi_{2}(b,T_{\phi}))=0.$ (58)

(This vanishes only with suitable boundary conditions, with subtleties like
the ones highlight in Gielen , e.g. singularities, etc.) We can also swap
$X_{\phi}$ and $T_{\phi}$ in this definition:

$\langle\psi_{1}|\psi_{2}\rangle=\int
dT_{\phi}\psi_{1}^{\star}(b,T_{\phi})\psi_{2}(b,T_{\phi})$ (59)

with the two definitions equivalent and amounting to:

$\langle\psi_{1}|\psi_{2}\rangle=\int d\phi{\cal A}_{1}^{\star}(\phi){\cal
A}_{2}(\phi),$ (60)

in view of (43). The normalizability condition $|\langle\psi|\psi\rangle|=1$
therefore supports (44) identifying the probability of $\phi$.

We stress that this argument is valid for each dominating fluid $i$, adopting
its associated $\alpha$, $X_{\alpha}$ and $T_{\alpha}$. The argument for
unitarity is far more complicated in the transition regions for multi-fluids,
or for the minority clock, i.e. clocks corresponding to sub-dominant
components, as we shall see later in this paper.

##  V Radiation and general perfect fluids

Our approach can be used to find the equivalent of the Chern-Simons state for
Universes filled with a single fluid with an equations of state other than
$w=-1$. To this purpose we add to the gravitational action (8) the “matter”
action:

$S_{m}=-6\kappa V_{c}\int dtNa\frac{m}{a^{1+3w}}$ (61)

valid for a perfect fluid with equation of state $w=p/\rho$ (seeGielen  and
refs therein). The factor of $6\kappa$ puts $m$ on the same footing as
$\Lambda$ (at least dimensionally), and it may need to be adapted if we target
$G_{N}$ for our procedure, as outlined in Myletter  [add to this].

Hence, the total Hamiltonian (matter and gravity) is:

$\displaystyle H$ $\displaystyle=$ $\displaystyle 6\kappa
V_{c}Na\left(-(b^{2}+k)+\frac{m}{a^{1+3w}}\right).$ (62)

so that the Hamiltonian constraint can be put in the equivalent form (34)
(i.e. ${\cal H}=h_{\alpha}(b)a^{2}-\alpha=0$, cf. Precept 3 ), with:

$\displaystyle h_{\alpha}(b)$ $\displaystyle=$
$\displaystyle(b^{2}+k)^{\frac{2}{1+3w}}$ (63) $\displaystyle\alpha$
$\displaystyle=$ $\displaystyle m^{\frac{2}{1+3w}}.$ (64)

Having identified the function of $m$ best suited to producing a Schrodinger
equation, we can then expand the phase space according to (13), so that the
total matter action is:

$S_{m}=6\kappa V_{c}\int
dt\left(\dot{\alpha}p_{\alpha}-Na\frac{m}{a^{1+3w}}\right).$ (65)

We now have commutation relations of the form (15) between $\alpha$ and its
momentum, $p_{\alpha}$, leading to a Schrodinger equation of the form (17),
just as before. Time $T_{\alpha}$ is now identified with $p_{\alpha}$. Its
monochromatic solutions, as before, are (18) with extra condition (19), that
is, we find that the WDW equation for a given fixed value of $m$ (or
equivalently $\alpha$) is:

$\left[-i\frac{l_{P}^{2}}{3V_{c}}h_{\alpha}(b)\frac{\partial}{\partial
b}-\alpha\right]\psi(b;\alpha)=0.$ (66)

By construction we therefore find plane waves in a “spatial” variable which,
instead of the Chern-Simons functional, has general form:

$X_{\alpha}(b)=\int\frac{db}{h_{\alpha}(b)}=\int\frac{db}{(b^{2}+k)^{\frac{2}{1+3w}}}.$
(67)

Notice how this reduces to the Chern-Simons functional (41) when $w=-1$. All
we need to do is change $X$ and the WDW equation still is:

$\left(-i\frac{l_{P}^{2}}{3V_{c}}\frac{\partial}{\partial
X_{\alpha}}-\alpha\right)\psi=0,$ (68)

with solution:

$\psi_{s}(b;\alpha)={\cal
N}\exp{\left[i\frac{3V_{c}\alpha}{l_{P}^{2}}X_{\alpha}(b)\right]}.$ (69)

This is the generalization of the (monochromatic) Chern-Simons state for
equations of state other than $w=-1$. Adding to it the unitary time evolution
(18) we have:

$\psi(b;\alpha)={\cal
N}\exp{\left[i\frac{3V_{c}}{l_{P}^{2}}\alpha(X_{\alpha}(b)-T_{\alpha})\right]},$
(70)

with $T_{\alpha}\equiv p_{\alpha}$. The most general solution superposes these
as:

$\psi(b)=\int\frac{d\alpha}{\sqrt{2\pi\mathfrak{h}}}{\cal
A}(\alpha)\exp{\left[\frac{i}{\mathfrak{h}}\alpha(X_{\alpha}(b)-T_{\alpha})\right]},$
(71)

or:

$\psi(b)=F(T_{\alpha}-X_{\alpha}).$ (72)

###  V.1 The example of radiation

A radiation dominated Universe ($w=1/3$) is a particularly simple case. Then
the Hamiltonian is:

$\displaystyle H$ $\displaystyle=$ $\displaystyle 6\kappa
V_{c}\frac{N}{a}\left(-(b^{2}+k)a^{2}+m\right)$ (73)

already compliant with Precept 3 , up to a redefinition of the lapse function:

$\tilde{N}=6\kappa\frac{N}{a}V_{c}$ (74)

even off-shell. Time, therefore, is the conjugate momentum to $m$ and this can
be identified with conformal time [more on this later]. The monochromatic
solutions to the time-dependent Sch equations are:

$\psi=\psi_{s}(b;m)\exp{\left[-\frac{i}{\mathfrak{h}}mp_{m}\right]}$ (75)

with:

$\left[i\mathfrak{h}(b^{2}+k)\frac{\partial}{\partial b}+m\right]\psi_{s}=0.$
(76)

so that we have plane waves in terms of:

$\displaystyle X_{r}(b)=\int\frac{db}{b^{2}+k}=$
$\displaystyle\frac{1}{\sqrt{k}}\arctan[\frac{b}{\sqrt{k}}]$
$\displaystyle\qquad{\rm if}\;k>0$ $\displaystyle=$
$\displaystyle-\frac{1}{b}$ $\displaystyle\qquad{\rm if}\;k=0$
$\displaystyle=$ $\displaystyle-\frac{1}{\sqrt{|k|}}{\rm
argtanh}[\frac{b}{\sqrt{|k|}}]$ $\displaystyle\qquad{\rm if}\;k<0$

to be seen as the equivalent of the Chern-Simons functional for a radiation
dominated Universe. The plane wave solutions at a generic time therefore are:

$\psi(b,T_{r};m)={\cal
N}\exp{\left[\frac{i}{\mathfrak{h}}m(X_{r}-T_{r})\right]}.$ (77)

(with $T_{r}=p_{m}$).

###  V.2 One exception: Milne or curvature domination

The general solution (63) breaks down for $w=-1/3$, an equation of state
degenerate with spatial curvature $k$ (or $kc^{2}$, to put it suggestively).
Backtracking to (62) we see that the problem is that we lose the spatial
differential operator contained in $a^{2}$. The spatial solution then becomes
$\psi_{s}=\delta(b^{2}-m)$, where $m$ can include, or indeed be just
$-kc^{2}$. The monochromatic solution is:

$\psi(b,T_{m};m)={\cal N}\delta(b^{2}-m)e^{-\frac{i}{\mathfrak{h}}mT_{m}}$
(78)

with $T_{m}=p_{m}$ as usual. Hence in this case there is no time evolution,
since for any superposition we have:

$\displaystyle\psi(b,T_{m})$ $\displaystyle=$ $\displaystyle\int dm{\cal
A}(m)\delta(b^{2}-m)e^{-\frac{i}{\mathfrak{h}}mT_{m}}$ (79) $\displaystyle=$
$\displaystyle{\cal A}(b^{2})e^{-\frac{i}{\mathfrak{h}}b^{2}T_{m}}$

so that $|\psi|^{2}=|{\cal A}(b^{2})|^{2}$. The reason why this happens will
be made clear in the next Section.

##  VI The classical limit

Given that “time evolution” is the most obvious feature of classical
cosmology, it is obvious that any quantum cosmology scheme lacking a “time”
will have trouble connecting with the classical world. Reciprocally, the
discovery of a quantum time should be used in the first instance to make sure
that the classical limit is sound, before exploring possible quantum
departures/corrections.

In this Section we first present the format in which the classical results are
best presented so that they can be recovered by appropriately (semi)-classical
wave-functions, within our scheme. We then prove that coherent states
reproduce the classical limit.

###  VI.1 The classical “time-formula”

We first find a classical expression for our physical times $T_{\alpha}$ as a
function of the non-physical coordinate time $t$ associated with lapse $N$.
From the second Hamilton equation (using (62), (14) and (64)) we can derive
the “time-formula”:

$\displaystyle\dot{T}_{\alpha}=\dot{p}_{\alpha}=\\{p_{\alpha},H\\}$
$\displaystyle=$ $\displaystyle-\frac{1+3w}{2}Na^{-3w}m^{\frac{3w-1}{1+3w}}$
(80) $\displaystyle=$
$\displaystyle-\frac{1+3w}{2}Na^{-3w}\alpha^{\frac{3w-1}{2}}.$

Note that we have used the original Hamiltonian, and not ${\cal H}$, to work
out the relation between $T_{\alpha}$ and time. We can now set $N=1$ to derive
the relation between $T_{\alpha}$ and cosmological proper time $t$:

$\frac{dT_{\alpha}}{dt}=-\frac{1+3w}{2}a^{-3w}m^{\frac{3w-1}{1+3w}}$ (81)

or else set

$N=N_{\alpha}=-\frac{2}{1+3w}a^{3w}m^{\frac{1-3w}{1+3w}}$ (82)

to ensure we are using a time coordinate coincident with the physical time
$T$.

Within this scheme (but note bruno1 ), we highlight the following facts:

  * •

Radiation is unique in that its time does not depend on $m$, so when this goes
to zero its time is still well defined.

  * •

Specifically, radiation time is minus conformal time, $T_{r}=-\eta$, since:

$\dot{T}_{r}=-N/a.$ (83)

This is in agreement with Gielen .

  * •

Dust time is proportional to minus proper cosmological time, with:

$T_{m}=-\frac{t}{2m}.$ (84)

  * •

Our Lambda time is proportional to Misner’s volume time misner  (as well as
unimodular time unimod1 ) :

$\dot{T}_{\phi}=N\frac{a^{3}}{\phi^{2}}=N\frac{\Lambda^{2}}{9}a^{3}.$ (85)

A canonical transformation relates the two: this is responsible for
linearizing the dispersion relations.

  * •

The only degenerate case in (80) is $w=-1/3$, but this case is exceptional, as
already discussed in Section V.2. In this case we should not transform from
$m$ to $\alpha$, so that $\dot{T}_{m}=-Na$.

The sign in the time-formula (80) is important and we note that it changes
from $w>-1/3$ to $w<-1/3$. This is a key feature of our formalism and we will
comment further on this below.

###  VI.2 The classical trajectory: a connection space picture

All classical descriptions are equivalent, so we may select whichever makes
better contact with our quantum theory. In our case we pick a description
which is unusual in that:

  * •

Instead of the expansion factor $a$, we take for dependent variable the
minisuperspace connection variable $b$. Recall that when torsion is zero, on-
shell this is the comoving inverse Hubble length $\dot{a}=b$. Quantum
mechanically $b$ is an independent and complementary variable to the metric
(or rather, the densitized inverse triad $a^{2}$).

  * •

Instead of using some coordinate time $t$ as independent variable, we use the
physical time(s) $T_{\alpha}$. These are classically (on-shell) a function of
$t$, as just calculated in (80). Fundamentally, and quantum mechanically,
there can be many $T$, but in the classical limit they all become functions of
$t$ (so that there is only one time classically, but there are several quantum
mechanical times).

Hence, the classical description we are aiming for takes the form $b=b(T)$,
possibly in the parametric form:

$\displaystyle b$ $\displaystyle=$ $\displaystyle b(t)$ (86) $\displaystyle
T_{\alpha}$ $\displaystyle=$ $\displaystyle T(t),$ (87)

rather than the textbook $a=a(t)$.

Then, we can show that the classical trajectory for a single fluid system is
given by:

$\dot{X}_{\alpha}=\dot{T}_{\alpha}.$ (88)

Indeed the full content of the classical equations can be obtained from the
first Friedman equation (equivalent to the Hamiltonian constraint $H=0$):

$b^{2}+k=\frac{m}{a^{1+3w}}$ (89)

which should be assumed throughout, as well as the two dynamical Hamilton
equations:

$\displaystyle\dot{a}$ $\displaystyle=$ $\displaystyle\\{a,H\\}=Nb$ (90)
$\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle\\{b,H\\}=-\frac{1+3w}{2}Na\frac{m}{a^{3(1+w)}}$ (91)

(where we have used (89) in the second equation). Together, these two
dynamical equations imply the Raychaudhuri (second Friedman) equation (for
$N=1$):

$\ddot{a}=-\frac{1+3w}{2}a\frac{m}{a^{3(1+w)}}.$ (92)

It is easy to see that (88) implies:

$\frac{\dot{b}}{h_{\alpha}(b)}=-\frac{1+3w}{2}a^{-3w}m^{\frac{3w-1}{1+3w}}$
(93)

which upon some manipulations reproduces the dynamical equation (91) (assuming
the constraint (89) throughout).

Using this unconventional description (i.e. (88)) may take some getting used
to, even though it is classically equivalent to the $a=a(t)$ description.
Points of note include:

  * •

Expanding and contracting Universes correspond to $b>0$ and $b<0$, with $b=0$
representing a static Universe (and its vicinity the loitering model).

  * •

Hence, the ekpyrotic ekp , or any such similar “bouncing” models, will see $b$
go through zero. Tunnelling between branches with different signs may also be
possible.

  * •

For a given matter content, $b$ can either only increase or only decrease in
parameter time $t$; the first if $w<-1/3$, the second if $w>-1/3$. For
$w=-1/3$ (or for the Milne Universe, for example) $b$ does not change (this
starts shedding light one the anomaly found in Section V.2).

  * •

Hence, the equivalent in of a “bounce” in $b$ space is a Universe undergoing a
transition from decelerated to accelerated expansion, such as we have
seemingly undergone recently. At the end of inflation the reverse happens, the
equivalent of a “turn-around” in $b$ space.

The fact that in this picture we have recently emerged from a $b$-bounce must
have quantum mechanical implications: quantum reflection always leaves its
traces. The matter will be studied in more detail in b-bounce .

###  VI.3 Parenthesis on the “arrow of time”

In view of what we said, the issue of the arrow of “time” merits a
parenthesis. In the metric representation flipping the time arrow inter-
converts expanding (increasing $a$) and contracting (decreasing $a$)
Universes. There are always two branches of solutions, as required by time-
reversal symmetry.

In the $b=b(T)$ description, face value, there is only one solution, for which
$b$ must increase with $T_{\alpha}$:

$\frac{db}{dT_{\alpha}}>0,$ (94)

as implied by (88). This is reflected in the quantum mechanical solutions (cf.
(28)): there can only be outgoing waves. Thus, there is a sense in which there
is only one arrow of time in the connection representation and using the times
$T_{\alpha}$. A Feynman absorber is not needed to set the physical arrow of
time.

This fact is actually an expression of the horizon (and ultimately also the
flatness) problem, as well as its standard solution, as we now show. Let us
first assume expanding Universes ($b>0$). Then, the horizon problem is that
for $w>-1/3$ the comoving Hubble length $1/b$ increases in time, whereas its
solution follows from that it decreases if $w<-1/3$. In our description this
is expressed by (94) and the fact that, using $t$ as the auxiliary arbitrary
time arrow, the sign in the time-formula (80) changes from $w>-1/3$ to
$w<-1/3$. Converting this into $b(T_{\alpha})$ we then get that (94) is a
statement of the horizon problem for $w>-1/3$ and its solution for $w<-1/3$.

The actual arrow of $t$ does not matter, because it cancels in its double
effect on $b=\dot{a}$ and on $\dot{b}$ (note the invariance of the
Raychaudhuri equation (92) under time reversal). Solutions to the horizon
problem based on a contracting phase (e.g. the ekpyrotic scenario ekp ) can be
understood in our description from (94) being independent of the sign of $b$;
however the statement of the problem and its solution involves $|b|$, not $b$.
So the criteria for problem and solution are reversed for models in a
contracting phase ($b<0$), and this is still expressed by (94).

.

###  VI.4 Coherent states and the classical limit

Having a quantum time variable and a larger space of solutions (as described
in Sec. IV) are the two reasons why contact with the classical limit is
possible. Monochromatic waves, such as those solving the fixed constant
theory, imply a uniform distribution (in $X(b)$), hardly a prediction, but
they are also not immediately physical. One needs both a time variable and the
ability to superpose plane waves into normalized peaked distributions to
recover something minimally physical.

In fact having a peak is not enough. For example, since
$\dot{X}_{\alpha}=\dot{T}_{\alpha}$ represents the classical trajectory (as
just discussed), one might think that the light-ray,
$\psi\propto\delta(X_{\alpha}-T_{\alpha})$, described in Section IV.1 would be
perfectly classical. But such a state would have a totally undefined $\alpha$,
and so the $T_{\alpha}=T_{\alpha}(t)$ part of the argument could not be true
(note that $\alpha$ generally appears in the RHS of (80)).

The semi-classical limit is only recovered for the coherent states
$\psi(b,T_{\alpha})$ described in Section IV.2. For these, the second Hamilton
equation (80) is true not only on average (an expression of Ehrenfest’s
theorem), but with minimal and balanced uncertainties in the complementary
$\alpha$ and $T_{\alpha}$ appearing on the two sides of (80). Both sides of
the argument implying that $\dot{X}_{\alpha}=\dot{T}_{\alpha}$ represents the
classical trajectory can now be reproduced, and so we have a truly semi-
classical state.

We can also understand the result in Section V.2. We do not have propagating
waves in this case. However, the classical equation of motion is $\dot{b}=0$,
i.e. the Universe is static in $b$, as already explained. Any coherent state
in $m$ therefore reproduces this result (as well as the time formula in terms
of $m$).

##  VII Multitime

Naturally, for all our pains, we now end up with the “curse of time” in
Quantum Gravity: either there is no time, or, if we succeed in defining one,
we are left spoilt for choice. The proposal in Myletter  was to accept a
democracy of times, with the adjustment of clocks across different “time
zones” to be seen as a physical feature of our world. We now examine further
how the handover between clocks can be made seamless for some states. But
there are also alternative states, which might be of great predictive
relevance, considering that the Universe is currently busy passing on the
baton from a matter (or $G_{N}$) clock to a $\Lambda$ clock.

How do we deal with multiple times and multiple fluids, even in situations
where there are epochs where one fluid dominates? Let $\bm{\alpha}$ be a
vector representing the whole set of relevant constants and $\bm{T}$ their
conjugates. The various components of $\bm{T}$ are a priori independent
variables, so we have a plethora of times instead of a single one. The
statements made in this paper imply that there is not one “Schrodinger”
equation (our examples being merely limiting cases) but a PDE in multiple
times running concurrently, obtained by taking the Hamiltonian in

$S=\frac{3V_{c}}{8\pi G_{0}}\int
dt\bigg{(}a^{2}\dot{b}-Na\left[-(b^{2}+kc^{2})+\sum_{i}\frac{m_{i}}{a^{1+3w_{i}}}\right]\bigg{)},$
(95)

and applying the replacements:

$H\left[b,a^{2};\bm{\alpha}\rightarrow
i\frac{l_{P}^{2}}{3V_{c}}\frac{\partial}{\partial\bm{T}}\right]\psi=0.$ (96)

Its general solutions are:

$\psi(b,\bm{T})=\int d\bm{\alpha}{\cal
A}(\bm{\alpha})\exp{\left[-i\frac{3V_{c}}{l_{P}^{2}}\bm{\alpha}\bm{T}\right]}\psi_{s}(b;\bm{\alpha}),$
(97)

where $\psi_{s}(b;\bm{\alpha})$ solves the WDW equation with constant
$\bm{\alpha}$.

As outlined in Myletter , for a Hamiltonian carving up phase space into
regions dominated by a single constant, the readjustment of quantum clocks
across such regions is seamless if we assume coherent states in all
$\alpha_{i}$ and factorization:

${\cal A}(\bm{\alpha})=\prod_{i}\sqrt{{\bf N}(\alpha_{0i},l_{P}^{2}/6V_{c})}.$
(98)

Then, the $\psi_{s}(b;\bm{\alpha})$ is a piecewise plane wave in the
$X_{i}(b)$ associated with each dominant $\alpha_{i}$. Each piece engages with
the phase associated with the corresponding $T_{i}$ (hence, the approximate
single-time Schrodinger equation), producing a wave-packet describing the
correct classical limit, as above. What happens to the minority clocks in each
phase will be studied in more detail later in this paper.

Indeed the rest of this paper will be devoted to making this argument more
explicit, whilst dealing with the unpleasant truth that we are not living in a
Universe where one component dominates, but quite the opposite. In order to
deal with multi-fluid situations we need to first re-examine the structure of
minisuperspace from a different angle.

##  VIII Minisuperspace as a dispersive medium

The arguments in Section II for variables $\alpha$, $X_{\alpha}$ and
$T_{\alpha}$ are persuasive, but realistic only in idealized situations, such
as when there is only one fluid filling the Universe. Unfortunately the real
world does not oblige. In some cases such variables cannot be found. For
example, at the crossover between 2 epochs dominated by different fluids, we
will find an $X$ variable which is a function not only of $b$ but also of
$\alpha$ (see next Section). In such cases it is more fruitful to revert to
the original variables (e.g. $b$ instead of any $X(b)$) and regard
minisuperspace as a dispersive medium. From this point of view the variables
$\alpha$, $X_{\alpha}$ and $T_{\alpha}$, when they exist, are the “linearizing
variables” of the dispersive medium, to use the terminology of DSR . They can
and should be used where they exist, but more generally we should face the
dispersive nature of minisuperspace head on.

In the general case we can still define times $\bm{T}$ for the various
$\bm{\alpha}$, impose the monochromatic ansatz (147), and find the spatial
solutions $\psi_{s}$, which in general will not be plane waves in any $X(b)$
variable independent of $\alpha$. The monochromatic solutions can still be
superposed into peaked wave-packets, as in (147). However it is important to
realize that, as with any other dispersive medium, the envelope of such
packets moves with a group speed that should not be confused with the phase
speed.

Specifically, writing:

$\psi_{s}(b,{\bm{\alpha}})=\exp{\left[i\frac{3V_{c}}{l_{P}^{2}}P(b,{\bm{\alpha}})\right]}$
(99)

we identify dispersion relations:

$\bm{\alpha}\cdot\bm{T}-P(b,\bm{\alpha})=0.$ (100)

Assuming that the amplitude ${\cal A}({\bm{\alpha}})$ is factorizable and
sufficiently peaked around ${\bm{\alpha}}_{0}$, we can expand:

$P(b,\bm{\alpha})=P(b;\bm{\alpha}_{0})+\sum_{i}\frac{\partial
P}{\partial\alpha_{i}}\biggr{\rvert}_{\bm{\alpha}_{0}}(\alpha_{i}-\alpha_{i0})+...$
(101)

to find that the wave function factorizes as:

$\psi\approx
e^{i\frac{3V_{c}}{l_{P}^{2}}(P(b;\bm{\alpha}_{0})-\bm{\alpha}_{0}\cdot\bm{T})}\prod_{i}\psi_{i}(b,T_{i}).$
(102)

The first factor is the monochromatic (generally non-plane) wave centered on
$\bm{\alpha}_{0}$. The other factors describe envelopes of the form:

$\psi_{i}(b,T_{i})=\int d\alpha_{i}{\cal
A}(\alpha_{i})e^{-i\frac{3V_{c}}{l_{P}^{2}}(\alpha_{i}-\alpha_{i0})(T_{i}-\frac{\partial
P}{\partial\alpha_{i}})}$ (103)

which therefore move according to:

$T_{i}=\frac{\partial
P(b)}{\partial\alpha_{i}}\biggr{\rvert}_{\bm{\alpha}_{0}}.$ (104)

We can also dot this equation, to find the group speed on $\\{b,T_{i}\\}$
space:

$c_{g}=\frac{db}{dT}\biggr{\rvert}_{peak}=\frac{\dot{b}}{\dot{T}}\biggr{\rvert}_{peak}=\frac{1}{\frac{\partial^{2}P}{\partial\alpha\partial
b}}.$ (105)

The motion of these envelopes (and so of the peak of the distribution) should
agree with the classical equations of motion. We will show in the rest of this
paper that indeed it does so, for coherent states, in a number of non-trivial
situations (such as for mixtures of fluids during transition periods when none
of them dominates, or for the sub-dominant clock).

This obviously generalizes the construction for single fluids, for which a
variable $X(b)$ can be found such that $P=\alpha X(b)$ for some $\alpha$.
Then, with a a suitable choice of $\alpha$ (and canonical $T_{\alpha}$) we can
always make the first term in the dispersion relations $\alpha T_{\alpha}$
(for example, in the case of Lambda by $\Lambda\rightarrow\phi=3/\Lambda$,
$T_{\Lambda}\rightarrow T_{\phi}=-T_{\Lambda}/\phi^{2}$). They are
“linearizing” variables because $c_{lin}={\dot{X}}/{\dot{T}}=1$.

##  IX Dealing with crossover regions

As it happens we are sitting right on a bounce in $b$. How do we deal with
such transitions? In this Section we show that the correct semi-classical
limit is still obtained assuming the wave function remains sharply peaked.
What actually happens to the wave function is left to future work b-bounce .
We also investigate the fate of the minority clock (i.e. the radiation and
Lambda clocks in the Lambda and radiation epochs) once the handover of clocks
is completed. We will use as a working model a mixture of radiation and
Lambda, because the algebra is clearer, but generalizations to the more
relevant case of dust and Lambda behave in the same way.

###  IX.1 Mono-chromatic solutions

Our working model has classical Hamiltonian:

$H=Na\left(-(b^{2}+k)+\frac{a^{2}}{\phi}+\frac{m}{a^{2}}\right)$ (106)

spanning a two-dimensional constant space with

${\bm{\alpha}}=\left(\phi\equiv\frac{3}{\Lambda},m\right).$ (107)

Its multi-time “Schroedinger” equation (96) has solutions of the form (147).
One way to find the spatial $\psi_{s}$ is to put $H=0$ in the form (2)/(12)
with $\alpha=\phi$, aware that $h_{\alpha}(b)$ will then not satisfy precept
2. To this end we solve the quadratic in $a^{2}$ equivalent to $H=0$ to find:

$a^{2}=\frac{g\pm\sqrt{g^{2}-4m/\phi}}{{2}/{\phi}}$ (108)

with $g(b)=b^{2}+k$. Since $a^{2}$ must be real (although not necessarily
positive) we have:

$g^{2}\geq g_{0}^{2}=\frac{4m}{\phi}=\frac{4}{3}\Lambda m.$ (109)

The plus branch contains Lambda domination when $g^{2}\gg g_{0}^{2}$; the
minus branch contains radiation domination, also with $g^{2}\gg g_{0}^{2}$.
The transition happens when $g^{2}\approx g_{0}^{2}$ (with $g^{2}>g_{0}^{2}$).
Thus, we have a “bounce” in $b$ space at $g=g_{0}$, i.e. a transition from
decelerated expansion (decreasing $b$) to accelerated expansion (increasing
$b$). The Hamiltonian constraint is therefore equivalent to two constraints of
the required form:

${\cal H}_{\pm}=h_{\pm}(b;\phi,m)a^{2}-\phi=0$ (110)

with the important novelty that $h$ (and so $H_{0}$) is “energy”-dependent
(dependent on the conjugate of time; i.e. the constants):

$h_{\pm}=\frac{2}{g\pm\sqrt{{g^{2}-4m\phi}}}.$ (111)

This is of course irrelevant for the $\psi_{s}$, which is given by:

$\psi_{s\pm}(b;\phi,m)={\cal N}\exp{\left[i\frac{3V_{c}}{l_{P}^{2}}\phi
X_{\pm}(b;\phi,m)\right]}.$ (112)

with:

$X_{\pm}(b;\phi,m)=\int db\frac{1}{2}\left(g\pm\sqrt{{g^{2}-4m/\phi}}\right).$
(113)

We see that for $g^{2}\gg m/\phi$ the $+/-$ branches have:

$\displaystyle X_{+}(b;\phi,m)$ $\displaystyle\approx$ $\displaystyle
X_{\phi}=\frac{b^{3}}{3}+kb$ (114) $\displaystyle X_{-}(b;\phi,m)$
$\displaystyle\approx$ $\displaystyle\frac{m}{\phi}X_{r}$ (115)

leading to the correct limits:

$\displaystyle\psi_{s+}(b;\phi,m)$ $\displaystyle\approx$ $\displaystyle{\cal
N}\exp{\left[i\frac{3V_{c}}{l_{P}^{2}}\phi X_{\phi}(b)\right]}$ (116)
$\displaystyle\psi_{s-}(b;\phi,m)$ $\displaystyle\approx$ $\displaystyle{\cal
N}\exp{\left[i\frac{3V_{c}}{l_{P}^{2}}mX_{r}(b)\right]}.$ (117)

This illustrates with a concrete example the comments made just after Eq.
(98): the $\psi_{s}(b;\bm{\alpha})$ is a piece-wise plane wave in the relevant
$\alpha$ and $X_{\alpha}$ in each region of single fluid domination. To
leading order it might seem that if ${\cal A}(\bm{\alpha})$ factorizes, then
all the other times factorize, too, and stop describing the $b$ evolution
since they became $b$-independent phases. However this is not the case, as we
now show by considering the next to leading order in the expansion.

###  IX.2 What happens to the minority clock(s)?

Before addressing the handover region itself, we first examine in more detail
what happens to the “minority” clock once the handover is finished. Expanding
(113) to the next order we find:

$\displaystyle X_{+}(b)$ $\displaystyle\approx$ $\displaystyle
X_{\phi}-\frac{m}{\phi}X_{r}+...$ (118) $\displaystyle X_{-}(b)$
$\displaystyle\approx$
$\displaystyle\frac{m}{\phi}X_{r}+\frac{m^{2}}{\phi^{2}}\int\frac{db}{g^{3}}+....$
(119)

####  IX.2.1 The radiation clock in the Lambda epoch

Including the next order term in $X_{+}$ we find that deep in the Lambda epoch
the monochromatic wave function is:

$\psi(b,{\bm{T}};{\bm{\alpha}})={\cal
N}\exp{\left[-i\frac{3V_{c}}{l_{P}^{2}}(\phi(T_{\phi}-X_{\phi})+m(T_{r}+X_{r}))\right]}.$

Inserting into (147) we find for any factorizable amplitude:

$\psi(b,{\bm{T}})=F_{1}(X_{\phi}-T_{\phi})F_{2}(X_{r}+T_{r}).$ (120)

In particular we could choose factorizable Gaussian amplitudes for $\phi$ and
$m$ leading to coherent $F_{1}$ and $F_{2}$ (of the form (50)). As we will see
this is more the exception than the rule.

We see that both factors reproduce the classical equations of motion. These
amount to the first and second Friedmann equations:

$\displaystyle b^{2}+k$ $\displaystyle=$
$\displaystyle\frac{a^{2}}{\phi}+\frac{m}{a^{2}}$ (121) $\displaystyle\dot{b}$
$\displaystyle=$ $\displaystyle\frac{a}{\phi}-\frac{m}{a^{3}}.$ (122)

In addition, the times formulae (replicated quantum mechanically by coherent
factorizable states, just as before) are:

$\displaystyle\dot{T}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{a^{3}}{\phi^{2}}$ (123) $\displaystyle\dot{T}_{r}$
$\displaystyle=$ $\displaystyle-\frac{1}{a}.$ (124)

Evaluating:

$\displaystyle\dot{X}_{\phi}$ $\displaystyle=$ $\displaystyle\dot{b}(b^{2}+k)$
(125) $\displaystyle\dot{X}_{r}$ $\displaystyle=$
$\displaystyle\frac{\dot{b}}{b^{2}+k}$ (126)

we can then recover the mono-fluid equations of motion in the appropriate
epochs with $\dot{X}_{\alpha}\approx\dot{T}_{\alpha}$, for $\alpha=\phi,m$.
But some more algebra also reveals that deep in the Lambda era we can write
the classical trajectory as:

$\dot{X}_{r}\approx-\dot{T}_{r}$ (128)

and this is equivalent to $\dot{X}_{\phi}\approx\dot{T}_{\phi}$. Hence, the
peak of both factors in (120) describes the classical trajectory.

This sheds light on what happens to our quantum “multi-time” in semi-classical
situations, given that classically only one time can exist. Quantum
mechanically the two $T_{i}$ are independent variables and fundamentally
remain so, even for semi-classical states. There is never a constraint between
the different $T_{i}$. What happens is that for peaked states the peak of the
joint distribution maps out a trajectory of $b$ in 2D space $\bm{T}$ (in this
case $X_{\phi}(b)=T_{\phi}$ and $X_{r}(b)=-T_{r}$). This implies a constraint
between the two times at the peak of the joint distribution, so classically
only one time exists. The quantum fluctuations or these different times, on
the other hand, would remain independent.

####  IX.2.2 The Lambda clock in the radiation epoch

Deep in the radiation epoch we have instead

$\displaystyle\psi(b,{\bm{T}};{\bm{\alpha}})=$ (129) $\displaystyle{\cal
N}\exp{\left[-\frac{i}{\mathfrak{h}}\left(m(T_{r}-X_{r})+\phi\left(T_{\phi}-\frac{m^{2}}{\phi^{2}}\int\frac{db}{g^{3}}\right)\right)\right]}$

with the novelty that the factor associated with the subdominant component
(Lambda) now depends on $m$ as well. As a result, the wave packets never
factorizes into separate radiation and Lambda factors, even if the amplitudes
${\cal A}(\bm{\alpha})$ do. In addition the minority factor no longer is a
plane wave in the original $X(b)$ and $\alpha=\phi$. Hence, even if the
original amplitudes were a diagonal Gaussian, the wave functions will be very
distorted. The simple arguments for unitarity for single fluids also break
down for a minority clock in this situation. This will be discussed further in
the next Section.

Nonetheless, we can show that for a peaked second factor, the motion of the
peak still reproduces the correct classical limit (the first one obviously
does). Using (105) and

$c_{g}^{-1}=\frac{\partial^{2}}{\partial\phi\partial
b}\frac{m^{2}}{\phi}\int\frac{db}{g^{3}}=-\frac{m^{2}}{\phi^{2}g^{3}}$ (130)

we find that for the peak:

$c_{g}=\frac{\dot{b}}{\dot{T}_{\phi}}\biggr{\rvert}_{peak}$ (131)

implies

$\dot{b}=-\frac{m}{a^{3}}$ (132)

which is nothing but approximately (122) in the radiation epoch.

Notice that within the same approximations used in Section VIII) to derive
(105), the wave-function effectively factorizes as:

$\psi(b,{\bm{T}};{\bm{\alpha}})\approx\psi_{1}(T_{r}-X_{r})\psi_{2}(b,T_{\phi};m_{0})$
(133)

Hence to this order, the arguments at the end of Section IX.2.1 in support of
a single classical time still apply.

###  IX.3 Sitting on the fence

We can also show that if the distribution remains peaked b-bounce , then the
peak follows the classical trajectory even during the $b$-bounce, for both
branches $\pm$. In this case the $P$ function defined in Section VIII is given
by:

$\displaystyle P_{\pm}(b,m,\phi)$ $\displaystyle=$ $\displaystyle\phi
X_{\pm}(b;m/\phi)$ (134) $\displaystyle=$ $\displaystyle\phi\int
db\frac{1}{2}\left(g\pm\sqrt{{g^{2}-4m/\phi}}\right).$

Within the approximations of Section VIII (see (102) in particular) the wave
function must be approximately given by the monochromatic solution times the
product of two envelopes $\psi_{1}(b,T_{\phi})\psi_{2}(b,T_{r})$. The latter
move with group speeds:

$\displaystyle c_{g1}$ $\displaystyle=$
$\displaystyle\frac{\dot{b}}{\dot{T}_{\phi}}\biggr{\rvert}_{peak}=\frac{1}{\frac{\partial^{2}P}{\partial\phi\partial
b}}$ (135) $\displaystyle c_{g2}$ $\displaystyle=$
$\displaystyle\frac{\dot{b}}{\dot{T}_{r}}\biggr{\rvert}_{peak}=\frac{1}{\frac{\partial^{2}P}{\partial
m\partial b}}.$ (136)

It is now a matter of algebra to shown that in both branches ($\pm$) these are
equivalent to the classical equation of motion (122) (with (121) assumed
throughout).

Indeed, for the Lambda wave packet factor we have:

${\frac{\partial^{2}P}{\partial\phi\partial
b}}=\frac{1}{h}\pm\frac{m}{\phi}\frac{1}{\sqrt{g^{2}-g^{2}_{0}}}.$ (137)

Using:

$\pm\sqrt{g^{2}-g^{2}_{0}}=\frac{2a^{2}}{\phi}-g=\frac{a^{2}}{\phi}-\frac{m}{a^{2}}$
(138)

and $h=\phi/a^{2}$ we have:

${\frac{\partial^{2}P}{\partial\phi\partial
b}}=\frac{a^{4}/\phi^{2}}{\frac{a^{2}}{\phi}-\frac{m}{a^{2}}}$ (139)

implying that the peak moves along:

$\dot{b}=\frac{\dot{T}_{\phi}}{a^{4}/\phi^{2}}\left(\frac{a^{2}}{\phi}-\frac{m}{a^{2}}\right)=\frac{a}{\phi}-\frac{m}{a^{3}}$
(140)

i.e. (122), as required.

Likewise, for the $m$ wave packet factor we have:

$\displaystyle{\frac{\partial^{2}P}{\partial m\partial b}}$ $\displaystyle=$
$\displaystyle\mp\frac{1}{\sqrt{g^{2}-g^{2}_{0}}}$ (141) $\displaystyle=$
$\displaystyle-\frac{1}{\frac{a^{2}}{\phi}-\frac{m}{a^{2}}}$

leading to:

$\dot{b}=-\dot{T}_{r}{\left(\frac{a^{2}}{\phi}-\frac{m}{a^{2}}\right)}$ (142)

or (122) .

Hence the correct classical limit is always obtained, assuming the wave
functions remain peaked. Whether this is a good approximation remains to be
seen b-bounce . In addition there are other issues regarding the semi-
classical limit, as we now explain.

##  X Why swap clocks?

In this Section we explain better why “a clock [should be] crafted with what
is at hand”, as proposed in Myletter . This is not just common sense: It
affects the semi-classical limit. As we have just seen in detail, the
probability peak’s motion has the correct classical limit (assuming
Ehrenfest’s theorem) even for the minority clock, but this hides the fact that
typically the state will not be coherent in such a set up, and so departures
from the semi-classical regime are expected.

The Lambda clock in the radiation epoch is a good illustration of this. As we
saw in Section IX.2.2 (cf. Eq. 129), to leading order (in the saddle
approximation of Section VIII), the Lambda factor in the radiation epoch
changes its dependence on $b$ from $X_{\phi}$ to:

$X=\int\frac{db}{g^{3}}$ (143)

and its $\alpha$ from $\phi$ to:

$\alpha=\frac{m_{0}}{\phi^{2}}.$ (144)

In Section IX.2.2 we studied in detail the peak of the wave function, but the
semi-classical limit requires also the arguments in Section VI for the correct
representation of the time-formula, and so we need more than a peaked
distribution: we should have a coherent state in $m_{0}/\phi^{2}$. But if we
chose a Gaussian amplitude in $\phi$, then this will not be Gaussian in
$m_{0}/\phi^{2}$, quite the contrary: strong distortions are expected. This is
representative of what usually happens to minority clocks. Indeed, the
radiation clock in the Lambda epoch (see Section IX.2.1) is the exception to
this rule. It is a rare case where a coherent majority clock remains coherent
in the subdominant phase.

We can also add the issue of unitarity and inner product to the discussion. We
can define a conserved inner product as in Section IV.4, but it only leads to
a simple conserved current and re-expression in terms of a measure in $b$ in
the dominant epoch. As we saw, in mono-fluid situations (and so using the
dominant clock in a multi-fluid situation) there is a range of options for
setting up the inner product and conserved current. These are all ultimately
equivalent: we can use Eq. (56), leading to general expression:

$d\mu(b)=dX_{\alpha}=\frac{db}{(b^{2}+k)^{\frac{2}{1+3w}}},$ (145)

we can trade $X_{\alpha}$ for $T_{\alpha}$ as in (59), or we can use (146)
(with $\phi$ replaced by the applicable $\alpha$).

However, only

$\langle\psi_{1}|\psi_{2}\rangle=\int d\alpha{\cal A}^{\star}(\alpha_{1}){\cal
A}(\alpha_{2}).$ (146)

generalizes to multi-fluid situations. Bearing in mind that the general
solution now is:

$\psi(b,T)=\int\frac{d{\alpha}}{\sqrt{2\pi}}{\cal
A}(\alpha)\exp{\left[-i\frac{3V_{c}}{l_{P}^{2}}\alpha
T\right]}\psi_{s}(b;\alpha),$ (147)

where $\psi_{s}(b;\alpha)$ solves the WDW equation with constant $\alpha$, it
is obvious that (146) reduces to (56) and (59) for single fluids. For multi-
fluids we recover (59) iff $\psi_{s}(b;\alpha)$ is a pure phase (so in cases
where there is no $b$ bounce), but not (56). For multi-times this is more
complicated. Whatever the case (146) appears to be a general time-independent
definition for the inner product. But only when the dominant clock and its $X$
are used does this lead to a simple inner product in terms of $b$.

##  XI Conclusions

In summary, in this paper we started off by proposing a number of aspirational
principles for defining clocks and rods in quantum cosmology, based on an
amplification of the standard theory allowing the constants of nature to be
non-constant off-shell. Classically, the constants remain good old constants,
but quantum mechanically it all changes. Each constant generates a space of
quantum states composed of superpositions of waves indexed by the value of the
constant (these can be seen as monochromatic partial waves). The waves
propagate in a fundamentally dispersive medium where the “space” is the
connection, the “time” is the momentum conjugate to the constant, and the
“energy” and “momentum” are functions of the constant. In some regions (or
“epochs”) we can find simple linearizing variables ($X_{\alpha}$, $\alpha$ and
$T_{\alpha}$), in terms of which the partial waves are plane waves moving at
fixed speed, conventionally set to 1. These implement our aspirational
principles. In such regions, for a given constant, our construction provides a
good clock and rod, leading to a simple inner product and definition of
unitarity, with coherent states providing the perfect definition of a semi-
classical states. However, this construction is never global, and hence the
need to change clocks at different “time zones” in our Universe.

Specifically, we showed that the dominant fluid always generates a good clock,
but the “minority clock” is generally problematic in that the linearizing
variables are not always available, and so the dispersive nature of the medium
has to be faced. Even if approximate linearizing variables can be found, they
will generally be different functions of the constant than in the dominant
phase. If the wave function was coherent in the original variable, it will not
be in the second. If the wave function is peaked, the peak follows the
classical equations of motion; however the quantum nature of the system can
never be erased. Unitarity can be defined in a universal way, but it becomes
cumbersome when written in terms of the minority clock. Changing clocks is
advisable.

The interesting point remains that in transition regions we may expect
anomalies, and so interesting phenomenology. To make matters more poignant, we
happen to be loosely sitting on the fence separating matter and Lambda
domination. Our clocks, therefore, have been likely non-aspirational for the
past few billion years. The fact that the transition from deceleration to
acceleration is a quantum bounce, with its inevitable ringing, only compounds
the issue. We should not regard this as an ontological tragedy but as an
epistemological opportunity. Is the Universe going quantum?

To this question we should attach another, more fundamental one: How would we
see a Universe going quantum? We are used to quantum systems as microscopical
sub-systems living inside larger classical macroscopical systems; but this is
just the opposite. What if our local classical world were encased in Universe
which on the very largest scales is behaving quantum mechanically? How would
we see it?

Even ignoring this million dollar question, other interesting problems remain.
Clock swapping is essential for keeping the classical description, but why
would the Universe choose to swap clocks? A selection principle seems to be at
play, and this may shed light on other fine tuning problems, such as the
cosmological constant problem (e.g. weinberg ; padilla ; pad ; pad1 ; lomb ).
Also, what implications are there for the constants of nature if they are not
allowed to be infinitely sharp in a classical world? In this framework, and
infinitely sharp constant implies total delocalization in the conjugate time.
In this sense, a perfect constant is a failed classical clock555One may also
appeal to Borges’ quote, ”[Eternity can be defined] as the simultaneous and
lucid possession of all the instants of time”. With this definition, an exact
constant of Nature is equivalent to eternity in one of the many possible times
(viz. the one dual to the infinitely sharp constant).. Would this have
implications, either for our local physics, or for our description of the
large-scale Universe (the two being essentially complementary)? Finally, one
may ask what happens if more than one dominant clock is at play? Preliminary
work suggests that they would get in each other’s way regarding classicality
bruno1 , but how problematic is this? Is this, instead, yet another
observational window of opportunity?

Acknowledgments. I would like to thank Bruno Alexandre, Steffen Gielen, Chris
Isham and Tony Padilla for discussions and advice in relation to this paper.
This work was supported by the STFC Consolidated Grant ST/L00044X/1.

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