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.