Time Observables in a Timeless Universe

Tommaso Favalli1,2 and Augusto Smerzi1

1QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy 2Universitá degli Studi di Napoli Federico II, Via Cinthia 21, I-80126 Napoli, Italy

Time in quantum mechanics is peculiar: 1 Introduction it is an observable that cannot be associ- ated to an Hermitian operator. As a conse- 1.1 Time and Clocks quence it is impossible to explain dynam- Observables in quantum theory are represented ics in an isolated system without invoking by Hermitian operators with the exception of an external classical clock, a fact that be- time [1,2] (and of a few more observables, in- comes particularly problematic in the con- cluding phases [3]). In Quantum Mechanics, as in text of quantum gravity. An unconven- Newtonian physics, time is an absolute “external” tional solution was pioneered by Page and real valued parameter that ﬂows continuously, in- Wootters (PaW) in 1983. PaW showed dependently from the material world. A change that dynamics can be an emergent prop- of perspective from this abstract Newtonian con- erty of the entanglement between two sub- cept was introduced in the theory of Relativity. systems of a static Universe. In this work Here time is an “internal” degree of freedom of the we ﬁrst investigate the possibility to intro- theory itself, operationally deﬁned by “what it is duce in this framework a Hermitian time shown on a clock”, with the clock being a wisely operator complement of a clock Hamilto- chosen physical system [4]. nian having an equally-spaced energy spec- An interesting question is if an operational ap- trum. An Hermitian operator complement proach would also be possible in Quantum Me- of such Hamiltonian was introduced by chanics by considering time as “what it is shown Pegg in 1998, who named it “Age”. We in a quantum clock” [5]. A proposal was for- show here that Age, when introduced in mulated in 1983 by Don N. Page and William the PaW context, can be interpreted as K. Wootters (PaW) [5,6]. Motivated by “the a proper Hermitian time operator conju- problem of time” in canonical quantization of gate to a “good”clock Hamiltonian. We gravity (see for example [7,8]) and considering therefore show that, still following Pegg’s a “Universe” in a stationary global state satis- formalism, it is possible to introduce in fying the Wheeler-DeWitt equation Hˆ |Ψi = 0, the PaW framework bounded clock Hamil- PaW suggested that dynamics can be consid- tonians with an unequally-spaced energy ered as an emergent property of entangled sub- spectrum with rational energy ratios. In systems, with the clock provided by a quan- this case time is described by a POVM tum spin rotating under the action of an applied and we demonstrate that Pegg’s POVM magnetic ﬁeld. This approach has recently at- states provide a consistent dynamical evo- tracted a large interest and has stimulated sev- lution of the system even if they are not arXiv:2003.09042v2 [quant-ph] 28 Oct 2020 eral extensions and generalisations (see for exam- orthogonal, and therefore partially un- ple [9, 10, 11, 12, 13, 14, 15, 16, 17]), including distinguishables. an experimental illustration [18]. A brief sum- mary of the Page and Wootters theory is given in Appendix A.

1.2 Time Observables in a Timeless Quantum System Tommaso Favalli: [email protected]ﬁ.it We consider here an isolated non-relativistic Augusto Smerzi: [email protected] quantum system that in the following we call

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 1 “Universe”. The Hilbert space of the “Universe” of the eigenvalues of Age. However, as Pegg em- is composed by a “clock-subspace” C that keeps phasized, and consistently with the Pauli objec- track of time and the “system-subspace” S of tion [1], the Age operator cannot be considered, the rest of the Universe. In absence of an ex- as a bona-ﬁde time operator. This because Age is ternal temporal reference frame we write the a property of the system itself and crucially de- Schrödinger equation of the Universe as: pends on its state. In particular, while for partic- ular states the rate of change of Age’s mean value ˆ ∂ (Hs − i~ ) |Ψi = 0 (1) can be constant, for an energy eigenstate its mean ∂tc value does not evolve: a system in a stationary state would not age as time goes on [20]. Pegg The ﬁrst term Hˆs is the Hamiltonian of the sub- system S. We interpret the second term as a also considered a larger set of Hamiltonians hav- (possibly approximate) time representation of the ing unequally-spaced energy eigenvalues. Also in clock-subspace Hamiltonian: this case it is possible to write down a comple- ment of the Hamiltonian whose eigenvalues can ∂ ˆ be used to construct a probability-operator mea- − i~ → Hc. (2) ∂tc sure (POVM). The central result of our work is to show that Under the implicit assumption that the two sub- the Pegg formalism ﬁnds a sound physical inter- systems C and S are not interacting, Eq. (1) pretation when incorporated in the Page-Wooters becomes: framework. We introduce as “good” clock a de- (Hˆ + Hˆ ) |Ψi = 0. (3) s c vice described by an Hamiltonian having equally The time representation of the clock Hamiltonian spaced eigenvalues. In this case time is described Eq. (2) would be correct, −i ∂ = Hˆ , only if Hˆ ~ ∂tc c c by an Hermitian operator. A device governed by has a continuous, unbounded spectrum. In this an Hamiltonian having rational energy diﬀerences case we could write the Hermitian time opera- can still provide a good clock mathematically de- tor in the energy representation as Tˆ = −i ∂ . ~ ∂Ec scribed by a POVM. We show that in both cases Since we consider here an isolated physical sys- we recover the Schrödinger dynamical evolution tem of ﬁnite size, the introduction of unbounded of the system S. Hamiltonians with a continuous spectrum would not be possible. As a consequence, an Hermi- tian time operator written in diﬀerential form as 2 Time from Entanglement in Eq. (1) cannot be introduced within standard 2.1 The Clock Subspace approaches [1]. In this work we explore the possibility to con- The ﬁrst problem to deal with is the introduc- struct a time Hermitian observable that is conju- tion of a good clock. We deﬁne as good clock gate to a clock Hamiltonian having an eigenvalue a physical system governed by a lower-bounded spectrum with a ﬁnite lower bound. It is clear, as Hamiltonian having discrete, equally-spaced, en- already mentioned, that such exploration would ergy levels, see also [25, 26] (a generalisation to be fruitless for the simple reason that its exis- non-equally spaced levels will be discussed in Sec- tence would contradict the Stone-von Neumann tion IV): theorem. A way out was considered by D. T. s X Pegg [20] (see also [21]) who suggested a proto- Hˆc = En |Eni hEn| , (4) col to construct an Hermitian operator, named n=0 “Age”, complement of a lower-bounded Hamilto- where s + 1 is the dimension of the clock space nian with equally-spaced energy eigenvalues. The that, following D. T. Pegg [20], we ﬁrst consider idea was to consider an Hamiltonian with an en- as ﬁnite. We now search for an Hermitian observ- ergy cut-oﬀ, calculate all quantities of interest, able τˆ in the clock space that is conjugated to the and eventually remove the cut-oﬀ by letting go clock Hamiltonian Hˆc. We deﬁne the time states to inﬁnity the upper bound. (we set ~ = 1) Age is conjugate to the Hamiltonian in the s sense that it is the generator of energy shifts while 1 X −iEnτm |τmic = √ e |Enic (5) the Hamiltonian is the generator of translations s + 1 n=0

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 2 T 2π with τm = τ0 +m s+1 , En = E0 +n T and m, n = 2.2 Dynamics 0, 1, ..., s. Eq. (5) provides an orthonormal and complete basis since We consider the total Hilbert space of the Uni- verse H = Hc ⊗ Hs, with Hc and Hs having hτm|τm0 i = δm,m0 (6) dimension dc = s + 1 and ds respectively. We require that our “good clock” has d d . A gen- and c s s X eral bipartite state of the Universe can be written |τmi hτm| = 1c. (7) as m=0

With the states (5) we can deﬁne the Hermitian dc−1 ds−1 X X time operator |Ψi = cn,k |Enic ⊗ |Ekis . (13) n=0 k=0 s τˆ = X τ |τ i hτ | (8) m m m ˆ m=0 We impose the constraint Eq. (3) H |Ψi = 0 and, under the assumption that the spectrum of the ˆ that is conjugated to the Hamiltonian Hc. It is clock Hamiltonian is suﬃciently dense (namely, ˆ indeed easy to show that Hc is the generator of that to each energy state of the system S there is shifts in τm values and, viceversa, τˆ is the gener- a state of the clock for which Eq. (3) is satisﬁed), ator of energy shifts: we obtain for the state of the Universe ˆ |τ i = e−iHc(τm−τ0) |τ i (9) m c 0 c d −1 Xs and |Ψi = c˜k |E = −Ekic ⊗ |Ekis (14) iτˆ(En−E0) k=0 |Enic = e |E0ic . (10) A second important property of the clock states P 2 with k |c˜k| = 1. With the resolution of the is their ciclic condition: |τm=s+1i = |τm=0i. The identity (7), we write time taken by the system to return to its initial state is d −1 d −1 2π 1 Xc Xs T = (11) |Ψi = √ |τ i ⊗ c˜ e−iEkτm |E i . δE m c k k s dc m=0 k=0 with δE being the spacing between two neigh- (15) bouring energy eigenvalues. Conversely, the By writing a generic state of the system as smallest time interval is Pds−1 −iEkτm |φmis = k=0 c˜ke |Ekis, the state (15) be- 2π comes δτ = τ − τ = . (12) m+1 m δE (s + 1) 1 dc−1 To summarise: the greater is the spectrum of X |Ψi = √ |τmic ⊗ |φmis . (16) the clock Hamiltonian, the smaller is the spac- dc m=0 ing δτ between two eigenvalues of the clock. The smaller is the distance between two eigenvalues It is interesting to note, and we emphasise, that of the clock energy, the larger the range T of the state |φmis is related to the the global |Ψi of the eigenvalues τm. The ﬁnal crucial step is to the Universe by choose the value of s. Following Pegg’s prescrip- tion [20], this has to be ﬁrst taken ﬁnite in order hτm|Ψi |φmis = √ (17) to allow the calculation of all quantities of inter- 1/ dc est, including the Schrödinger equation, that will therefore functionally depend on s. The physical that is the Everett relative state deﬁnition of the values of the observables are eventually obtained subsystem S with respect to the clock system C in the limit s −→ ∞. Obviously, this limit im- [19]. As pointed out in [11], this kind of pro- plies a continuous ﬂow of time, but nothing for- jection has nothing to do with a measurement. bids, in principle, to choose s large but ﬁnite so Rather, |φmis is a state of S conditioned to the to preserve a discrete time evolution. The two clock C in the state |τmic. prescriptions would give diﬀerent predictions for Now, following the PaW framework and using measured values of observables. Eq. (17), the constraints Eq. (3) and Eq. (9), we

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 3 have: PaW mechanism. As clearly pointed out by Pegg, τˆ has dimensions of time but it is a property of p iHˆc(τm−τ0) |φmis = dc hτ0| e |Ψi = the quantum system, and it strongly depends on the state of the system. With a quantum system p ˆ ˆ with Hamiltonian Hˆ we would be forced to con- = d hτ | ei(H−Hs)(τm−τ0) |Ψi = (18) c 0 sider τˆ deﬁned on the space of the system itself. So the evolution of the mean value of τˆ operator ˆ −iHs(τm−τ0) with respect to an external time has to be con- = e |φ0is stant or at least not zero, otherwise the dynamics hτ0|Ψi Pds−1 −iE τ0 where |φ0i = √ = c˜ke k |Eki . would freeze: s 1/ dc k=0 s The Eq. (18) provides the Schrödinger evolution dhτˆi h i = −i hψ| τ,ˆ Hˆ |ψi of S with respect to the clock time. dt Now we can also consider the global state writ- (21) ten in the form (16) and, through (18), we can X ∝ (E 0 − E ) hψ|E 0 i hE |ψi consider the unitary operator Uˆ (τ − τ ) = n n n n s m 0 n,n0 ˆ e−iHs(τm−τ0) [9]. With this choice the state of the global system can be written as where |ψi is a generic state of the system . If we consider the system in an energy eigenstate (that dc−1 is |ψi = |E i), we obtain 1 X ˆ i |Ψi = √ |τmic ⊗ Us(τm − τ0) |φ0is (19) dc dhτˆi m=0 = 0 (22) dt where explicitly is included the entire time his- which means that the τ values stops running tory of the Universe. We conclude this Section m over time. So, outside the PaW framework, the τˆ by noticing that the conditional probability of operator can not be considered as a time observ- obtaininig the outcome a for the system S when able, but as a property of the system that has measuring the observable A at a certain time τ m dimension of time. Conversely, within the PaW is given, as expected, by the Born rule: framework, we have a global stationary state that includes the whole time history of S with re- P (a on S , τm on C) spect to C. An energy eigenstate of the system P (a on S|τm on C) = = P (τm on C) S evolves with an unobservable global phase

−iEkτm |φmis = e |Ekis . (23) 2 ˆ = ha| Us(τm − τ0) |φ0i . However, this does not mean that the Universe (20) stops. Indeed, from the fact that in the clock space τˆ and Hˆc are conjugated operators, it fol- 3 The Hermitian Time Operator lows that, even if taking the system S in an energy eigenstate |Eki forces the clock in an Here we show that within the PaW framework eigenstate of Hˆc, all time states exist (indeed, the operator τˆ has the expected properties of a thanks to the fact that τˆ and Hˆc are incompati- Hermitian time observable. It is well known that ble observables, for construction we have |Eki ∝ P −iEkτm Pauli objected about the existence of a time Her- m e |τmi). The τˆ operator that Pegg’s mitian operator because time is continuous and deﬁned as complement of the Hamiltonian be- unbounded in the past and in the future while comes a proper time operator when included in general Hamiltonians have a lower bounded (con- the PaW framework. This happens in general tinuous or discrete) spectrum [27]. Pegg’s Age with any choice of the clock Hamiltonian, as dis- operator overcome the energy objection [1] since cussed by Leon and Maccone in [23], because in τˆ has a discrete spectrum and cyclical boundary the Page and Wootter theory the concept of ex- conditions while the appropriate limits are taken ternal time is eliminated (or in any case becomes only after calculating whatever of interest. The irrelevant), and time is an emerging property of question we address here is why τˆ can not be entanglement between the system S and the clock considered as a proper time operator outside the C imposed by the Wheeler-DeWitt constraint.

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 4 4 Unequally Spaced Energy Levels for and require that it satisﬁes the PaW constraint. the Clock Hamiltonian By writing

dc−1 ds−1 With the perspective to extend the set of Hamil- X X |Ψi = cn,k |Enic ⊗ |Ekis (29) tonians useful to describe a clock, we now con- n=0 k=0 sider the case in which the clock Hamiltonian ˆ does not have equally spaced energy levels, but and imposing H |Ψi = 0 (considering dc ds), non-degenerate eigenstates having rational en- we obtain ergy diﬀerences. In this case we cannot deﬁne ds−1 an Hermitian operator but we can still introduce X |Ψi = c˜k |E = −Ekic ⊗ |Ekis (30) a probability-operator measure, complement of k=0 such Hamiltonian [20]. Pds−1 2 with k=0 |c˜k| = 1. We can now apply the resolution of the identity (28) to the state |Ψi 4.1 Discrete Flow of Time and obtain (D = s + 1):

We consider a quantum system described by p+1 D−1 dc X energy states |Eii and Ei energy levels with i = |Ψi = |α i hα |Ψi = D m m 0, 1, 2, ..., p such that m=0

Ei − E0 Ci √ = , (24) D−1 ds−1 dc X X E1 − E0 Bi = |α i ⊗ c˜ e−iEkαm |E i . D m c k k s m=0 k=0 where Ci and Bi are integers with no common (31) factors. We can write

2π We notice here that the states |αmic are not or- Ei = E0 + ri (25) thogonal. This introduces a possible conceptual T warning that needs to be discussed. It is clear 2πr1 Ci that by considering time states that are not or- where T = E −E , ri = r1 B for i > 1 (with r0 = 1 0 i thogonal implies that these are partially indistin- 0) and r1 equal to the lowest common multiple of guishable with a single measurement, the prob- the values of Bi. In this space we deﬁne the states ability of indistinguishability being proportional 2 to | hαm0 |αmi | . The partial indistinguishability d −1 1 c of the states |αmic implies an overlap between X −iEiαm |αmi = √ e |Eii (26) diﬀerent times. We show, however, that also in c d c c i=0 this case the time evolution is described by the Schrödinger equation. Considering where dc = p + 1 and

hαm|Ψi T |φmi = √ , (32) α = α + m (27) s 1/ d m 0 s + 1 c

Pds−1 −iαmE we obtain |φmi = c˜ke k |Eki for the with m = 0, 1, 2, ..., s and s + 1 = D ≥ rp. The s k=0 s state of the system S (see Appendix C). There- number of |αmi states is therefore greater than fore, even if time states are partially indistin- the number of energy states in Hc and the s + 1 guishable, the state of the system S, conditioned values of αm are uniformly distributed over T . The resolution of the identity (7) is now replaced on a given |αmic, evolves with αm. Indeed, by (see Appendix B) thanks to the fact that ˆ |α i = e−iHc(αm−α0) |α i , (33) p + 1 s m c 0 c 1 = X |α i hα | . (28) c s + 1 m m m=0 using once again the constraint (3) and the (32), we obtain As in the previous discussion, we can now con- −iHˆs(αm−α0) sider a general state in the space H = Hc ⊗ Hs |φmis = e |φ0is (34)

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 5 that is the Schrödinger evolution for the state The global state is ˆ |φmis with the Hamiltonian Hs. We notice here 1 Z α0+T that POVMs generalizing the one introduced by |Ψi = dα |α˜i hα˜|Ψi = Pegg have been discussed in [28]. The conse- T α0 quence of using POVM is that time states are not fully indistinguishable. This suggests a possible d −1 1 Z α0+T Xs extension of the deﬁnition of the Everett relative = dα |α˜i ⊗ c e−iEkα |E i = T c k k s states where is still possible to have a consistent α0 k=0 dynamical evolution of the system S. To conclude this Section we notice that, 1 Z α0+T = dα |α˜i ⊗ |φ(α)i through equation (34), we can again deﬁne the T c s ˆ α0 ˆ −iHs(αm−α0) unitary operator Us(αm − α0) = e . (39) With this choice the state of the global system can be written as and, since |φ(α)is ∝ hα˜|Ψi, we derive the Schrödinger equation for the state |φ(α)is D−1 dc X ˆ |Ψi = √ |αmic ⊗ Us(αm − α0) |φ0is (35) ∂ ∂ D i |φ(α)i = i hα˜|Ψi = m=0 ∂α s ∂α and the conditional probability of obtaining the outcome a for the system S when measuring the dc−1 ∂ X = i hE | eiEkα |Ψi = observable A at a certain time αm is given again ∂α k by the Born rule (see Appendix D): k=0 (40)

P (a on S , αm on C) dc−1 P (a on S|αm on C) = = X iEkα P (αm on C) = − hEk| Eke |Ψi = k=0

2 ˆ = ha| Us(αm − α0) |φ0i . ˆ ˆ = − hα˜| Hc |Ψi = Hs |φ(α)is . (36) To conclude this Section we brieﬂy discuss the The Eq. (36) shows that the conditioned state of case of a clock Hamiltonian with a discrete spec- S to a certain clock value αm has no contributions trum and arbitrary (not rational) energy level ra- from diﬀerent times αm0 6= αm, and so interfer- tios. Also in this scenario a Schrödinger-like evo- ence phenomena are not present even if the time lution is recovered for the state of the system S states are not orthogonal. with respect to the clock. A caveat is that, in this case, the resolution of the identity (28) is no 4.2 Continuous Flow of Time longer exact and time states |αmic do not provide an overcomplete basis in C. Nevertheless, since So far we have considered a discrete ﬂow of time. any real number can be approximated with ar- A continuous can be obtained in the limit s → ∞ bitrary precision by a ratio between two rational [20]. We deﬁne numbers, the residual terms in the resolution of the identity and consequent small corrections can p X −iEiα be arbitrarily reduced. |α˜i = e |Eii (37) i=0 4.3 Non-Observable Universe as Clock and the where again p + 1 is the number of energy eigen- Arrow of Time states and α can now take any real value from α0 to α0 +T . In this framework the resolution of the Is there an arrow of time in the PaW formalism? identity (28) becomes The answer in clearly negative. However, it is possible to introduce an emergent arrow of time. 1 Z α0+T Following [11] one can consider for simplicity that 1 = dα |α˜i hα˜| . (38) c the system S consists only of two subsystems, T α0

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 6 “the observer”(Σ1) and “the observed”(Σ2) which In addition we have shown that it is possible to are initially in a product state. The arrow of extend this framework to any Hamiltonian with time (with respect to clock time) can be provided a discrete spectrum having rational ratios of the by the increase in entanglement between the two energy levels. We have demonstrated that even subsystems within S, as the observer learns more if in this case the time states are not fully distin- and more about the observed. We can ask now guishable, the system S still evolves with respect where to ﬁnd a good clock for the Universe [29]. to the clock time according to the Schrödinger We have considered as a “good”clock a device de- equation. Finally, we have considered the contin- ﬁned in a Hilbert space larger than the Hilbert uous limit of the ﬂow of time. space of the system S, that is dc ds. Indeed, We can read the PaW approach as a general if dc ≤ ds, it would not be possible to connect “internalization protocol" where, beside time, it every energy state of the system S to an energy is possible to internalise spatial reference frames state of C satisfying the constraint (3), and some where space is “what is shown on a meter”. In the states of S would be excluded from the dynamics. current formalism of quantum mechanics there is Therefore, the clock introduced here has essen- an evident asymmetry in how space and time are tially two properties: it has to be larger than the treated: the spatial degrees of freedom are typ- system S and it has to interact only weakly with ically a quantum property of the system under S or, in the ideal case, it should not interact at investigation while time on the contrary appears all. Does such a clock exist? A possible choice is as a classical parameter external to the theory to consider the non-observable Universe (namely, [30]. Perhaps a PaW formulation of spacetime the Universe laying outside the light cone centred may represent a ﬁrst step towards removing this in the Earth) as a clock for the observable Uni- asymmetry. This might help to develop relativis- verse. Indeed, in this case, the clock and the ob- tic generalizations of the PaW formalism, see also servable Universe S are not interacting but can [31, 32, 33, 34]. still be fully entangled, with the Hilbert space Acknowledgements : We acknowledge fund- of the clock that can be quite larger that the ing from the project EMPIR-USOQS, EMPIR Hilbert space of S. With this choice (which of projects are co-funded by the European Unions course is just one among several equally specu- Horizon2020 research and innovation programme lative choices) the two requirements for a good and the EMPIR Participating States. We also clock are satisﬁed. We can then consider that acknowledge ﬁnancial support from the H2020 during the evolution with respect to such a clock, QuantERA ERA-NET Cofund in Quantum Tech- inside the observable Universe (that is inside the nologies projects QCLOCKS. interacting subsystems Σ1 and Σ2 of S) there is an increasing entanglement generated by Hˆs and, therefore, an increasing relative entropy and the References emergence of a thermodynamic arrow of time. [1] W. Pauli, General Principles of Quantum Me- chanics, Springer Berlin Heidelberg (1980). 5 Conclusions [2] P. Busch, The Time-Energy Uncertainty Re- lation, (2008) on this topic. In: Muga J., May- In this work we have elaborated on the Page and ato R. S., Egusquiza Í., Time in Quantum Me- Wootters theory. PaW provides a consistent pic- chanics. Lecture Notes in Physics, vol. 734. ture of quantum time as an emerging property Springer, Berlin, Heidelberg. of entanglement among subsystems of the Uni- [3] R. Lynch, Physics Reports 256, 367 (1995). verse. We have considered a protocol introduced by Pegg for the construction of a “Age”operator [4] C. Rovelli, Phys. Rev. D 43, 442 (1991). complement of a bounded Hamiltonian having a [5] D. N. Page and W. K. Wootters, Phys. Rev. spectrum of equally spaced energy levels. By in- D 27, 2885 (1983). corporating Pegg’s formalism in the PaW theory we have shown that Age can be interpreted as an [6] W. K. Wootters, International Journal of Hermitian time operator providing the dynamical Theoretical Physics, 23, 701–711 (1984). evolution of the system. [7] B. S. DeWitt, Phys. Rev. 160, 1113 (1967).

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Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 8 A Summary of PaW Theory

We give here a brief summary of PaW theory, following [11]. Page and Wootters consider the whole Universe as being in a stationary state with zero eigenvalue (consistently with the the Wheeler-DeWitt equation), that is

Hˆ |Ψi = 0 (41) where Hˆ and |Ψi are the Hamiltonian and the state of the Universe, respectively. They divide the Universe into two non-interacting subsystems, the clock C and the rest of the Universe S, and thus the total Hamiltonian can be written as

Hˆ = Hˆc ⊗ 1s + 1c ⊗ Hˆs (42) where Hˆc and Hˆs are the Hamiltonians acting on C and S respectively, and 1c, 1s are unit operators. The condensed history of the system S is written through the entangled global stationary state |Ψi ∈ H = Hc ⊗ Hs (which satisﬁes the constraint (41)) as follows: X |Ψi = ct |tic ⊗ |φtis (43) t where the states {|tic} are eigenstates of the operator choosen to be the clock observable. The relative state (in Everett sense [19]) of the subsystem S with respect to the clock C can be deﬁned

h (c) i T rc Pt ρ ρ(s) = = |φ i hφ | (44) t h (c) i t t T r Pt ρ

(c) where ρ = |Ψi hΨ| and Pt is the projector on a certain time state in the clock subspace. Note that equation (44) is the Everett relative state deﬁnition of the subsystem S with respect to the clock system

C. As pointed out in [11], this kind of projection has nothing to do with a measurement. Rather, |φtis is a state of S conditioned to the clock C being in the state |tic. From equations (41), (42) and (44), it is then possible to derive the Schrödinger equation for the relative state of the subsystem S with respect to the clock C:

(s) ∂ρ h i t = i ρ(s), Hˆ . (45) ∂t t s Since the subsystem S experiences a Schrödinger-like evolution with respect to the clock C, the param- eter t can be interpreted as time and the evolution of S has been recovered within a globally stationary Universe. The last point concerns conditional probabilities. In the PaW framework the probability to obtain the outcome a when measuring the observable Aˆ on the subspace S “at a certain time”t˜ can be written as: P (a on S , t˜ on C) P (a on S | t˜ on C) = (46) P (t˜ on C) that is the conditional probability of obtaining a on S given that the clock C shows t˜. That’s why the PaW mechanism is sometimes called “conditional probability interpretation of time”. The PaW approach to time has not been without criticism. For instance Kuchar [22] questioned the possibility of constructing a two-time propagator and Albrecht and Iglesias [23] stressed how the possibility for diﬀerent choices of the clock inexorably leads to an ambiguity in the dynamics of the rest of the Universe. These objections were addressed by Giovannetti, Lloyd and Maccone [9] (see also [10, 24]) and Marletto and Vedral [11], respectively.

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 9 B Proof of Equation (28)

Ps We prove that m=0 |αmi hαm| is equal to the identity:

s s X 1 X X X |α i hα | = e−iαmEi eiαmEk |E i hE | = m m p + 1 i k m=0 m=0 i k (47) 1 s s X X X X iαm(rk−ri)2π/T = |Eki hEk| + e |Eii hEk| . p + 1 m=0 k k6=i m=0

For (Ek −E0)/(E1 −E0) rational, and thus rk −ri integer, the second term of the right side of equation (47) will be zero and then we have

p + 1 s X |α i hα | = 1 . (48) s + 1 m m c m=0

C Relative State Deﬁnition for S in case of non-orthogonal Time States

We start considering the global state |Ψi written as

d −1 Xs |Ψi = c˜k |E = −Ekic ⊗ |Ekis (49) k=0 and we apply in sequence the resolutions of the identity on the clock subspace

d D−1 c X |α i hα | = 1 (50) D m m c m=0 and

d −1 Xc |Eni hEn| = 1c. (51) n=0

We obtain

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 10 d D−1 |Ψi = c X |α i hα |Ψi = D m m m=0

d D−1 ds−1 = c X |α i ⊗ X c˜ hα |E = −E i |E i = D m c k m k k s m=0 k=0

√ D−1 ds−1 dc X X = |α i ⊗ c˜ e−iαmEk |E i = D m c k k s m=0 k=0

√ dc−1 D−1 ds−1 X dc X X = |E i hE | |α i ⊗ c˜ e−iαmEk |E i = (52) n n D m c k k s n=0 m=0 k=0

√ dc−1 D−1 ds−1 dc X X X = |E i ⊗ hE |α i c˜ e−iαmEk |E i = D n c n m k k s n=0 m=0 k=0

d −1 D−1 d −1 1 Xc X Xs = |E i ⊗ e−iαmEn c˜ e−iαmEk |E i = D n c k k s n=0 m=0 k=0

d −1 d −1 D−1 Xc Xs 1 X = |E i ⊗ c˜ e−iαm(En+Ek) |E i n c k D k s n=0 k=0 m=0

from which we have

D−1 X −iαm0 (En+Ek) e = DδEn,−Ek . (53) m0=0

Considering now the deﬁnition of the |φmis state as

p |φmis = dc hαm|Ψi , (54)

we have:

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 11 √ D−1 ds−1 dc X X −iα 0 E hα |Ψi = hα | |α 0 i ⊗ c˜ e m k |E i = m m D m k k s m0=0 k=0

√ D−1 dc−1 ds−1 dc X X X 1 iEnαm −iE 0 α 0 −iE α 0 = e e n m hE |E 0 i c˜ e k m |E i = D d n n k k s m0=0 n,n0=0 k=0 c (55) √ D−1 dc−1 ds−1 dc X X X 1 = eiEnαm e−iEn0 αm0 δ c˜ e−iEkαm0 |E i = D d En,En0 k k s m0=0 n,n0=0 k=0 c

1 dc−1 ds−1 D−1 X iEnαm X X −iαm0 (En+Ek) = √ e c˜k e |Ekis , D dc n=0 k=0 m0=0 and considering (53) we obtain

1 ds−1 X −iαmEk hαm|Ψi = √ c˜ke |Ekis . (56) dc k=0

Then the deﬁnition (54) implies

ds−1 X −iαmEk |φmis = c˜ke |Ekis . (57) k=0

D Proof of Equation (36)

We start considering the global state written as

√ d D−1 |Ψi = c X |α i ⊗ |φ i (58) D m c m s m=0

Pds−1 −iEkαm where |φmis = k=0 c˜ke |Ekis. We can now calculate the conditional probability as follows

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 12 P (a on S , αm on C) P (a on S|αm on C) = = P (αm on C)

|(hα | ha|) |Ψi|2 = m = P 2 a |(hαm| ha|) |Ψi|

√ 2 dc PD−1 (hαm| ha|) D m0=0 |αm0 ic |φm0 is = √ 2 = P dc PD−1 a (hαm| ha|) D m0=0 |αm0 ic |φm0 is

d −1 2 (59) PD−1 P s −iEkαm0 (hαm| ha|) m0=0 |αm0 ic k=0 c˜ke |Ekis = = D−1 d −1 2 P P P s −iEkαm0 a (hαm| ha|) m0=0 |αm0 ic k=0 c˜ke |Ekis

d −1 d −1 2 PD−1 P c iEn(αm−αm0 ) P s −iEkαm0 m0=0 n=0 e ha| k=0 c˜ke |Ekis = = D−1 d −1 d −1 2 P P P c iEn(αm−αm0 ) P s −iEkαm0 a m0=0 n=0 e ha| k=0 c˜ke |Ekis

d −1 d −1 2 P c iEnαm P s PD−1 −i(Ek+En)αm0 n=0 e ha| k=0 c˜k m0=0 e |Ekis = . d −1 d −1 D−1 2 P P c iEnαm P s P −i(Ek+En)αm0 a n=0 e ha| k=0 c˜k m0=0 e |Ekis

PD−1 −iαm0 (En+Ek) Thanks to equation (53), that is m0=0 e = DδEn,−Ek , we have

2 Pds−1 −iEkαm ha| k=0 c˜ke |Ekis P (a on S|αm on C) = = d −1 2 P P s −iEkαm a ha| k=0 c˜ke |Ekis (60)

|ha|φ i|2 = m = |ha|φ i|2 P 2 m a |ha|φmi|

ˆ −iHs(αm−α0) ˆ and considering that |φmis = e |φ0is = Us(αm − α0) |φ0i we obtain 2 ˆ P (a on S|αm on C) = ha| Us(αm − α0) |φ0i . (61)

Accepted in Quantum 2020-10-28, click title to verify. Published under CC-BY 4.0. 13