###### Abstract

Entertaining the possibility of time travel will invariably challenge dearly
held concepts of fundamental physics. It becomes relatively easy to construct
multiple logical contradictions using differing starting points from various
well-established fields of physics. Sometimes, the interpretation is that only
a full theory of quantum gravity will be able to settle these logical
contradictions. Even then, it remains unclear if the multitude of problems
could be overcome. Yet as definitive as this seems to the notion of time
travel in physics, such a recourse to quantum gravity comes with its own,
long-standing challenge to most of these counter-arguments to time travel:
These arguments rely on time, while quantum gravity is (in)famously stuck with
and dealing with the _problem of time_. One attempt to answer this problem
within the canonical framework resulted in the Page–Wootters formalism, and
its recent gauge-theoretic re-interpretation—as an emergent notion of time.
Herein, we will begin a programme to study toy models implementing the
Hamiltonian constraint in quantum theory, with an aim towards understanding
what an emergent notion of time can tell us about the (im)possibility of time
travel.

###### keywords:

time travel; quantum gravity; minisuperspace; Page–Wootters formalism;
emergent time; relational dynamics

10 2 73 14/12/2023 26/01/2024 28/01/2024 02/02/2024
https://doi.org/10.3390/universe10020073 10.3390/universe10020073 Emergent
Time and Time Travel in Quantum Physics Emergent Time and Time Travel in
Quantum Physics Ana Alonso-Serrano 1,2,†, Sebastian Schuster 3,*,† and Matt
Visser 4,† Ana Alonso-Serrano, Sebastian Schuster and Matt Visser Alonso-
Serrano, A.; Schuster, S.; Visser, M. Correspondence:
<EMAIL_ADDRESS>These authors contributed equally to this
work.

##  1 Introduction

As fascinating as the notion of time travel is, its long list of problematic
issues—ranging from the classical Roman (1993); Krasnikov (2002) & (Visser,
1996, p.212ff) to the quantum theoretical Hawking (1992); Klinkhammer (1992);
Visser (1993); Emparan and Tomašević (2022)—presents a very solid case against
it, and its absence is effectively a feature of the universe accessible to
human experience. Yet if one wants to look closely at these arguments, cracks
and loopholes start to appear—many of which shall be mentioned in the
following. To an extent, it almost seems as if the best evidence against any
notion of time travel is that of our daily experienced notion of time and
causality, our intuition. Still, the known universe is bigger than our day-to-
day experience, and particularly the field of high energy physics is ripe with
examples experimentally, observationally, or theoretically open to inquiry—in
which various concepts or preconceived notions of human-scale physics have to
be let go. In no field of theoretical physics is this more apparent than in
the quest for quantum gravity. The field of contenders is large, and still
growing: String theory, loop quantum gravity, quantum geometrodynamics,
asymptotic safety, causal dynamical triangulation, causal set theory,
Hořava–Lifshitz gravity, …Each comes with its own set of new notions beyond
what can be experienced directly by human senses. Yet, these proposals still
have to address, in one way or another, an issue plaguing quantum gravity from
its beginning: Time. This _problem of time_ , and its eventual resolution,
then naturally will also impact what these theories have to say about (or
against) time travel.

The ‘problem of time’ in _classical_ canonical gravity is settled by a
careful, gauge-theoretic analysis Pons et al. (2010) that distinguishes the
different rôles played by the Hamiltonian, either in the constraint/gauge
fixing or in the definition of observables, respectively. In quantum gravity,
however, the _problem of time_ remains (at the very least) a topic of active
discussion and research Anderson (2017); Gioia et al. (2021); Kiefer and Peter
(2022). The reason for this goes back even before a theory of quantum gravity
was the focus: While there are ways to give position and momentum a meaning in
terms of operators on (rigged) Hilbert spaces in quantum mechanics, this
cannot be carried over to time and energy. This was concerning to researchers
in quantum mechanics early on, as the time-energy uncertainty relations proved
to be both reliable and instructive, while not fitting into the standard
paradigm of the Robertson–Schrödinger uncertainty relations for (generic)
Hermitian operators. Only the proof of Mandelstam and Tamm in 1945 improved
this uncertainty relation’s foundation, while begging the question of why a
different approach was needed Mandelstam and Tamm (1945). Essentially, a
series of impossibility results for time operators canonically conjugate to a
Hamiltonian was presented: Schrödinger used an approach based on
normalizability Schrödinger (1931), similar to the argument for why plane
waves require a rigged Hilbert space. Pauli, in a footnote, used a simpler
argument juxtaposing discrete spectra and the continuous translation in energy
such a time operator would induce Pauli (1990, 1980). Finally (and in reply to
what will be discussed shortly), Unruh and Wald gave a general argument for
semi-bounded Hamiltonians linked to unitarity Unruh and Wald (1989). Two at
first glance quite distinct approaches were developed in reply to these no-go
results.

One approach was similar in spirit to the answer to early counterarguments to
plane waves, _e.g._ , eigenstates of definite position and momentum. While
with position and momentum one relaxes the idea of being restricted to Hilbert
space and instead works in a rigged Hilbert space Böhm (1993); de la Madrid et
al. (2002); Thiemann (2008), here, when faced with the above-cited no-go
results concerning time operators, one relaxes the idea of Hermiticity.
Instead, one introduces the notion of a positive, operator-valued measure
(POVM). It can represent not just perfect measurements yielding a definite
eigenstate, but also imprecise measurements Busch et al. (2016). Besides
allowing one to address the measurement problem in quantum field theory
Fewster and Verch (2020), phase observables Busch et al. (1995), or the
already mentioned, imprecise measurement processes Auletta et al. (2009);
Busch et al. (2016)—and more important for the present context—this allows a
notion of time measurement Busch et al. (1994). Since the no-go theorems
relate to Hermitian operators, the more general POVMs are not ruled out as
time observables. As we will see, the context of time travel allows one to
avoid these no-go results by other means, too. Still, the bigger picture of
gauge theory provides the most pertinent background for extensions of the
model to be described in this paper.

Another approach that developed independently from POVMs was to carefully
distinguish between a clock and time. Paraphrasing Einstein Einstein (1905),
time is meaningful only after specifying the clock measuring it. Early on, in
developing canonical quantum gravity, DeWitt DeWitt (1967) pointed out how
different subsystem’s operators in a given Hamiltonian constraint could serve
this purpose of an effective clock. Page and Wootters then took this idea and
made it more general and workable, by phrasing time evolution in terms of
conditional probabilities linking different subsystems Page and Wootters
(1983); Wootters (1984).111With a bit of hindsight, already Schrödinger
anticipated this in his footnote on page 245 of Schrödinger (1931) [here, in
our translation and keeping the somewhat convoluted grammar of the original]:
“An interesting application of this is the following: if one knows of a system
composed of several, coupled subsystems only the total energy, then it is
impossible to know more about the distribution of energy across the subsystems
than the statistical, time-independent data, which already follows from the
knowledge of the total energy. Except for the case that individual subsystems
are in truth fully decoupled, energetically isolated from the others.” This
Page–Wootters (PW) formalism was soon challenged on various grounds by Unruh,
Wald Unruh and Wald (1989) and Kuchař Kuchař (2011), leading to this approach
lying dormant for some time.

Recent developments then made the latter approach re-emerge Giovannetti et al.
(2015); Marletto and Vedral (2017); Diaz and Rossignoli (2019); Diaz et al.
(2019, 2021); Gemsheim and Rost (2023), and both approaches converge Smith and
Ahmadi (2019); Höhn et al. (2021a). This also allowed one to address and re-
contextualize the earlier criticisms of the PW formalism Höhn et al. (2021a,
b). In essence, the clocks chosen within the PW formalism become part of a
gauge-theoretic picture of clocks; the ‘physical Hilbert space’ being a
‘clock-neutral’ picture, and different clocks representing different ‘gauge
conditions’ Höhn et al. (2021a). These results form part of a larger research
programme that aims to ground the rulers and clocks at least implicitly
employed in external symmetries with an operational underpinning Krumm et al.
(2021); Goeller et al. (2022); Adlam (2022); Höhn et al. (2023); Hausmann et
al. (2023); Höhn et al. (2023). For our purposes below, the key takeaway is
that the PW formalism still works, if its defects and the criticisms of it are
understood as related to issues already familiar from electromagnetism: One
has to choose whether a calculation should be gauge-fixed or gauge-
independent. Depending on the question at hand, one or the other will be
better suited to calculations or understanding. Our model below will be too
simple to fully showcase these developments, but given the criticisms the PW
formalism faced, it is important to keep the resolution in mind.

Within this special issue’s scope, many other pertinent links of time and time
travel to quantum physics could be made. A selected, non-exhaustive list would
include: Deutsch CTCs and (non/retro-)causal quantum processes Deutsch (1991);
Bishop et al. (2021); Baumann et al. (2022); Vilasini (2021); Adlam (2022),
simulation of time (travel) in analogues Jannes (2015); Barceló et al. (2022);
Sabín (2022), semi-classical/quantum stability Hawking (1992); Klinkhammer
(1992); Visser (1993, 1997); Emparan and Tomašević (2022), _et cetera_. For
the purposes of this article, however, we would like to highlight only two
further research avenues. One is that of different _notions_ of time. Yet
another, non-exhaustive list would include at least cosmological time,
psychological time, parameter time, thermodynamic time, dynamical _versus_
kinematical time, …While this distinction is studied mostly in the context of
‘the arrow of time’ Zeh (2007), it is worth pointing out that many
counterarguments to time travel have to rely at least on some conflation of
these concepts. To give just one example, if thermodynamic arguments are
brought forth, one needs to establish a more or less direct relation between
the thermodynamic notion implied by the second law222One would also have to
brush aside that standard thermodynamics has to artificially graft time onto
its formalism in the first place, leading to the very name ‘thermo _dynamics_
’ being rather unintuitive compared to other occurrences of ‘dynamics’ in
physical terminology Truesdell (1971). and the notion of time invoked in time
travel. (This itself does not have to be the same notion as that of general
relativity (GR), where time travel can be identified with closed, time-like
curves (CTCs) or related concepts Minguzzi and Sánchez (2006). In theories
other than GR, a different notion of time might be relevant to describe time
travel!)

The other important point to make is that despite all the counterarguments,
time travel is just one notion of many with questionable ‘physicality’
Schuster (2023). Even within the context of GR, other notions of physicality
vie for validity with the absence of time travel, for example, various kinds
of completeness (geodesic, hole free, …) or the validity of various kinds of
energy conditions. These notions, however, are not all compatible with each
other Barceló and Visser (2002); Earman et al. (2009); Manchak (2009, 2018,
2021); Santiago et al. (2022). One might find oneself in the uncomfortable
situation that a dearly held and important property (stability, completeness,
fulfilled energy conditions, …) will require time travel to be permitted; and
this just within GR. Other theories are unlikely to be free from such
problems, especially as many of the comforting no-go theorems are specifically
proven within the context of GR. It it therefore not surprising that in the
context of classical field theories of gravity alone, there exist many
different ways to realize time travel. This includes wormholes Visser (1996);
González-Díaz and Alonso-Serrano (2011), warp drives Everett (1996),
cosmologies Gödel (1949); Hawking and Ellis (1974); Griffiths and Podolský
(2012); Gray et al. (2017), maximally extended space-times (such as Kerr near
the central ring singularity) Wald (1984), and various more mathematical
construction techniques Penrose (1972); Krasnikov (1998); Manchak (2011). The
latter can also be employed to distinguish between, for example, time travel
(CTCs) and time machines (whatever structure creates or causes CTCs) Earman et
al. (2009); Manchak (2009, 2011).

This finally brings us to the goal of this paper: We want to study what can be
said about the viability of time travel if time itself is only an emergent
concept, as in the PW formalism. Concretely, we will be studying an example of
two non-interacting harmonic oscillators similarly constrained to a fixed
total energy of zero as in a Wheeler–DeWitt (WDW) equation, _i.e._ , it will
mimic a minisuperspace model of time travel. This toy model is by no means
meant to exhibit an exhaustive description of whether and how time travel can
arise in the PW formalism. Nor should this model be taken literally as a
minisuperspace model, as no gravitational model is canonically quantized to
arrive at it; rather it demonstrates possible phenomenology. We will see that
the results in our toy model point to an extraordinarily bland version of
Novikov’s self-consistency conjecture Novikov (1992); Sklar (1990); Friedman
et al. (1990).

###  1.1 Outline

Following the preceding introduction to and overview of the related research
question in section 1, we will provide an introduction into the methods
employed: In section 2, we briefly summarize the PW formalism. In section 3 we
demonstrate POVM’s utility for implementing time observables in the context of
a single harmonic oscillator. The main part of the paper is employing these
concepts then, in section 4, to two harmonic oscillators subjected to a WDW-
like Hamiltonian constraint equation. We close by discussing our results and
pointing out future directions and open questions in section 5. Finally, the
appendices collect additional details and context: Appendix A reviews the
mathematical definition of a POVM, appendix B collects results concerning
periodic time in the quantum harmonic oscillator that supplement the
discussion of section 3.1, while appendix C gives additional details
concerning the construction of the paper’s figures.

We will use natural units in which $G=\hbar=c=1$.

##  2 The Page–Wootters Formalism

Let us consider a quantum system (‘the universe’) that separates into two
subsystems, a ‘clock’ system and a ‘residual’ system. The PW formalism is our
first, and most important ingredient for studying time travel without time (or
rather: time travel with only an emergent notion of time). It is a proposal to
give an operational meaning to measuring the time evolution of the residual
subsystem in the larger quantum system (the ‘universe’) with respect to the
other, (mostly) decoupled ‘clock’ subsystem. Partly, this separation into
‘clock’ and ‘residual’ (in the literature often ‘rest’) systems is motivated
by the problem of time encountered in quantum gravity. In terms of the total
Hilbert space $\mathcal{H}$, the clock Hilbert space
$\mathcal{H}_{\textsc{c}}$, and $\mathcal{H}_{\textsc{r}}$, the residual
Hilbert space, this means:

$\mathcal{H}=\mathcal{H}_{\textsc{c}}\otimes\mathcal{H}_{\textsc{r}}.$ (1)

Subsequent developments and refinements sometimes further separated what we
call $\mathcal{H}_{\textsc{r}}$ into evolving and
ancillary/memory/supplementary/…systems, but for our purposes, we will not
follow this line of thinking—at least in this current paper. In future work,
it could be interesting to see whether such an approach could provide a notion
of time travel that—contrary to GR’s CTCs—only allows for a finite number of
temporal round trips. Each of these Hilbert spaces is equipped with a
Hamiltonian acting on these separate Hilbert spaces, such that

$\hat{H}\ket{\Psi}=\left(\hat{H}_{\textsc{c}}\otimes\mathds{1}_{\textsc{r}}+\mathds{1}_{\textsc{c}}\otimes\hat{H}_{\textsc{r}}\right)\ket{\Psi}=E\ket{\Psi},$
(2)

where $\ket{\Psi}$ is a state in the total Hilbert space $\mathcal{H}$. In
line with Schrödinger’s 1931 footnote, this state $\ket{\Psi}$ is required to
be in a definite, fixed eigenstate of $\hat{H}$ with total energy $E$. More
generally, the PW formalism remains viable even for density matrices
$\hat{\rho}$, with the above case regained by setting
$\hat{\rho}=\ket{\Psi}\bra{\Psi}$.

Examples of such systems usually are constructed by symmetry-reducing the WDW
equation of quantum gravity, the quantized Hamiltonian constraint of GR:

$\hat{H}\ket{\Psi}=0.$ (3)

In this expression, we gloss over many technicalities (such as operator
ordering, domain issues, topological concerns, configuration space
considerations, …) DeWitt (1967); Thiemann (2008); Kiefer (2012), as our focus
lies less on quantum gravity and more on the emergent notion of time that the
PW formalism provides. Still, we will occasionally take inspiration from
quantum gravity. In particular, one can consider our final model to be a crude
example of a minisuperspace model that has _not_ been found through symmetry
reduction. Already here, we therefore want to point out that many questions
the minisuperspace models of quantum black holes or quantum cosmology try to
answer, will not and cannot appear in our context. Most notably will be an
absence of gravitational singularities, whose avoidance has been a mainstay of
quantum gravity.

The key step for formulating the PW formalism now lies in describing the
evolution of the system encoded through a tuple
$(\mathcal{H}_{\textsc{r}},\hat{H}_{\textsc{r}})$ in terms of the evolution of
$(\mathcal{H}_{\textsc{c}},\hat{H}_{\textsc{c}})$. For this, one picks some
initial state $\ket{\psi(0_{\ast})}_{\textsc{c}}\in\mathcal{H}_{\textsc{c}}$,
which is then evolved using the clock Hamiltonian according to

$\ket{\psi(\theta_{\ast})}_{\textsc{c}}\mathrel{\mathop{:}}=e^{-i\hat{H}_{\textsc{c}}\theta_{\ast}}\ket{\psi(0_{\ast})}_{\textsc{c}}.$
(4)

This corresponds to the familiar idea of unitary time evolution of states in
the Schrödinger picture through a Hamiltonian, but—at this level of the
discussion, anyway—is a definition. The Schrödinger or Heisenberg equations of
motion would then be a _result_ of this approach Page and Wootters (1983);
Smith and Ahmadi (2019).

The evolution of the subsystem $\hat{H}_{\textsc{r}}$ is then evaluated in
terms of relative probabilities with respect to these clock states. To do
this, let us first define the projector

$\hat{P}_{\theta_{\ast}}\mathrel{\mathop{:}}=\left(\ket{\psi(\theta_{\ast})}_{\textsc{c}}\
{\vphantom{\bra{\psi({{\theta}_{\ast}})}}}_{\textsc{c}}\\!\bra{\psi({{\theta}_{\ast}})}\right)\otimes\mathds{1}_{\textsc{r}}$
(5)

of a given state onto particular ‘clock time states’
$\ket{\psi(\theta_{\ast})}_{\textsc{c}}$ in $\mathcal{H}_{\textsc{c}}$,
leaving states in $\mathcal{H}_{\textsc{r}}$ unchanged. Then the evolution in
$\mathcal{H}_{\textsc{r}}$ by $\hat{H}_{\textsc{r}}$ with respect to clock
states is given by the expression

$E(\hat{H}_{\textsc{r}}|\theta_{\ast})=\frac{\operatorname{tr}\left(\hat{H}_{\textsc{r}}\hat{P}_{\theta_{\ast}}\hat{\rho}\right)}{\operatorname{tr}\left(\hat{P}_{\theta_{\ast}}\hat{\rho}\right)}.$
(6)

This gives the conditional expectation value of the residual Hamiltonian
$\hat{H}_{\textsc{r}}$ when the clock (state) ‘reads’ $\theta_{\ast}$. The
technical requirements behind this interpretation are stationarity,

$[\hat{H},\hat{\rho}]=0,$ (7)

and commutativity of $\hat{H}_{\textsc{r}}$ and $\hat{H}_{\textsc{c}}$, which
we enforced by the block structure of $\hat{H}$ encoded in equation (2). These
assumptions are quite strong, especially once one wants to include an
interaction Hamiltonian coupling $\mathcal{H}_{\textsc{c}}$ and
$\mathcal{H}_{\textsc{r}}$ on the right hand side of equation (2). Given the
recent clarification on this issue provided by Giovannetti et al. (2015);
Marletto and Vedral (2017); Smith and Ahmadi (2019); Höhn et al. (2021a), we
want to return to such generalizations at a later point, but stick to these
more restrictive assumptions in the present context.

Some general comments are appropriate at this point: First, fixing the
initial, _total_ state $\ket{\Psi}$ can, in general, greatly change the time
evolution as described by equation (6). Second, to reiterate: This is not the
most sophisticated or modern implementation of the PW formalism, see the
previous paragraph. Third, without further specifying a particular system
comprised of the various Hamiltonians present in equation (2), we cannot even
begin our investigation. This holds true for the sort of time evolution people
expect—one without time travel—just as well as for the more unexpected time
evolution—showcasing a notion of time travel. The next step will be to lay the
groundwork for our toy model, specifying the state and the evolution.

###  2.1 Time Travel in the Page–Wootters Formalism

In order to prepare this next step, let us first connect the PW formalism to
some of the notions mentioned in the introduction. In particular, we would
like to be able to talk about time travel in this setting. The simplest way to
achieve this is to demand that the domain from which our time parameter
$\theta$ is taken to be $S^{1}$, _i.e._ , periodic. By itself, this will not
guarantee a useful notion of time travel. A simple way to see this is by
looking at everyday clocks: They are all periodic. Only a _calendar_ , also
keeping track of years, would be a truly non-periodic clock. One does not
travel back in time when one wakes up at the same time each morning, just
because the alarm clock is periodic.

Instead, we will try to find a way to implement the simplest possible version
of time travel. This is (Novikov) self-consistency. When the clock returns to
a state, so will the remainder system return to the same state it had when the
clock last was in this state. This situation can (at least in some systems and
models) also yield existence results, though usually at the expense of
uniqueness Friedman et al. (1990); Echeverria et al. (1991); Fewster et al.
(1996); Bachelot (2002); Dolanský and Krtouš (2010). So far, however, the PW
formalism as presented here does not allow for this. The evolution of the
clock as prescribed by equation (4) is ignorant of the domain from which to
draw $\theta_{\ast}$.

In order to alleviate this problem, we will borrow concepts from the POVM
approach to time operators already mentioned in the introduction, and to be
discussed in more detail in the next section. In order to incorporate this in
the PW formalism, we note that the key ingredient for comparing the evolution
of the clock with that of the remainder system is encoded in the definition of
the projector onto clock states, equation (5). The goal, thus, will be to not
only influence the clock evolution through a convenient choice of the global
state $\ket{\Psi}$. Rather, we also will have to make a judicious choice of
the clock states themselves, and how to identify them in a meaningful way.

##  3 POVMs, the Harmonic Oscillator, and Time

The second ingredient of our investigation is a particularly simple example of
time operators and POVMs: The harmonic oscillator Susskind and Glogower
(1964); Garrison and Wong (1970); Galindo (1984); Busch et al. (1994, 1995).
As an ubiquitous model system in all fields of classical and quantum physics,
it is likely to be an instructive ingredient even in our systems following a
Hamiltonian constraint similar to the WDW equation (3). Our discussion will
follow the presentation of the two just-mentioned articles, adapted to the
notation chosen for our application. Concretely, it is possible in the POVM
framework to introduce time with a periodic domain through a phase POVM.
Similar to time operators, also an operator representing a phase variable has
been difficult to construct in a mathematically rigorous manner. Here, the
problems are tied to issues of boundary conditions, hermiticity, and the
domain of operators not given in standard notation. As with time operators,
the more general setting of POVMs provides a way around this. Since a time
operator ideally would be canonically conjugate to the Hamiltonian (as
suggested by the energy-time uncertainty relation), and a phase operator
ideally would be canonically conjugate to the number operator, the connection
to periodic time is easily made.

In our toy model of harmonic oscillators (further specified below in section
4), we will identify phase and time. This is also provides the easiest way to
rigorously define time travel, making use of Novikov’s self-consistency
conjecture: A system comprised of clock and dynamical system with respect to
clock time undergoes time travel, if the dynamical system state is the same
after each full clock period. This definition could, in principle, be relaxed.
For example, this could happen only a finite number of times. An emergent
notion of time furthermore allows one to make such a statement merely
approximate. This will allow us to study rigorously notions of time travel
that not fully fall into this framework of self-consistency; for example, only
parts of the system could undergo time travel. We hope that such models
including the clock itself can shed light on when a system is ‘merely periodic
in time’ _versus_ when it actually undergoes time travel. In our final section
5, we will further expound on this. Importantly, a requirement of self-
consistency will preempt any of the traditional paradoxa constructed out of
it.

###  3.1 Time and Phase Operators for the Harmonic Oscillator

Leaving the mathematical formalities (mostly) aside, we will now introduce the
specific POVM used in our simple model of time travel. Formal, supplementary
details regarding the definition of POVMs can be found in appendix A. For the
purposes of this section, we will drop all label indices, and only look at the
simple, standard harmonic oscillator with Hamiltonian

$\hat{H}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}+\frac{1}{2}.$ (8)

We want to construct a time operator for this. In order to do so, we look at
the _polar decomposition_ of the ladder operator $\hat{a}$, (Reed and Simon,
1980, Thm. VIII.32):

$\hat{a}=\hat{W}\widehat{\left\lvert a\right\rvert}.$ (9)

Here, $\widehat{\left\lvert a\right\rvert}$ is easy to define as

$\widehat{\left\lvert
a\right\rvert}\mathrel{\mathop{:}}=\hat{n}^{1/2}=\sqrt{\hat{a}^{\dagger}\hat{a}}.$
(10)

The ‘operator’ $\hat{W}$, however, is less simple than its occurrence in a
‘polar decomposition’ might suggest: It will not be unitary. More concretely
Busch et al. (1995):

$\hat{W}\hat{W}^{\dagger}=\mathds{1},$ (11)

but

$\hat{W}^{\dagger}\hat{W}=\mathds{1}-\ket{0}\negthickspace\bra{0}\neq\mathds{1}.$
(12)

The phase or time states to be used are now constructed as eigenstates
$\ket{\theta}$ of $\hat{W}$.

$\hat{W}\ket{\theta}=e^{i\theta}\ket{\theta}.$ (13)

While these eigenstates do exist, given the properties of $\hat{W}$, they are
now only an _over_ complete set of eigenstates.333Such overcomplete states
find ample application in, for example, the context of coherent states Auletta
et al. (2009). So different eigenstates $\ket{\theta}$ and
$\ket{\theta^{\prime}}$ will not be orthogonal, and expressing a general state
$\ket{\psi}$ in terms of these eigenstates will not be unique anymore. Each of
these states $\ket{\theta}$ can be written as

$\ket{\theta}=\sum_{n\geq 0}e^{in\theta}\ket{n}.$ (14)

Now is the time to define a concrete POVM. In this instance, let us define for
all Borel sets $X$ of $[0,2\pi)$

$\displaystyle B_{0}(X)$
$\displaystyle\mathrel{\mathop{:}}=\frac{1}{2\pi}\int_{X}\ket{\theta}\negthickspace\bra{\theta}\operatorname{d}\\!\theta,$
(15) $\displaystyle=\sum_{n,m\geq
0}\frac{1}{2\pi}\int_{X}e^{i(n-m)\theta}\ket{n}\negthickspace\bra{m}\operatorname{d}\\!\theta.$
(16)

The index $0$ is a first indication of what is to come: Different points on
the unit circle $S^{1}$ can be chosen as starting points. In the above
definition, one starts on the positive real axis, _i.e._ , we choose our angle
$\theta$ from $[0,2\pi)$, as opposed to say from
$[\theta_{\ast},\theta_{\ast}+2\pi)$ for some $\theta_{\ast}\in[0,2\pi)$.

Using

$\frac{1}{2\pi}\int_{o}^{2\pi}e^{i(n-m)\theta}\operatorname{d}\\!\theta=\delta_{nm}$
(17)

and

$\sum_{n=0}^{\infty}\ket{n}\negthickspace\bra{n}=\mathds{1}$ (18)

one can show that

$B_{0}([0,2\pi))=\mathds{1},$ (19)

which means that $B_{0}$ is a normalized POVM. So $B_{0}$ represents an
observable in the sense of POVMs. Note that in the context of the definition
of a POVM, in the present example we have that $\Omega=S^{1}$.

From the POVM side of things, this gives all ingredients needed for the
calculations to follow. However, for a more comprehensive understanding of the
physics of time involved in the harmonic oscillator, it is worthwhile to point
out additional considerations concerning such periodic time variables. We have
collected these observations in appendix B.

##  4 Time Travel in Two Harmonic Oscillators Subject to a Hamiltonian
Constraint

The goals for the present article are modest. We want to study as simple a
model as possible that is subject to a constraint reminiscent of the WDW
equation (3). As we also want to capture time travel as defined above, we need
for this two ingredients: A periodic clock system
$(\mathcal{H}_{\textsc{c}},\hat{H}_{\textsc{c}})$ and a system
$(\mathcal{H}_{\textsc{r}},\hat{H}_{\textsc{r}})$ that evolves in the sense of
the PW formalism (at least) as periodic as the clock itself. Then it would be
an instance of a quantum system fulfilling the Novikov self-consistency
conjecture. Based on the discussion of harmonic oscillators above, the
simplest such system would be to use a single harmonic oscillator for each
$\hat{H}_{\textsc{r}}$ and $\hat{H}_{\textsc{c}}$, respectively. To allow for
some more generality, we will not demand their respective frequencies
$\omega_{\textsc{r}}$ and $\omega_{\textsc{c}}$ to be the same, _a priori._
Then, our model will have the following Hamiltonian in the full Hilbert space
$\mathcal{H}$:

$\hat{H}\ket{\Psi}=\left(\vphantom{\omega_{\textsc{c}}\,\hat{n}+\omega_{\textsc{c}}\frac{1}{2}\mathds{1}_{\textsc{c}}}\smash{\underbrace{\omega_{\textsc{c}}\,\hat{n}_{\textsc{c}}+\frac{\omega_{\textsc{c}}}{2}\,\mathds{1}_{\textsc{c}}}_{\hat{H}_{\textsc{c}}}\underbrace{-\omega_{\textsc{r}}\,\hat{n}_{\textsc{r}}-\frac{\omega_{\textsc{r}}}{2}\,\mathds{1}_{\textsc{r}}}_{-\hat{H}_{\textsc{r}}}}\right)\ket{\Psi}=0.\vphantom{\underbrace{\omega_{\textsc{c}}\hat{n}+\omega_{\textsc{c}}\frac{1}{2}\mathds{1}_{\textsc{c}}}_{\hat{H}_{\textsc{c}}}}$
(20)

This means that the system Hamiltonian $\hat{H}_{\textsc{r}}$ and the clock
Hamiltonian $\hat{H}_{\textsc{c}}$ are part of a WDW-like equation. Naturally,
this implies that it falls into the framework of the PW formalism for a
stationary Hamilton equation $\hat{H}\ket{\Psi}=E\ket{\Psi}$ with energy
$E=0$.

The previous two sections gave us the two most important ingredients in place:
First, the way to measure time evolution with respect to a clock system using
the PW formalism’s conditional probability (6). Second, the required clock
states given by equation (13) with respect to which we will measure evolution,
and its representation in terms of the harmonic oscillator eigenstates,
equation (14). Now, we can make use of it by implementing the Hamiltonian
constraint (20) into this model. What remains to be done, though, is to
specify more explicitly the type of total state $\ket{\Psi}\in\mathcal{H}$ the
‘universe’ (as described by $\hat{H}$) is in.

###  4.1 Important Note on the Difference to Quantum Cosmology

Having introduced the model, it is easy to recognize it as a variant of a
model already used in the context of quantum cosmology Kiefer (1989, 1990);
Page (1991); Brunetti et al. (2010). There is, however, an important
distinction both mathematically and interpretationally. One of the goals of
quantum cosmology is to use symmetry reduction to arrive at minisuperspace
models testing aspects of quantum gravity. In the context of (quantum)
cosmology, this is in particular often the status of Hawking-type singularity
theorems in quantum gravity: Can a cosmological singularity be avoided or not
Hawking and Ellis (1965); DeWitt (1967); O’Neill (1983); Kiefer (2012)? In
many, if not most situations this leads to the demand that the configuration
space $\mathds{R}^{2}$ of the two harmonic oscillators is limited to the half-
plane $\mathds{H}^{2}$, as one of the variables corresponds to the non-
negative scale factor $a$. The scale factor at $a=0$ corresponds to a
gravitational singularity. Restricting one harmonic oscillator to only a half-
plane for its configuration space variable, however, inevitably leads to the
inability to use the very convenient, powerful, and simple algebraic methods
in terms of ladder operators used so far in the present paper.

Our present goals, however, are different. We want to study time travel in a
quantum-theoretic framework, but in a manner that is as theory-agnostic as
possible. Hence, we will simply not assume that equation (20) necessarily has
its origins in a Hamiltonian formulation of general relativity or similar
gravitational theories. This means that equation (20) with the domain used is
an explicit input. It very well could be derived from a specific gravitational
model in a specific gravitational theory through symmetry-reduction. Should
this not be possible, it still may serve as a toy model. Especially, as the
process of symmetry-reduction (to arrive at minisuperspace models) itself may
be a fraught undertaking. Our assumption will be that such an equation with
the _full_ configuration space $\mathds{R}^{2}$ _is_ available, and, likewise,
so are our corresponding trusty ladder operators. This also bypasses issues
such as the precise nature of the inner product on the quantum gravitational
wavefunctions, as already discussed in (DeWitt, 1967, §9). Two extremal
options for inner products (with many in between) are a quantum harmonic
oscillator on the one side and a wave equation on the other. We assume the
inner product structure as the one familiar from and resulting in standard
quantum mechanics. Naturally, this leads to many directions into which to
extend our present model, and we will come back to these in our concluding
section 5.

In conclusion and in conscious distinction to quantum cosmology, our toy model
(20) for time travel will be treated using a ladder operator approach of
quantum mechanics. This allows us to use the methods introduced in the
previous two section, and hopefully also provides easy extensions in diverse
directions.

###  4.2 A Boring Model of Time Travel

Looking for a square-integrable solution to equation (20) introduces a new,
additional condition on this equation. Only if

$\frac{\omega_{\textsc{r}}}{\omega_{\textsc{c}}}=\frac{2n_{\textsc{c}}+1}{2n_{\textsc{r}}+1},\qquad\text{where\leavevmode\nobreak\
}n_{\textsc{c}},n_{\textsc{r}}\in\mathds{N}_{0},$ (21)

will this be possible. This _commensurability condition_ will have far-
reaching consequences for our simple toy model in what is to come.444Deviating
from the commensurability condition even a little would lead to non-
normalizable wave functions. In the context of quantum cosmology such wave
functions can be encountered, such as the no-boundary proposal (Kiefer, 2012,
p.282). For the sake of our (not necessarily gravitational) toy model, we
believe that this would add another layer of interpretational problems best
left for a separate study. In fact, these results will (for this toy model, at
least) be so far reaching, that the precise nature of the time operators
employed or their respective states used as initial state of the clock becomes
immaterial. The interested reader can find more details on a possible choice
of time operators useful for the current purposes in appendix B.

Quite generally, the wave function of the ‘universe’, $\ket{\Psi}$, will have
the following form:

$\ket{\Psi}=\sum_{n,n^{\prime}}A_{n,n^{\prime}}\ket{n}_{\textsc{c}}\otimes\ket{n^{\prime}}_{\textsc{r}}.$
(22)

Our calculation will proceed in three steps based on this general form.

  1. 1.

The simplest possibility is a diagonal $A_{n,n^{\prime}}$. We will demonstrate
that this leads, at least according to the PW formalism, to the simplest time
evolution possible. We will demonstrate how this time evolution can be
interpreted as a particularly boring case of the Novikov self-consistency
conjecture.

  2. 2.

We derive a quite general statement about $A_{n,n^{\prime}}$ when it is non-
diagonal. Concretely, introducing two sparse matrices $\Delta_{1},\Delta_{2}$
and a diagonal matrix $D$, it will take the form

$A=\Delta_{1}^{T}D\Delta_{2}.$ (23)

  3. 3.

A general identity for the conditional probabilities (6) in our toy model is
derived.

  4. 4.

Using the previous two steps, we will prove that the results of the first step
carry over to non-diagonal $A_{n,n^{\prime}}$, too.

In the first step, let us calculate the components of equation (6). This will
tell us straightforwardly what we want to know.

$\displaystyle\operatorname{tr}\left(\hat{H}_{\textsc{r}}\hat{P}_{\theta_{\ast}}\hat{\rho}\right)$
$\displaystyle=\operatorname{tr}\left(\left[\mathds{1}_{\textsc{c}}\otimes\hat{H}_{\textsc{r}}\right]\;\left[\left(\ket{\psi(\theta_{\ast})}_{\textsc{c}}{\vphantom{\bra{\psi({{\theta}_{\ast}})}}}_{\textsc{c}}\\!\bra{\psi({{\theta}_{\ast}})}\right)\otimes\mathds{1}_{\textsc{r}}\right]\;\left[\ket{\Psi}\negthickspace\bra{\Psi}\right]\right),$
(24)
$\displaystyle=\Bra{{\vphantom{{\left[\mathds{1}_{\textsc{c}}\otimes\hat{H}_{\textsc{r}}\right]}_{\textsc{c}}}}\Psi}\left[\left({\ket{\psi(\theta_{\ast})}_{\textsc{c}}}\
{{\vphantom{\bra{\psi({{\theta}_{\ast}})}}}_{\textsc{c}}\\!\bra{\psi({{\theta}_{\ast}})}}\right)\otimes\mathds{1}_{\textsc{r}}\right]\;\left[\mathds{1}_{\textsc{c}}\otimes\hat{H}_{\textsc{r}}\right]\Ket{{\vphantom{{\left[\mathds{1}_{\textsc{c}}\otimes\hat{H}_{\textsc{r}}\right]}_{\textsc{c}}}}\Psi}.$
(25)

With the currently assumed diagonal form for $A_{n,n^{\prime}}$, we can
evaluate

$(\mathds{1}_{\textsc{c}}\otimes\hat{H}_{\textsc{r}})\ket{\Psi}=\sum_{n}A_{n}\left(n+\frac{1}{2}\right)\ket{n}_{\textsc{c}}\otimes\ket{n}_{\textsc{r}},$
(26)

which in turn—using
${\vphantom{\braket{m}{n}}}_{\textsc{r}}\\!\braket{m}{n}_{\textsc{r}}=\delta_{mn}$—turns
equation (25) into

$\displaystyle\operatorname{tr}\left(\hat{H}_{\textsc{r}}\hat{P}_{\theta_{\ast}}\hat{\rho}\right)$
$\displaystyle=\sum_{n}|A_{n}|^{2}\left(n+\frac{1}{2}\right)\left\lvert{\vphantom{\braket{\psi({\theta}_{\ast})}{n}}}_{\textsc{c}}\\!\braket{n}{\psi(\theta_{\ast})}_{\textsc{c}}\right\rvert^{2}.$
(27)

Going back to the definition of $\ket{\psi(\theta_{\ast})}_{\textsc{c}}$,
equation (4), we note that

${\vphantom{\braket{\psi({\theta}_{\ast})}{n}}}_{\textsc{c}}\\!\braket{n}{\psi(\theta_{\ast})}_{\textsc{c}}=e^{-i(n+\frac{1}{2})\theta_{\ast}}\
{\vphantom{\braket{\psi({0}_{\ast})}{n}}}_{\textsc{c}}\\!\braket{n}{\psi(0_{\ast})}_{\textsc{c}}.$
(28)

This finally leads to

$\left\lvert{\vphantom{\braket{\psi({\theta}_{\ast})}{n}}}_{\textsc{c}}\\!\braket{n}{\psi(\theta_{\ast})}_{\textsc{c}}\right\rvert^{2}=\left\lvert{\vphantom{\braket{\psi({0}_{\ast})}{n}}}_{\textsc{c}}\\!\braket{n}{\psi(0_{\ast})}_{\textsc{c}}\right\rvert^{2},$
(29)

from which we can deduce that

$\operatorname{tr}\left(\hat{H}_{\textsc{r}}\hat{P}_{\theta_{\ast}}\hat{\rho}\right)=\operatorname{tr}\left(\hat{H}_{\textsc{r}}\hat{P}_{0_{\ast}}\hat{\rho}\right),$
(30)

and, after a similar lead-up, likewise for the denominator of equation (6)
that

$\operatorname{tr}\left(\hat{P}_{\theta_{\ast}}\hat{\rho}\right)=\operatorname{tr}\left(\hat{P}_{0_{\ast}}\hat{\rho}\right).$
(31)

It follows immediately that

$E(\hat{H}_{\textsc{r}}|\theta_{\ast})=E(\hat{H}_{\textsc{r}}|0_{\ast}).$ (32)

Put differently, the result (32) means that with respect to such diagonal
states of the ‘universe’, no time will pass, irrespective of the chosen clock
states. As was frequently pointed out in the literature, much of the evolution
is fixed by the choice of the full state $\ket{\Psi}\in\mathcal{H}$, so at
this point this might just be an artefact of an overly specific type of
states. Still, even for these states it is interesting that changing clock
states cannot change the result. It is important to note that we make no
claims concerning more general models; this analysis heavily relies on the
structure of our two-harmonic-oscillator model. We will postpone the
discussion of the implications to time travel for until after the more general
results of ‘step three’ for non-diagonal $A_{n,n^{\prime}}$.

Two examples of states $\ket{\Psi}$ with diagonal $A_{n,n^{\prime}}$ and
isotropic oscillators (_i.e._ ,
$\omega_{\textsc{c}}=\omega_{\textsc{r}}=\mathrel{\mathop{:}}\omega$) are
visualized in figure 1, with different choices for a dispersion parameter
$\beta$. The coordinates $a,\chi$ can in this case be assigned to either
system (clock or residual), and were chosen to allow easy comparison with the
closely related examples of quantum cosmology found in Kiefer (1990). More
details about the construction of the wave packets is relegated to appendix C.

(a) $\beta=$1$$

(b) $\beta=$0.03$$

Figure 1: Illustration of wave packets solving our toy model for time travel
with $\omega_{\textsc{c}}=\omega_{\textsc{r}}$. Details regarding the
construction of these solutions to the Hamiltonian constraint equation (20)
can be found in appendix C. In comparison, quantum cosmological applications
of this equation constrain themselves to only a half-plane $a\geq 0$. More
general wave functions with $\omega_{\textsc{c}}\neq\omega_{\textsc{r}}$ will
have less symmetry than the pictured examples. Left: Dispersion-less packets,
corresponding to a parameter choice $\beta=1$. Right: Wave packets with strong
dispersion, $\beta=0.03$.

As a preliminary, second step for the discussion of non-diagonal
$A_{n,n^{\prime}}$, it is instructive to reexamine the commensurability
condition. Plugging the ansatz (22) into the Hamiltonian constraint (20) we
get for all values of $n$ and $n^{\prime}$ that

$\left(\omega_{\textsc{c}}n+\frac{\omega_{\textsc{c}}}{2}-\omega_{\textsc{r}}n^{\prime}-\frac{\omega_{\textsc{r}}}{2}\right)A_{n,n^{\prime}}=0.$
(33)

This implies that either $A_{n,n^{\prime}}=0$ or if $A_{n,n^{\prime}}\neq 0$
that

$\left(\omega_{\textsc{c}}n+\frac{\omega_{\textsc{c}}}{2}-\omega_{\textsc{r}}n^{\prime}-\frac{\omega_{\textsc{r}}}{2}\right)=0,$
(34)

which is equivalent to the original commensurability condition (21). Given
that this condition yields a ratio of two odd numbers, we can rephrase $n$ and
$n^{\prime}$ as

$n=s(2p+1)+p;\qquad n^{\prime}=s(2q+1)+q;\qquad s,p,q\in\mathds{N}_{0}.$ (35)

This equation gives additional insight about the coefficients
$A_{n,n^{\prime}}$. In components, we can write

$A_{n,n^{\prime}}=\sum_{s=0}^{\infty}\delta_{n,s(2p+1)+p}\;d_{s}\;\delta_{n^{\prime},s(2q+1)+q}$
(36)

which in ‘matrix form’ appears as follows:

$A=\Delta_{1}^{T}D\Delta_{2},$ (37)

where $D$ really is diagonal and $\Delta_{1},\Delta_{2}$ are sparse matrices
connecting $s$ with $n$ and $n^{\prime}$, respectively.

The third step takes a look at the numerator of the conditional probability of
the PW formalism, equation (6). Note that equation (25) is fully general, and
needed no diagonality of $A_{n,n^{\prime}}$ in its derivation. Its last two
factors can now be evaluated with our general ansatz (22):

$(\mathds{1}_{\textsc{c}}\otimes\hat{H}_{\textsc{r}})\ket{\Psi}=\sum_{n,n^{\prime}}A_{n,n^{\prime}}\left(n^{\prime}+\frac{1}{2}\right)\ket{n}_{\textsc{c}}\otimes\ket{n^{\prime}}_{\textsc{r}}.$
(38)

As in the first step, we can then use orthonormality of the Fock states
$\ket{n^{\prime}}_{\textsc{c,r}}$ and cyclicity of the trace to show that

$\operatorname{tr}\left(\hat{H}_{\textsc{r}}\hat{P}_{\theta_{\ast}}\hat{\rho}\right)=\sum_{m,n,n^{\prime}}\left({\vphantom{\braket{m}{\psi({\theta}_{\ast})}}}_{\textsc{c}}\\!\braket{m}{\psi(\theta_{\ast})}_{\textsc{c}}A_{m,n^{\prime}}^{*}\right)\left(A_{n,n^{\prime}}\left[n^{\prime}+\frac{1}{2}\right]\
{\vphantom{\braket{\psi({\theta}_{\ast})}{n}}}_{\textsc{c}}\\!\braket{\psi(\theta_{\ast})}{n}_{\textsc{c}}\right).$
(39)

Next, let us employ the definition of
$\ket{\psi(\theta_{\ast})}_{\textsc{c}}$, equation (4):

$\displaystyle{\vphantom{\braket{m}{\psi({\theta}_{\ast})}}}_{\textsc{c}}\\!\braket{m}{\psi(\theta_{\ast})}_{\textsc{c}}$
$\displaystyle={\vphantom{\ket{\psi({\theta}_{\ast})}}}_{\textsc{c}}\\!\bra{m}e^{-i\hat{H}_{\textsc{c}}\theta_{\ast}}\ket{\psi(0_{\ast})}_{\textsc{c}},$
(40) $\displaystyle=e^{-i(m+\frac{1}{2})\theta_{\ast}}\
{\vphantom{\braket{m}{\psi({0}_{\ast})}}}_{\textsc{c}}\\!\braket{m}{\psi(0_{\ast})}_{\textsc{c}}.$
(41)

This allows us to rearrange equation (39) to

$\operatorname{tr}\left(\hat{H}_{\textsc{r}}\hat{P}_{\theta_{\ast}}\hat{\rho}\right)=\sum_{m,n}Q_{m,n}e^{+i(n-m)\theta_{\ast}}$
(42)

where

$Q_{m,n}\mathrel{\mathop{:}}=\left(\left[\sum_{n^{\prime}}A_{m,n^{\prime}}^{*}A_{n,n^{\prime}}(n^{\prime}+1/2)\right]\
{\vphantom{\braket{m}{\psi({0}_{\ast})}}}_{\textsc{c}}\\!\braket{m}{\psi(0_{\ast})}_{\textsc{c}}\
{\vphantom{\braket{\psi({0}_{\ast})}{n}}}_{\textsc{c}}\\!\braket{\psi(0_{\ast})}{n}_{\textsc{c}}\right)$
(43)

This effectively demonstrates how the choice of the universe’s state
$\ket{\Psi}$ and the chosen initial clock state
$\ket{\psi(0_{\ast})}_{\textsc{c}}$ together fully determine the time
evolution of our model. Equation (43) can equivalenty be expressed in ‘matrix
form’ as

$Q=D_{1}^{\dagger}A^{\dagger}D_{0}AD_{1},$ (44)

where $D_{1}$ is a complex diagonal ‘matrix’, and $D_{0}$ a real diagonal
‘matrix’.

A closely related identity can also be found—using similar arguments—for the
denominator of equation (6), reading:

$\operatorname{tr}\left(\hat{P}_{\theta_{\ast}}\hat{\rho}\right)=\sum_{m,n}\left(\left[\sum_{n^{\prime}}A_{m,n^{\prime}}^{*}A_{n,n^{\prime}}\right]\;{\vphantom{\braket{m}{\psi({{0}_{\ast}})}}}_{\textsc{c}}\\!\braket{m}{\psi(0_{\ast})}_{\textsc{c}}\
{\vphantom{\braket{\psi({0}_{\ast})}{n}}}_{\textsc{c}}\\!\braket{\psi(0_{\ast})}{n}_{\textsc{c}}e^{+i(n-m)\theta_{\ast}}\right).$
(45)

Note that this is again of the form (42), though with a simpler ‘$Q$’ for
which $D_{0}=\mathds{1}$. This means any general statement that can be shown
for $Q$ will hold for the corresponding part appearing in equation (45).

In our fourth and final step, we shall bring together our result (37) for
$A_{n,n^{\prime}}$ and the result (44) for $Q_{m,n}$ to greatly constrain the
conditional probabilities of equation (6). Concretely, inserting the former
into the latter gives:

$\displaystyle Q$
$\displaystyle=D_{1}^{\dagger}(\Delta_{1}^{T}D\Delta_{2})^{\dagger}D_{0}(\Delta_{1}^{T}D\Delta_{2})D_{1},$
(46)
$\displaystyle=D_{1}^{\dagger}\Delta_{2}^{\dagger}D^{*}(\Delta_{1}^{*}D_{0}\Delta_{1}^{T})D\Delta_{2}D_{1}.$
(47)

In this expression, let us investigate the central, bracketed term,
$\Delta_{1}^{*}D_{0}\Delta_{1}^{T}$:

$\displaystyle\left(\Delta_{1}^{*}D_{0}\Delta_{1}^{T}\right)_{ss^{\prime}}$
$\displaystyle=\sum_{n^{\prime}}\delta_{n^{\prime},s(2q+1)+q}[D_{0}]_{n^{\prime}}\delta_{n^{\prime},s^{\prime}(2q+1)+q},$
(48) $\displaystyle=\mathrel{\mathop{:}}(D_{3/2})_{ss^{\prime}},$ (49)

another diagonal matrix. This in turn means that even more terms in the middle
of equation (47) reduce to yet another diagonal matrix $D_{2}$, defined as

$D_{2}\mathrel{\mathop{:}}=D^{*}(\Delta_{1}^{*}D_{0}\Delta_{1}^{T})D=D^{*}D_{3/2}D.$
(50)

This, then, allows us to greatly simplify the structure of $Q$ itself:

$Q=D_{1}^{\dagger}(\Delta_{2}^{\dagger}D_{2}\Delta_{2})D_{1},$ (51)

whose middle part in brackets can, using the same approach as in equation
(48), be shown to be diagonal. Finally, it follows that $Q$ itself is
diagonal.

Essential for this result was the specific form of $A$, equation (37).
Modifying the underlying model will therefore likely change this result. For
the PW formalism in our context, though, the result is far-reaching: If we
take equation (42), we see that for diagonal $Q$, the term
$e^{i(n-m)\theta_{\ast}}=1$, as $n=m$. The same observation holds for the
denominator of the PW conditional probabilities (6), equation (45). So, as in
our first step when considering diagonal $A_{n,n^{\prime}}$, even the general
case produces trivial time evolution.

At first glance, this seems a very unremarkable result, and therefore of
little use for any interesting physics question. Ironically, this is _not_ the
case in our present situation, in which we are discussing time travel: An
evolution that is simply constant is certainly still trivially a case of
Novikov’s self-consistency conjecture. Given that nothing is happening, one
might even go so far as to say that it gives an odd entry to the intersection
of the Novikov self-consistency conjecture and the ‘boring physics conjecture’
Visser (1996). It is simultaneously both time travel and the absence of all
evolution.

This result is also a marked departure from results in quantum cosmology
Conradi and Zeh (1991); Brunetti et al. (2010). The reason for this disparity
is the limited domain of the scale parameter in the models of the form (20).
Most importantly, limiting a harmonic oscillator to half of its configuration
space will prevent the application of ladder operators as we used them in the
above analysis. This again demonstrates how ‘simple’ changes to the model can
quite radically change results.

##  5 Interpretation, Discussion and Outlook

Despite the disappointingly boring outcome for our particular model, our toy
model serves its purpose as a proof-of-concept study. It demonstrates the
feasibility for future model building of quantum systems with periodic,
emergent notions of time as a tool to study time travel. The obvious model-
dependence of the PW formalism through its dependence on $\ket{\Psi}$, the
‘state of the universe’, means that different models might have very different
conclusions as to what sort of time evolution is possible. As even collections
of harmonic oscillators become increasingly complex once interactions start to
enter, necessitating many different heuristics or approaches across fields
Auletta et al. (2009); Zee (2010), it is unlikely that the PW formalism can be
fully general even when discussing whether time travel is possible based on
first principles of quantum theory.

In the absence of a fully general result, we still believe that some general
conclusions can nevertheless be achieved. Analytically tractable models beyond
the one presented in our study should be possible, especially given that POVMs
and the PW formalism feature prominently in the modern, gauge theoretic
consolidation of emergent time in quantum theory. The use of POVMs in future
extensions was part of the motivation to highlight them as much in our
discussion as we did, despite the final result not relying on them.

The goals of such a future model building task would contain the following:

  * •

More and different subsystems; more in the sense of more harmonic oscillators,
different in adding various combinations of other systems. Also ancillary
systems could be used. For example, in the context of time travel, one can ask
the following question: Can ancillary or memory systems allow for (more or
less self-consistent) time travel being limited to just a fixed, finite number
of round trips as opposed to the infinite traversability of GR’s CTCs?

  * •

In a closely related step, one would introduce various interactions among the
subsystems. These will greatly change the outcome of our above analysis.

  * •

Ultimately, the combination of these two model building drives can try to
investigate entropic arguments in the context of time travel. If self-
consistent time travel in quantum systems with an emergent notion of time is
possible, are such configurations potentially favoured or disfavoured on
thermodynamic grounds? In particular, entropy concepts geared towards
constrained or fixed energy systems, like ‘observational entropy’ Šafránek et
al. (2019a, b, 2021), promise access to new arguments for or against time
travel without having to rely on either mixing different concepts of time
(GR’s dynamic time _vs._ thermodynamic time _vs._ parameter time _vs._ …) or,
in fact, without relying on any kind of explicit notion of time; the entropic
arguments in such systems would similarly be based on emergent notions, just
as the notion of time itself is.

  * •

Similarly, this exercise should be able to further clarify the difference
between a periodic clock time and time travel as such. Obviously, some kind of
synchronicity between residual system and clock system has to hold for the
latter. Yet the above mentioned research directions would allow one to turn
concepts like self-consistency into less of a binary. After all, if time is
emergent, its absence might be ill-suited for human survival but still allow a
valid physical system that describes something ‘close’ to a quantum system
with self-consistent, observable time travel. In a sense, this would be a
first step towards rigorously testing Hawking’s chronology protection.

  * •

Much of the current research which led to a reimagining of the PW formalism as
a gauge fixed picture aims to ask what happens when one changes frames, _i.e._
, clocks. Once the model systems for time travel are complex enough (and,
thus, more complex than our equation (20)), a natural question is: What
happens to time travel in such quantum reference frame transformations? And,
linking to the previous point, does this differ from the situation for merely
periodic clocks?

  * •

Lastly, the previous two points hint at the possibility of making time travel
more a local, emergent notion in a large system. This is, for one, the
explicit way to study time travel classically as CTCs in a space-time
manifold, see, for example, Earman et al. (2009); Dolanský and Krtouš (2010).
For another, this is similarly done in quantum physics, ranging from the
rigorously and explicitly local Bishop et al. (2021); Alonso-Serrano et al.
(2021) to the more imprecise and only implicitly local Deutsch (1991); Lloyd
et al. (2011). Yet, so far, these approaches all had to rely on an _ad-hoc_
introduction of ‘background’ time.

There are other avenues along similar lines that warrant a closer look in the
framework of quantum gravity. In the canonical approach to it, for instance,
different inner products can be considered, so at least changing the
mathematical framework drastically—begging the question how much of the
physical interpretation changes or has to change in tandem. Other approaches
to quantum gravity besides the canonical ones can also test emergent notions
of time in their respective frameworks. Besides a question of tractability,
this was part of the motivation to keep our toy model somewhat agnostic and
conservative with respect to a theory of quantum gravity.

Independent of which direction one would want to take, we believe that our
first result shows that emergent notions of time in quantum physics can lead
to a multitude of questions and interpretational conundra pertaining to time
travel. The very introduction of such an emergent time concepts undermines
most traditional arguments against time travel. The concepts evoked in these
arguments simply have to rely on notions that are not valid in more _ab
initio_ approaches.

A.A.-S. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) — Project ID 516730869. This work was also partially supported by
Spanish Project No. MICINN PID2020-118159GB-C44. S.S. was financially
supported by Czech Science Foundation grant GACR 23-07457S. M.V. was directly
supported by the Marsden Fund, via a grant administered by the Royal Society
of New Zealand.

No new data were created or analyzed in this study. Data sharing is not
applicable to this article.

###### Acknowledgements.

S.S. acknowledges support from the technical and administrative staff at the
Charles University. He thanks Emily Adlam, Antonis Antoniou, Julian Barbour,
Martin Bojowald, Leonardo Chataignier, Fabio Costa, Ricardo Faleiro, Klaus
Fredenhagen, Philipp A. Höhn, Claus Kiefer, Jorma Louko, Jessica Santiago,
Alexander R. H. Smith, Reinhard Werner, Magdalena Zych, and the participants
of the 781st WE-Heraeus-Seminar ‘Time and Clocks’ in Bad Honnef, the Geometric
Foundations of Gravity 2023 conference in Tartu, the 13th RQI-North meeting in
Chania, and the Odborné soustředění ÚTF in Světlá pod Blaníkem for many
valuable discussions and their interest. The authors declare no conflict of
interest. The funders had no rôle in the design of the study; in the
collection, analyses, or interpretation of data; in the writing of the
manuscript; or in the decision to publish the results. Abbreviations The
following abbreviations are used in this paper:
CTC | Closed, time-like curves
---|---
GR | General relativity
POVM | Positive, operator-valued measure
PVM | Projection-valued measure
PW | Page–Wootters
WDW | Wheeler–DeWitt
yes

##  Appendix A Formal Definition of POVMs

Let us present in this appendix the formal definition of a POVM. An operator
$\hat{A}$ on a Hilbert space $\mathcal{H}$ is called _positive_ , if

$\forall\ket{\psi}\in\mathcal{H}:\qquad\braket{\psi}{\hat{A}}{\psi}\geq 0.$
(52)

Positivity of an operator is often denoted as ‘$\hat{A}\geq 0$.’ An operator
$B$ is called a _POVM_ if and only if it is an _additive_ map from a Borel
$\sigma$-algebra $\mathcal{A}$ of subsets of a set $\Omega$ to operators on
$\mathcal{H}$ such that

  * (i)

for all sets $X\in\mathcal{A}$, $B(X)\geq 0$ (‘positivity’),

  * (ii)

for disjoint $X,Y\in\mathcal{A}$, $B(X\cup Y)=B(X)+B(Y)$ (‘additivity’),

  * (iii)

for all sequences of sets $X_{n}\in\mathcal{A}$ such that
$\cap_{n=1}^{\infty}X_{n}=\emptyset$, all operators $B(X_{n})$ have $\hat{0}$
(the zero operator) as greatest lower bound.

For more details, see (Busch et al., 2016, p.71, Def.4.5). A POVM that fulfils
$B(\Omega)=\mathds{1}$ is called _normalized_.

Normalized POVMs are a more general notion of observable compared to that of
observables being unbounded, self-adjoint operators on $\mathcal{H}$. The
(set-theoretic) ‘universe’ $\Omega$ corresponds to possible measurement
outcomes. In this sense, additivity and being normalized links measurements as
described by POVMs to the theory of probablity as set down in the Kolmogorov
axioms. They allow for more and wider notions of imprecision than those
encoded in density matrices Auletta et al. (2009); Busch et al. (2016). In
‘standard quantum mechanics,’ The measurement process of an observable
$\hat{A}$ according to the Born rule projects onto a subspace formed by
eigenvectors of $\hat{A}$ with set eigenvalue $A_{i}$. In the terminology of
POVMs this corresponds to ‘projection-valued measures’ (PVMs), and PVMs
$\subsetneq$ POVMs.

##  Appendix B Time Operators for a Harmonic Oscillator

In this appendix, we want to draw attention to some additional background
regarding POVMs and phase or time operators in the context of harmonic
oscillators.

With the POVM $B_{0}$ of equation (15) in place, we can now define a large
number of new, self-adjoint operators Busch et al. (1994). To do this, we
associate to each real-valued, bounded, and measurable function $f$ on
$[0,2\pi)$ the operator $\hat{B}_{0}[f]$ defined as

$\displaystyle\hat{B}_{0}[f]$
$\displaystyle\mathrel{\mathop{:}}=\frac{1}{2\pi}\int_{0}^{2\pi}f(\theta)\ket{\theta}\negthickspace\bra{\theta}\operatorname{d}\\!\theta,$
(53) $\displaystyle=\sum_{n,m\geq
0}\frac{1}{2\pi}\int_{0}^{2\pi}e^{i(n-m)\theta}f(\theta)\ket{n}\negthickspace\bra{m}\operatorname{d}\\!\theta.$
(54)

This procedure fixes the set $X$ in the original POVM $B_{0}$, and then uses
it to construct various Toeplitz operators. Different choices for $f$ can be
found in the literature (see, for example, Garrison and Wong (1970)), but we
will focus on one of the simplest options:

$f(\theta)=\theta.$ (55)

This will give us a time operator $\hat{T}_{0}$, which we calculate with the
above expressions to be

$\hat{T}_{0}=\hat{B}_{0}[\theta]=\sum_{n\neq m\geq
0}\frac{1}{i(n-m)}\ket{n}\negthickspace\bra{m}+\pi\mathds{1}.$ (56)

However, this operator is not unique. First of all, we could have chosen a
different starting point for our angles, changing the index $0$ at that point.
There is also a different, but more physically motivated point of arriving at
additional time operators. The time operator $\hat{T}_{0}$ can be covariantly
shifted to a new operator $\hat{T}_{\theta_{\ast}}$:

$\hat{T}_{\theta_{\ast}}\mathrel{\mathop{:}}=e^{i\hat{H}\theta_{\ast}}\hat{T}_{0}e^{-i\hat{H}\theta_{\ast}}.$
(57)

This _covariant shift property_ is a general feature of such time operators
from POVMs, see Busch et al. (1994), that is increasingly prominent and clear
in their modern implementations Höhn et al. (2021a); Smith (2019).
Importantly, while all such $\hat{T}_{\theta_{\ast}}$ fulfil canonical
commutation relations with $\hat{H}$ (when the resulting operator is well-
defined), different time operators will not commute with each other:

$\theta_{\ast}\neq\theta_{\ast}^{\prime}\mod
2\pi\qquad\Longrightarrow\qquad[\hat{T}_{\theta_{\ast}}\hat{T}_{\theta_{\ast}^{\prime}}]\neq
0.$ (58)

These results deserve some commentary. The reason why these time operators are
still self-adjoint, despite the illustrious list of no-go theorems given in
the introduction, is two-fold Busch et al. (1994): One, they are hardly
unique. Any change in starting point (phase) will yield a different time
operator. A covariant shift similarly yields new time operators. Two, the
domain of time is now periodic. The no-go theorems usually aim for
$\mathds{R}$ as domain for time $\theta$, or that the time would be unique in
some sense. In the present context, it is more a question of when one started
the periodic clock, and what period the clock has. It is also worth pointing
out that our time operators subtly fail to satisfy the expected time-energy
uncertainty: It will not hold for all states in $\mathcal{H}$, but only for a
densely defined set Garrison and Wong (1970); Galindo (1984). This latter
restriction is not an issue for our purposes, at least in the present article,
and the other reasons for evading the no-go theorems are what allows our
application in the context of time travel.

##  Appendix C Detailed Construction of Depicted Wave Packets

For our figure 1, we adapted the plots of Kiefer (1990) to our present context
with a larger configuration space. For ease of comparison, we kept the
notation for the wave functions the same. We do, however, emphasize that in
our context one should _not_ interpret $a$ as a scale factor. Our
investigation is not quantum cosmology, just inspired by a particular model of
it due to its simplicity.

A general solution of equation (20) can be expressed in configuration space
variables $a,\chi$ instead of the Fock space picture adapted in the remainder
of the article. For illustration purposes, this is preferable. We will
restrict attention to symmetric wavefunctions, in which $a$ and $\chi$ can be
assigned equally well to the clock system $\hat{H}_{\textsc{c}}$ or the
residual system $\hat{H}_{\textsc{r}}$. While our enlarged configuration space
(compared to that of quantum cosmology) might change the overall
normalization, it will not change the qualitative behaviour of the
wavefunctions. We therefore also copy the normalization of reference Kiefer
(1990).

A wavefunction with two Gaussians centered around $\chi=\pm\chi_{0}$, each of
width $\beta$, at time/position $a=0$, is given by

$\Psi(a,\chi)=\sum_{m=0}^{\infty}c_{2m}\frac{H_{2m}(a)H_{2m}(\chi)}{\sqrt{2^{2m}(2m)!}H_{2m}(0)}\exp\left(-\frac{1}{2}a^{2}-\frac{1}{2}\chi^{2}\right),$
(59)

where $H_{n}(x)$ is the $n$-th Hermite polynomial and the expansion
coefficients are

$c_{n}=\frac{1}{\sqrt{2^{n}n!}}\sqrt{\frac{2\beta}{1+\beta^{2}}}\exp\left(-\frac{\chi_{0}^{2}}{2(1+\beta^{2})}\right)\left(\sqrt{\frac{1-\beta^{2}}{1+\beta^{2}}}\right)^{n}H_{n}\left(\frac{\chi_{0}}{\sqrt{1-\beta^{4}}}\right).$
(60)

These coefficients $c_{n}$ simplify further for the ‘coherent case’, _i.e._ ,
when $\beta=1$, to

$c_{n}=\exp\left(-\frac{1}{4}\chi_{0}^{2}\right)\frac{\chi_{0}^{n}}{\sqrt{2^{n}n!}}.$
(61)

Note that the index $m$ in equation (59) is meant to make more explicit that
the sum runs only over even $n$.

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