Quantizing time: Interacting clocks and systems
Alexander R. H. Smith
1
2
1
Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA
2
Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053, USA
July 3, 2019
probability interpretation of time in which
time evolution is realized through entan-
glement between a clock and a system of
interest. This formalism is based upon
conditioning a solution to the Wheeler-
DeWitt equation on a subsystem of the
Universe, serving as a clock, being in a
state corresponding to a time t. Do-
ing so assigns a conditional state to the
rest of the Universe |ψ
S
(t)i, referred to as
the system. We demonstrate that when
the total Hamiltonian appearing in the
Wheeler-DeWitt equation contains an in-
teraction term coupling the clock and sys-
tem, the conditional state |ψ
S
(t)i satis-
ﬁes a time-nonlocal Schrödinger equation
in which the system Hamiltonian is re-
placed with a self-adjoint integral opera-
tor. This time-nonlocal Schrödinger equa-
tion is solved perturbatively and three ex-
amples of clock-system interactions are ex-
amined. One example considered supposes
that the clock and system interact via
Newtonian gravity, which leads to the sys-
tem’s Hamiltonian developing corrections
on the order of G/c
4
and inversely propor-
tional to the distance between the clock
and system.
1 Introduction
In quantum theory, time enters through its
appearance as a classical parameter in the
Schrödinger equation, as opposed to other phys-
ical quantities, such as position or momentum,
which are associated with self-adjoint operators
and treated dynamically. Operationally, this no-
tion of time is what is measured by the clock on
Alexander R. H. Smith: alexander.r.smith@dartmouth.edu
the wall of an experimenter’s laboratory. The
clock is considered to be a large classical object,
not subject to quantum ﬂuctuations, and does
not interact with the system whose evolution it
is tracking. Quantum theory describes the evo-
lution of physical systems with respect to such a
clock.
However, the canonical quantization of grav-
ity leads to the Wheeler-DeWitt equation: phys-
ical states are annihilated by the Hamiltonian of
the theory. In other words, the wave function of
the Universe which includes the experimenter’s
clock, the system the experimenter is interested
in, and everything else is in an eigenstate of
its Hamiltonian. Combined with the Schrödinger
equation, the Wheeler-DeWitt equation dictates
that the physical states of the theory do not
evolve in time (i.e., the Hamiltonian of the the-
ory does not generate time translations of the
physical states with respect to an external time).
How then do we explain the time evolution we
see around us? This dilemma constitutes one as-
pect of the problem of time, and is sometimes
phrased as the problem of ﬁnding a background-
independent quantum theory [1, 2].
A necessary requirement for any quantum the-
ory of gravity is to answer this question and
explain how the familiar Schrödinger equation
comes about from the Wheeler-DeWitt equation.
The conditional probability interpretation of time
oﬀers an answer. As introduced by Page and
Wootters [36], the conditional probability in-
terpretation deﬁnes the state of a system at a
time t as a solution to the Wheeler-DeWitt equa-
tion conditioned on a subsystem of the Universe,
serving as a clock, to be in a state correspond-
ing to the time t. Given an appropriate choice
for the Hamiltonian of the Universe and choice
of clock, one ﬁnds this conditional state of the
system satisﬁes the Schrödinger equation.
This interpretation of time was initially criti-
Accepted in Quantum 2019-06-04, click title to verify 1
arXiv:1712.00081v3 [quant-ph] 2 Jul 2019
cized by Kuchař [2, 6], who argued that it was
unable to reproduce the correct two-time corre-
lation functions, i.e., supposing a system was ini-
tially prepared in some state, what is the prob-
ability of ﬁnding the system in a diﬀerent state
at a later time? This criticism has since been
overcome in two diﬀerent ways: ﬁrst, by correctly
formulating the two-time correlation functions in
terms of physical observables [7], and second, by
modelling the measurement of the two-time cor-
relation function as two successive von Neumann
measurements [8].
Recently, the conditional probability interpre-
tation of time has received considerable attention.
Gambini et al. [911] have demonstrated that the
conditional probability interpretation can result
in a fundamental decoherence mechanism and ex-
plored the consequences of this fact in relation
to the black hole information loss problem [12].
Leon and Maccone [13] have shown that this in-
terpretation overcomes Pauli’s objection to con-
structing a time operator in quantum mechanics.
More recently, the the conditional probability in-
terpretation of time has been studied in the con-
text of quantum reference frames and quantum
resource theory of coherence [14, 15]. Further-
more, conditional probabilities were employed to
illustrate a novel quantum time dilation eﬀect in-
duced by nonclassical states of relativistic parti-
cles whose internal degrees of freedom serve as
clocks [16]. Others have applied the formalism to
a number of diﬀerent systems and commented on
various aspects of the proposal [1721].
ditional probability interpretation of time to take
into account the possibility that the system be-
ing employed as a clock interacts with a system
whose evolution the clock is tracking. As grav-
ity couples everything, including clocks, this ex-
tension is necessary if the conditional probability
interpretation of time is to be applied in a quan-
tum gravitational setting. We ﬁnd that taking
into account a possible clock-system interaction
within the conditional probability interpretation
of time results in a time-nonlocal modiﬁcation to
the Schrödinger equation.
We begin in Sec. 2 by reviewing the timeless
formulation of classical mechanics [22, 23] and its
quantization. In Sec. 3 we introduce the con-
ditional probability interpretation of time and
its generalization to interacting clocks and sys-
Figure 1: A clock depicted in green is used to track the
evolution of a system depicted in red. The conditional
probability interpretation is extended to allow for the
clock and system to interact, this interaction is depicted
by the grey springs.
tems. This results in a modiﬁed time-nonlocal
Schrödinger equation, for which we derive a series
solution. In Sec. 4 we give three explicit exam-
ples of clock-system interactions: the ﬁrst leads
to a time-dependent modiﬁcation to the system
Hamiltonian, the second is of a clock and sys-
tem interacting gravitationally, and the third ex-
amines the case when the clock and system are
two-level systems. We conclude in Sec. 5 with a
summary of our results and comment on future
directions of research.
2 The Hamiltonian constraint in clas-
sical and quantum mechanics
Both classical and quantum mechanics describe
the evolution of the state, or equivalently the ob-
servables, of a physical system in time. How-
ever, general relativity demands that time itself
be treated like any other physical system, that is,
time should be treated dynamically. In this con-
text, the dynamics of both classical and quantum
mechanics describe relations between two physi-
cal systems: a clock, which indicates the time,
and everything else. In this section we present a
formulation of classical mechanics in which time
is treated dynamically and on equal footing with
Accepted in Quantum 2019-06-04, click title to verify 2
the system whose evolution we are interested in.
We then pass over to the corresponding quantum
theory à la Dirac [24].
Consider a system S described by the action
S =
Z
t
2
t
1
dt L
S
q, q
0
,
where L
S
(q, q
0
) is the Lagrangian associated with
S, q = q(t) denotes a set of generalized coordi-
nates describing S, and q
0
= q
0
(t) denotes the
diﬀerentiation of these coordinates with respect
to t.
Let us introduce an integration parameter τ
and promote t to a dynamical variable t(τ),
which we associate with the reading of a clock C.
Through application of the chain rule, the action
may be expressed as
S =
Z
τ
2
τ
1
˙
tL
S
q, ˙q/
˙
t
=
Z
τ
2
τ
1
L
q, ˙q,
˙
t
,
where L
q, ˙q,
˙
t
:
=
˙
tL
S
q, ˙q/
˙
t
is the Lagrangian
describing both C and S and the dot denotes dif-
ferentiation with respect to τ.
The Hamiltonian associated with L
q, ˙q,
˙
t
is
obtained by a Legendre transformation with re-
spect to both ˙q and
˙
t
˜
H = p
t
˙
t + p
q
˙q L(q, ˙q,
˙
t) =
˙
t (p
t
+ H
S
) , (1)
where H
S
:
= p
q
q
0
L
S
(q, q
0
) is the Hamiltonian
associated with L
S
(q, q
0
) and we have used the
fact that the momentum conjugate to q deﬁned
by L
q, ˙q,
˙
t
is
p
q
:
=
L
q, ˙q,
˙
t
˙q
=
˙
t
L
S
q, ˙q/
˙
t
˙q/
˙
t
1
˙
t
=
L
S
(q, q
0
)
q
0
,
which coincides with the momentum conjugate to
q deﬁned by L
S
(q, q
0
). The momentum conjugate
to t is
p
t
:
=
L
q, ˙q,
˙
t
˙
t
= L
S
q, q
0
q
0
p
q
= H
S
. (2)
In light of Eq. (2), we see that the term inside
the brackets in Eq. (1) is constrained to vanish
H
:
= p
t
+ H
S
0, (3)
where indicates weak equality in the sense of
Dirac [24]. We will refer to H as the total Hamil-
tonian (a.k.a. the super Hamiltonian or Hamilto-
nian constraint) as it describes both C and S.
It is natural to ask if the total Hamiltonian
given in Eq. (3) is the most general possible. The
answer is no. The total Hamiltonian can diﬀer in
two important ways:
int
= H
int
(t, p
t
, q, p
q
)
may be included in the total Hamiltonian,
which couples C and S; this term will be
referred to as the interaction Hamiltonian.
2. The momentum p
t
may be replaced by a
function of the conjugate variables associ-
ated with C, which we will refer to as the
clock Hamiltonian and denote by H
C
=
H
C
(t, p
t
). Note that in general C may be
a composite system and have more than just
one pair of conjugate variables.
Accounting for these generalizations, the most
general total Hamiltonian is
H = H
C
+ H
S
+ H
int
0. (4)
Motivating these generalizations is our best
theory of time: general relativity. The Hamil-
tonian formulation of general relativity does not
admit a total Hamiltonian of the form given in
Eq. (3). Total Hamiltonians that are linear in
one of the conjugate momenta, like Eq. (3), in-
dicate there is a preferred time variable in the
theory [25], and this structure is not present in
general relativity. Further, gravity couples every-
thing, including a clock and the system whose
evolution it is tracking. Therefore, in a gravi-
tational setting, we should expect an interaction
Hamiltonian H
int
to appear in the total Hamil-
tonian H coupling C and S. We note that the
inclusion of such an interaction Hamiltonian may
result in a total Hamiltonian that is nonlinear
in the conjugate momentum associated with the
clock, even if the clock Hamiltonian is propor-
tional to this momentum.
We now wish to quantize the theory described
by the total Hamiltonian given in Eq. (4). To do
so, we follow the prescription given by Dirac [24].
We associate with C and S the Hilbert spaces
H
C
and H
S
, respectively. The total Hamiltonian
H becomes an operator acting on the kinematical
Hilbert space H
kin
' H
C
⊗H
S
, and the constraint
in Eq. (4) becomes
H |Ψii =
H
C
I
S
+ I
C
H
S
+ H
int
|Ψii
= 0, (5)
where I
C
and I
S
denote the identity operators
on H
C
and H
S
, respectively. Within the con-
text of quantum gravity, Eq. (5) is referred to
as the Wheeler-DeWitt equation. The double
Accepted in Quantum 2019-06-04, click title to verify 3
ket notation is used to remind us that |Ψii is
a state of both the clock and system. States
|Ψii satisfying the constraint are in the physical
Hilbert space H
phy
; states in the physical Hilbert
space |Ψii H
phy
will be referred to as physical
states. To completely specify the physical Hilbert
space H
phy
one must also choose an inner prod-
uct on H
phy
, which we will do in the following
section.
In general, the physical states evolve unitar-
ily with respect to an external time, this evo-
lution being generated by the total Hamilto-
nian. However, in totally constrained theories,
such as the one deﬁned by Eq. (5), the physical
states are annihilated by the total Hamiltonian,
H |Ψii = 0, and therefore do not evolve with re-
spect to any external time. The question then
arises, how do we recover the dynamics we see
around us from the frozen state |Ψii? How does
the Schrödinger equation come about from the
constraint H |Ψii = 0?
These questions constitute one aspect of the
problem of time in quantum gravity [1, 2]. The
conditional probability interpretation of time of-
fers a way to reconcile the fact that the phys-
ical states are frozen with the time evolution
described by quantum theory. In this formal-
ism the dynamics of S are encoded in entangle-
ment shared between C and S in the physical
state |Ψii. The outcome of a measurement of an
observable on S at a speciﬁc time t, is interpreted
as a measurement of the physical state |Ψii con-
ditioned on the clock being in a state correspond-
ing to the time t.
3 The conditional probability interpre-
tation
We now introduce the conditional probability in-
terpretation of time for theories described by the
general total Hamiltonian given in Eq. (5), which
includes the possibility of an interaction Hamil-
tonian H
int
coupling C and S. We will intro-
duce the state of the system at time t by condi-
tioning a solution to the constraint |Ψii on the
clock being in a state corresponding to the time t.
This state of the system will be seen to satisfy
the Schrödinger equation in the limit where H
int
vanishes.
3.1 The modiﬁed Schrödinger equation
In the classical theory speciﬁed by the total
Hamiltonian given in Eq. (3), time is deﬁned op-
erationally as the outcome of a measurement of
the phase space variable t associated with a clock
governed by the Hamiltonian H
C
= p
t
. In this
case, the variable t is canonically conjugate to
the clock Hamiltonian.
The quantized version of this notion of time is
to deﬁne time as a measurement of a time op-
erator T on the clock Hilbert space H
C
' L
2
(R)
which is canonically conjugate to the clock Hamil-
tonian H
C
, (~ = 1)
[T, H
C
] = i. (6)
In other words, states of the clock indicating dif-
ferent times correspond to eigenstates |ti of the
time operator T , and the associated eigenvalue t
is the time indicated by the clock. Employing the
Baker-Campbell-Hausdorﬀ formula, we see that
as a consequence of the commutation relation in
Eq. (6), H
C
generates translations of T
e
iH
C
s
T e
iH
C
s
= T sI
C
. (7)
Resolving the identity on H
C
as I
C
=
R
dt |tiht|
and making use of the spectral representation of
the time operator T =
R
dt t |tiht|, Eq. (7) may
be expressed as
Z
dt te
iH
C
s
|tiht|e
iH
C
s
=
Z
dt (t s) |tiht|
=
Z
dt t |t + siht + s|,
or
e
iH
C
s
|ti = |t + si, (8)
up to an overall phase.
Following Page and Wootters [3, 4], we will take
Eq. (8) to deﬁne the states of the clock corre-
sponding to a particular instant of time t, that
is, |ti
:
= e
iH
C
t
|t
0
i where |t
0
i H
C
; all other
states of the clock are not to be associated with
a deﬁnite value of t. We refer to the set of states
{|ti |t} as the clock states. In what follows,
we will also demand that the clock states form a
resolution of the identity I
C
R
dt |tiht|, which
ensures that they deﬁne a normalized positive op-
erator valued measure. The translation property
expressed in Eq. (8) insures that this POVM is
Accepted in Quantum 2019-06-04, click title to verify 4
covariant with respect to the clock Hamiltonian
H
C
[2629]
This deﬁnition of T does not necessarily im-
ply that the clock states are eigenstates of a time
operator T canonically conjugate to the clock
Hamiltonian H
C
. However, when this is the
case, the time operator T and clock Hamilto-
nian H
C
are, by the Stone-von Neumann theo-
rem [30], unitarily equivalent to the position and
momentum operators on the real line. In this case
the clock behaves perfectly in the sense that the
clock states are orthogonal, ht|t
0
i = δ(t t
0
), so
as they are perfectly distinguishable, and there
is no chance that the clock will run backwards
in time, i.e., ht
0
|e
iH
C
˜
t
|ti = 0 for all
˜
t > 0 if
t > t
0
[1, 7, 31]. The ﬁrst two examples consid-
ered in Sec. 4 employ such a perfect clock, while
the third example does not.
We now deﬁne the state of the system at time t
as a solution to the constraint in Eq. (5) condi-
tioned on the clock being in the state |ti
|ψ
S
(t)i
:
=
ht| I
S
|Ψii, (9)
where |ψ
S
(t)i H
S
. The state |ψ
S
(t)i should be
thought of as the time-dependent state of the sys-
tem in the conventional formulation of quantum
mechanics. Note that with this deﬁnition of the
system state we may express the physical state
|Ψii as
|Ψii =
Z
dt |tiht| I
S
|Ψii
=
Z
dt |ti|ψ
S
(t)i. (10)
This state is expressed pictorially in Fig. 2.
As mentioned above, we need to choose an in-
ner product for the physical Hilbert space H
phy
.
We will choose this inner product to be
hhΨ|Φii
phy
:
= hhΨ|
|tiht| I
S
|Φii, (11)
for two states |Ψii and |Φii in H
phy
, and de-
mand that the inner product is independent of
the choice
1
of t. We will also require that states
1
We should emphasize this is a choice of inner prod-
uct and normalization of the physical states, which may
severely reduce the size of the physical Hilbert space. Fur-
thermore, it may not be necessary to preserve the prob-
abilistic interpretation of the system state. For example,
the probabilistic interpretation may only be applicable in
some limit, and it is the task of the physicist to explain
|ψ⟩⟩
|ψ
S
(t
0
)
t
t
0
Figure 2: The rectangular prism is a pictorial representa-
tion of the state |Ψii =
R
dt |ti|ψ
S
(t)i. The horizon-
tal axis represents the Hilbert space associated with the
clock H
C
and the directions orthogonal to the horizontal
axis represent the Hilbert space of the system state H
S
.
The system state |ψ
S
(t
0
)i at the time t
0
is obtained by
conditioning |Ψii on the clock being in the state |t
0
i
and pictorially represented by a slice of the rectangular
prism. Adapted from Giovannetti et al. [8].
in H
phy
are normalized with respect to this inner
product
1 = hhΨ|Ψii
phy
=
Z
dt
0
ht
0
|hψ
S
(t
0
)|
|tiht| I
S
×
Z
dt
00
|t
00
i|ψ
S
(t
00
)i
= hψ
S
(t)|ψ
S
(t)i. (12)
From the above equation, we see that this choice
of inner product and normalization of the physi-
cal states ensures that the system state |ψ
S
(t)i is
properly normalized at all times. Consequently,
we can maintain the usual probabilistic interpre-
tation of |ψ
S
(t)i.
Let us observe how |ψ
S
(t)i changes with the
parameter t labeling the clock states by acting
how this limit comes about. To quote DeWitt on this
point [23, 32]:
“. . . one learns that time and probability are
phenomenological concepts.
And Kiefer’s clariﬁcation of DeWitt’s statement [23]:
“The reference to probability refers to the
‘Hilbert-space’ problem, which is intimately
connected with the ‘problem of time’. If time
is absent, the notion of a probability conserved
in time does not make much sense; the tradi-
tional Hilbert-space structure was designed to
implement the probability interpretation, and
its fate in a timeless world thus remains open.
Accepted in Quantum 2019-06-04, click title to verify 5
on both sides of Eq. (9) with id/dt:
i
d
dt
|ψ
S
(t)i = i
d
dt
ht| I
S
|Ψii
= ht|
H
C
I
S
|Ψii
= ht|
H I
C
H
S
H
int
|Ψii.
Using the fact that H |Ψii = 0 we ﬁnd |ψ
S
(t)i
satisﬁes
i
d
dt
|ψ
S
(t)i = H
S
|ψ
S
(t)i + ht|H
int
|Ψii. (13)
Inserting a resolution of the identify on H
C
in
terms of the clock states I
C
=
R
dt |tiht| between
H
int
and |Ψii in the second term of Eq. (13) and
using the deﬁnition of the system state in Eq. (9),
we ﬁnd
i
d
dt
|ψ
S
(t)i = H
S
|ψ
S
(t)i +
Z
dt
0
K(t, t
0
) |ψ
S
(t
0
)i,
(14)
where K(t, t
0
)
:
= ht|H
int
|t
0
i is an operator acting
on H
S
. We will refer to Eq. (14) as the modi-
ﬁed Schrödinger equation. When the interaction
Hamiltonian vanishes, the modiﬁed Schrödinger
equation reduces to the usual Schrödinger equa-
tion.
The second term on the right-hand side of
Eq. (14) is a linear integral operator on H
S
with
integration kernel K(t
0
, t); we will denote this in-
tegral operator as H
K
and its action as
H
K
|ψ
S
(t)i
:
=
Z
dt
0
K(t, t
0
) |ψ
S
(t
0
)i.
Note
2
, H
K
is a self-adjoint operator if and only if
K(t, t
0
)
= K(t
0
, t), which is seen to be true from
the deﬁnition of K(t, t
0
)
K(t, t
0
)
=
ht|H
int
|t
0
i
= ht
0
|H
int
|ti
= K(t
0
, t),
and therefore H
K
With the deﬁnition of H
K
, let us write the
modiﬁed Schrödinger equation in a more sugges-
tive form
i
d
dt
|ψ
S
(t)i =
H
S
+ H
K
|ψ
S
(t)i.
Expressed this way, the modiﬁed Schrödinger
equation can be seen as the ordinary Schrödinger
2
A proof of this can be found on pages 197-198 of [33].
equation with the system Hamiltonian H
S
re-
placed with the self-adjoint integral operator
H
S
+ H
K
.
The modiﬁed Schrödinger equation is nonlo-
cal in time, which means that to verify whether
|ψ
S
(t)i is a solution requires knowledge of |ψ
S
(t)i
at all times t, rather than a small neighbour
around t as is the case for the usual Schrödinger
equation. This is due to the presence of the inte-
gral operator H
K
whose kernel K(t, t
0
) does not
vanish when t 6= t
0
.
To maintain the usual probabilistic interpre-
tation of the system state |ψ
S
(t)i, the evolution
described by the modiﬁed Schrödinger equation
must preserve its norm
hψ
S
(t)|ψ
S
(t)i = 1, t
d
dt
hψ
S
(t)|ψ
S
(t)i = 0.
In other words, the time evolution described by
the modiﬁed Schrödinger equation is an isom-
etry [34]. Evaluating
d
dt
hψ
S
(t)|ψ
S
(t)i using
Eq. (14) yields the condition
Z
dt
0
Im
h
hψ
S
(t)|K(t, t
0
)|ψ
S
(t
0
)i
i
= 0. (15)
Since K(t, t
0
)
:
= ht|H
int
|t
0
i, Eq. (15) is a con-
dition on the interaction Hamiltonian such that
the modiﬁed Schrödinger equation preserves the
norm of the system state.
We now show that Eq. (15) is satisﬁed so long
as the physical state associated with the clock
and system is normalized according to Eq. (12).
Using Eq. (10) and the deﬁnition of K(t, t
0
), the
left-hand side of Eq. (15) can be expressed as
hhΨ|[H
int
, |tiht| I
S
] |Ψii
= hhΨ|[H, |tiht| I
S
] |Ψii
hhΨ|[H
C
I
S
, |tiht| I
S
] |Ψii
hhΨ|[I
C
H
S
, |tiht| I
S
] |Ψii
= hhΨ|
[H
C
, |tiht|] I
S
|Ψii
= i
d
dt
hhΨ|
|tiht| I
S
|Ψii
= i
d
dt
hhΨ|Ψii
phy
,
which vanishes if the physical inner product is
independent of t. Therefore, the condition given
in Eq. (15) is equivalent to demanding that the
physical inner product is independent of t, as de-
manded by Eq. (12).
From Eq. (10) we see that |Ψii is an entangled
state of the clock and system. This entanglement
Accepted in Quantum 2019-06-04, click title to verify 6
encodes the time evolution of the system state
|ψ
S
(t)i generated by the modiﬁed Schrödinger
equation. This is somewhat analogous to the sit-
uation in general relativity. The state |Ψii is
analogous to the 4-dimensional spacetime met-
ric neither evolve with respect to an external
time. However, one can foliate the 4-dimensional
spacetime by spacelike hypersurfaces (by choos-
ing a clock), and then the 4-dimensional metric
encodes the evolution from one hypersurface to
the next of the induced 3-metric and its conju-
gate momentum on these hypersurfaces. This
evolution is analogous to the evolution of the
system state |ψ
S
(t)i governed by the modiﬁed
Schrödinger equation.
3.2 Solving the modiﬁed Schrödinger equation
To further explore the consequences of an inter-
action Hamiltonian H
int
:
= λ
¯
H
int
, where we have
made the interaction strength λ R explicit,
we seek a series solution in λ to the modiﬁed
Schrödinger equation.
Suppose the modiﬁed Schrödinger equation can
be solved for |ψ
S
(t)i in terms of a time evolu-
tion operator V (t, t
0
), so that the solution may
be given as
|ψ
S
(t)i = V (t, t
0
) |ψ
S
(t
0
)i,
where |ψ
S
(t
0
)i H
S
is the state of the system
at the time t = t
0
and V (t
0
, t
0
) = I
S
. Suppose
V (t, t
0
) may be expanded in powers of λ as
V (t, t
0
) =
X
n=0
λ
n
V
n
(t, t
0
).
Upon substituting |ψ
S
(t)i = V (t, t
0
) |ψ
S
(t
0
)i into
the modiﬁed Schrödinger equation and equating
terms at equal order in λ, we ﬁnd the operator
V
0
(t, t
0
) satisﬁes
i
d
dt
V
0
(t, t
0
) = H
S
V
0
(t, t
0
),
which implies that
V
0
(t, t
0
) = e
iH
S
(tt
0
)
, (16)
and we see that V
0
(t, t
0
) is the usual Schrödinger
time evolution operator U(t, t
0
)
:
= e
iH
S
(tt
0
)
.
The higher order operators V
n
(t, t
0
) satisfy
i
d
dt
V
n
(t, t
0
) = H
S
V
n
(t, t
0
) +
Z
dt
0
¯
K(t, t
0
)V
n1
(t
0
, t
0
), (17)
where
¯
K(t, t
0
)
:
= ht|
¯
H
int
|t
0
i. The solution to Eq. (17) is given by the recurrence relation
V
n
(t, t
0
) = iU(t, t
0
)
Z
t
t
0
ds U(s, t
0
)
Z
du
¯
K(s, u)V
n1
(u, t
0
). (18)
Using Eqs. (16) and (18), the time evolution operator V (t, t
0
) may be expanded to leading order in λ
as
V (t, t
0
) = U(t, t
0
)
I
S
Z
t
t
0
ds U(s, t
0
)
Z
du
¯
K(s, u)U (u, t
0
) + O
λ
2
. (19)
Equation (19) is analogous to the Dyson series,
and reduces to the Dyson series when the inte-
gration kernel is of the form given in Eq. (21).
Before closing this section we note that given a
solution to the modiﬁed Schrödinger equation in
terms of the time evolution operator V (t, t
0
), the
modiﬁed Schrödinger equation may be expressed
as
i
d
dt
|ψ
S
(t)i =
H
S
+ H
mod
(t)
|ψ
S
(t)i,
where we have deﬁned the modiﬁed Hamiltonian
H
mod
(t)
:
=
Z
dt
0
K(t, t
0
)V (t
0
, t).
It immediately follows from the condition ex-
pressed in Eq. (15) that H
mod
(t) is self-
adjoint, and thus the evolution generated by
Accepted in Quantum 2019-06-04, click title to verify 7
the modiﬁed Schrödinger equation is unitary.
The consequence of this is that the modiﬁed
Schrödinger equation can be recast as the usual
Schrödinger equation deﬁned by the Hamiltonian
H
S
+ H
mod
(t). Employing the series solution for
V (t, t
0
) given above, H
mod
(t) can be evaluated
perturbatively.
4 Examples of clock-system interac-
tions
In this section we examine three diﬀerent clock-
system interactions and examine the ensuing re-
lational dynamics of the system that result.
system Hamiltonians
Consider the following interaction Hamiltonian
H
int
=
X
i
f
i
(T ) S
i
, (20)
where f
i
(T )
:
=
R
dt f
i
function of the time operator T satisfying Eq. (6)
and S
i
is a self-adjoint operator on H
S
. The
operator K(t, t
0
) associated with this interaction
Hamiltonian is
K(t, t
0
)
:
= ht|H
int
|t
0
i
=
X
i
ht|f
i
(T )|t
0
iS
i
=
X
i
ht|
Z
dt
00
f
i
(t
00
) |t
00
iht
00
|
|t
0
iS
i
= δ(t t
0
)
X
i
f
i
(t)S
i
. (21)
One can easily verify that this K(t, t
0
) satisﬁes
the condition in Eq. (15). Substituting Eq. (21)
into the modiﬁed Schrödinger equation and sim-
plifying, we ﬁnd
i
d
dt
|ψ
S
(t)i =
"
H
S
+
X
i
f
i
(t)S
i
#
|ψ
S
(t)i,
which we recognize as the ordinary Schrödinger
equation with the system Hamiltonian H
S
re-
placed with the time-dependent Hamiltonian
H
S
(t) = H
S
+
X
i
f
i
(t)S
i
.
Therefore, an interaction Hamiltonian of the form
given in Eq. (20) results in a time-dependent evo-
lution of the system state.
This example demonstrates that interactions of
the form given in Eq. (20) result in a relational
dynamics generated by a time dependent Hamil-
tonian H
S
(t). The usual interpretation of such
Hamiltonians is that they arise from an exter-
nal agent exerting some classical control over the
system (e.g., turning a nob which increases the
strength of a magnetic ﬁeld). This demonstrates
that clock-system couplings of the form given in
Eq. (20) can model such time-dependent classical
control.
4.2 Gravitationally interacting clocks and sys-
tems
Let us now consider the case in which the clock
and system are coupled through Newtonian grav-
ity as described by the interaction Hamiltonian
H
int
=
Gm
C
M
S
d
=
G
c
4
d
H
C
H
S
,
where we have associated the mass of the clock
m
C
and system m
S
with their respective Hamil-
tonians. Suppose that the clock Hamiltonian is
given by its momentum operator H
C
= P
C
, as
discussed at the beginning of Sec. 3.1. In this
case the integration kernel appearing in the mod-
iﬁed Schrödinger equation deﬁned by the above
interaction Hamiltonian evaluates to
K(t, t
0
)
:
= ht|H
int
|t
0
i
=
G
c
4
d
ht|P
C
|t
0
iH
S
=
G
c
4
d
Z
dp p ht|pihp|t
0
i
H
S
=
G
c
4
d
1
2π
Z
dp pe
ip(t
0
t)
H
S
=
G
c
4
d
0
(t
0
t)H
S
. (22)
where |pi are eigenkets of P
C
with eigenvalue p
and we have made use of the spectral decomposi-
tion of the clock Hamiltonian P
C
=
R
dp p |pihp|
in arriving at the third equality above.
Substituting Eq. (22) into the modiﬁed
Accepted in Quantum 2019-06-04, click title to verify 8
Schrödinger, Eq. (14), yields
i
d
dt
|ψ
S
(t)i = H
S
|ψ
S
(t)i
G
c
4
d
i
Z
dt
0
δ
0
(t
0
t)H
S
|ψ
S
(t
0
)i
= H
S
|ψ
S
(t)i
+
G
c
4
d
i
Z
dt
0
δ(t
0
t)H
S
d
dt
0
|ψ
S
(t
0
)i
= H
S
|ψ
S
(t)i +
G
c
4
d
iH
S
d
dt
|ψ
S
(t)i,
which upon rearranging gives
i
d
dt
|ψ
S
(t)i =
H
S
I
S
G
c
4
d
H
S
|ψ
S
(t)i
=
H
S
+
G
c
4
d
H
2
S
+ O
G
2
c
8
d
2
|ψ
S
(t)i.
(23)
From Eq. (23) we see that the eﬀect of a gravita-
tional interaction coupling the clock and system is
such that the system Hamiltonian gets corrected
at order G/c
4
and the strength of this correc-
tion is inversely proportional to the distance d
between the clock and system.
4.3 Qubit clock and system
In this example we consider both the clock and
system to be two-level systems, H
C
' C
2
and
H
S
' C
2
, with Hamiltonians H
C
= σ
z
and
H
S
= σ
z
. As deﬁned below Eq. (8), the clock
state indicating the time t is |ti
:
= e
iH
C
t
|t
0
i,
where we choose |t
0
i
:
=
1
2
(|0i + |1i). The iden-
tity on H
C
may be resolved in terms of these clock
states as
I
C
=
π
Z
2π
0
dt
0
|t
0
iht
0
|.
Suppose that the clock and system are coupled
through the interaction Hamiltonian
H
int
= a (σ
x
σ
x
σ
y
σ
y
)
+ b (σ
z
σ
z
+ I
C
I
S
) , (24)
where a, b R.
In this case, the conditional state of the system
|ψ
S
(t)i satisﬁes the modiﬁed Schrödinger equa-
tion
i
d
dt
|ψ
S
(t)i = H
S
|ψ
S
(t)i
+
π
Z
dt
0
K(t, t
0
) |ψ
S
(t)i,
where
K(t, t
0
)
:
= ht|H
int
|t
0
i
= a
e
iΩ(t+t
0
)
|0ih1| + e
iΩ(t+t
0
)
|1ih0|
+ b
e
iΩ(tt
0
)
|0ih0| + e
iΩ(tt
0
)
|1ih1|
.
(25)
This modiﬁed Schrödinger equation can be solved
perturbatively using Eq. (18). The ﬁrst order cor-
rection to the propagator is
V
1
(t, t
0
)
:
= iU(t, t
0
)
×
Z
t
t
0
ds U(s, t
0
)
Z
du K(s, u)U(u, t
0
),
where U (t, t
0
)
:
= e
iσ
z
(tt
0
)
. Given K(t, t
0
) in
Eq. (25) the ﬁrst integrand can be evaluated and
shown to vanish. Thus V
1
(t, t
0
) = 0, and then
by Eq. (18) the nth order contribution to the
propagator also vanishes, V
n
(t, t
0
) = 0. We con-
clude that the coupling between the clock and
and system described by the interaction Hamil-
tonian in Eq. (24) has no eﬀect on the rela-
tional dynamics between the clock and system,
and thus the conditional state |ψ
S
(t)i also satis-
ﬁes the usual Schrödinger equation, which in this
case is i
d
dt
|ψ
S
(t)i = σ
z
|ψ
S
(t)i.
In this example we have the luxury of solv-
ing the associated constraint equation, Eq. (5),
for the physical state |Ψii directly. With the
clock, system, and interaction Hamiltonians spec-
iﬁed above, the most general solution to Eq. (5)
is
|Ψii =
2
cos θ |10i + sin θe
|01i
,
where θ [0, π], φ [0, 2π), and the physical
state |Ψii is normalized according to Eq. (12).
Then, the conditional state at time t, as deﬁned
in Eq. (9), is
|ψ
S
(t)i
:
=
ht| I
S
|Ψii
=
1
2
e
it
h0| + e
it
h1|
I
S
×
2
cos θ |10i + sin θe
|01i
= e
iH
S
t
|ψ
S
(t
0
)i, (26)
where |ψ
S
(t
0
)i
:
= cos θ |0i + sin θe
|1i is an ar-
bitrary state in H
S
. Note that in this exam-
ple the fact that |ψ
S
(t
0
)i can be an arbitrary
Accepted in Quantum 2019-06-04, click title to verify 9
state in H
S
follows from the fact that the dimen-
sion of the null space of the total Hamiltonian
H = H
C
I
S
+ I
C
H
S
+ H
int
is equal to the
dimension of the system Hilbert space H
S
' C
2
,
dim [Null (H)] = dim [H
S
] = 2. Equation (26)
conﬁrms the conclusion stated in the previous
paragraph.
5 Conclusions and outlook
probability interpretation of time to account for
an interaction between a clock and the system
whose evolution it is tracking. This is a neces-
sary consideration if the conditional probability
interpretation is to be applied to any model of
quantum gravity because gravity couples every-
thing, including any clock and system of interest.
In the case of an interaction between the clock
and system, we ﬁnd the conditional state of the
system |ψ
S
(t)i satisﬁes a modiﬁed time-nonlocal
Schrödinger equation. A series solution in the in-
teraction strength to this modiﬁed Schrödinger is
derived.
In Sec. IV we gave three explicit examples
of clock system interactions. In the ﬁrst exam-
ple we found that when the interaction Hamil-
tonian H
int
is of the form given in Eq. (20), the
modiﬁed Schrödinger equation becomes the usual
Schrödinger equation with a time-dependent
Hamiltonian dependent on H
int
. In the limit
when the interaction between the clock and sys-
tem vanishes, H
int
= 0, the modiﬁed Schrödinger
equation reduces to the ordinary Schrödinger
equation. In the second example the clock and
system are coupled through a Newtonian gravi-
tational interaction. In such a case we ﬁnd that
the modiﬁed Schrödinger equation reduces to the
usual Schrödinger equation with a modiﬁed sys-
tem Hamiltonian. This example illustrates one
way in which a gravitational interaction between
the clock and system could aﬀect the induced re-
lational dynamics and oﬀers a toy model in which
the primary motivation for our investigation is
realized (gravity couples to everything). Finally,
the third example examines the situation when
both the clock and system are two-level systems.
We consider a clock-system interaction for which
the modiﬁed Schrödinger equation is equivalent
to the usual Schrödinger equation, demonstrating
that an interaction between the clock and system
doesn’t necessarily modify the relational dynam-
ics governing the conditional state. This example
also illustrates the case when the clock states are
not eigenstates of an operator canonically conju-
gate to the clock Hamiltonian.
As it stands, the conditional probability inter-
pretation of time does not specify a unique choice
of clock states, and thus does not address the
multiple choice problem [2] (although [19] oﬀers
a possible resolution). In case of a perfect clock,
as discussed below Eq. (8), the clock states are
completely delocalized in the energy basis of the
clock, that is, they are eigenstates of the operator
canonically conjugate to the clock Hamiltonian.
These clock states are maximally asymmetric un-
der the action of the group generated by the clock
Hamiltonian, which suggests that an appropri-
ate ﬁgure of merit for choosing the clock states
may come from the resource theory of asymme-
try [35, 36].
Future work will focus on realizing speciﬁc ex-
amples of the developed formalism. It should be
noted that although the results presented were in
the context of nonrelativistic quantum mechan-
ics, in principle, there is nothing stopping the
application of the conditional probability inter-
pretation to relativistic quantum ﬁeld theory and
theories of quantum gravity.
Another avenue to explore is the possibility of
replacing the inﬁnite dimensional clock Hilbert
space with a ﬁnite dimensional one. The canon-
ical commutation relations between the clock
Hamiltonian and time operator in Eq. (6) will no
longer be satisﬁed. However, it is still possible to
deﬁne a self-adjoint time operator that satisﬁes
an approximate canonical commutation relation
with the clock Hamiltonian [3739]. It will be in-
teresting to explore the role the dimension of the
clock Hilbert space plays in a classical limit.
Another task will be to generalize the above
formalism to mixed states. This will allow for
the investigation of how an interaction between
the system and clock eﬀects the fundamental de-
coherence mechanism discussed by Gambini et al.
[912].
Acknowledgments
We wish to thank Felix Beaudoin, Leigh Norris,
Lorenza Viola, Edward Vrscay, and the attendees
at the 2017 Spacetime and Information workshop
Accepted in Quantum 2019-06-04, click title to verify 10
for useful discussions. We also thank Hilary Sny-
der for producing Figure 1. This work was sup-
ported by the Natural Sciences and Engineering
Research Council of Canada and the Dartmouth
College Society of Fellows.
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