# Flow of time during energy measurements and the resulting time-energy
uncertainty relations

I. L. Paiva  Faculty of Engineering and the Institute of Nanotechnology and
Advanced Materials, Bar-Ilan University, Ramat Gan 5290002, Israel  A. C. Lobo
Institute of Exact and Biological Sciences, Federal University of Ouro Preto,
Ouro Preto, Minas Gerais 35400-000, Brazil  E. Cohen  Faculty of Engineering
and the Institute of Nanotechnology and Advanced Materials, Bar-Ilan
University, Ramat Gan 5290002, Israel

###### Abstract

Uncertainty relations play a crucial role in quantum mechanics. Well-defined
methods exist for the derivation of such uncertainties for pairs of
observables. Other approaches also allow the formulation of time-energy
uncertainty relations, even though time is not an operator in standard quantum
mechanics. However, in these cases, different approaches are associated with
different meanings and interpretations for these relations. The one of
interest here revolves around the idea of whether quantum mechanics inherently
imposes a fundamental minimum duration for energy measurements with a certain
precision. In our study, we investigate within the Page and Wootters timeless
framework how energy measurements modify the relative “flow of time” between
internal and external clocks. This provides a unified framework for discussing
the subject, allowing us to recover previous results and derive new ones. In
particular, we show that the duration of an energy measurement carried out by
an external system cannot be performed arbitrarily fast from the perspective
of the internal clock. Moreover, we show that during any energy measurement
the evolution given by the internal clock is non-unitary.

##  1 Introduction

In standard quantum mechanics, while the canonical conjugate variables of
position and momentum are treated as observables, the same does not hold for
the pair time-energy. While energy (associated with a Hamiltonian) is an
observable, time is treated as an external parameter. Since time and position
are part of the same object in relativistic theories, the lack of symmetry in
their status in quantum mechanics seems to make the incompatibility of this
theory with relativity deeper than the incompatibility of earlier non-
relativistic theories. An answer to this conundrum was the development of the
relativistic quantum field theory, which is, to this day, the main tool to
describe the standard model of elementary particles. This approach
“downgrades” the status of the position observable to a parameter, putting it
alongside time, while “upgrading” quantum particles to the status of fields.
In turn, this gives place to the arena of the “classical” Minkowski spacetime,
upon where the quantum relativistic fields are defined. Yet, this move does
not solve many fundamental issues present in quantum mechanics. In particular,
the problem of constructing a consistent quantum theory of gravity remains.

A more recent venue to reconsider this problem is to look back at Einstein’s
operational approach that led to the discovery of special relativity in the
first place. He achieved a better understanding of space and time not by
asking deep metaphysical questions, such as what space and time are, but by
answering much more seemingly mundane questions, like how physicists in
relative motion synchronize the measurements of length and time between events
with rods and clocks. An approach to quantum mechanics that abides by this
operational perspective has been coined relational quantum mechanics. It may
become instrumental in describing spacetime and gravity quantum mechanically
[1]. The main idea is to recognize that clocks and rulers are physical objects
and, as such, they must obey quantum mechanical laws. Also, physical
quantities are always given relative to some reference object or quantity, and
so one is led to the concept of quantum reference frames [2, 3, 4]. As it will
be seen, this approach is central to the discussion here.

A related question to the aforementioned relativistic asymmetry of quantum
mechanics concerns the existence of a time-energy uncertainty relation. On the
one hand, this type of relation is expected to exist in view of relativistic
ideas together with Heisenberg’s uncertainty principle, which establishes an
uncertainty relation for position and momentum. However, standard approaches
for obtaining uncertainty relations, which are appropriate for the study of
the relation between two observables [5, 6, 7, 8, 9], do not work for time-
energy. As a result, there exist multiple ways to address this problem [10,
11, 12, 13]. For instance, Mandelstam and Tamm introduced a time-energy
uncertainty relation based on the fact that, in many cases, for a general
observable $O$, the ratio $\Delta O/\langle\dot{O}\rangle$, where $\Delta O$
is the variance of $O$ and $\dot{O}$ is its time derivative, defines an
interval of time [14]. Also, Anandan and Aharonov presented a geometric time-
energy uncertainty relation based on the concept of orthogonal-time, defined
as the time that it takes for the state of a system to become orthogonal to
its initial state [15]. A similar idea to the latter was later examined by
Margolus and Levitin in Ref. [16].

However, the approach that is of particular interest here and whose first
studies precede the aforementioned two has to do with time-energy relations
associated with the process of measurements of energy. More specifically,
relations that determine how fast such measurements can be made given a
certain desirable precision. First, Landau and Peierls argued in favor of an
uncertainty relation with a particular model for measuring energy [17].
However, Aharonov and Bohm showed that their model did not provide an optimal
energy measurement scheme and, in fact, it is possible to design energy
measurements that can be performed arbitrarily fast [18]. Nevertheless, a vast
number of measurements are bounded by an uncertainty relation. Investigating
the reasons behind it, Aharonov, Massar, and Popescu showed that arbitrarily
fast measurements require prior knowledge about the Hamiltonian [19]. In case
the Hamiltonian is unknown, as they showed, some time must be spent in the
estimation of it. Only after that, the measurement of energy can be performed
arbitrarily fast.

In common, these discussions considered measurements carried out by external
systems to the system of interest. However, as analyzed by Aharonov and
Reznik, one can also consider the case where the system’s energy is measured
by a part of it [20]. They suggested that, in this case, the measurement can
never be performed arbitrarily fast. However, Massar and Popescu showed that
if one considers the proper time (i.e., internal time) of the system of
interest, there exists no limit on the speed of the measurement [21].

These studies, directly or indirectly, involve the idea of quantum frames of
reference and, in particular, clock frames. The point in common is that they
all use a physical system to represent a clock. However, the way these
approaches were constructed makes it non-trivial to discuss the different
scenarios (i.e., internal or external measurements with an analysis of the
internal or external time) in a unified way. In fact, each of these results
relies on specific measurement schemes. Their approach included a single clock
system. Because of that, in the works that studied both internal and external
times (Refs. [20, 21]), an analysis of the composed state of the relevant
systems after the end of the von Neumann interaction was carried out to draw
conclusions about the duration of the measurement according to the two
distinct notions of time.

Regarding the idea of quantum clocks, Page and Wootters introduced a framework
for the study of timeless quantum mechanics [22]. Assuming the state of the
joint system composed by a clock and a system of interest satisfy a certain
constraint, known as the Wheeler-DeWitt equation, they showed that the unitary
Schrödinger dynamics of the system of interest could be recovered in their
timeless framework. This approach was further developed in Refs. [23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] and is similar to other
approaches in the literature, e.g., in Refs. [38, 39]. It has also been
realized experimentally [40, 41].

Among the studies in this area, multiple clocks and how quantum systems evolve
from their perspective were considered [27, 31, 32]. Inspired by it, we
introduce the discussion about the passage of time during a von Neumann
measurement (or pre-measurement since the final “collapse” is not involved) of
energy in the timeless framework. This allows us to analyze how the “flow of
time” changes during the measurement in an internal or external clock and from
the perspective of each of them. Besides the interesting aspects of this
analysis on its own, it also provides a unified and simplified framework for
the study and understanding of time-energy uncertainty relations associated
with the process of energy measurements, as formulated by Landau and Peierls
[17] and further developed in Refs. [18, 19, 21].

One of the main differences proposed by our approach is the explicit use of
two physical clocks: one internal and the other external to the system of
interest. As a result, we do not need to rely on analysis of phase factors
multiplying the wave function of the system after the measurement is
completed, as done in previous works. In addition, as already mentioned, the
model we consider allows us to observe how the measurement interaction changes
the relative flow of time between the different clocks during the measurement.
Furthermore, with this unified framework, we can also draw conclusions about
the existence or absence of a time-energy uncertainty relation for the
internal time when the measurement is carried out by an external system and
for the external time when the measurement is carried out by an internal
subpart of the system of interest. To the best of our knowledge, the existence
or not of a time-energy uncertainty relation in these two scenarios was not
known prior to the present work.

Another feature revealed by our results is the appearance of non-unitarity
from the internal’s clock perspective regardless of the measurement being
conducted by an external or an internal system. This is another contribution
of this article. In previous discussions of time-energy uncertainty relations,
the particularities of the models and the way they were solved did not allow
for this property to be observed. Furthermore, in the context of the Page and
Wootters framework, to the best of our knowledge, this work is the first to
point out that the effective Hamiltonian of a system from a clock’s
perspective is not always Hermitian.

The article is organized as follows. In Section 2, we present an overview of
Page and Wootters framework. Following that, in Section 3, we introduce our
model for the measurement of energy carried out by both an external system or
an internal part of the system of interest. This allows to analyze the
relative flow of time between the different clocks. This leads to a two-fold
study: the subsequent time-energy uncertainty relations in Section 4 and the
emergence of non-unitarity in Section 5. Finally, we discuss the results
presented here as well as some remaining questions that deserve further
examination in Section 6.

##  2 Quantum mechanics in the timeless framework

The timeless framework consists of a clock system, whose state is given by a
vector in a Hilbert space $\mathcal{H}_{A}$, and the rest of the system,
represented by a state in a Hilbert space $\mathcal{H}_{R}$, whose evolution
is studied. The joint system
$|\Psi\rangle\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{R}$ is assumed to be
closed and, hence,

$H_{T}|\Psi\rangle\rangle=0,$ (1)

where $H_{T}$ is the total Hamiltonian acting on systems $A$ and $R$. This
equation is known as the Wheeler-DeWitt equation. Observe that the imposition
of $|\Psi\rangle\rangle$ being an eigenstate with null eigenvalue by Eq. (1)
is not as restrictive as it seems [24]. In fact, Hamiltonians that differ by
constant terms are physically equivalent, which is associated with quantum
states being defined up to a global phase. Then, Eq. (1) just reads as
$|\Psi\rangle\rangle$ not evolving with respect to an external time.

The clock system should have an observable $T_{A}$ associated with its time.
It is desirable that such operator is covariant under translations generated
by the Hamiltonian $H_{A}$, i.e.,
$|t_{0}+t_{A}\rangle=e^{-iH_{A}t_{A}/\hbar}|t_{0}\rangle$, where
$|t_{0}\rangle$ and $|t_{0}+t_{A}\rangle$ are taken to be clock states. This
does not require $T_{A}$ to be a self-adjoint canonical conjugate to $H_{A}$.
In fact, it is possible to construct $T_{A}$ as a positive operator-valued
measure (POVM), which means that the clock states are not necessarily
eigenstates of $T_{A}$ [42, 43, 44]. Such operators obtained from these
constructions are symmetric but need not be self-adjoint. Here, however, for
simplicity we assume an ideal clock, i.e., $[T_{A},H_{A}]=i\hbar I$,
$T_{A}|t_{A}\rangle=t_{A}|t_{A}\rangle$, and $H_{A}=-i\hbar\partial/\partial
t_{A}$. Although the Hamiltonians of these clocks are unbounded from bellow
and, hence, unrealistic, they help us avoid technicalities associated with
real clocks [45, 46] while providing approximations of them [47, 48]. Thus,
corrections to the results presented here are expected when dealing with non-
ideal clocks.

Moreover, let $H_{R}$ denote the Hamiltonian of the system of interest and
$H_{int}(T_{A})$ represent the time dependent term of the evolution of system
$R$ set by clock $A$, which is an interaction between $A$ and $R$, we have

$H_{T}=H_{A}+H_{R}+H_{int}(T_{A}).$ (2)

Observe that, in particular, $H_{int}$ is being assumed to be independent of
$H_{A}$. Replacing Eq. (2) in Eq. (1), applying a scalar product by an
eigenstate $|t_{A}\rangle$ of $T_{A}$ on the left, and defining
$|\psi(t_{A})\rangle\equiv\langle t_{A}|\Psi\rangle\rangle$, it holds that

$i\hbar\frac{\partial}{\partial
t_{A}}|\psi(t_{A})\rangle=\left[H_{R}+H_{int}(t_{A})\right]|\psi(t_{A})\rangle,$
(3)

which is the Schrödinger equation that gives the evolution of system $R$ with
respect to the time measured by clock $A$. Then, the usual unitary evolution
of a quantum system is recovered from the static picture introduced by Page
and Wootters.

As a result, $|\Psi\rangle\rangle$ can be written as

$|\Psi\rangle\rangle=\int dt_{A}\,|t_{A}\rangle\otimes|\psi(t_{A})\rangle.$
(4)

Because $|\Psi\rangle\rangle$ contains information about $|\psi(t_{A})\rangle$
at every $t_{A}$, it is referred to as the history state.

The results in Refs. [27, 31, 32] are central to this work. In their approach,
system $R$ is assumed to contain a clock $B$, and the rest of it is simply
referred to as system $S$. Then, while system $B$ gives the internal time of
the system $R=B+S$, system $A$ provides time as observed by an external
system. In this case, the total Hamiltonian is

$H_{T}=H_{A}+H_{B}+H_{S}+H_{int}(T_{B}),$ (5)

where $T_{B}$ is the time operator associated with clock $B$. Although the
time dependent term $H_{int}$ can be taken to be a function of both $T_{A}$
and $T_{B}$ in a more general scenario, we assume it does not depend on
$T_{A}$ for simplicity. However, nothing significant changes in our analysis
if $H_{int}$ is also a function of $T_{A}$.

With the previous $H_{T}$, the analog of the Schrödinger equation becomes

$i\hbar\frac{\partial}{\partial
t_{A}}|\psi(t_{A})\rangle=\left(H_{B}+H_{S}+H_{int}(T_{B})\right)|\psi(t_{A})\rangle.$
(6)

This implies that the effective Hamiltonian acting on system $R$ is

$H_{eff}^{A}\equiv H_{B}+H_{S}+H_{int}(T_{B}).$ (7)

Even though this result is a particular case of Eq. (3), it is of special
interest here because it allows us to study the relation between the “flow of
time” in clocks $A$ and $B$.

While the designation of which clock is internal or external to the system of
interest seems to be arbitrary up until now, they will acquire a more concrete
meaning in the next section, where measurements of energy are studied.
Furthermore, in the next section, the evolution will be considered not only
from clock $A$’s perspective but also in the frame of clock $B$.

##  3 Measurements of energy with quantum clocks

###  3.1 Measurement carried out by an external system

In this subsection, we study the von Neumann measurement of the total energy
of system $R$ in the Page and Wootters timeless framework. Specifically, we
consider the case where the measurement is carried out by an external system.
Then, in addition to the systems already introduced up until now, we also
consider an external pointer system $E$, and the history state
$|\Psi\rangle\rangle$ is an element of the space
$\mathcal{H}_{A}\otimes\mathcal{H}_{R}\otimes\mathcal{H}_{E}$. This scenario
is illustrated in Fig. 1. For simplicity, the free evolution of system $E$
will be neglected.

Figure 1: Representation of the relevant parts in a measurement of energy
carried out by an external system. The external apparatus $E$ uses an external
clock $A$ for time reference during the measurement of energy of system $R$,
composed of an internal clock $B$ and system $S$.

The measurement interaction can be represented by the von Neumann interaction
$H_{VN}\equiv g(T_{A})H_{R}P_{E}$, where $H_{R}\equiv
H_{B}+H_{S}+H_{int}(T_{B})$, $P_{E}$ is the conjugate momentum of the pointer
$E$, and $g$ is a non-negative function that differs from zero exclusively
during the duration of the measurement. For notation purposes, we assume the
measurement starts in clock $A$ at $t_{A}=0$ and ends at $t_{A}=\tau$.
Moreover, we let

$\int_{-\infty}^{\infty}g(t)dt=\int_{0}^{\tau}g(t)dt=K,$ (8)

where $K$ is a positive real constant associated with the strength (hence, the
precision) of the measurement.

In addition, we assume that $P_{E}$ only takes non-negative values. As we
shall see, the duration of the measurement in clock $B$ will linearly depend
on $P_{E}$. Thus, this condition will assure that the clocks considered here
are “good” clocks in the sense that they do not move backward in time and, in
particular, that the duration of the measurement is non-negative. While this
requirement on $P_{E}$ means that the measurement device’s pointer cannot be
well-localized in an ideal way, it does not forbid it to be localized up to
some order. In fact, if $P_{E}$ has a sufficient large average value, it is
possible for the measurement device’s pointer to have a small variance since
its state can be built in such a way that $P_{E}$’s variance is large.

With that, the total Hamiltonian of the composed system is
$H_{T}=H_{A}+H_{R}+g(T_{A})H_{R}P_{E}$. Then, in a similar manner Eqs. (3) and
(6) were derived in the previous section, we obtain the Schrödinger equation

$i\hbar\frac{\partial}{\partial
t_{A}}|\psi(t_{A})\rangle=\left[H_{R}+g(t_{A})H_{R}P_{E}\right]|\psi(t_{A})\rangle,$
(9)

where $|\psi(t_{A})\rangle\equiv\langle t_{A}|\Psi\rangle\rangle$. This means
that, from the perspective of clock $A$, the evolution is generated by the
effective Hamiltonian

$H_{1}\equiv H_{R}+g(t_{A})H_{R}P_{E},$ (10)

which is the usual Hamiltonian of a time-independent system during a
measurement. This is expected since, in standard quantum mechanics, time is an
external parameter, as it is in this case.

Using the Heisenberg equation of motion to study the evolution of $T_{B}$ with
respect to clock $A$, we conclude that

$\frac{d}{dt_{A}}T_{B}=-\frac{i}{\hbar}[T_{B},H_{1}]=I+g(t_{A})P_{E}.$ (11)

This shows that, when $g$ vanishes, the flow of time in both clocks is the
same. In fact, in this case, from the perspective of clock $A$, the variance
of $T_{B}$ at any instant originates from its initial variance. In particular,
if both clocks start localized and synchronized, they remain localized and
synchronized.

Yet, clock $B$ ticks faster than clock $A$ whenever $g$ is non-null, i.e.,
during the measurement of energy. If $g$ is a function whose integral over
time grows smoothly, then the transition to a faster ticking also happens
smoothly. However, this is not always the case. A dramatic example can be
observed by assuming a highly idealized case where $g$ is the delta function.
In this case, from the perspective of clock $A$, there is a sudden “jump” of
the pointer of clock $B$.

While the dependence of $T_{B}$ on $P_{E}$ might seem surprising, it reflects
the fact that clock $B$ interacts with the pointer. However, not every
interaction with a clock leads to a change in its flow of time. To understand
the change observed in the case of interest here, it suffices to analyze the
term $g(t_{A})H_{B}P_{E}$ from the von Neumann interaction. Although it is
usually thought of as a term that causes a shift (generated by $P_{E}$) of a
quantity associated with $H_{B}$ in the measurement device’s pointer, it can
be also understood as a term causing a shift (generated by $H_{B}$) of a
quantity associated with $P_{E}$ in the pointer (i.e., time) of clock $B$.
Generally, by this reasoning, interactions with clocks intermediated by their
Hamiltonian should affect their flow of time by quantities associated with the
interacting system.

Now, using Eq. (1) and defining $|\phi(t_{B})\rangle\equiv\langle
t_{B}|\Psi\rangle\rangle$, we obtain the following Schrödinger equation from
the perspective of clock $B$:

$\displaystyle i\hbar[I+g$ $\displaystyle(T_{A})P_{E}]\frac{\partial}{\partial
t_{B}}|\phi(t_{B})\rangle=H_{A}|\phi(t_{B})\rangle$ (12)
$\displaystyle+\left[I+g(T_{A})P_{E}\right][H_{S}+H_{int}(t_{B})]|\phi(t_{B})\rangle.$

Moreover, recalling that $P_{E}$ is assumed to only take non-negative values,
which implies that $I+g(T_{A})P_{E}$ is invertible, the effective Hamiltonian
is

$H_{2}\equiv\left[I+g(T_{A})P_{E}\right]^{-1}H_{A}+H_{S}+H_{int}(t_{B}).$ (13)

Observe that the function $g$ that controls the measurement continues to be a
function of the operator $T_{A}$, while it was a function of the parameter
$t_{A}$ when considering $A$’s perspective. In a scenario like this, a well-
localized event in clock $A$ has a variance from clock $B$’s perspective, a
result illustrated in Fig. 2 and previously discussed, for instance, in Ref.
[32]. This means, in particular, that the start and end of the measurement,
which are well-localized in $A$, have variance in $B$.

Figure 2: Localizability of events with multiple clocks. A well-localized
event in clock $A$ has a variance in clock $B$, and vice-versa.

Moreover, in a sense, Hamiltonian $H_{2}$ corresponds to a measurement of
$H_{A}$. However, the measurement function $g$ is controlled by the time in
clock $A$, which is an observable from the perspective of $B$. Because of it,
the measurement Hamiltonian would have been symmetrized in more standard
treatments of the process, which is not the case here. In fact, $H_{2}$ is
non-Hermitian and, as a consequence, the evolution of clock $A$ from the
perspective of clock $B$ is non-unitary, a characteristic that will be further
discussed later in this work. For now, observe that the Heisenberg equation of
motion for $T_{A}$ with respect to $t_{B}$ gives

$\frac{d}{dt_{B}}T_{A}=-\frac{i}{\hbar}[T_{A},H_{2}]=[I+g(T_{A})P_{E}]^{-1}.$
(14)

This result is expected in view of Eq. (11) since it gives the inverse of the
passage of time on clock $B$ with respect to clock $A$. However, once again,
it should be noticed that here the argument of $g$ is an operator, and no
longer a parameter. Then, with respect to clock $B$, if $T_{A}$ starts with a
variance, the duration of the measurement will have a variance as well.
Nevertheless, on average, the “flow of time” during the measurement of energy
is expected to be smaller in clock $A$, from the perspective of both clock $A$
and clock $B$.

Even though Eq. (14) shows consistency across different clock perspectives, it
raises questions about the use of the Heisenberg equation here. While
Schrödinger’s and Heisenberg’s representations are unitarily equivalent, since
the evolution from clock $B$’s perspective is, in general, non-unitary, it is
not trivial to obtain the Heisenberg equation from the effective Hamiltonian
in the Schrödinger equation. Nevertheless, this can be done with the
introduction of an “indefinite metric” in the Hilbert space, a method
introduced by Dirac [49] and briefly explained in Appendix A.

###  3.2 Measurement carried out by an internal system

This subsection considers, once more, the total energy measurement of system
$R$, although, this time, the measurement in question is carried out by an
internal system. A concrete example of this type of measurement of total
energy of an isolated system was given by Aharonov and Reznik [20]. They
considered an isolated planet with a large radius $R$ ejecting a tiny portion
$m$ of its total mass radially outwards with a known speed. Then, using the
relativistic correspondence of mass and energy, they showed that it is
possible to infer the total energy of the planet by measuring how long it
takes for $m$ to fall back.

Here, we use an abstract model for these measurements, similarly to what was
done in the previous subsection. We assume that system $R$ has access to an
apparatus $I$, and the history state $|\Psi\rangle\rangle$ is an element of
$\mathcal{H}_{A}\otimes\mathcal{H}_{R}\otimes\mathcal{H}_{I}$, as represented
in Fig. 3. Note that the apparatus may be assumed to be an internal degree of
freedom of the system whose energy is left out of the measurement.

Figure 3: Representation of the relevant parts in a measurement of energy
carried out by an internal part. In this case, the internal apparatus $I$
measures the energy of the remaining parts, i.e., system $R$, composed of
clock $B$ and system $S$. Clock $A$ is an external clock, which does not
interact with any other system during this process. For the measurement, the
internal clock $B$ is used as a reference for time.

Similarly to what was done in the previous section, the free evolution of the
pointer $I$ will be neglected. As a result, the measurement can be represented
by the following von Neumann interaction

$H_{VN}\equiv\frac{1}{2}[g(T_{B})H_{R}+H_{R}g(T_{B})]P_{I},$ (15)

where $H_{R}=H_{B}+H_{S}+H_{int}(T_{B})$, $P_{I}$ is the conjugate momentum of
apparatus $I$, presupposed to only take non-negative values, and $g$ is the
same function introduced in the previous section. Observe the necessary
symmetrization of the product $g(T_{B})H_{R}$ due to the lack of commutativity
between $T_{B}$ and $H_{B}$, as previously considered, for instance, in Ref.
[21]. Also, we assume that, in clock $B$, the measurement starts at $t_{B}=0$
and ends at $t_{B}=\tau$.

Since the total Hamiltonian of the composed system is
$H_{T}=H_{A}+H_{R}+H_{VN}$, in a similar manner Eqs. (3) and (6) were derived
in the previous section, we obtain the Schrödinger equation

$i\hbar\frac{\partial}{\partial
t_{A}}|\psi(t_{A})\rangle=H_{3}|\psi(t_{A})\rangle,$ (16)

where $|\psi(t_{A})\rangle\equiv\langle t_{A}|\Psi\rangle\rangle$ and

$H_{3}\equiv H_{R}+\frac{1}{2}[g(T_{B})H_{R}+H_{R}g(T_{B})]P_{I}$ (17)

is the effective Hamiltonian from the perspective of clock $A$. With that, the
Heisenberg equation of motion for $T_{B}$ with respect to $t_{A}$ is

$\frac{d}{dt_{A}}T_{B}=-\frac{i}{\hbar}[T_{B},H_{3}]=I+g(T_{B})P_{I}.$ (18)

On the other hand, using Eq. (1) and defining
$|\phi(t_{B})\rangle\equiv\langle t_{B}|\Psi\rangle\rangle$, we obtain

$\displaystyle i\hbar$ $\displaystyle[I+g(t_{B})P_{I}]\frac{\partial}{\partial
t_{B}}|\phi(t_{B})\rangle=$ (19)
$\displaystyle=\left[H_{A}-\frac{i\hbar}{2}g^{\prime}(t_{B})P_{I}\right]|\phi(t_{B})\rangle$
$\displaystyle\ \ \
+[I+g(t_{B})P_{I}][H_{S}+H_{int}(t_{B})]|\phi(t_{B})\rangle,$

where it was used the fact that $I+g(t_{B})P_{I}$ can be inverted. This means
that, from the perspective of clock $B$, the evolution is generated by the
effective Hamiltonian

$\displaystyle H_{4}\equiv$
$\displaystyle[I+g(t_{B})P_{I}]^{-1}\left[H_{A}-\frac{i\hbar}{2}g^{\prime}(t_{B})P_{I}\right]$
(20) $\displaystyle+H_{S}+H_{int}(t_{B}),$

which, similarly to $H_{2}$, is a non-Hermitian operator.

Then, using the Heisenberg equation of motion to study the evolution of
$T_{A}$ with respect to $t_{B}$, we conclude that

$\frac{d}{dt_{B}}T_{A}=-\frac{i}{\hbar}[T_{A},H_{4}]=[I+g(t_{B})P_{I}]^{-1},$
(21)

which, as expected, up to the difference that the argument of $g$ is, now, a
parameter, the inverse of the relation between the passage of time in the two
clocks given by Eq. (18). Here, again, the use of the Heisenberg equation is
justified with the introduction of an indefinite metric in the Hilbert space,
as discussed in Appendix A.

##  4 Time-energy uncertainty relations

Now, we shall discuss how the comparison between the flow of interior and the
exterior time during an energy measurement provides insights into time-energy
uncertainty relations associated with the von Neumann measurement of energy.
As already mentioned, knowing the Hamiltonian beforehand makes a difference
when studying these relations since, otherwise, the process of effectively
measuring the Hamiltonian includes a preliminary step of estimating it [19].
Here, we focus on the cases the Hamiltonian of system $R$ is known and leave
the study of estimation of Hamiltonians to future work.

It should be noticed that, because of the explicit inclusion of the frames
associated with the internal and external clocks, our analysis becomes
slightly more subtle than others presented up until now. While, like the
previous works, we can consider which time (internal or external) one wants to
optimize when an internal or external system carries out the measurement, we
have to also specify with respect to which clock frame (internal or external)
the question is being asked. However, our approach has the advantage of
introducing a unified framework to analyzing time-energy uncertainty
relations, which qualifies us to recover previous results and determine the
existence or absence of such relations in scenarios where it is still unknown.

###  4.1 Measurement carried out by an external system

First, we consider measurements carried out by an external system from the
perspective of $A$. Observe that the measurement function $g$ is controlled by
clock $A$ and, then, nothing in the von Neumann measurement procedure we
follow imposes a limit on how small the interval where $g$ does not vanishes
is, i.e., how fast the measurement can be performed. As a consequence, the
external time can be arbitrarily minimized without requiring any compromising
of the measurement accuracy, i.e., without imposing any restriction on the
constant $K$. This corresponds to the result obtained by Aharonov and Bohm in
Ref. [18], which shows the nonexistence of an uncertainty relation in this
scenario.

Now, a question that has not been dealt with in the literature thus far
concerns the possibility of optimizing the internal time in this scenario. For
that, observe that Eq. (11) lead to

$T_{B}(\tau)-T_{B}(0)=\tau I+KP_{E}.$ (22)

This result is represented in Fig. 4a. It shows that the internal duration of
the measurement decreases with $\tau$, i.e., with the external duration, which
can, in principle, be as small as desirable. However, this can only reduce the
internal duration up to $KP_{E}$. Thus, in order to obtain an arbitrarily
small internal duration, it is necessary to decrease the value of the constant
$K$. More precisely,

$\langle T_{B}(\tau)-T_{B}(0)\rangle\geq K\langle P_{E}\rangle>0.$ (23)

Here, the inequality refers to statistics associated with repeated
implementations of the measurement protocol. The average on the left-hand side
refers to the average duration of the experiment in clock $B$. On the other
part of the inequality, $\langle P_{E}\rangle$ is the average value of the
conjugate momentum of the pointer $E$.

Inequality (23) means that there exists a trade-off between the precision of
the measurement (which is associated with $K$) and how long the measurement
lasts in the internal clock, i.e., there exists a time-energy uncertainty
relation in the sense of Landau and Peierls [17] discussed in the
Introduction, i.e., quantum mechanics imposes a minimum duration for
measurements of energy with a desired precision. Note that such a relation is
independent of the variance of the operator $T_{B}$. Also, the existence of
such an inequality cannot be attributed to disturbances on clock $B$ caused by
the measurement since they vanish, as explained in Appendix B.

Figure 4: Representation of the duration of energy measurements in internal
and external clocks. Clock $A$ is external and clock $B$ is internal to the
measured system. (a) When an external system carries out the measurement, its
duration in $B$ is longer on average. (b) However, when an internal part of
the system is responsible for the measurement, its duration in the external
clock is shorter on average.

Moreover, taking variances into consideration, we obtain a different type of
uncertainty relation. In fact, observe that the variance $\Delta T_{B}$ in the
duration of the experiment in clock $B$ from clock $A$’s perspective is
associated with the variance $\Delta P_{E}$ of $P_{E}$. More precisely,
$\Delta T_{B}$ grows with $\Delta P_{E}$. Then, in order to obtain a smaller
$\Delta T_{B}$, it is necessary to decrease $\Delta P_{E}$. However, $\Delta
P_{E}$ is also associated with the precision of the measurement since a
smaller $\Delta P_{E}$ implies that pointer of the measurement device is more
delocalized. As a result, a smaller $\Delta T_{B}$ leads to an increase in the
precision of the measurement, implying an uncertainty relation of the type
$\Delta E\Delta T_{B}\geq 1$, where $\Delta E$ is the uncertainty of the
energy measurement.

Now, from the perspective of clock $B$, the analysis of the measurement
carried out by an external system leads mostly to the same conclusions. In
fact, with some probability, we can say that the measurement starts in a
certain instant $t^{B}_{i}$ and ends at $t^{B}_{f}$ in clock $B$. Then, Eq.
(14) implies that

$t^{B}_{f}-t^{B}_{i}=\int_{t^{B}_{i}}^{t^{B}_{f}}[I+g(T_{A}(t_{B}))P_{E}]\,\frac{dT_{A}}{dt_{B}}(t_{B})\,dt_{B}$
(24)

or, equivalently,

$t^{B}_{f}-t^{B}_{i}=T_{A}(t_{f}^{B})-T_{A}(t_{i}^{B})+KP_{E}.$ (25)

Note that, on average, $T_{A}(t_{f}^{B})-T_{A}(t_{i}^{B})$ gives $\tau$. Then,
the only difference in the analysis from $B$’s perspective is that the time in
clock $A$ is given by an operator and, hence, has a variance associated with
it, which, in turn, leads to a variance in the duration of the measurement.
This leads to an uncertainty relation for the duration of the measurement and
its precision associated with the variances of both clocks $A$ and $B$.
However, in terms of the uncertainty relations of main interest here,
conclusions about their existence or not are independent of perspective.

###  4.2 Measurement carried out by an internal system

We consider now the measurement carried out by an internal system from the
perspective of clock $B$. Again, we start by pointing out that nothing in our
description disallows the measurement to be conducted arbitrarily fast. Then,
there exists no uncertainty relation in this case, a conclusion that
corroborates the result obtained by Massar and Popescu in Ref. [21].

Furthermore, still from the perspective of clock $B$, we can address the
question about the existence of such a relation for the external time, which,
to the best of our knowledge, has not been answered in the literature prior to
this work. Note that Eq. (21) implies that

$T_{A}(\tau)-T_{A}(0)=\int_{0}^{\tau}[I+g(t_{B})P_{I}]^{-1}dt_{B},$ (26)

as represented in Fig. 4b. With the hypothesis that the evaluation of $P_{I}$
is non-negative, it holds that the norm of the last integral is bounded by
$\tau$ (independently of $K$), i.e., the the duration of the measurement in
clock $A$ can be as small as its duration in clock $B$ — in fact, it is, in
general, smaller since the evaluation of $I+g(t_{B})P_{I}$ is always greater
or equal one. Considering that the duration can be arbitrarily small in clock
$B$, there exists no time-energy uncertainty relation for clock $A$ from the
perspective of clock $B$.

This is a rather surprising result. In fact, Massar and Popescu briefly
consider this scenario in Ref. [21], where they presented a conjecture stating
that an uncertainty relation should hold.

Before jumping into $A$’s perspective, it should be noted once again the
existence of an uncertainty relation for the duration of the measurement in
clock $A$ associated with the variance of $T_{A}$.

Now, from the perspective of clock $A$, the conclusions are, once again,
similar. Suppose the measurement starts and ends at $t_{i}^{A}$ and
$t_{f}^{A}$, respectively. Then, Eq. (18) implies that

$t_{f}^{A}-t_{i}^{A}=\int_{t_{i}^{A}}^{t_{f}^{A}}[I+g(T_{B}(t_{A}))P_{I}]^{-1}\,\frac{dT_{B}}{dt_{A}}(t_{A})\,dt_{A}.$
(27)

Since the evaluation of $g(T_{B})P_{I}$ is assumed to be non-negative, the
average of the right-hand is not greater than the average of the integral of
$dT_{B}/dt_{A}$, which is $\tau$. Hence, the only difference in the analysis
from $A$’s frame is the variance associated with $T_{B}$, which leads to an
uncertainty relation for the duration of the measurement in both clocks $A$
and $B$. Here, again, it should be noted that disturbances on clock $B$ do not
affect this analysis since they can be taken to be null, as discussed in
Appendix B.

##  5 Emergence of non-unitarity

As noticed earlier in the context of energy measurements, an interesting
feature of the dynamics generated by the measurement of energy is that, from
the perspective of the internal clock, it is non-unitary regardless of whether
an external or an internal system carries out the measurement.

In relativistic theories, energy and mass are tightly related. Then, while
measuring the energy of a system that includes an internal clock, there is a
sense in which the mass of the system and, in particular, the mass of the
clock is being weighed. Could, then, the non-unitarity observed in this work
be related to gravitational effects? After all, non-unitarity is expected to
play a role in quantum gravity, see e.g. [50, 51, 52, 53]. However,
gravitationally interacting clocks have been considered in the literature [54,
26, 32], and it was shown that they lead to unitary dynamics. In fact, in
these studies, the interaction term between two clocks $A$ and $B$ was of the
type $\lambda H_{A}H_{B}$. Then, it can be checked that the effective
Hamiltonian of $B$ with respect to $A$ is $(I+\lambda H_{B})^{-1}H_{B}$.
Similarly, the effective Hamiltonian of $A$ with respect to $B$ is $(I+\lambda
H_{A})^{-1}H_{A}$. Since these two effective Hamiltonians are Hermitian, the
evolution generated by them is unitary.

Then, why is the result different here? The answer to this question lies in
the fact that the von Neumann interaction includes products of the time
operator of a clock by the individual Hamiltonian of the same or other clocks.
In fact, if such a condition is matched, the dynamics of the clocks associated
with the time operator in the interaction is non-unitary with respect to the
clock whose individual Hamiltonian appears multiplying it. To see that,
consider a system composed of $n$ clocks $C_{1},C_{2},\ldots,C_{n}$ and let
the total Hamiltonian be

$H_{T}=\sum_{k}H_{C_{k}}+f(T_{C_{r_{1}}},\ldots,T_{C_{r_{m}}})H_{C_{s}}+H_{int},$
(28)

where $m<n$, the indices $r_{1},r_{2},\ldots,r_{m}$, and $s$ are different
elements of $\\{1,2,\ldots,n\\}$, and $H_{int}$ is not a function of $H_{s}$.
Then, from the perspective of $C_{s}$ we can write

$\displaystyle i\hbar[I+f($ $\displaystyle
T_{C_{r_{1}}},\ldots,T_{C_{r_{m}}})]\frac{\partial}{\partial
t_{C_{s}}}|\psi(t_{C_{s}})\rangle=$ (29) $\displaystyle=\left[\sum_{k\neq
s}H_{C_{k}}+H_{int}\right]|\psi(t_{C_{s}})\rangle,$

i.e., the effective Hamiltonian, which is to be compared with $H_{2}$, is

$H_{eff}^{C_{s}}\equiv[I+f(T_{C_{r_{1}}},\ldots,T_{C_{r_{m}}})]^{-1}\left[\sum_{k\neq
s}H_{C_{k}}+H_{int}\right],$ (30)

assuming that $I+f(T_{C_{r_{1}}},T_{C_{r_{2}}},\ldots,T_{C_{r_{m}}})$ is
invertible. This Hamiltonian is manifestly non-Hermitian. More specifically,
the parts associated with clocks $C_{r_{1}},C_{r_{2}},\ldots,C_{r_{m}}$ are
non-Hermitian and, hence, have non-unitary evolution because of the lack of
commutativity between their individual time and Hamiltonian operators.

The only scenario that remains to be analyzed now corresponds to the case
where an interaction between the time operator and the individual Hamiltonian
of the same clock exists. For that, suppose the total Hamiltonian is given by

$\displaystyle H_{T}$
$\displaystyle=\sum_{k}H_{C_{k}}+\frac{1}{2}[f(T_{C_{s}})H_{C_{s}}+H_{C_{s}}f(T_{C_{s}})]+H_{int}$
(31)
$\displaystyle=\sum_{k}H_{C_{k}}+f(T_{C_{s}})H_{C_{s}}-\frac{i\hbar}{2}f^{\prime}(T_{C_{s}})+H_{int},$

where, once again, $H_{int}$ is assumed to not depend on $H_{C_{s}}$. From
clock $C_{s}$’s perspective, this leads to

$\displaystyle i\hbar[I+$ $\displaystyle f(t_{C_{s}})]\frac{\partial}{\partial
t_{C_{s}}}|\psi(t_{C_{s}})\rangle=$ (32) $\displaystyle=\left[\sum_{k\neq
s}H_{C_{k}}-\frac{i\hbar}{2}f^{\prime}(t_{C_{s}})+H_{int}\right]|\psi(t_{C_{s}})\rangle,$

which shows that the effective Hamiltonian is

$H_{eff}^{C_{s}}=[I+f(t_{C_{s}})]^{-1}\left[\sum_{k\neq
s}H_{C_{k}}-\frac{i\hbar}{2}f^{\prime}(t_{C_{s}})+H_{int}\right],$ (33)

assuming that $I+f(t_{C_{s}})$ is invertible. This Hamiltonian is be compared
with $H_{4}$. Although there is no commutation problem, the Hamiltonian in
this case is also non-Hermitian because of the imaginary unit multiplying
$f^{\prime}(t_{C_{s}})$.

Despite the differences between the discussion in the present work and the
gravitationally interacting clocks studied in Ref. [32], there exists an
aspect that leads to a similar analysis. In Ref. [32], factors like
$(I+\lambda H_{A})^{-1}$ and $(I+\lambda H_{B})^{-1}$ were interpreted as
“redshift” operators. Also, these operators were combined with the time
derivative present in their respective Schrödinger equation, resulting in
derivatives with respect to new time operators:
$\partial/\partial\hat{t}_{A}\equiv(I+\lambda H_{B})\partial/\partial t_{A}$
and $\partial/\partial\hat{t}_{B}\equiv(I+\lambda H_{A})\partial/\partial
t_{B}$. In a similar manner, the term between brackets on the left-hand side
of Eq. (29) can be interpreted as a sort of “redshift” operator, which,
combined with the $t_{C_{s}}$-derivative operator, results in a derivative
with respect to a time operator $\hat{t}_{s}$. This allows us to write

$i\hbar\frac{\partial}{\partial\hat{t}_{s}}|\psi\rangle=\left[\sum_{k\neq
s}H_{C_{k}}+H_{int}\right]|\psi\rangle.$ (34)

Observe that, with these coordinates, the unitarity of the dynamics is
restored. However, a complete change to clock $C_{s}$’s referential leads to
the observation of the “redshift,” an analogy that leads to the conclusion
that clock $C_{s}$ is not an inertial frame of reference.

A similar analysis can be made with Eq. (32). However, in this case, the non-
unitarity of the dynamics persists even when the derivative with respect to an
operator $\hat{t}_{s}$, which consists of the $t_{C_{s}}$-derivative operator
combined with the “redshift” factor, is considered.

##  6 Discussion

Using the Page and Wootters framework, we have studied how the relation
between internal and external “flow of time” in quantum clocks changes during
energy measurements performed either by an external or internal system. Some
of these measurements, as was discussed, cause disturbances in the energy of
the systems, which are independent of the duration of the measurements. While
there is no penalty for performing a fast energy measurement (in terms of
disturbances added to it), the change in the flow of time between an external
and an internal clock reveals a minimum duration in the internal clock of a
measurement with a given precision carried out by an external system. Any
other scenario considered here led to the conclusion of an absence of a time-
energy uncertainty relation in the sense of Landau and Peierls [17].

Importantly, the time-energy uncertainty relations considered here assume the
Hamiltonian being measured to be known beforehand. However, as mentioned in
the Introduction, if this is not the case, the Hamiltonian has to be
estimated. While Ref. [19] shows the existence of a minimum duration in an
external clock for an external system to estimate the Hamiltonian of the
system of interest, a discussion about the relation of this time interval and
the interval observed in an internal clock remains to be conducted. Moreover,
the scenario where an internal system attempts to estimate its total
Hamiltonian has yet to be investigated. An interesting aspect of the proof
provided by Aharonov, Massar, and Popescu in Ref. [19] is that it involves the
time necessary for the state of a system to become orthogonal to its initial
state under a designed evolution. Then, the type of uncertainty relation that
arrives from it is of the geometric type, briefly mentioned in the
Introduction and originally discussed by Anandan and Aharonov in Ref. [15] —
followed by Margolus and Levitin in Ref. [16].

While we concentrated on the measurement’s unitary interaction in this work,
it is well-known that von Neumann’s description of a measurement includes a
“collapse” (or post-selection) in case the pointer is found in a superposition
at the end of the process. The inclusion of this final step in our analysis
adds various new subtleties. On the one hand, since time and energy of a clock
are canonically conjugated, there exists an uncertainty relation for a
measurement carried out by an internal system, meaning that time in the
internal clock can only be known up to a certain precision given a desired
precision for the measurement of energy. This is a Heisenberg-like uncertainty
relation and is captured by entropic time-energy uncertainty relations
recently introduced by Boette et al. [55] and Coles et al. [56]. On the other
hand, the inclusion of a “collapse” in our description would necessarily
violate Eq. (1). This is a known problem in the measurement of quantum clocks
and, more generally, of any quantum system satisfying the Wheeler-DeWitt
equation. A typical solution to this problem consists of the inclusion of
ancilla systems that act as the measurement device and register the
measurement outcome [39, 24, 32, 35]. This perspective solves at least the
operational problem of computing the probability of the outcome of (possibly
many) measurements. Since our approach includes measurement devices, this
problem has been, then, already dealt with (at least from this perspective). A
different approach consists of updating the state with a “collapse” and, then,
applying a map to transform the resulting state into a state that satisfies
the Wheeler-DeWitt equation. This procedure, however, cannot be formulated in
a perspective-neutral manner [57]. We have not addressed this approach here.

Observe that the system $S$ introduced in Section 3 served exclusively to
enlarge the generality of our measurement model, not affecting the time-energy
uncertainty relations nor the non-unitarity of the evolution. This is the case
because the interaction between system $S$ and clock $B$ was assumed to be
mediated by $T_{B}$. If interactions mediated by $H_{B}$ were allowed, they
would be given by a term that contains the product $O_{S}H_{B}$, where $O_{S}$
is an observable of $S$. This, in turn, causes a shift (generated by $H_{B}$)
associated with $O_{S}$ in the pointer (i.e., time) of clock $B$, as discussed
in Section 3. Although this effect alone would not necessarily lead to non-
unitary dynamics, it could in some cases. Further analysis of generic
interactions that lead to non-unitarity is left for a future work. Here, for
simplicity, as already stated, we have restricted our study to interactions
between $S$ and $B$ modeled by a function of $T_{B}$ on the clock space.

Finally, there still exist many remaining questions concerning the emergence
of non-unitarity observed in our study. An immediate analysis can attribute it
to the asymmetry imposed by the von Neumann measurement, as it was already
discussed throughout the text. However, since this type of evolution always
concerns the evolution of the external clock from the perspective of the
internal one, it may be connected with the decoherence of the internal clock
during the measurement. This is a perspective that deserves further
examination. More precisely, one may investigate the connection between non-
unitarity emergent in this work with decoherence and, in particular, with
Refs. [58] and [59], where time dilation was associated with decoherence.
Moreover, following the approach of Refs. [60] and [61], the concepts of
shared and mutual asymmetry may shed some light on the non-unitary dynamics
manifested in this work. Also, it could be that the clock is not an inertial
frame of reference. In this case, deviations of the metric $\eta$ discussed in
Appendix A from the identity could be used as an indicator of non-inertiality.

## Acknowledgements

We thank the anonymous referees for the thorough reading of this work and the
constructive comments. We are also grateful to Yakir Aharonov, Renato Moreira
Angelo, Pedro Ruas Dieguez, Cristhiano Duarte, Luis Pedro García-Pintos, and
Marcin Nowakowski for helpful discussions and constructive remarks regarding
this text. This research was supported by grant number FQXi-RFP-CPW-2006 from
the Foundational Questions Institute and Fetzer Franklin Fund, a donor-advised
fund of Silicon Valley Community Foundation, by the Israeli Innovation
authority (grants 70002 and 73795), by the Pazy foundation, and by the Quantum
Science and Technology Program of the Israeli Council of Higher Education.

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##  Appendix A Non-unitarity and the Heisenberg picture

Although most of our analysis is centered around the unitary dynamics given by
the external clock $A$, the dynamics — and, in particular, the Heisenberg
equation — from the perspective of the internal clock $B$ was used to show
some level of consistence across the different perspectives. However, the
Heisenberg and the Schrödinger pictures are unitarily equivalent and one may
question whether the use of the Heisenberg equation in the non-Hermitian
scenario is valid. The main goal of this appendix is to justify this use.

When analyzing non-Hermitian Hamiltonians, like the effective ones from clock
$B$’s perspective found in Section 3, a method introduced by Dirac [49] and
further studied, for instance, in Refs. [62, 63, 64] comes in handy. It
entails the replacement of the usual inner product in the Hilbert space by a
product associated with an indefinite Hermitian metric $\eta$, i.e.,

$\langle\psi|\phi\rangle_{\eta}\equiv\langle\psi|\eta|\phi\rangle.$ (35)

The spectrum of $\eta$ is simply composed of $-1$ and $1$.

Then, in addition to the Hermitian conjugate $O^{\dagger}$ of an operator $O$,
it is also possible to define its conjugate $O^{*}$ as

$O^{*}\equiv\eta^{-1}O^{\dagger}\eta.$ (36)

Observables that are self-adjoint in this sense are called $\eta$-pseudo-
adjoint. Physical observables and, in particular, the non-Hermitian
Hamiltonian of interest $H$, must be $\eta$-pseudo-adjoint. With that, it can
be observed that the $\eta$-norm of a state vector is kept constant throughout
the evolution generated by $H$. As a result, the expected value

$\langle O\rangle=\langle\psi|O|\psi\rangle_{\eta}=\langle\psi|\eta
O|\psi\rangle$ (37)

of a non-explicitly time-dependent operator $O$ is such that

$\frac{d}{dt}\langle O\rangle=-\frac{i}{\hbar}\langle[O,H]\rangle,$ (38)

which allows us to recover the Heisenberg picture.

It should be emphasized that the metric $\eta$ diverges from the identity only
if the Hamiltonian is non-Hermitian, like Hamiltonians $H_{2}$ and $H_{4}$. As
a result, Hamiltonians $H_{1}$ and $H_{3}$ lead to the familiar structure of
quantum mechanics.

The main concern about the introduction of indefinite metrics that takes into
consideration some symmetries of the Hamiltonian is the fact that it typically
introduces ghost states, i.e., states with negative norm, which lacks a clear
physical interpretation [62]. However, it could be the case that the system
has some extra symmetry and, when this symmetry is considered, the metric
becomes positive-definite, which transforms the ghost states into standard
states, i.e., states with positive norm.

A particular example of the above is given by $\mathcal{PT}$-symmetric
Hamiltonians, which are invariant under the composition of parity inversion
time reversal [65] — see also [66, 67, 68, 69]. The former is represented by
the parity operator $\mathcal{P}$ and the latter by the time reversal operator
$\mathcal{T}$. A remarkable characteristic of $\mathcal{PT}$-symmetric
Hamiltonians is that their eigenvalues are real. Now, if we consider the
conjugation with respect to the indefinite metric $\mathcal{PT}$, the
resultant theory contains ghost states. This is a problem that puzzled many in
the community for decades. However, it was later observed that systems with
unbroken $\mathcal{PT}$-symmetry have a third symmetry $\mathcal{C}$ such that
the resulting $\mathcal{CPT}$ metric is positive-definite [70].

While Hamiltonians $H_{2}$ and $H_{4}$ are, in general, not
$\mathcal{PT}$-symmetric and their induced metric may lead to ghost particles
a priori, one may wonder if an extended analysis could lead to the
introduction of a positive-definite metric given a measurement function $g$.
If this turns out to indeed be the case, there will be a sense in which the
resultant evolution is still unitary.

##  Appendix B Measurement and disturbance of the system’s dynamics

In this appendix, we study disturbances on system $R$, and in particular on
clock $B$, caused by measurements of energy. The goal is to show that they do
not influence the time-energy uncertainty relations analyzed in Section 4.

To start, we observe that the standard von Neumann measurement of energy does
not disturb the evolution of the observable of interest of the measurement.
This can be seen with the measurement of $H_{R}$ given by $H_{1}$, which
corresponds to the standard treatment of the measurement in quantum mechanics.
In fact, it can be checked that the $t_{A}$-derivative of $H_{R}$ vanishes
and, as a consequence, $[dH_{R}/dt_{A},H_{R}]=0$, which means that the
evolution of $H_{R}$ is not disturbed by the measurement interaction.

Moving on, we study whether the measurement of energy carried out by the
internal system disturbs system $R$’s dynamics from the external clock’s
perspective. In this regard, it can be obtained that

$\frac{d}{dt_{A}}H_{R}=-\frac{i}{\hbar}[H_{R},H_{3}]=-\frac{1}{2}\left\\{g^{\prime}(T_{B}),H_{R}\right\\}P_{I}.$
(39)

and

$\left[\frac{d}{dt_{A}}H_{R},H_{3}\right]=-\frac{i\hbar}{2}\left\\{g^{\prime\prime}(T_{B}),H_{R}\right\\}P_{I},$
(40)

where $\\{\cdot,\cdot\\}$ denotes the anticommutator. This means that the
dynamics of system $R$ is affected by the measurement of its energy by an
internal system. Despite this, by using the Heisenberg equation of motion for
the position $Q_{I}$ of the pointer, it holds that

$Q_{I}(t_{f}^{A})-Q_{I}(t_{i}^{A})=\int_{t_{i}^{A}}^{t_{f}^{A}}\left[\frac{1}{2}\left\\{g(T_{B}(t_{A})),H_{B}\right\\}+g(T_{B}(t_{A}))H_{S}+g(T_{B}(t_{A}))H_{int}(T_{B}(t_{A}))\right]dt_{A},$
(41)

where the measurement was assumed to start and end at $t_{i}^{A}$ and
$t_{f}^{A}$, respectively, depending on the uncertainty of $T_{B}$ from the
perspective of $A$. Then, the shift in the pointer is proportional to the
weighted average energy of system $R$ during the measurement’s interaction had
it not affected the system’s dynamics. To be more precise, the previous result
can be rewritten as

$Q_{I}(t_{f}^{A})-Q_{I}(t_{i}^{A})=K[H_{B}+H_{S}+\bar{H}_{int}],$ (42)

where

$\bar{H}_{int}\equiv\frac{1}{K}\int_{t_{i}^{A}}^{t_{f}^{A}}g(T_{B}(t_{A}))H_{int}(T_{B}(t_{A}))\,dt_{A}.$
(43)

Before concluding, observe that the disturbance to the evolution of system $R$
depends on derivatives of the measurement control function $g$. Most
importantly, $g$ can be chosen in such a way that the net disturbance is null.