aa r X i v : . [ qu a n t - ph ] J un Realistic clocks for a Universe without time
K.L.H. Bryan (1) , A.J.M. Medved (1 , (1) Department of Physics & Electronics, Rhodes University, Grahamstown 6140, South Africa(2) National Institute for Theoretical Physics (NITheP), Western Cape 7602, South Africa [email protected], [email protected] Abstract
There are a number of problematic features within the currenttreatment of time in physical theories, including the “timelessness” ofthe Universe as encapsulated by the Wheeler-DeWitt equation. Thispaper considers one particular investigation into resolving this issue; aconditional probability interpretation that was first proposed by Pageand Wooters. Those authors addressed the apparent timelessness bysubdividing a faux Universe into two entangled parts, “the clock” and“the remainder of the Universe”, and then synchronizing the effectivedynamics of the two subsystems by way of conditional probabilities.The current treatment focuses on the possibility of using a (some-what) realistic clock system; namely, a coherent-state description of adamped harmonic oscillator. This clock proves to be consistent withthe conditional probability interpretation; in particular, a standardevolution operator is identified with the position of the clock playingthe role of time for the rest of the Universe. Restrictions on the damp-ing factor are determined and, perhaps contrary to expectations, theoptimal choice of clock is not necessarily one of minimal damping.
Although the concept of time in physical theories is usually taken for granted,it soon produces many problems when attempts are made to clearly define the1henomenon or provide a reason behind its features. This so-called “prob-lem of time” is typically referred to in the singular, but it is a complex issuewhich can be separated into at least three major concerns. The first of theseis the apparent necessity for an “arrow of time”, in spite of the contradictionsthat arise when the arrow is confronted with the time-reversal invariance ofquantum mechanics. This preferred direction of time is normally attributedto the second law of thermodynamics and still awaiting a convincing expla-nation for its origin. The second is the concern over two different treatmentsof time in physical theories: One within the theory of gravity, where timeis dynamical, and the other within quantum mechanics, where time is abso-lute and akin to time in classical physics. For a working theory of quantumgravity, this difference must presumably be resolved.Last but certainly not least is the apparent “timelessness” of the Universe.This aspect of the problem of time was introduced by Wheeler and DeWitt,who formalized the problem into a relation that has become to be known asthe Wheeler–DeWitt equation [1],ˆ H | ψ i = 0 . (1)Here, ˆ H is the Hamiltonian constraint of general relativity but elevated tothe status of a quantum operator, and | ψ i represents the state of the Uni-verse. This expression along with the Schr¨odinger equation make it clearthat physical states experience no time evolution. This may seem to be arather strange requirement to impose on the Universe — that no time shallpass — but it is the starting point for an interpretation of time that is in-vestigated by the current paper. This notion of timelessness inspired Page It has also been a major influence on some other interpretations of time; for example, C and the (remaining)system S respectively.Along with the viability of dividing up the Universe into C and S , thisinterpretation comes with two further conditions which are well summarizedby Marletto and Vedral in [6] (also see [7]). One of these is that the formerof the two subsystems can be viewed not only as a clock but as a “good” one.In this interpretation of time, a good clock is one which has a large numberof distinguishable states and also, to some approximation, shares no interac-tions with the rest of the Universe. The distinguishable-states requirementallows the clock to be measured multiple times, while the no-interactions re-quirement ensures that C and S are not interfering with one another as theyeffectively evolve. A mathematical description of the latter can be stated interms of Hamiltonians and identity operators for the respective subsystems:ˆ H = ˆ H C ⊗ ˆ I S + ˆ I C ⊗ ˆ H S , (2)Although any real-world scenario would naturally include some interactionsbetween the two subsystems, C can always be chosen to be small enough sothat any influence it has on S is negligible, but then the converse would notbe true. This point is clarified in Section 3. Rovelli’s so-called relational time [2, 3]. C and S (but notice that this would follow automati-cally if the Universe is attributed with being in a pure state). Importantly, itis this quantum entanglement that essentially “sources” the PW descriptionof time evolution [4]. With these three conditions in tow, the PW proposalcan provide a meaningful interpretation of time, for which the key aspect isa joint description of a measurement of S and a corresponding measurementof C . The two measurements are formally linked by way of a conditionalprobability; hence, the PW framework sometimes goes by the name of theconditional probability interpretation.It is worthwhile to elaborate further on the logistics: In standard quantummechanics, the probability of S being in a certain state at a certain time takesthe time variable for granted as time is treated as an absolute. This is much inline with the treatment of time in Newtonian mechanics. The crucial changein the PW interpretation is that time should now be viewed as an explicitconsequence of a measurement on the clock and, as such, can no longer betaken for granted as an absolute quantity. Specifically, the probability for acertain measurement of S is conditioned on the probability of measuring acertain time which is, in reality, a particular measurement of C . In this way,a specific moment in time can always be assigned to any given measurementon S .It is important to note that there is an integration variable which inher-its the role of classical time within the PW description of evolution. Thisintegration variable, denoted here by n , provides an “abstract time” for de-scribing the evolution of C . To be clear, n takes the usual place of t in4he evolution operator for C and, then vicariously (via the systems’ mu-tual entanglement), in the evolution operator for S . However, as n has noreal physical significance, it is necessarily integrated out of the description;leaving only the measurement of some clock observable to provide a timeparameter for S . The procedure will be made clearer later in the paper whenexplicit descriptions of the probabilities are given. An alternative approachto the manual inclusion of n is to introduce an ancillary system which iseventually traced out as in [7].PW’s approach to time has not been without criticism. Most noteworthyis that of Kuchar, who questioned the ability of the conditional probabili-ties to display any evolution at all [8]. This was essentially a question ofwhether a dynamical description could be obtained using this interpretationor whether such a description was inevitably static and unable to describechange. These concerns of Kuchar have since been investigated by Dolby [9].By presenting a more explicit definition of the PW conditional probabilities,Dolby showed that measurements at successive time intervals (and so dy-namical descriptions) were in fact possible. Further investigation has morerecently been carried out by Giovannetti, Lloyd and Maccone [7]. Throughan independent approach, Giovannetti et al showed that using the conditionsof PW’s interpretation allowed for a description of measurements at succes-sive points in time within the framework of modern quantum-informationtheory. These two responses individually put to rest the concerns of Kuchar;the conditional probability approach does indeed allow for a dynamical de-scription of subsystems, even within the confines of a timeless description ofthe Universe. 5ince the PW interpretation requires specifying a clock system, it is ofinterest to investigate what types of clocks are consistent with such a frame-work. Indeed, this was pursued by Cornish and Corbin (CC) in [10], wherethey built upon the formalism that was already developed by Dolby. WhereasDolby’s specific example was a toy model with the Hamiltonian for the clockgiven by H C = p , CC rather employed a free particle as the clock so thatthe Hamiltonian is given by H C = p / m ( p and m are the momentumand mass of the clock respectively). At this point, it is worth recalling thatthe clocks in PW’s interpretation do not measure time directly. Instead, an-other clock variable (such as position) is used as a time parameter for thesystem. For instance, by using classical relations between time, position, andmomentum, CC translate the position of the particle into a correspondingtime measurement. The more conventional notion of time remains an “un-observable” in the sense that no operator can be used to directly measurethis abstract parameter.In order to further assess the applicability of the conditional probabilityframework, an investigation into realistic clocks would be helpful. Thechoice of clock for the current study is a damped harmonic oscillator, as thisis a reasonable facsimile of the types of clocks that are encountered in “real-world” scenarios. The analysis shows that the evolution of S can be faithfullydescribed in terms of a specific measurement of time, which is actually theposition of the oscillator clock C . This treatment, following that of CC,also considers the accuracy of the clock and the rate at which such a system In this regard, the free-particle clock, as proposed by CC, is not a realistic choice asit can only ever be subjected to one measurement. S in terms of the clock’s positionis explored in Section 3. The outcomes are assessed in the final section of themain text, while Appendix A provides some additional mathematical details. In order to motivate the current choice of clock, the PW criteria for a goodclock will first be considered. As discussed in the previous section, a goodclock implies that C and S are approximately non-interacting systems —meaning zero interactions between them in an idealized case. If a clock istruly ideal, then the measurement of some clock observable would exactlymatch the abstract time variable n at any given instance. The goal of thispaper, however, is to further assess the use of PW’s interpretation of time fora more realistic scenario. Therefore, the ability of the clock to approximate areal-world system rather than the ideal case becomes a crucial considerationin making a choice. In this regard, a well-chosen clock should somehowaccount for interactions between it and the system. Another considerationis the ability of the clock to be subjected to multiple measurements (in other The inclusion of interactions was also considered by [7], but the approach of that paperdiffers significantly from the current one.
As already discussed, the clock in the current treatment will measure timeindirectly through another variable, which is taken as the position of theoscillator x for concreteness. The wavefunction of the clock Ψ C will thenbe given by the coherent-state wavefunction of a damped oscillator in theposition representation, which is expressible as [11]Ψ C ( x, n ) = h x | α ( n ) i = Ae − e − rnmω ~ ( x −h x i ) + iφ , (3)8here A is a normalization constant, h x i = e − rn/ q ~ mω R ( α ) cos (Ω n ) is theexpectation value of position, Ω/ ω is the frequency with/without damping, r is the damping coefficient, α is the usual coherent-state (complex) param-eter and iφ is an irrelevant phase. Notice that the evolution is described interms of the abstract time n , which will eventually be integrated out of theconditional-probability formalism (see Section 3).Equation (3) can be obtained by starting with the standard Hamiltonianfor a damped oscillator and then performing a particular canonical trans-formation [12]. The resulting Hamiltonian is formally the same as that ofa simple harmonic oscillator — and so the usual techniques for derivingthe coherent-state wavefunction can be applied — but now with a time-dependent mass. The position and momentum operators ˆ x , ˆ p for this systemare related through [11] ˆ x = e − rn/ s Em ω + ˆ p ω , (4)where E is the energy of the oscillator and m is its unevolved mass. Thisrelation illustrates how the time parameter n can be linked to a measurementof position in this context. Further details can be found in [11].The quantity of immediate interest is the probability of the clock measur-ing a particular position x at (abstract) time n . From Eq. (3), this probabilityis simply | ψ C ( x, n ) | . The accuracy of such a position measurement can beparametrized by the width of the relevant Gaussian δ , δ = e − rn/ r ~ mω . (5)Another measure of accuracy for this choice of clock would be the rateof change of the width. As a damped system, the clock C is expected to9ose energy and so decohere. This suggests a nonlinear behavior which is notdescribed by the usual brand of quantum evolution but rather by a Lindblad-like evolution equation. In a case such as this, a proposed measure for thedecoherence rate is given by [13] σ ( n ) = ∂ (cid:16) δ ( n ) min (cid:17) ∂n . For the dampedharmonic oscillator, this works out as σ ( n ) = r ~ e − rn mω . (6)This result allows an assessment of how quickly the damped oscillator losesaccuracy; in other words, on what time scale it decoheres. Perhaps contrary to one’s expectations, it will be shown that a minimallydamped ( r →
0) oscillator is not necessarily a preferential choice of clock inthe current context. But first it is recalled that the frequency of a dampedoscillator is given by Ω = p ω − r / r , the under-damping condition of r < ω .Another relevant consideration is the need for “resetting” or “winding up”the clock after some specified time n reset so that the damping factor is notallowed to become too large. As made clear in Appendix A, a constraint of rn reset < n ; a relation that turns out to be essential for the condition probabilityinterpretation to succeed (see Section 3). This limit on the running time, n < n reset . r , (7)can now be adopted to impose a second condition on the clock. It follows10hat minimizing r will achieve the longest possible running time. But, on thecontrary, maximizing r (subject to the under-damping condition) realizes amaximally accurate clock. That this is the case will be clarified below usingthe results from the previous section.The minimization of the decoherence rate, as determined from ∂σ ( n ) ∂r = 0 ,produces the result rn = 1 . (8)Given that the inequality (7) is in play and that the accuracy of the clock isat a premium, this outcome suggests choosing r = n reset , as this is the onlyway Eq. (8) could ever be realized.The very same conclusion is reached when the width δ ( n ) min is minimizedwith respect to r . In this case, the condition becomes − n r ~ mω e − rn/ = 0 . (9)The only value of r which would satisfy this equation is r −→ ∞ , whichobviously fails the already stipulated conditions. The largest possible valuefor r should then be considered, which leads back to the r = n reset , thesame as before. This discussion makes it clear that the desired choice of r is a compromise between minimizing uncertainty (large r ) and maximizingrunning time (small r ). The optimal choice then depends on what one’sparticular requirements are for a clock.11 Finding the time for S
The next aspect to be investigated is the question of whether or not thedamped-oscillator clock C is indeed able to keep track of time for its purifyingsystem S . This query is best addressed by phrasing the evolution of C andthen of S in terms of conditional probabilities. What is first required isthe probability of the clock being in an eigenstate | x i at some referencetime n ′ given that the clock is in an unevolved coherent state at n = 0 .Mathematically, this can be described as P (cid:16) Ψ C = Ψ x ; n = n ′ | Ψ C = Ψ α (0); n = 0 (cid:17) = |h x | α ( n ′ ) i | , (10)where Ψ C , Ψ α and Ψ x represent the clock state, a coherent state and a stateof definite position respectively, and the wavefunction on the right-hand sideis that from Eq. (3).Now, as will become evident after a reading of Subsection 3.2, the evo-lution operator for the system can be expressed in a form like e i ˆ H S T , wherethe system time T can be identified with the clock position x (to some levelof approximation). In view of this and Eq. (10), there is also a conditionalprobability for the system S which is of the form P (cid:16) Ψ S ( T = x ); Ψ C = Ψ x | Ψ S = Ψ in ; Ψ C = Ψ α (0) (cid:17) = Z /r dn ′ |h x | α ( n ′ ) i | , (11)where Ψ S is the state of the system and Ψ in is some preselected initial statefor S . The meaning of Eq. (11) is the probability of system S being in stateΨ S at time T = x when the clock has a position of x given that the clock12nd system have been initially synchronized such that Ψ S = Ψ in whenΨ C = Ψ α (0) .It is worth noting that the amplitude of the square of the wavefunctionfor an ideal clock (as discussed in Subsection 2.1) would be given by |h x | α ( n ′ ) i | = δ ( n ( x ) − n ′ ) , (12)where n ( x ) is obtained by inverting the expectation value x ( n ) = h Ψ C ( n ) | ˆ x | Ψ C ( n ) i .Then P (cid:16) Ψ S ( T = x ); Ψ C = Ψ x | Ψ S = Ψ in ; Ψ C = Ψ (0) α (cid:17) = Z /r dn ′ δ ( n ( x ) − n ′ )= 1 , (13)provided that 0 < n ( x ) < /r . What is left is to confirm the previous claim about the evolution operatorfor the system. This entails relating the Hamiltonians ˆ H C and ˆ H S to oneanother, but some care must be taken. Although the constraint ˆ H | Ψ i = 0is in place, it is a requirement on physical states only. The relation betweenthe individual Hamiltonians can therefore be expressed as H C ˙ ≈ − H S , (14)with the operator symbolism and the associated identity operators now im-plied, and where ˙ ≈ represents a weakly vanishing constraint as defined inDirac’s constraint formalism [14]. 13quation (14) would also generally include an interaction term H int ac-counting for the influence of the clock and the system on one another. How-ever, for current purposes, the clock can be considered small enough so thatits influence on the system can be considered negligible. But the conversestatement is not applicable, and the missing interactions would indeed greatlyinfluence the clock. Nonetheless, this picture is fully consistent with the cur-rent analysis, as the interaction term can be traced out of the final descrip-tion, leaving H C with a damping factor and H S (approximately) unaffected.The utility of Eq. (14) in this discussion lies in its ability to allow a re-lationship between the evolution operators for S and C , which implicitlyfollows from the entanglement between the two subsystems. Since the po-sition of C is supposed to be used as a time marker for a measurement on S , it is important to see if the evolution operator for S can be describedin terms of this clock variable x . To this end, one can start by consider-ing the total (combined) system, whose state can always be expressed in aSchmidt-decomposed form: | Ψ i = X j | Ψ j i = X j c j | Ψ j i C | Ψ j i S , (15)where the c ’s are numerical coefficients.Now, given that the clock is evolving in abstract time, this should ratherbe expressed as | Ψ j i = c j (cid:16) e − iH C n | Ψ j i C (cid:17) | Ψ j i S = c j e − i ( H − H S ) n | Ψ j i C | Ψ j i S ˙ ≈ c j | Ψ j i C (cid:16) e iH S n | Ψ j i S (cid:17) , (16)for each value of the index j . This allows the evolution operator for S tobe identified as U S ( n ) = e iH s n . However, since n is to be integrated out,14he goal is to replace this abstract time with the position of the clock andthen determine whether the resulting expression is an acceptable descriptionof evolution.To accomplish this last task, the semiclassical relationship between po-sition and time can be employed to relate n to a position measurement ofthe clock. In other words, expectation values will be used to determine n ( x ).The expectation value for position is then recalled [11] h x i = Ae − rn/ cos (Ω( ω, r ) n ) , (17)where A represents the amplitude for an oscillator without damping. Ex-panding the cosine component in terms of Ω n and retaining terms only upto linear order, one finds that h x i = Ae − rn/ → n ( x ) = 2 r ln (cid:16) Ax (cid:17) . (18)A more careful approach would be to find n ( x ) by accurately invertingEq. (17). However, given that n < n reset < r ∼ ω is in play, the higher-order terms produce corrections which are considered small enough to ignorein the current treatment (which is meant to illustrate a method and notprovide a rigorous analysis). A fuller motivation for this approximation isgiven in Appendix A.The substitution of n ( x ), as provided by Eq. (18), can be used in assessingthe evolution of system S , as described by the operator U S ( n ) = e iH s n . Forthis discussion, there is no need to specify an exact form of H S (see the veryend of the section). To proceed, one should then consider Eq. (18) with thesubstitution y = A − x and notice that x ≤ A due to the decay in time of15 , and so y ≥ n ( x ) = − r ln (cid:16) A − yA (cid:17) = − r ln (cid:16) − yA (cid:17) ≈ r (cid:16) yA + · · · (cid:17) , (19)which provides a form of n ( x ) that can be inserted into the description of U S . Along with the redefinition ˜ H S = rA H S , the evolution operator nowbecomes U S ≈ e i ˜ H S y . (20)The form of Eq. (20) is that of a standard evolution operator. The con-clusion is therefore that the position measurement of the clock does indeedprovide a useful measure of time for S , as long as n < n reset < r ∼ ω . Thisalso confirms the claims that were put forth regarding Eq. (11).One last point remains to be made. Although y (and so x ) represents ameasurable observable of the clock, x is a time parameter from the perspec-tive of the system S . This comes down to the fact that, from the viewpointof system S , the clock position is not an operator of any sort. This can bemotivated by the (approximate) commutation relation [ˆ x, ˆ O S ] = 0 for alloperators O S ∈ S , which follows from the (approximate) separability of theHamiltonian. Then, in terms of the previous Schmidt decomposition, h x | ˆ x (cid:16) | Ψ j i S | Ψ j i C (cid:17) = h x | ˆ x | Ψ j i C | Ψ j i S = x Ψ C j ( x ) | Ψ j i S , (21)indicating that x is only a number from the perspective of the system S .16 Conclusions
The damped harmonic oscillator appears to stand up well as a clock withinPW’s interpretation of time. As was shown here, the evolution of a system S is describable in such a way that the role of time is played by the positionof the clock.As far as resolving the problems of time goes, one can hope that animproved understanding of the conditional probability interpretation mightthen account for the disconnect between the treatments of time in relativityand in quantum mechanics. For instance, a quantum measurement could beassociated with a time which is measured by a clock experiencing relativisticeffects rather than a time which has no dynamics of its own. The problemof timelessness in already taken care of in such an interpretation since it isbuilt in as a assumption from the outset. This leaves only the problem ofa preferred direction for the arrow of time as dictated by the second law ofthermodynamics.The logical next step in analyzing PW’s interpretation of time would beto use an actual clock which is found in the real world. An investigation thatuses atomic clocks for this purpose is currently underway [15]. Acknowledgments
The research of AJMM received support from an NRF Incentive FundingGrant 85353 and NRF Competitive Programme Grant 93595. KLHB is sup-ported by an NRF bursary through Competitive Programme Grant 93595and a Henderson Scholarship from Rhodes University.17
Relating n and x
Given the definition Ω = q ω − r , the assumption r ∼ ω/ n < n reset < r (as proposed in the maintext), then the following approximation can be madeΩ n ∼ rn ≪ . (22)The task here is to verify if these inputs are sufficient to attain a linearrelation between abstract time n and clock position x .To obtain an explicit form for n ( x ), the first-order approximation ofEq. (17) is taken. After the substitution of Eq. (22), the equation of in-terest (17) can be expanded as follows: rn h Ax cos ( rn ) i = ln (cid:16) Ax (cid:17) + ln h − r n O ( r n ) i = ln (cid:16) Ax (cid:17) − r n O ( r n ) . (23)It is apparent that the limit Ax → n = 0 , as it otherwiseimplies that rn < n > A > x , thenln (cid:16) Ax (cid:17) ≈ rn ≫ r n . (24)It follows that rn ≈ ln (cid:16) Ax (cid:17) will be a good approximation for a descrip-tion of n ( x ) up to linear order in rn . Solving for n and defining y = A − x ,one finds that, to first order, n = 2 rA y , (25)as already claimed in Eq. (19). 18 eferences [1] B. S. DeWitt, “Quantum theory of gravity. I. The canonical theory,”Phys. Rev. no. 5, 1113 (1967) [doi:10.1103/PhysRev.160.1113].[2] C. Rovelli, “Time in quantum gravity: An hypotheis,” Phys. Rev. D no. 2, 442 (1991) [doi.org/10.1103/PhysRevD.43.442].[3] C. Rovelli, “Quantum mechanics without time: A model,” Phys. Rev. D no. 8, 2638 (1991) [doi.org/10.1103/PhysRevD.42.2638].[4] D. N. Page and W. K. Wootters, “Evolution without evolution: Dynamicsdescribed by stationary observables,” Phys. Rev. D no. 12, 2885 (1983)[doi:10.1103/PhysRevD.27.2885].[5] Y. Aharoniv and T. Kaufherr, “Quantum frames of reference,” Phy. Rev.D no. 2, 368 (1984) [doi.org/10.1103/PhysRevD.30.368].[6] C. Marletto and V. Vedral, “Evolution without evolution andwithout ambiguities,” Phys. Rev. D no. 4, 043510 (2017)[doi:10.1103/PhysRevD.95.043510] [arXiv:1610.04773 [quant-ph]].[7] V. Giovannetti, S. Lloyd and L. Maccone, “Quantum Time,” Phys.Rev. D no.4, 045033 (2015). [doi:10.1103/PhysRevD.92.045033][arXiv:1504.04215 [quant-ph]].[8] K. V. Kuchar, “Time and interpretations of quantum gravity,” Proc. 4thCanadian Conference on General Relativity and Relativistic Astrophysics(1992), Edited by G. Kunstatter, D. E. Vincent and J. G. Williams[doi:10.1142/S0218271811019347].199] C. E. Dolby, “The Conditional probability interpretation of the Hamilto-nian constraint,” [gr-qc/0406034].[10] V. Corbin and N. J. Cornish, “Semi-classical limit and minimum deco-herence in the Conditional Probability Interpretation of Quantum Me-chanics,” Found. Phys. no. 5, 474 (2009) [doi:10.1007/s10701-009-9298-5] [arXiv:0811.2814 [gr-qc]].[11] S. K. Bose and U. B. Dubey and V. N. Tewari, “Coherent states of adamped harmonic oscillator,” Pramana Jour. Phys. no. 4, 591-594(1985) [doi:10.1007/BF02846767].[12] H. Gzyl, “Quantization of the damped harmonic oscillator,” Phys. Rev.A no. 5, 2297 (1983).[13] Y. J. Ng and H. van Dam, “Limitation to quantum measurements ofspace-time distances,” Annals N. Y. Acad. Sci. no. 1, 579 (1995)[doi:10.1111/j.1749-6632.1995.tb38998.x] [hep-th/9406110].[14] P. A. M. Dirac, Lectures on Quantum Mechanics , Belfar GraduateSchool of Science (Yeshiva University, New York, 1964).[15] K. L. H. Bryan and A. J. M. Medved, work in progresswork in progress