Relativistic quantum clocks observe classical and quantum time dilation

Alexander R. H. Smith1, ∗ and Mehdi Ahmadi2, †

1Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA2Department of Mathematics and Computer Science,

Santa Clara University, Santa Clara, California 95053, USA(Dated: August 22, 2019)

We consider quantum clocks in curved spacetimes described by relativistic particles carryinginternal degrees of freedom that serve as clocks. The proper time associated with such a clock isdefined in terms of an observable TC (positive operator valued measure) that transforms covariantlywith respect to the clock’s free Hamiltonian. This definition ensures that TC is an unbiased estimatorof the elapsed proper time observed by such a clock. The probability that one clock reads a givenproper time conditioned on another clock reading a different proper time is derived. From thisconditional probability distribution it is shown that when the external degrees of freedom of theseclock particles are described by Gaussian wave packets localized in momentum space, the clocksobserve classical time dilation in accordance with special relativity. We then illustrate a novelquantum time dilation effect induced by nonclassical states of the clock particles realized by theirexternal degrees of freedom being in a superposition of localized momentum wave packets and givean order of magnitude estimation of such effects. Moreover, we use the Helstrom-Holevo lower boundto derive a time-energy uncertainty relation between the proper time recorded by these clocks andtheir internal energy.

I. INTRODUCTION

What allowed Einstein to transcend Newton’s concep-tion of an absolute time was his insistence on an opera-tional notion of time defined in terms of a measurementof a physical system serving as a clock [1]:

‘[Time is] considered measurable by a clock(ideal periodic process) of negligible spatial ex-tent. The time of an event taking place at apoint is then defined as the time shown on theclock simultaneous with the event.’

However, quantum mechanics has yet to be liberatedfrom the Newtonian conception of time. Standard quan-tum theory is formulated in terms of an external back-ground time, which is not quantized and with respect towhich quantum states are normalized. This fact leads tounitary evolution or, more generally, quantum dynamicsdefined in terms of completely positive trace preservingmaps.

The external time inherent in quantum theory is in ten-sion with the principle of background independence onwhich general relativity is built. This tension becomes aconceptual hurdle in the construction of a quantum the-ory of gravity, which leads to the problem of time [2–4].One aspect of this problem is that the Hamiltonian as-sociated with a generally covariant theory is constrainedto vanish, leading to the Wheeler-DeWitt equation, andconsequently this Hamiltonian does not generate evolu-tion with respect to an external notion of time. One is

∗ alexander.r.smith@dartmouth.edu† mahmadi@scu.edu

then tasked with introducing a notion of quantum dy-namics that in a particular limit recovers the unitary dy-namics of standard quantum theory, a task which has yetto be fully completed in a quantum gravity setting [2–6].

Given the above, we seek to formulate a relativisticnotion of time defined operationally in terms of a mea-surement of a physical clock. We do so by consideringfree relativistic particles moving through curved space-time which carry both external and internal degrees offreedom. The external degrees of freedom of such a parti-cle are associated with its position and momentum, whilethe internal degrees of freedom (e.g. energy levels of anatom) will be used as a clock which tracks the particle’sproper time. We begin by formulating the classical the-ory of such relativistic particles in terms of an actionprinciple and derive the associated Hamiltonian, whichwe show is constrained to vanish. This constraint formu-lation of the classical theory is analogous to the Hamil-tonian formulation of general relativity in the sense thatin both cases the constraints are quadratic in the rele-vant canonical momentum. Therefore the investigationof the former serves as an example that may inform thelatter [7].

We then canonically quantize this system based onthe procedure outlined by Dirac [8], which results in aWheeler-DeWitt type equation. Solutions to this equa-tion are projected onto a constant time hypersurfaceand normalized accordingly. The construction that isfollowed is based on the Page and Wootters mecha-nism [9] (i.e. the conditional probability interpretationof time [10–25]), which allows one to recover a relationaldynamics between a physical clock and system of inter-est from solutions to a Wheeler-DeWitt type equation.We demonstrate that in the context considered here thisquantization scheme is equivalent to the standard for-mulation of quantum theory in terms of a background

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coordinate time. Given this, the results we present canbe viewed as a generalization of the Page and Woottersmechanism to systems described by relativistic quantummechanics. It is in the spirit of Feynman1 that the Pageand Wootters mechanism is employed because it offers acoordinate independent description of conditional proba-bilities associated with measurements of different clocks(see Eq. (30)).

After quantization we consider the internal degreesof freedom of these relativistic particles to constitutea clock which tracks their proper time. Based on thework of Holevo [27, 28] and Busch et al. [29], we definea proper time observable as a positive operator valuedmeasure (POVM) on these internal clock degrees of free-dom that is covariant with respect to their Hamiltonian(as later defined in Eq. (23)). This covariance propertyensures that the proper time observable gives an unbi-ased estimate of the elapsed proper time experienced bythe clock particle, the variance of which is independentof the proper time to be estimated. We then use theHelstrom-Holevo lower bound to state a time-energy un-certainty relation between the proper time measured bythese clocks and their internal energy and discuss opti-mal clocks which saturate this uncertainty relation. Hav-ing introduced a proper time observable, we derive the

1 The following text is an excerpt that has been edited for clarityfrom Feynman’s 1964 Messenger lectures delivered at CornellUniversity [26]:

Consider two identical theories A and B, which lookcompletely different psychologically and have differ-ent ideas in them, but all their consequences are ex-actly the same. A thing that people often say is howare we going to decide which one is right?

No way! Not by science because both theory A andtheory B agree with experiment to the same extentso there is no way to distinguish one from the other.So if two theories, though they may have deeply verydifferent ideas behind them, can be shown to be math-ematically equivalent then people usually say in sci-ence that the theories can not be distinguished.

However, theories A and B for psychological rea-sons, in order to guess new theories, are very far fromequivalent because one gives the scientist very differ-ent ideas than the other. By putting a theory in agiven framework you get an idea of what to change.It may be the case that a simple change in theory Amay be a very complicated change in theory B. Inother words, although theories A and B are identi-cal before they’re changed, there are certain ways ofchanging one that look natural which don’t look nat-ural in the other.

Therefore, psychologically we must keep all the the-ories in our head and every theoretical physicist thatis any good knows six or seven different theoreti-cal representations for exactly the same physics, andknows that they are all equivalent, and that nobodyis every going to be able to decide which one is rightat that level. But they keep these representations intheir head hoping they will give them different ideasfor guessing.

probability distribution associated with one clock read-ing a particular proper time conditioned on a anotherclock reading a different proper time. It is shown thatthis probability distribution depends on the state of theexternal degrees of freedom of these clock particles.

We then specialize to two such clock particles A and Bmoving through Minkowski space and evaluate the lead-ing order relativistic correction to the conditional prob-ability that clock A reads the proper time τA given thatclock B reads the proper time τB . We show that whenthe external degrees of freedom of clocks A and B are pre-pared in Gaussian wave packets localized around a partic-ular momentum, their associated clocks agree on averagewith the classical time dilation effect predicted by specialrelativity. However, we illustrate a novel quantum timedilation effect stemming from the clock particles beingprepared in a Schrodinger cat state realized by a super-position of two Gaussian momentum wave packets. Anoptimistic order of magnitude estimate is given for thisquantum time dilation effect. We conclude that such aneffect may be observable in near future experiments, al-though the feasibility of such experiments remains to beexplored.

This paper is organized as follows. In Sec. II the classi-cal and quantum theory of N free relativistic particles ina curved spacetime carrying internal degrees of freedom isderived. In Sec. III we introduce a covariant proper timeobservable on these internal degree of freedom, which aretaken to constitute a clock. The properties of this propertime observable are discussed and the conditional prob-ability relating the proper times observed by two suchclocks is derived. In Sec. IV we specialize to the caseof two clocks moving through Minkowski space and re-cover the classical special relativistic time dilation effectand illustrate a novel quantum time dilation effect. Weconclude in Sec. V with a summary of our results, a dis-cussion of their relation with other literature, and givean outlook to future questions.

Throughout we will work in units where ~ = 1 andadopt the (− + + +) convention for the metric signature.The greek letters µ, ν indicate spacetime indices and theLatin letters i, j indicate spatial indices. For clarity, clas-sical phase space functions and their corresponding quan-tum operators will be denoted by the same symbols, themeaning of them will be clear from the context in whichthey appear. Further, S(H) and E(H) will denote re-spectively the space of density operators2 and the spaceof effect operators3 acting on the Hilbert space H.

2 S(H) := {ρ ∈ T (H) | ρ � 0 and tr ρ = 1}, where T (H) is thespace of trace class operators acting on H.

3 E(H) := {E ∈ B(H) | 0 � E � I and E = E†}, where B(H) isthe space of bounded operators acting on H.

3

II. RELATIVISTIC QUANTUM CLOCKPARTICLES IN CURVED SPACETIME

We begin in Sec. II A by developing the classical the-ory of N relativistic particles moving through a curvedspacetime each carrying an internal degree of freedom be-ginning from a local Lorentz invariant action principle.We then quantize this theory via the Dirac prescriptionin Sec. II B and define in Sec. II C the conditional statenecessary to implement the Page and Wootters mecha-nism [9].

A. Classical theory of free relativistic particles

Consider a system of N free relativistic particles eachcarrying a set of internal degrees of freedom, labeledcollectively by the configuration variables qn and theirconjugate momentum Pqn (n = 1, . . . , N), and supposethese particles are moving through a curved spacetimedescribed by the metric gµν . The action describing sucha system is S =

∑n

∫dτn Ln(τn), where

Ln(τn) := −mnc2 + Pqn

dqndτn−Hclock

n , (1)

is the Lagrangian associated with the nth particle, τnand mn denote respectively this particle’s proper timeand rest mass, and Hclock

n = Hclockn (qn, Pqn) is the Hamil-

tonian governing its internal degrees of freedom. In thefollowing sections these internal degrees of freedom willbe used as a clock which tracks the nth particle’s propertime, hence the superscript label. It should be notedthat Eq. (1) specifies that Hclock

n generates an evolu-tion of the internal degrees of freedom of the nth par-ticle with respect to its proper time. Let xµn denote thespacetime position of the nth particle; we will refer to x0

n

and xin as the nth particle’s temporal and external (spa-tial/momentum) degrees of freedom, respectively. Fig-ure 1 depicts the described situation.

The differential proper time dτn along the nth parti-cle’s world line xµn(tn), parametrized in terms of an ar-bitrary parameter tn, is given by dτn =

√−gµν xµnxνndtn,

where the over dot denotes differentiation with respectto tn. In terms of the parameters tn the action takes theform

S =∑

n

∫dtn

√−gµν xµnxνn Ln(tn). (2)

The action in Eq. (2) is invariant under changes of theworld line parameters tn, as long as there is a one-to-one correspondence between tn and τn. This invarianceallows for the action to instead be parameterized in termsof a single parameter t, which is connected to the nthparticles proper time through a monotonically increasingfunction fn(τn) := t. Expressed in terms of the single

External degrees of freedom

Temporal degree of freedom�

x0n, P 0

n

�

�xi

n, P in

�

Internal degrees of freedom(qn, Pqn

)

FIG. 1. The nth clock particle described by its temporal de-grees of freedom (x0n, P

0n), external position and momentum

degrees of freedom (xin, Pin), and internal clock degrees of free-

dom (qn, Pqn).

parameter t, the action in Eq. (2) is S =∫dtL(t), where4

L(t) :=∑

n

√−x2

n

(−mnc

2 +Pqn qn√−x2

n

−Hclockn

), (3)

in which we have used the shorthand x2n := gµν x

µnx

νn and

the over dot now denotes differentiation with respect to t.This Lagrangian treats the temporal, spatial, and inter-nal degrees of freedom as dynamical variables on equalfooting.

The Hamiltonian associated with L(t) is constructedby a Legendre transform of Eq. (3), which yields

H :=∑

n

[gµνPµn x

νn + Pqn qn]− L(t)

=∑

n

[gµνP

µn x

νn +

√−x2

n

(mnc

2 +Hclockn

)], (4)

where Pµn is the momentum conjugate to the nth parti-cle’s spacetime position xµn and defined as

Pµn :=∂L(t)

∂xµn=

xµn√−x2

n

(mnc

2 +Hclockn

). (5)

4 A similar trick was employed in Ref. [30] to formulate the theoryof N relativistic particles which do not carry internal degrees offreedom.

4

Upon substituting Eq. (5) into the above Hamiltonian H,we see that each term in Eq. (4) is constrained to vanish

Hn := gµνPµn x

νn +

√−x2

n

(mn +Hclock

n

)

=gµν x

µnx

νn√

−x2n

(mnc

2 +Hclockn

)

+√−x2

n

(mnc

2 +Hclockn

)

=x2n√−x2

n

[(mnc

2 +Hclockn

)−(mnc

2 +Hclockn

)]

≈ 0, (6)

where ≈ means Hn vanishes as a constraint [8]. Fur-thermore, the velocity appearing in the definition of Hn

above may be replaced with the momentum Pµn usingEq. (5), which allows the N constraints in Eq. (6) to beexpressed in the form

CHn := gµνPµnP

νn +

(mnc

2 +Hclockn

)2 ≈ 0, ∀n, (7)

which we will refer to as the Hamiltonian constraints.These are a collection of primary first class constraintswhich are quadratic in the particles’ momentum. Onemight heuristically interpret Eq. (7) as requiring the to-tal energy shared between each particles external andinternal degrees of freedom to be conserved.

Each of these constraints may be factorized5 as CHn =C+n C−n , where C±n is defined as

C±n := P 0n ± hn, (8)

and

hn :=1√−g00

√gijP inP

jn + (mnc2 +Hclock

n )2

(9)

In Eq. (9) we have assumed the time-space componentsof the metric vanish, g0i = 0. Such an assumption is notnecessary, although in the sections that follow this willbe the case and thus we make this assumption now forsimplicity.

By construction, the momentum conjugate to the nthparticle’s spacetime coordinates satisfy the canonicalPoisson relations {xµm, P νn} = δµνδmn. This implies thatthe canonical momentum Pµn generates translations inthe spacetime coordinate xµn. Moreover, if it is the casethat C±n ≈ 0, it follows that P 0

n = ±hn, which is thegenerator of translations in the nth particle’s time coor-dinate. Said another way, ±hn is the Hamiltonian forboth the external and internal degrees of freedom of thenth particle, generating an evolution of these degrees offreedom with respect to the particle’s temporal coordi-nate x0

n.

5 Similar constraint factorization is described in more detail inRefs. [31, 32].

B. Canonical quantization

In what follows we will make use of the canonical quan-tization scheme outlined by Dirac [7, 8, 33], which is ap-plicable to the quantization of the theory of N relativis-tic particles described above. We begin by promotingthe phase space variables associated with the nth parti-cle to operators acting on appropriate Hilbert spaces: x0

n

and P 0n become canonically conjugate self-adjoint opera-

tors acting on the Hilbert space H0n ' L2 (R) associated

with the nth particle’s temporal degree of freedom; xinand P in become canonically conjugate operators actingon the Hilbert space Hext

n ' L2 (R3) associated with thenth particle’s external degrees of freedom; and qn andPqn become canonically conjugate operators acting onthe Hilbert spaceHclock

n associated with the nth particle’sinternal degrees of freedom. The Hilbert space describingthe nth particle is thus Hn ' H0

n ⊗Hextn ⊗Hclock

n .The constraint functions in Eqs. (7) and (8) become

operators CHn and C±n acting on Hn. The quantumanalogue of the constraints is to demand that physicalstates of the theory are annihilated by these constraintoperators

CHn |Ψ〉〉 = C+n C−n |Ψ〉〉 = 0, ∀n, (10)

where |Ψ〉〉 ∈ H is a physical state that is an element ofthe physical Hilbert space H [5, 6]. The reason for intro-ducing the physical Hilbert space is because the spectrumof CHn is continuous around zero, which implies solu-tions to Eq. (10) are not normalizable in the kinematicalHilbert space K '

⊗nHn. To fully specify H a physi-

cal inner product must be defined, which is done belowin Eq. (13). Note that because [C+

n , C−n ] = 0, it follows

CHn |Ψ〉〉 = 0 if either C+n |Ψ〉〉 = 0 or C−n |Ψ〉〉 = 0.

C. The conditional state and the recovery ofrelativistic quantum mechanics

In this subsection we recover the standard formulationof relativistic quantum mechanics with respect to an ex-ternal (coordinate) time by implementing the Page andWootters mechanism. To do so, the physical state |Ψ〉〉describing all N particles is normalized on a spatial hy-persurface. This normalization is realized by projecting aphysical state |Ψ〉〉 onto a subspace in which the tempo-ral degree of freedom of each particle is in an eigenstatestate |tn〉 of the operator x0

n associated with the eigen-value t ∈ R in the spectrum of x0

n, that is, x0n |tn〉 = t |tn〉.

Explicitly, this projection takes the form

Πt ⊗ IS |Ψ〉〉 = |t〉 |ψS(t)〉 , (11)

where IS denotes the identity on HS '⊗

nHextn ⊗Hclock

n ,Πt := |t〉〈t| is a projector onto the subspace ofH in whichthe temporal degree of freedom of each particle is in adefinite temporal state |t〉 :=

⊗n |tn〉 associated with

5

the eigenvalue t. Equation (11) defines the conditionalstate

|ψS(t)〉 :=(〈t| ⊗ IS

)|Ψ〉〉 ∈ HS , (12)

which describes the state of the external and internaldegrees of freedom of all N particles conditioned ontheir temporal degree of freedom being in the state |tn〉.We will interpret the conditional state as describing theexternal and internal degrees of freedom of the parti-cles on a spatial hypersurface defined by their tempo-ral degrees of freedom being in |tn〉 and demand thatthe conditional state is normalized on this hypersurface〈ψS(t)|ψS(t)〉 = 1 for all t ∈ R. The normalization ofthe conditional state implies that the physical states arenormalized with respect to the inner product [20]

〈〈Ψ|Ψ〉〉PW := 〈〈Ψ|Πt ⊗ IS |Ψ〉〉= 〈ψS(t)|ψS(t)〉= 1, (13)

for all choices of t ∈ R.Note that the set {Πt | t ∈ R} constitutes a projective

valued measure (PVM) on the Hilbert space⊗

nH0n,

since ΠtΠt′ = δ(t − t′) and∫dtΠt = I0, where I0 is

the identity on⊗

nH0n. Given this observation and the

definition of the conditional state in Eq. (12), it is seenthat the physical state |Ψ〉〉 is an entangled state betweenthe temporal degrees of freedom of the particles describedby H0

n and their external and internal degrees of freedomdescribed by Hext

n ⊗Hclockn

|Ψ〉〉 =

(∫dtΠt ⊗ IS

)|Ψ〉〉 =

∫dt |t〉 |ψS(t)〉 . (14)

In what follows we consider physical states that satisfy

C+n |Ψ〉〉 =

(P 0n + hn

)|Ψ〉〉 = 0, (15)

for all values of n, where hn is now the operator equiv-alent of Eq. (9). This amounts to demanding that theconditional state of the system has positive energy asmeasured by hn.

We now show that the conditional state |ψS(t)〉, asdefined in Eq. (12), satisfies the Schrodinger equation inthe parameter t. Recall that x0

n and P 0n are canonically

conjugate variables satisfying [x0n, P

0n ] = i, from which it

follows that the operators P 0n generate translations in x0

n,

|t′n〉 = e−i(t′−t)P 0

n |tn〉 , (16)

where |tn〉 and |t′n〉 are eigenkets of the operator x0n with

respective eigenvalues t and t′. Now consider how |ψS(t)〉changes with the parameter t:

id

dt|ψS(t)〉 = i

d

dt

(⊗

n

〈t|n ⊗ IS

)|Ψ〉〉

= −

(∑

m

〈t|P 0m ⊗ I0

n 6=m ⊗ IS

)|Ψ〉〉 , (17)

where I0n 6=m denotes the identity operator on all of the

Hilbert spaces H0n for which n 6= m and the derivative

with respect to t was evaluated using Eq. (16). Re-call that |Ψ〉〉 satisfies the constraint C+

n |Ψ〉〉 = 0, fromwhich it follows that

P 0m ⊗ I0

n 6=m ⊗ IS |Ψ〉〉 = −I0 ⊗ hm ⊗ ISm 6=n |Ψ〉〉 , (18)

for all m where hm is now the operator equivalent ofEq. (9) acting on Hext

n ⊗ Hclockn and ISm 6=n is the iden-

tity on⊗

m 6=nHextm ⊗Hclock

m . Substituting Eq. (18) into

Eq. (17) yields

id

dt|ψS(t)〉 =

(∑

m

〈t| ⊗ hm ⊗ ISm6=n

)|Ψ〉〉

=∑

m

∫dt′ 〈t|t′〉hm ⊗ ISm6=n |ψS(t′)〉

=∑

m

hm ⊗ ISm6=n |ψS(t)〉 , (19)

where the second equality is obtained from the first usingEq. (14). Equation (19) asserts that the conditional state|ψS(t)〉 satisfies the Schrodinger equation

id

dt|ψS(t)〉 = HS |ψS(t)〉 , (20)

where HS :=∑m hm ⊗ ISm6=n is the relativistic Hamil-

tonian describing the evolution of all N particles. Thesolution to Eq. (20) may be expressed in terms of aunitary operator |ψS(t)〉 = U(t − t0) |ψS(t0)〉, whereU(t− t0) := e−iHS(t−t0).

Given Eq. (20), we are justified in identifying the con-ditional state |ψS(t)〉 as the usual time-dependent stateof the N particles in the standard formulation of rela-tivistic quantum mechanics with respect to an externalcoordinate time as described for example in Ref. [34]. Itis thus seen that in this context the implementation of thePage and Wootters mechanism above is equivalent to thestandard formulation of relativistic quantum mechanics.However, the former does not require a priori the notionof a background coordinate time and, as will be shownin the following section, provides a natural frameworkin which conditional probabilities between measurementoutcomes of different particles may be computed fromthe physical state |Ψ〉〉 using the Born rule.

Had we instead considered physical states satisfyingthe constraints C−n |Ψ〉〉 = 0 for all n, the conditionalstate would still satisfy the Schrodinger equation inEq. (20) under the replacement HS → −HS . However,a general solution to the constraints in Eq. (10) will notresult in a conditional state that satisfies a Schrodingerequation. As shown in Appendix A, if the physical statewas chosen to satisfy the constraints given in Eq. (10) andconsidering the background spacetime to be Minkowskispace, gµν = ηµν , the conditional state would satisfy theKlein-Gordon equation

[� +

(mc2 +Hclock

)2] |ψS(t,x)〉 = 0, (21)

6

where |ψS(t,x)〉 := (〈x| ⊗ IC) |ψS(t)〉 is a further con-ditioning of the physical sate, |x〉 := |x1〉 |x2〉 |x3〉 with|xi〉 denoting an eigenstate of the operator xi, and � :=ηµν∂µ∂ν is the d’Alembertian operator.

III. PROPER TIME OBSERVABLES

A. Defining a proper time observable

We define a clock to be the quadruple{Hclock, ρC , H

clock, TC}, where the clock is de-scribed by the Hilbert space Hclock, the fiducialstate ρC ∈ S(Hclock), a Hamiltonian Hclock, and timeobservable TC with measurement outcomes τ containedin the set G ⊆ R, which correspond to the time read bythe clock. The time observable is defined as a POVM

TC : G→ E(Hclock),

τ 7→ E(τ) s.t.

∫

G

dτ E(τ) = IC , (22)

where each measurement outcome τ ∈ G is associatedwith an effect operator E(τ) ∈ E(Hclock). The definingproperty of a time observable is that it is covariant withrespect the group of time translations generated by theclock Hamiltonian Hclock [27–29], which amounts to thefollowing condition on the effect operators

E(τ + τ ′) = UC(τ)E(τ ′)UC(τ)†, (23)

where UC(τ) := e−iHclockτ is the unitary representation

of the group element parametrized by τ ∈ G.

The physical significance of the covariance condition inEq. (23) is that it implies that a covariant time observablesatisfies the following two physical properties commonlyassociated with a clock, which we state in the followingtheorem.

Theorem (Desiderata of physical clocks). Let TC be acovariant time observable satisfying Eq. (23), the fiducialstate ρC be such that on average a measurement of TCyields 〈TC〉ρC = 0, and define ρC(τ) := UC(τ)ρCUC(τ)†.The following two physical properties of such a time ob-servable follow:

1. TC is an unbiased estimator of the parameter τ sothat on average TC yields the value 〈TC〉ρC(τ) = τ

on the state ρC(τ).

2. The variance 〈∆T 2C〉ρC(τ) of the time observable

TC is independent of the parameter τ on thestate ρC(τ).

Proof. The first statement follows from a direct compu-

tation of the average of TC on the state ρC(τ)

〈TC〉ρC(τ) =

∫

G

dτ ′ tr [E(τ ′)ρC(τ)] τ ′

=

∫

G

dτ ′ tr[U(τ)†E(τ ′)U(τ)ρC

]τ ′

=

∫

G

dτ ′ tr [E(τ ′ − τ)ρC ] τ ′

=

∫

G

dτ ′ tr [E(τ ′)ρC ] (τ ′ + τ)

= τ, (24)

where the third equality follows from Eq. (23), the fourthequality follows from a change of variables τ ′ → τ ′ − τ ,and in arriving at the last equality we used the fact thatby construction 〈TC〉ρC = 0.

The second statement follows in a similar manner

〈∆T 2C〉ρC(τ) =

∫

G

dτ ′ tr [E(τ ′)ρC(τ)] τ ′2 − 〈TC〉2ρC(τ)

=

∫

G

dτ ′ tr [E(τ ′ − τ)ρC ] τ ′2 − τ2

=

∫

G

dτ ′ tr [E(τ ′)ρC ] (τ ′ + τ)2 − τ2

= 〈∆T 2C〉ρC . (25)

It is this theorem that justifies interpreting TC as ameasurement of proper time — we expect a system func-tioning as a clock to estimate its proper time on averagewhen measured (i.e. the clock should be unbiased) andthat the variance of this measurement is independent ofthe proper time being estimated.

B. The uncertainty relation between proper timeand clock energy

For an unbiased estimator, the Helstrom-Holevo lowerbound [28, 35] places the fundamental limit on the vari-ance of the proper time measured by the clock

〈∆T 2C〉ρC ≥

1

4 〈(∆Hclock)2〉ρC, (26)

where 〈(∆Hclock)2〉ρC is the variance of Hclock on the

fiducial state ρC . Equation (26) is a time-energy uncer-tainty relation between the proper time estimated by TCand a measurement of the clock’s energy Hclock.

This inequality can be saturated and the optimalproper time observable constructed provided we makethe additional assumption that the effect operators E(τ)defining TC in Eq. (22) are proportional to ‘projection’operators

E(τ) = µ |τ〉〈τ | ∈ E(Hclock

), (27)

7

for µ ∈ R such that Eq. (22) is satisfied. We will refer to|τ〉 as the clock state corresponding to the proper timeτ ∈ G, which may in general be unnormalizable. Themotivation for this assumption is that measurements notdescribed by one-dimensional projectors have less resolu-tion [36]. Note that the clock states |τ〉 are not necessar-ily orthogonal, 〈τ |τ ′〉 6= 0, and form a possibly overcom-plete basis for Hclock.

Braunstein and Caves [36] proved that covariant ob-servables that satisfy Eq. (27), like the proper time ob-servable TC considered here, constitute an optimal mea-surement to estimate the parameter τ unitarily encodedin the state ρC(τ) := UC(τ)ρCUC(τ)†, provided that thefiducially state is pure ρC = |ψclock〉〈ψclock| and

|ψclock〉 =

∫

G

dτ∣∣ψclock(τ)

∣∣ eiτ〈Hclock〉ρC |τ〉 , (28)

where ψclock(τ) :=√µ 〈τ |ψclock〉 is the wave function of

the fiducial state in the basis furnished by the clock states{|τ〉 , ∀ τ ∈ G} and

∣∣ψclock(τ)∣∣ is an arbitrary real func-

tion of τ such that 〈TC〉ρC = 0. Such a proper time ob-servable TC is optimal in the sense that it maximizes theFisher information6 over all estimates of τ ; the Fisherinformation in this context is given by

F (τ ; ρC(τ)) = 4 〈(∆Hclock)2〉ρC . (29)

The covariance condition in Eq. (23) ensures that theFisher information is independent of τ .

C. Time dilation between quantum clocks incurved spacetime

Let us now consider two relativistic particles A and B,each with an internal degree of freedom serving as a clock,{Hclock

A , ρA, HclockA , TA} and {Hclock

B , ρB , HclockB , TB}. To

probe time dilation effects between these clocks we con-sider the probability that clock A reads the proper timeτA conditioned on clock B reading the proper time τB .This conditional probability is evaluated using the phys-

6 The Fisher information is defined as

F (x; ρ(x)) :=

∫dξ p(ξ|x)

(d ln p(ξ|x)

dx

)2

,

where p(ξ|x) := tr (E(ξ)ρ(x)) is the probability of getting themeasurement result ξ given that the parameter x is encoded stateof the quantum system ρ(x). For an unbiased estimator, theFisher information places a lower bound on the variance of theestimator, which is known as the Cramer-Rao bound [37].

ical state |Ψ〉〉 and the Born rule as follows

Prob [TA = τA | TB = τB ]

=Prob [TA = τA & TB = τB ]

Prob [TB = τB ]

=〈〈Ψ |EA(τA)EB(τB) |Ψ〉〉〈〈Ψ |EB(τB) |Ψ〉〉

=

∫dt tr [EA(τA)ρA(t)] tr [EB(τB)ρB(t)]∫

dt tr [EA(τB)ρB(t)], (30)

where we have made use of Eq. (14) in arriving at the lastequality, assumed the state of the clock particles A andB is unentangled |ψS〉 = |ψSA〉 |ψSB 〉, and defined thereduced state of the internal clock degrees of freedom as

ρn(t) := trHS\Hclockn

[U(t) |ψSn〉〈ψSn |U(t)†

], (31)

where U(t−t0) := e−iHS(t−t0), as defined below Eq. (20),and the partial trace is over the complement of the clockHilbert space HS\Hclock

n .

IV. PROBABILISTIC TIME DILATION INMINKOWSKI SPACE

We now apply the formalism developed in Secs. IIand III to investigate time dilation effects betweentwo quantum clocks in relative motion moving throughMinkowski space. It is demonstrated that when the ex-ternal degrees of freedom of these particles are localizedin momentum space these clocks observe on average timedilation described by special relativity. Then a quantumtime dilation effect is exhibited by considering nonclas-sical states of the external degrees of freedom of theseclock particles realized by a superposition of localizedmomentum wave packets.

A. Quantum clocks and time dilation

Consider the case where two clock particles A andB are moving through Minkowski space and have equalmass m. From Eq. (14) it is seen that the physical statetakes the form

|Ψ〉〉 =

∫dt |t〉 |ψS(t)〉

=

∫dt |t〉

⊗

n∈{A,B}

Un(t) |ψextn 〉 |ψclock

n 〉 , (32)

where Un(t) := e−ihnt and in writing Eq. (32) we havesupposed that the conditional state at t = 0 is unen-tangled, |ψS(0)〉 = |ψSA〉 |ψSB 〉 ∈ HA ⊗ HB , and thatthe external and internal degrees of freedom of bothparticles are unentangled, |ψSn〉 = |ψext

n 〉 |ψclockn 〉, where

|ψextn 〉 ∈ Hext

n is the state of the external degrees of free-dom and |ψclock

n 〉 ∈ Hclockn is the state of the internal

clock.

8

Suppose that the Hilbert space of both clocks isHclockA ' L2(R) ' Hclock

B and their Hamiltonians areequal to the momentum operator acting on these spaces,HclockA = pA and Hclock

B = pB . Equation (27) and thecovariance condition in Eq. (23) fix the time observablesTA and TB to be equal to the position operator on L2(R)so that G = R. Such clocks represent a commonly usedidealization in which the time observable is sharp, that is,the clock states are orthogonal 〈τ |τ ′〉 = δ(τ − τ ′) and sooutcomes of different clock measurements are perfectlydistinguishable. It follows that the time observable isa PVM that can be associated with a self-adjoint op-erator canonically conjugate to the clock Hamiltonian,[TA, H

clockA

]=[TB , H

clockB

]= i. We employ such clocks

for their mathematical simplicity in illustrating the quan-tum time dilation effect, however we stress that for anycovariant time observable, on account of Eq. (30), a quan-tum time dilation effect is expected.

Finally, suppose that the fiducial state of the internalclock degrees of freedom of each particle is prepared inthe state

|ψclockn 〉 :=

1

π1/4√σ

∫dτ e−

τ2

2σ2 |τn〉 , (33)

corresponding to a Gaussian wave packet with a spreadσ > 0 in the basis formed by the clock states {|τn〉 , τ ∈R}. The above suppositions define clock A by thequadruple {Hclock

A , |ψclockA 〉 , Hclock

A , TA} and clock B bythe quadruple {Hclock

B , |ψclockB 〉 , Hclock

B , TB}.To compute the conditional probability in Eq. (30) we

require the reduced states of both clocks, ρA(t) and ρB(t),as defined in Eq. (31) with U(t) = e−ihAt ⊗ e−ihBt. Spe-cializing to Minkowski space in Eq. (9), the nth particle’sHamiltonian is

hn = mc2

√P2n

m2c4+

(1 +

Hclockn

mc2

)2

= Hclockn +Hext

n +H intn , (34)

where P2n := ηijP

inP

jn and ηij is the Minkowski metric; in

the last equality we have dropped an overall constant mc2

and defined the external Hamiltonian Hextn := P2

n/2mand the interaction Hamiltonian

H intn := − 1

mc2

(Hextn ⊗Hclock

n +(Hextn

)2)

+O((

Hextn

mc2

)2

,(Hclockn

mc2

)2), (35)

which is arrived at by expanding the square root inEq. (34) in both Pn/mc and Hclock

n /mc2 and retain-ing only the leading-order relativistic correction. UsingEqs. (31) - (35) the conditional probability in Eq. (30)may be evaluated perturbatively. This is done explicitlyin Appendix B with the result that the probability thatclock A reads the proper time TA = τA conditioned on

clock B reading the proper time TB = τB is given by

Prob [TA = τA | TB = τB ]

=e−

(τA−τB)2

2σ2

√2πσ

(1 +〈Hext

A 〉 − 〈HextB 〉

2mc2

(1− τ2

A − τ2B

σ2

))

+O(

1σ

(〈Hext

n 〉〈Hclockn 〉σ

mc2

)2), (36)

where 〈Hextn 〉 := 〈ψext

n |Hextn |ψext

n 〉. As described by thisprobability distribution, the average proper time read byclock A conditioned on clock B indicating the time τB is

〈TA〉 =

∫dτ Prob [TA = τ | TB = τB ] τ

=

(1− 〈H

extA 〉 − 〈Hext

B 〉mc2

)τB

+O((〈Hext

n 〉〈Hclockn 〉σ

mc2

)2

τB

), (37)

and the variance in such a measurement is

〈∆T 2A〉 =

∫dτ Prob [TA = τ | TB = τB ] τ2 − 〈TA〉2

=

(1− 〈H

extA 〉 − 〈Hext

B 〉mc2

)σ2

+O((〈Hext

n 〉〈Hclockn 〉σ

mc2

)2). (38)

As might have been anticipated, the variance in a mea-surement of TA is proportional to σ2, which quantifiesthe spread in the fiducially clock state given in Eq. (33).Further, we note that this variance is modified at leading-order in 〈Hext

n 〉 /mc2.

B. Recovering special relativistic time dilation forclassical clocks

Suppose that the external degrees of freedom of bothclock particles A and B are prepared in a Gaussianstate localized around an average momentum pn withspread ∆n > 0,

|ψextn 〉 =

1

π1/4√

∆n

∫dp e

− (p−pn)2

2∆2n |pn〉 =: |pn〉 , (39)

for which 〈Hextn 〉 =

p2n

2m +∆2n

4m . It follows that the observedaverage time dilation between two such clocks is

〈TA〉 =

[1−

p2A − p2

B + 12 (∆2

A −∆2B)

2m2c2

]τB

+O((〈Hext

n 〉〈Hclockn 〉σ

mc2

)2

τB

). (40)

If instead the two clocks were classical moving at ve-locities vA = pA/m and vB = pB/m corresponding to

9

the average velocity of the momentum wave packets ofthe clocks just considered, then the proper time τA readby clock A given that clock B reads the proper time τB is

τA =γBγA

τB

=

[1− p2

A − p2B

2m2c2

]τB +O

((〈Hext

n 〉mc2

)2), (41)

where γn := 1/√

1− v2n/c

2. Therefore, upon comparisonwith Eq. (40) and supposing that ∆A = ∆B , quantumclocks whose external degrees of freedom are preparedin Gaussian wave packets localized around a particularmomentum agree on average with the time dilation effectdescribed by special relativity.

C. Nonclassical states of clocks and quantum timedilation effects

In the previous subsection we showed that quantumclocks prepared in momentum wave packets behave clas-sically in the sense that they observe on average specialrelativistic time dilation. It is natural to ask: Do novelquantum time dilation effects arise if instead the externaldegrees of freedom of the clock particles are prepared innonclassical states?

To answer this question we consider two clocks, A andB, and suppose that the state of the external degree offreedom of clock A begins in a nonclassical Schrodingercat state realized by a superposition of two Gaussianwave packets with average momenta pA and p′A,

|ψextA 〉 :=

1√N

(cos θ |pA〉+ sin θeiφ |p′A〉

)∈ Hext

A , (42)

where θ ∈ [0, 2π), φ ∈ [0, π], |pA〉 and |p′A〉 are definedby Eq. (39), and

N := 1 + cos(φ) sin(2θ)e− (pA−p′A)2

4∆2A , (43)

is a normalization constant. Further, suppose that theexternal degrees of freedom of clock B are prepared in aGaussian wave packet with average momentum pB , likein Eq. (39), and that the internal degrees of freedom ofboth clocks are again given by Eq. (33).

In such a setup, the probability that clock A reads theproper time τA conditioned on clock B reading the propertime τB is given by Eq. (30). The difference betweenthis case and the case considered in Sec. IV B is that therelativistic correction is now proportional to

〈HextA 〉 − 〈Hext

B 〉mc2

= Kclassical +Kquantum, (44)

where Kclassical is equal to the contribution correspond-ing to particle A being prepared in a classical (incoher-ent) mixture of the momentum wave packets with aver-age momentum pA with probability cos2 θ and p′A with

0.00 0.01 0.02 0.03 0.04 0.05 0.06

0

-2.× 10-5

-4.× 10-5

2.× 10-5

FIG. 2. The strength of the quantum time dilation ef-fect Kquantum is plotted as a function of the difference|p′A − pA| /mc = (p′A − pA)/mc, where pA = (pA, 0, 0) andp′A = (p′A, 0, 0) denote the average momentum of the wavepackets comprising the superposition state in Eq. (42). Dif-ferent values of average total momentum (p′A + pA)/mc areshown, in all cases θ = π/8 and ∆A/mc = 0.01. The thinblack line traces the trajectory of the optimal momentum dif-ference popt of for different total momentum (p′A + pA) /mc.

probability sin2 θ,

Kclassical :=p2A cos2 θ + p′2A sin2 θ − p2

B

2m2c2+

∆2A −∆2

B

4m2c2,

(45)

and Kquantum quantifies the contribution from the non-classical aspect of the state of the external degrees offreedom of clock A,

Kquantum :=sin 2θ cosφ

8m2c2Ne− (p′A−pA)2

4∆2A

×[2(p′2A − p2

A

)cos 2θ − (p′A − pA)

2]. (46)

It is clear from Eq. (46) that if either θ ∈ {0, π2 }or pA = p′A, the quantum contribution vanishes,Kquantum = 0. This is expected given that in these casesthe external state of the clock particle is no longer asuperposition of momentum wave packets; see Eq. (42).From Eq. (45) it is clear that choosing ∆A = ∆B andp2B = p2

A cos2 θ + p′2A sin2 θ results in Kclassical = 0 andany observed time dilation between clock A and B insuch a case would result from the quantum contributionKquantum. Moreover, note that the average and varianceof the observed time dilation of clock A with respect toclock B will be given by Eqs. (37) and (38) using Eq. (44).

To illustrate the behaviour of the quantum timedilation effect stemming from the nonclassicality ofthe external state of clock particle A, the quantityKquantum is plotted in Figs. 2 and 3; for simplicity the

10

0 π

8π

43 π

8π

2

0

-2.× 10-5

-4.× 10-5

0

2.× 10-5

FIG. 3. The strength of the quantum time dilation effectKquantum is plotted as a function of the parameter θ, whichquantifies the weight of the momentum wave packets withaverage momentum pA = (pA, 0, 0) and p′A = (p′A, 0, 0) com-prising the superposition state in Eq. (42). Different values ofthe sum of these average momentum (p′A+pA)/mc are shown,in all cases (p′A − pA)/mc = 0.017 and ∆A/mc = 0.01.

one-dimensional case is exhibited by supposing pA =(pA, 0, 0) and p′A = (p′A, 0, 0) with p′A > pA.

Figure 2 shows the behaviour of Kquantum as a func-tion of the difference (p′A − pA) /mc in the average mo-mentum of each wave packet comprising the momentumsuperposition in Eq. (42) for different values of their to-tal momentum (p′A + pA) /mc. It is seen that there is anoptimal difference in the average momentum of the twowave packets popt, which increases as the total averagemomentum of the wave packets (p′A + pA) /mc increases;this is depicted by the thin black line in Fig. 2. Further,from Fig. 2 it is seen that Kquantum increases as the totalaverage momentum of the wave packets (p′A + pA) /mcincreases.

Figure 3 is a plot of Kquantum as a function of theparameter θ quantifying the weight of each momentumwave packet comprising the superposition in Eq. (42)for a fixed value of the difference in average momentumof each wave packet (p′A − pA) /mc. It is observed thatwhen (p′A − pA) /mc = 0, Kquantum is negative for all val-ues of θ and reaches its most negative value at θ = π/4.As the total average momentum increases, Kquantum in-creases for 0 < θ < π/4 and decreases for π/4 < θ < π/2with the largest positive value at θ ≈ π/8 and largestnegative value at θ ≈ 3π/8. Recall that the plot in Fig. 2is for p′A ≥ pA, so it is not unexpected that the largest(negative) value occurs at θ ≈ 3π/8 corresponding tomore weight being attributed to the wave packet withlarger momentum in Eq. (42).

D. Quantum time dilation in experiment?

Consider the two clock particles to be 87Rb atoms,which have a mass of m = 1.4 × 10−25kg and atomicradius of r = ~/∆A = 2.5 × 10−10m. Suppose theseclock particles can be prepared in a superposition ofmomentum wave packets such that (p′A + pA) /mc =(p′A − pA) /mc = 6.7 × 10−8, corresponding each branchof the momentum superposition moving at average ve-locities of vA = 5 m/s and v′A = 15 m/s. We notethat (classical) special relativistic time dilation has beenobserved with atomic clocks moving at these velocities[38, 39], and perhaps the momentum superposition canbe prepared by a momentum beam splitter realized us-ing coherent momentum exchange between atoms andlight [40, 41]. Supposing that θ = 3π/4 and φ = 0 resultsin Kquantum ≈ 10−15. Assuming that the resolution ofthe clock formed by the internal degrees of freedom ofthe 87Rb atoms is 10−14s, corresponding to the resolu-tion of 87Rb atomic clocks [42], it is seen from Eq. (37)that the coherence time of the momentum superpositionmust be on the order of 10s to observe a quantum timedilation effect. We note that the required coherence timeis comparable to coherence times of the superpositionscreated in the experiments of Kasevich et al. [43], whichare on the order of seconds; however, in these experi-ments spatial superpositions are constructed as opposedto the momentum superpositions considered here.

Alternatively, Bushev et al. [44] have proposed to usethe spin precession of a single electron in a Penning trapas a clock to observe relativistic time dilation, and it isconceivable that such a clock might be able to witnessthe quantum time dilation effect.

V. CONCLUSIONS AND OUTLOOK

We began by formulating the classical theory of Nfree relativistic particles with internal degrees of free-dom moving through a curved spacetime in terms of aLorentz invariant action principle. The importance offormulating the theory in terms of an action principle,as remarked by Dirac [8], is that it is easy to implementthe conditions for the theory to be relativistic: one sim-ply has to require that the action integral be invariant.With an action comes a Lagrangian from which we for-mulated the theory in terms of a Hamiltonian constraintas given in Eqs. (7) and (8). We then passed over tothe quantum theory via the canonical quantization pro-cedure and implemented the constraints in Eq. (10) as aWheeler-DeWitt type equation.

We then considered the internal degrees of freedom ofthese relativistic particles to function as clocks that tracktheir proper time. In doing so we constructed an optimalcovariant proper time observable which gives an unbi-ased estimate of the clock’s proper time. It was shownthat the Helstrom-Holevo lower bound [28, 35] impliesa time-energy uncertainty relation between the proper

11

time read by such a clock and a measurement of its en-ergy. From the Born rule the conditional probability dis-tribution that one such clock reads the proper time τAconditioned on another clock reading the proper time τBwas derived in Eq. (30).

We then specialized to two such clock particles mov-ing through Minkowski space and evaluated the leading-order relativistic correction to this conditional probabil-ity distribution. It was shown that on average thesequantum clocks measure a time dilation consistent withspecial relativity when the state of their external degreesof freedom is localized in momentum space. However,when the state of their external degrees of freedom isin a superposition of such localized momentum states,we demonstrated that a quantum time dilation effect oc-curs. We exhibited how this quantum time dilation effectdepends on the parameters defining the momentum su-perposition and gave an order of magnitude estimate forthe size of this effect. We conclude that such a quantumtime dilation effect may be observable with present daytechnology, but note that the experimental feasibility ofobserving this effect remains to be explored.

It should be noted that the conditional probability dis-tribution in Eq. (30) associated with clocks reading dif-ferent times was a nonperturbative expression for clocksin arbitrary nonclassical states in a curved spacetime. Itthus remains to investigate the effect of other nonclassi-cal features of the clock particles such as shared entan-glement among the clocks and spatial superpositions. Inregard to the latter, it will be interesting to recover previ-ous relativistic time dilation effects in quantum systemsrelated to particles prepared in spatial superpositions andeach branch in the superposition experiencing a differentproper time due to gravitational time dilation [45–48];this work has largely focused on gravitational decoher-ence effects. We emphasize that the quantum time dila-tion effect described in Sec. IV C differs from these resultsin that it is a consequence of a momentum superpositionrather than gravitational time dilation. Nonetheless, itwill be interesting to examine such gravitational time di-lation effects in the framework developed above and makeconnections with previous literature on quantum aspectsof the equivalence principle [49–51]. We also note thatwhile we exhibited the quantum time dilation effect fora specific clock model in Sec. IV, based on the precedinganalysis in terms covariant time observables it is expectedthat any clock will witness quantum time dilation. Giventhis, it will be fruitful to examine our results in relationto other models of quantum clocks that have been con-sidered [52–57].

Another avenue of exploration is the construction ofrelativistic quantum reference frames from the relativis-tic clock particles considered here [58–62]. In particu-lar, one might define relational coordinates with respectto a reference particle and examine the correspondingrelational quantum theory and the possibility of chang-

ing between different reference frames [63–70]. Relatedis the perspective-neutral interpretation of the Hamil-tonian constraint in terms of which a formalism forchanging clock reference systems has recently been de-veloped [31, 32, 71, 72]. Further, one might also wishto consider interactions among the clock particles whichhas been shown to lead to interesting effects in relatedcontexts [20, 73, 74].

We note that the results presented here complementthose of Greenberger [75] and his desire for proper timeand rest mass to be treated as quantum observables. Theapproach adopted here differs in that we construct aproper time observable in terms of a covariant POVMrather than a self-adjoint operator. This has the ad-vantage of overcoming Pauli’s objection to the construc-tion of a canonically conjugate self-adjoint time observ-able [76]. Like Greenberger [75] we obtain in Eq. (26) anuncertainty relation between proper time and rest mass,provided we interpret M := mIC + Hclock/c2 as a massoperator [50].

As remarked in the introduction, the results presentedhere constitute the generalization of the Page and Woot-ters mechanism into the regime of relativistic quantummechanics. The advantage of employing this formalismwas three fold: it a quantization scheme independentof a background coordinate time as described by theWheeler-DeWitt equation in Eq. (10), the recovery of thestandard formulation of relativistic quantum mechanicsas described by the Schrodinger equation in Eq. (20),and provided a framework in which to compute the con-ditional probability that different clocks read differentproper times that is independent of a background co-ordinate time (see Eq. (30)). We view these results ascontributing to the program of extending the Page andWootters mechanism into the relativistic regime, includ-ing relativistic quantum field theory and quantum grav-ity [13–15, 25]. In this regard, once a clear field-theoreticformulation of the Page and Wootters mechanism is avail-able, it will be interesting to make connections with otherstudies of physical clocks in such settings [77–79]. Push-ing the Page and Wootters mechanism into the relativis-tic regime is an important program to pursue because ithas the potential to address aspects of the problem oftime by providing a way in which a relational dynamicscan emerge from solutions to the Wheeler-DeWitt equa-tion.

ACKNOWLEDGMENTS

This work was supported by the Natural Sciences andEngineering Research Council of Canada and the Dart-mouth College Society of Fellows. We’d like thank andPhilipp A. Hohn and Maximilian P. E. Lock for usefuldiscussions.

12

Appendix A: Derivation of the Klein-Gordon equation in the Page and Wootters formalism

For simplicity, let us consider a single particle situated in Minkowski space so that the constraint in Eq. (7) becomes

CH |Ψ〉〉 =(ηµνP

µP ν ⊗ IC + I0 ⊗ Iext ⊗(mc2 +Hclock

)2) |Ψ〉〉 = 0, (A1)

where ηµν denotes the Minkowski metric and we have suppressed the subscript n. Given a physical state satisfyingthis constraint, Eq. (12) defines the conditional state of the external and internal degrees of freedom of the particle

|ψS(t,x)〉 :=(〈t| 〈x| ⊗ IC

)|Ψ〉〉 , (A2)

where x0 |t〉 = t |t〉 and |x〉 := |x1〉 |x2〉 |x3〉 with |xi〉 denoting an eigenstate of the operator xi. Now consider theaction of the d’Alambertian operator � := ηµν∂µ∂ν on the conditional state

� |ψS(t,x)〉 = ηµν∂µ∂ν(〈t| 〈x| ⊗ IC

)|Ψ〉〉

= ηµν∂µ∂ν

(〈t0| eiP

0t 〈x0| eiP·x ⊗ IC)|Ψ〉〉

=(〈t| 〈x| ⊗ IC

)(ηµνP

µP ν ⊗ IC)|Ψ〉〉

= −(〈t| 〈x| ⊗ IC

) (I0 ⊗ Iext ⊗

(mc2 +Hclock

)2) |Ψ〉〉

= −(I0 ⊗ Iext ⊗

(mc2 +Hclock

)2) |ψS(t,x)〉 , (A3)

where the third equality is obtained using Eq. (A1). Upon rearranging Eq. (A3) we find that the conditional statesatisfies

[� +

(mc2 +Hclock

)2] |ψS(t,x)〉 = 0, (A4)

where we have suppressed the identity operators I0, IC , and Iext. If one suppose Hclock vanishes, then Eq. (A4)reduces to the usual Klein-Gordon equation.

Appendix B: Derivation of the conditional probability distribution given in Eq. (36).

Let us define the free evolution of the external and internal clock degrees of freedom respectively as

ρextn (t) := e−iH

extn tρext

n eiHextn t ∈ S(Hext

n ) and ρn(t) := e−iHclockn tρne

iHclockn t ∈ S(Hclock

n ), (B1)

for n ∈ {A,B}. Then the reduced state of of the clock is given by

ρn(t) = trext

(e−iH

intn tρext

n (t)⊗ ρn(t)eiHintn t)

= trext

(ρn(t)⊗ ρext

n (t)− it[H intn , ρext

n (t)⊗ ρn(t)]

+O((H int

n t)2) )

= ρn(t)− it trext

([H int

n , ρextn (t)⊗ ρn(t)]

)+O

((H int

n t)2)

= ρn(t) + it〈Hext

n 〉mc2

[Hclockn , ρn(t)

]+O

((〈Hext

n 〉Hclockn

mc2 t)2). (B2)

Using Eq. (B2) the integrands defining the conditional probability distribution in Eq. (30) may be evaluated pertur-batively

tr(En(τn)ρn(t)

)= tr

(En(τn)ρn(t)

)+ it〈Hext

n 〉mc2

tr(En(τn)

[Hclockn , ρn(t)

] )+O

((〈Hext

n 〉〈Hclockn 〉

mc2 t)2). (B3)

Suppose that the fiducial state |ψclockn 〉 ∈ Hclock

n of the clock is Gaussian with a spread σ as given by Eq. (33), thenthe first term in Eq. (B3) is given by

tr [En(τn)ρn(t)] =∣∣∣〈τn| e−iH

clockn t |ψclock

n 〉∣∣∣2

=

∣∣∣∣∣∣

∫

Rdτ ′ 〈τn|τ ′n〉

e−(τ′−t)2

2σ2

π14√σ

∣∣∣∣∣∣

2

=e−

(τ−t)2

σ2

√πσ

. (B4)

13

Defining |ψclockn (t)〉 := e−iH

clockn t |ψclock

n 〉, the trace in the second term is given by

tr[ETn(τn)

[Hclockn , ρn(t)

]]= 〈τn|

[Hclockn , ρn(t)

]|τn〉

= 〈τn|Hclockn |ψclock

n (t)〉 〈ψclockn (t)|τn〉 − 〈τn|ψclock

n (t)〉 〈ψclockn (t)|Hclock

n |τn〉

=e−

(τ−t)2

2σ2

π14√σ

(〈τn|Hclock

n |ψclockn (t)〉 − 〈ψclock

n (t)|Hclockn |τn〉

). (B5)

It follows from the covariance relation in Eq. (23) that the clock states satisfy

|τ + τ ′〉 = e−iHclockn τ ′ |τ〉 , (B6)

which implies that Hclockn ≡ −i∂/∂τ is the displacement operator in the |τ〉 representation [37]. This observation

allows us to evaluate the probability amplitudes in Eq. (B5)

〈τn|Hclockn |ψclock

n (t)〉 = −i ∂∂τ

e−(τ−t)2

2σ2

π14√σ

= ie−

(τ−t)2

2σ2

π14√σ

τ − tσ2

. (B7)

This allows for the simplification of Eq. (B5)

tr[ETn(τn)

[Hclockn , ρn(t)

]]= 2i

e−(τ−t)2

σ2

√πσ

τ − tσ2

, (B8)

and together with Eq. (B4), this implies Eq. (B3) simplifies to

tr [ETn(τn)ρn(t)] = tr [ETn(τn)ρn(t)] + it〈Hext

n 〉mc2

tr[ETn(τn)

[Hclockn , ρn(t)

]]+O

((〈Hext

n 〉〈Hclockn 〉

mc2 t)2)

=e−

(τ−t)2

σ2

√πσ

+ it〈Hext

n 〉mc2

2i

e−(τ−t)2

σ2

√πσ

τ − tσ2

+O

((〈Hext

n 〉〈Hclockn 〉

mc2 t)2)

=e−

(τ−t)2

σ2

√πσ2

[1− 2

〈Hextn 〉

mc2t(τ − t)σ2

+O((〈Hext

n 〉〈Hclockn 〉

mc2 t)2)]

. (B9)

Finally, with Eq. (B9) the conditional probability defined in Eq. (30) maybe evaluated, yielding the result statedin Eq. (36)

Prob [TA = τA | TB = τB ] =

∫dt tr [EA(τA)ρA(t)] tr [EB(τB)ρB(t)]∫

dt tr [EB(τB)ρB(t)]

=

e− (τA−τB)2

2σ2√

2πσ

[1 +

〈HextA 〉+〈H

extB 〉

2mc2 − 〈HextA 〉−〈H

extB 〉

2mc2τ2A−τ

2B

σ2 +O((〈Hext

n 〉〈Hclockn 〉

mc2 σ)2)]

1 +〈Hext

B 〉mc2 +O

((〈Hext

B 〉mc2

)2)

=e−

(τA−τB)2

2σ2

√2πσ

[1 +〈Hext

A 〉 − 〈HextB 〉

mc2σ2 − τ2

A + τ2B

2σ2+O

((〈Hext

n 〉〈Hclockn 〉

mc2 σ)2)]

. (B10)

[1] A. Einstein, in Out of My Later Years (Wings Books,New York, 1996) pp. 39–46.

[2] K. V. Kuchar, in Proceedings of the 4th Canadian Confer-ence on General Relativity and Relativistic Astrophysics(World Scientific, Singapore, 1992).

[3] C. J. Isham, in Integrable Systems, Quantum Groups, and

Quantum Field Theories, edited by L. A. Ibort and M. A.Rodrıguez (Springer Netherlands, Dordrecht, 1993) pp.157–287.

[4] E. Anderson, The Problem of Time, Fundamental Theo-ries of Physics, Vol. 190 (Springer, 2017).

[5] C. Rovelli, Quantum Gravity (Cambridge University

14

Press, Cambridge, 2004).[6] C. Kiefer, Quantum Gravity, 3rd ed. (Oxford University

Press, Oxford, 2012).[7] A. Ashtekar, Lectures on Non-Perturbative Canonical

Gravity, Physics and Cosmology, Vol. 6 (World Scientific,Singapore, 1991).

[8] P. A. M. Dirac, Lectures on Quantum Mechanics (BelferGraduate School of Sciencem Yeshiva University, NewYork, 1964).

[9] D. N. Page and W. K. Wootters, Phys. Rev. D 27, 2885(1983).

[10] W. K. Wootters, Int. J. Theor. Phys. 23, 701 (1984).[11] D. N. Page, NSF-ITP-89-18 (1989).[12] D. N. Page, in Physical Origins of Time Asymmetry,

edited by J. J. Halliwell, J. Perez-Mercader, and W. H.Zurek (Cambridge University Press, Cambridge, 1994).

[13] R. Gambini, R. A. Porto, and J. Pullin, New J. Phys. 6,45 (2004).

[14] R. Gambini, R. A. Porto, and J. Pullin, Phys. Rev. Lett.93, 24041 (2004).

[15] R. Gambini, R. A. Porto, S. Torterolo, and J. Pullin,Phys. Rev. D 79, 041501 (2009).

[16] V. Corbin and N. J. Cornish, Found. Phys. 39, 474(2009).

[17] C. E. Dolby, arXiv:0406034 [gr-qc] (2004).[18] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev.

D 79, 945933 (2015).[19] A. Boette, R. Rossignoli, N. Gigena, and M. Cerezo,

Phys. Rev. A 93, 062127 (2016).[20] A. R. H. Smith and M. Ahmadi, Quantum 3, 160 (2019).[21] C. Marletto and V. Vedral, Phys. Rev. D 95, 043510

(2017).[22] J. Leon and L. Maccone, Found. Phys. 47, 1597 (2017).[23] K. L. H. Bryan and A. J. M. Medved, Found. Phys. 48,

48 (2018).[24] K. L. H. Bryan and A. J. M. Medved, arXiv:1803.02045

[quant-ph] (2018).[25] M. Rotondo and Y. Nambu, Universe 5, 66 (2019).[26] R. Feynman, “Lecture 2: The Relation of Mathematics

to Physics — The Character of Physical Law,” (1964).[27] A. S. Holevo, Rep. Math Phys. 16, 385 (1979).[28] A. S. Holevo, Probabilistic and Statistical Aspects of

Quantum Theory, Statistics and Probability, Vol. 1(North-Holland, Amsterdam, 1982).

[29] P. Busch, M. Grabowski, and P. J. Lahti, OperationalQuantum Physics, Lecture Notes in Physics Monographs,Vol. 31 (Springer, Verlag Berlin Heidelberg).

[30] R. Marnelius, Phys. Rev. D 10, 2535 (1974).[31] P. A. Hoehn and A. Vanrietvelde, arXiv:1810.04153 [gr-

qc] (2018).[32] A. Vanrietvelde, P. A. Hoehn, F. Giacomini, and

E. Castro-Ruiz, arXiv:1809.00556 [quant-ph] (2018).[33] D. Marolf, in The Ninth Marcel Grossmann Meeting

(World Scientiffic, Singapore, 2002) pp. 1348–1349.[34] M. Zych, Quantum Systems under Graviational Time Di-

lation, Springer Theses (Springer International Publish-ing, 2017).

[35] C. W. Helstrom, Quantum Detection and EstimationTheory, Mathematics in Science and Engineering, Vol.123 (Academic Press, New York, 1976).

[36] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72,3439 (1994).

[37] H. M. Wiseman and G. J. Milburn, Quantum Measure-ment and Control (Cambridge University Press, Cam-

bridge, UK, 2010).[38] S. Reinhardt, G. Saathoff, H. Buhr, L. A. Carlson,

A. Wolf, D. Schwalm, S. Karpuk, C. Novotny, G. Hu-ber, M. Zimmermann, R. Holzwarth, T. Udem, T. W.Hansch, and G. Gwinner, Nat. Phys. 3, 861 (2007).

[39] C. W. Chou, D. B. Hume, T. Rosenband, and D. J.Wineland, Science 329, 1630 (2010).

[40] P. R. Berman, Atom Interferometry (Academic Press,1997).

[41] P. Clade, S. Guellati-Khelifa, F. Nez, and F. Biraben,Phys. Rev. Lett. 102, 240402 (2009).

[42] J. Camparo, Phys. Today 60, 33 (2007).[43] T. Kovachy, P. Asenbaum, C. Overstreet, C. A. Donnelly,

S. M. Dickerson, A. Sugarbaker, J. M. Hogan, and M. A.Kasevich, Nature 528, 530 (2015).

[44] P. A. Bushev, J. H. Cole, D. Sholokhov, N. Kukharchyk,and M. Zych, New J. Phys. 18, 093050 (2016).

[45] M. Zych, F. Costa, I. Pikovski, T. C. Ralph, andC. Brukner, Class. Quantum Grav. 29, 224010 (2012).

[46] I. Pikovski, M. Zych, F. Costa, and C. Brukner, Nat.Phys. 11, 668 (2015).

[47] M. Zych, F. Costa, I. Pikovski, and C. Brukner, Nat.Commun. 2, 505 (2011).

[48] S. Khandelwal, M. P. E. Lock, and M. P. Woods,arXiv:1904.02178 [quant-ph] (2019).

[49] L. Viola and R. Onofrio, Phys. Rev. D 55, 455 (1997).[50] M. Zych and C. Brukner, Nat. Phys. 14, 1027 (2018).[51] L. Hardy, arXiv:1903.01289 [quant-ph] (2019).[52] H. Salecker and E. P. Wigner, Phys. Rev. 109, 571

(1958).[53] A. Peres, Am. J. Phys. 48, 552 (1980).[54] V. Buzek, R. Derka, and S. Massar, Phys. Rev. Lett. 82,

2207 (1999).[55] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods,

N. Brunner, and M. Huber, Phys. Rev. X 7, 031022(2017).

[56] A. J. Paige, A. D. K. Plato, and M. S. Kim,arXiv:1809.01517 [quant-ph] (2018).

[57] M. P. Woods, R. Silva, and J. Oppenheim, Ann. HenriPoincare (2018).

[58] S. D. Bartlett and D. R. Terno, Phys. Rev. A 71 (2005).[59] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev.

Mod. Phys. 79, 555 (2007).[60] S. D. Bartlett, T. Rudolph, R. W. Spekkens, and P. S.

Turner, New J. Phys. 11, 063013 (2009).[61] M. Ahmadi, A. R. H. Smith, and A. Dragan, Phys. Rev.

A 92, 062319 (2015).[62] L. Loveridge, T. Miyadera, and P. Busch, Found. Phys.

48, 135 (2018).[63] D. Poulin, Int. J. Theor. Phys. 45, 1189 (2006).[64] R. M. Angelo, N. Brunner, S. Popescu, A. J. Short, and

P. Skrzypczyk, J. Phys. A 44, 145304 (2011).[65] M. C. Palmer, F. Girelli, and S. D. Bartlett, Phys. Rev.

A 89, 052121 (2014).[66] D. Safranek, M. Ahmadi, and I. Fuentes, New Journal

of Physics 17, 033012 (2015).[67] A. R. H. Smith, M. Piani, and R. B. Mann, Phys. Rev.

A 94, 012333 (2016).[68] A. R. H. Smith, Phys. Rev. A 99, 052315 (2019).[69] F. Giacomini, E. Castro-Ruiz, and C. Brukner,

arXiv:1811.08228 [quant-ph] (2018).[70] F. Giacomini, E. Castro-Ruiz, and C. Brukner, Nat.

Commun. 10, 494 (2019).[71] P. A. Hoehn, Universe 5, 116 (2019).

15

[72] A. Vanrietvelde, P. A. Hoehn, and F. Giacomini,arXiv:1809.05093 [quant-ph] (2018).

[73] P. A. Hoehn, E. Kubalova, and A. Tsobanjan, Phys.Rev. D 86, 065014 (2012).

[74] E. C. Ruiz, F. Giacomini, and C. Brukner, PNAS 114,E2303 (2017).

[75] D. M. Greenberger, arXiv:1011.3709 [quant-ph] (2010).[76] W. Pauli, General Principles of Quantum Mechanics

(Springer-Verlag, Berlin Heidelberg New York, 1980).[77] K. Lorek, J. Louko, and A. Dragan, Class. Quantum

Grav. 32, 175003 (2015).[78] M. P. E. Lock, I. Fuentes, R. Renner, and S. Stupar, in

Time in Physics, Tutorials, Schools, and Workshops inthe Mathematical Sciences (Springer International Pub-lishing, Cham, 2017) pp. 51–68.

[79] M. P. E. Lock and I. Fuentes, arXiv:1901.08000 [quant-ph] (2019).