ORIGINAL RESEARCH published: 18 June 2021 doi: 10.3389/fphy.2021.587083

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

Philipp A. Höhn 1,2*, Alexander R. H. Smith 3,4* and Maximilian P. E. Lock 5*

1 Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan, 2 Department of Physics and Astronomy, University College London, London, United Kingdom, 3 Department of Physics, Saint Anselm College, Manchester, NH, United States, 4 Department of Physics and Astronomy, Dartmouth College, Hanover, NH, United States, 5 Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Vienna, Austria

We have previously shown that three approaches to relational quantum dynamics— relational Dirac observables, the Page-Wootters formalism and quantum

Edited by: deparametrizations—are equivalent. Here we show that this “trinity” of relational Daniele Oriti, quantum dynamics holds in relativistic settings per frequency superselection sector. Ludwig-Maximilians-University Time according to a clock subsystem is defined via a positive operator-valued measure Munich, Germany (POVM) that is covariant with respect to the group generated by its (quadratic) Reviewed by: Mehdi Assanioussi, Hamiltonian. This differs from the usual choice of a self-adjoint clock observable University of Warsaw, Poland conjugate to the clock momentum. It also resolves Kuchar’sˇ criticism that the Page- Zhenbin Wu, University of Illinois at Chicago, Wootters formalism yields incorrect localization probabilities for the relativistic particle United States when conditioning on a Minkowski time operator. We show that conditioning instead Steffen Gielen, on the covariant clock POVM results in a Newton-Wigner type localization probability The University of Sheffield, United Kingdom commonly used in relativistic quantum mechanics. By establishing the equivalence *Correspondence: mentioned above, we also assign a consistent conditional-probability interpretation to Philipp A. Höhn relational observables and deparametrizations. Finally, we expand a recent method of [email protected] Alexander R. H. Smith changing temporal reference frames, and show how to transform states and observables [email protected] frequency-sector-wise. We use this method to discuss an indirect clock self-reference Maximilian P. E. Lock effect and explore the state and temporal frame-dependence of the task of comparing [email protected] and synchronizing different quantum clocks. Specialty section: Keywords: relational quantum dynamics, problem of time, relational Dirac observables, Page-Wootters formalism, This article was submitted to quantum deparametrizations, quantum clocks, quantum reference frames, relativistic quantum clocks High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics 1. INTRODUCTION Received: 24 July 2020 In general relativity, time plays a different role than in classical and quantum mechanics, or Accepted: 22 March 2021 quantum field theory on a Minkowski background. General covariance dispenses with a preferred Published: 18 June 2021 choice of time and introduces instead a dynamical notion of time which depends on solutions Citation: to the Einstein field equations. In the canonical approach to quantum gravity this leads to the Höhn PA, Smith ARH and Lock MPE infamous problem of time [1–3]. Its most well-known facet is that, due to the constraints of the (2021) Equivalence of Approaches to Relational Quantum Dynamics in theory, quantum states of spacetime (and any matter contained in it) do not at first sight appear to Relativistic Settings. undergo any time evolution, in seeming contradiction with everyday experience. Front. Phys. 9:587083. The resolution comes from one of the key insights of general relativity: any physical notion of doi: 10.3389/fphy.2021.587083 time is relational, the degrees of freedom of the Universe evolve relative to one another [4–6]. This

Frontiers in Physics | www.frontiersin.org 1 June 2021 | Volume 9 | Article 587083 Höhn et al. Relational Dynamics in Relativistic Settings insight has led to three main relational approaches to the problem quantum constraints. Furthermore, the trinity resolves Kuchaˇr’s of time, each of which seeks to extract a notion of time from criticism (c) that the Page-Wootters formalism would yield within the quantum degrees of freedom, relative to which the wrong propagators, by showing that the correct propagators others evolve: always follow from manifestly gauge-invariant conditional probabilities on the physical Hilbert space [7]. This resolution (i) a Dirac quantization scheme, wherein relational observables of criticism (c) differs from previous resolution proposals which are constructed that encode correlations between evolving relied on ideal clocks [25, 43, 68] and auxiliary ancilla systems and clock degrees of freedom [1, 2, 4, 7–38], [43] and can be viewed as an extension of [46]. (ii) the Page-Wootters formalism, which defines a relational The transformations between (i)–(iii) of the trinity also dynamics in terms of conditional probabilities for clock and allowed us to address the multiple choice problem in Höhn evolving degrees of freedom [7, 25, 39–57], and et al. [7] by extending a previous method for changing temporal (iii) classical or quantum deparametrizations, which result in reference frames, i.e., clocks, in the quantum theory [30, 31, 47] a reduced quantum theory that only treats the evolving (see also [32–34, 69]). The resolution to the problem lies in degrees of freedom as quantum [1, 2, 7, 10, 30, 31, 58–65]. part in realizing that a solution to the Wheeler-DeWitt equation These three approaches have been pursued largely independently encodes the relations between all subsystems, including the with the relation between them previously unknown. They have relations between subsystems employed as clocks to track the also not been without criticism, especially the Page-Wootters dynamics of other subsystems; there are multiple choices of formalism. For example, Kuchaˇr [1] raised three fundamental clocks, each of which can be used to define dynamics. Our criticisms against this approach, namely that it: proposal is thus to turn the multiple choice problem into a feature by having a multitude of quantum time choices at (a) leads to wrong localization probabilities in relativistic our disposal, which we are able to connect through quantum settings, temporal frame transformations. This is in line with developing (b) is in conflict with the constraints of the theory, and a genuine quantum implementation of general covariance [7, 30, (c) yields wrong propagators. 31, 38, 70–74]. This proposal is part of current efforts to develop a Concern has also been voiced that there is an inherent ambiguity general framework of quantum reference frame transformations in terms of which clock degrees of freedom one should choose, (and study their physical consequences [75–85]), and should be also known as the multiple choice problem [1–3, 66, 67]. Indeed, in contrasted with other attempts at resolving the multiple choice generic general relativistic systems there is no preferred choice of problem by identifying a preferred choice of clock [53] (see [7] relational time variable and different choices may lead to a priori for further discussion of this proposal). different quantum theories. We did not address Kuchaˇr’s criticism (a) that the Page- In our recent work [7] we addressed the relation between Wootters formalism yields the wrong localization probabilities these three approaches (i)–(iii) to relational quantum dynamics, for relativistic models in Höhn et al. [7] as they feature demonstrating that they are, in fact, equivalent when the clock clock Hamiltonians which are quadratic in momenta and Hamiltonian features a continuous and non-degenerate spectrum thus generally have a degenerate spectrum, splitting into and is decoupled from the system whose dynamics it is used to positive and negative frequency sectors. This degeneracy is describe. Specifically, we constructed the explicit transformations not covered by our previous construction. While quadratic mapping each formulation of relational quantum dynamics into clock Hamiltonians are standard in the literature on relational the others. These maps revealed the Page-Wootters formalism observables [approach (i)] and deparametrizations [approach (ii) and quantum deparametrizations (iii) as quantum symmetry (iii)], see e.g., [4, 10, 11, 19, 29, 31], relativistic particle reductions of the manifestly gauge-invariant formulation (i). In models have only recently been studied in the Page- other words, the Page-Wootters formalism (ii) and quantum Wootters formalism [approach (ii)] [45, 49–51]. However, deparametrizations (iii) can be regarded as quantum analogs Kuchaˇr’s criticism (a) that the Page-Wootters approach of gauge-fixed formulations of gauge-invariant quantities (i). yields incorrect localization probabilities in relativistic Conversely, the formulation in terms of relational Dirac settings has yet to be addressed. Since the Page-Wootters observables (i) constitutes the quantum analog of a gauge- formalism encounters challenges in relativistic settings, given invariant extension of the gauge-fixed formulations (ii) and (iii). the equivalence of relational approaches implied by the More physically, these transformations establish (i) as a clock- trinity, one might worry about relational observables and choice-neutral (in a sense explained below), (ii) as a relational deparametrizations too. Schrödinger, and (iii) as a relational Heisenberg picture of In this article, we show that these challenges can be overcome, the dynamics. Constituting three faces of the same quantum and a consistent interpretation of the relational dynamics can be dynamics, we called the equivalence of (i)–(iii) the trinity of provided. To this end, we extend the trinity to quadratic clock relational quantum dynamics. Hamiltonians, thus encompassing many relativistic settings; we This equivalence not only provides relational Dirac show that all the results of Höhn et al. [7] hold per frequency observables with a consistent conditional probability sector associated to the clock due to a superselection rule induced interpretation, but also resolves Kuchaˇr’s criticism (b) that by the Hamiltonian constraint. Frequency-sector-wise, the the Page-Wootters formalism would be in conflict with the relational dynamics encoded in (i) relational observables, (ii) the

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Page-Wootters formalism, and (iii) quantum deparametrizations reference frame would yield, one must contract the tensor with are thus also fully equivalent. the vectors corresponding to that choice of frame. In this way, The key to our construction, as in Höhn et al. [7], is the the description of the same tensor looks different relative to use of a Positive-Operator Valued Measure (POVM) which different frames, but the tensor per se, as a multilinear map, is here transforms covariantly with respect to the quadratic clock reference-frame-neutral. It is this reference-frame-neutrality of Hamiltonian [86–89] as a time observable. This contrasts with tensors which results in the frame-independence of physical laws. the usual approach of employing an operator conjugate to the The notion of reference frame as a vector frame is usually clock momentum (i.e., the Minkowski time operator in the taken to define the orientation of a local laboratory of some case of a relativistic particle). This covariant clock POVM is observer. In practice, one often implicitly identifies the local lab instrumental in our resolution of Kuchaˇr’s criticism (a) that the (i.e., the reference system relative to which the remaining physics Page-Wootters formalism yields wrong localization probabilities is described) with the reference frame. This is an idealization for relativistic systems. We show that when conditioning on which ignores the lab’s back-reaction on spacetime, interaction this covariant clock POVM rather than Minkowski time, one with other physical systems and possible internal dynamics, obtains a Newton-Wigner type localization probability [90, 91]. while at the same time assuming it to be sufficiently classical While a Newton-Wigner type localization is approximate and not so that superpositions of orientations can be ignored. Such an fully Lorentz covariant, due to the relativistic localization no- idealization is appropriate in general relativity where the aim go theorems of Perez-Wilde [92] and Malament [93] (see also is to describe the large-scale structure of spacetime. However, [94, 95]), it is generally accepted as the best possible localization in quantum gravity, where the goal is to describe the micro- in relativistic quantum mechanics (In quantum field theory structure of spacetime, this may no longer be appropriate [98]. localization is a different matter [90, 94]). This demonstrates More generally, we may ask about the fate of general covariance the advantage of using covariant clock POVMs in relational when we take seriously the fact that physically meaningful quantum dynamics [7, 44, 45, 96, 97]. The trinity also extends reference frames are in practice always associated with physical the probabilistic interpretation of relational observables: a Dirac systems, and as such are comprised of dynamical degrees of observable describing the relation between a position operator freedom that may couple with other systems, undergo their own and the covariant clock POVM corresponds to a Newton-Wigner dynamics and will ultimately be subject to the laws of quantum type localization in relativistic settings. theory. What are then the reference-frame-neutral structures? Finally, we again use the equivalence between (i) and (iii) In regard to this question, we note that the classical notion to construct temporal frame changes in the quantum theory. of general covariance for reference frames associated to idealized On account of superselection rules across frequency sectors, local labs is deeply intertwined with invariance under general temporal frame changes can only map information contained in coordinate transformations, i.e., passive diffeomorphisms. In the overlap of two frequency sectors, one associated to each clock, moving toward non-idealized reference frames (or rather from one clock “perspective” to another. We apply these temporal systems), we shift focus from coordinate descriptions to frame change maps to explore an indirect clock self-reference dynamical reference degrees of freedom, relative to which the and the temporal frame and state dependence of comparing and remaining physics will be described. In line with this, we shift synchronizing readings of different quantum clocks. the focus from passive to active diffeomorphisms, which directly While completing this manuscript, we became aware act on the dynamical degrees of freedom. This is advantageous of Chataignier [38], which independently extends some results for quantum gravity, where classical spacetime coordinates are a of Höhn et al. [7] on the conditional probability interpretation priori absent. A quantum version of general covariance should be of relational observables and their equivalence with the Page- formulated in terms of dynamical reference degrees of freedom Wootters formalism into a more general setting. However, a [7, 30, 31, 38, 70–74]. different formalism [37] is used in Chataignier [38], which does The active symmetries imply a redundancy in the description not employ covariant clock POVMs and therefore the two works of the physics. A priori all degrees of freedom stand on an equal complement one another. footing, giving rise to a freedom in choosing which of them shall Throughout this article we work in units where h 1. be treated as the redundant ones. The key idea is to identify = this choice with the choice of reference degrees of freedom, i.e., those relative to which the remaining degrees of freedom will be 2. CLOCK-NEUTRAL FORMULATION OF described1. Accordingly, choosing a dynamical reference system CLASSICAL AND QUANTUM MECHANICS amounts to removing redundancy from the description. As such, we may interpret the redundancy-containing description Colloquially, general covariance posits that the laws of physics (in both the classical and quantum theory) as a perspective- are the same in every reference frame. This is usually interpreted neutral description of physics, i.e., as a global description of as implying that physical laws should take the form of tensor equations. Tensors can be viewed as reference-frame-neutral 1 objects: they define a description of physics prior to choosing Indeed, we do not want to describe the reference degrees of freedom directly relative to themselves in order to avoid the self-reference problem [99, 100]. a reference frame. They thereby encode the physics as “seen” Nevertheless, through the perspective-neutral structure it is possible to construct by all reference frames at once. If one wants to know the indirect self-reference effects of quantum clocks through temporal frame changes, numbers which a measurement of the tensor in a particular see Höhn et al. [7] and section 7.3.

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physics prior to having chosen a reference system, from whose generated by CH on C when the clock T reads τ?”[4, 7, 11–19, 19– perspective the remaining degrees of freedom are to be described 24, 26, 30, 31]. We will denote such an observable by Ff ,T(τ). As [7, 30, 31, 71, 72]. This perspective-neutral structure is thus shown in Dittrich [21, 22] and Dittrich and Tambornino [23, 24], proposed as the reference-frame-neutral structure for dynamical these observables can be constructed by solving αu T τ for CH = (i.e., non-idealized) reference systems. u yielding the solution u u (τ) and defining = T In this article we focus purely on temporal diffeomorphisms τ : αu and thus on temporal reference frames/systems, or simply clocks. Ff ,T( ) CH f = u uT (τ) In this case, we refer to the perspective-neutral structure as = ∞ (τ T)n C a clock-(choice-)neutral structure [7, 30, 31], which we briefly − f , H , (4) review here in both the classical and quantum theory. It is a ≈ n! T, C n 0  { H}n description of the physics, prior to having chosen a temporal X= : th reference system relative to which the dynamics of the remaining where f , g n f , g n 1, g is the n -nested Poisson bracket { } = {{ } − } degrees of freedom are to be described. subject to f , g : f . The F (τ) satisfy Equation (3) and thus { }0 = f ,T constitute a family of Dirac observables parametrized by τ. Such 2.1. Clock-Neutral Classical Theory relational observables are so-called gauge-invariant extensions of Consider a classical theory described by an action gauge-fixed quantities [7, 21–24, 37, 102]. S du L(qa, dqa/du), where qa denotes a collection = R In generic models there is no preferred choice for the clock of configuration variables indexed by a. Such a theory P R function T among the degrees of freedom on kin, which is exhibits temporal diffeomorphism invariance if the action sometimes referred to as the multiple choice problem [1, 2]. a a S is reparametrization invariant; that is, L(q , dq /du) Different choices of T will lead to different relational Dirac a a → L(q , dq /du′)du′/du transforms as a scalar density under observables, as can be seen in Equation (4). All these different u u′(u). The Hamiltonian of such a theory is of the form C → choices are encoded in the constraint surface and stand a priori H N(u) C , where N(u) is an arbitrary lapse function and = H on an equal footing. This gives rise to the interpretation of C as a clock-neutral dqa C p L 0, (1) structure. The temporal diffeomorphism symmetry leads to a H = a − ≈ a du redundancy in the description of C: thanks to the Hamiltonian X constraint the kinematical canonical degrees of freedom are the so-called Hamiltonian constraint, is a consequence of the not independent and due to its gauge flow there will only temporal diffeomorphism symmetry. This equation defines the be dim P 2 independent physical phase space degrees of kin − constraint surface C inside the kinematical phase space Pkin, freedom. In particular, relative to any choice of clock function which is parametrized by the canonical coordinates qa, p . The b T one can construct dim Pkin 2 independent relational Dirac denotes a weak equality, i.e., one which only holds on C − ≈ observables using Equation (4) [101, 102]. Hence, the relational [101, 102]. Dirac observables relative to any other clock choice T′ can The Hamiltonian generates a dynamical flow on C, which be constructed from them. Consequently, there is redundancy transforms an arbitrary phase space function f according to among the relational Dirac observables relative to different clock choices. Thus C yields a description of the physics prior df : f , CH (2) to choosing and fixing a clock relative to which the gauge- du = { } invariant dynamics of the remaining degrees of freedom can be and integrates to a finite transformation αu f , where for described. Specifically, no choice has been made as to which of the CH simplicity the lapse function has been chosen to be unity, kinematical and physical degrees of freedom are to be considered N(u) 1. Owing to the reparametrization invariance, this flow as redundant. In analogy to the tensor case, C still contains the = should be interpreted as a gauge transformation rather than true information about all clock choices and their associated relational evolution [4, 11], and thus for an observable F to be physical, it dynamics at once; it yields a clock-neutral description. Being of odd dimension dim P 1, C is also not a phase must be invariant under such a transformation, i.e., kin − space. A proper phase space description can be obtained, e.g., F, CH 0. (3) through phase space reduction by gauge-fixing [7, 30, 31, 38, 58]. { } ≈ Given a choice of clock function T, we may consider the gauge- Observables satisfying Equation (3) are known as (weak) fixing condition T const, which may be valid only locally on = Dirac observables. C. Since Ff ,T(τ) is constant along each orbit generated by CH In order to obtain a gauge-invariant dynamics, we have for each value of τ, we do not lose any information about the to choose a dynamical temporal reference system, i.e., a relational dynamics by restricting to T const and leaving τ a = clock function T(q , pa), to parametrize the dynamical flow, free. By restricting to the relational observables Ff ,T(τ) relative to Equation (2), generated by the constraint. We can then describe clock T and by solving the two conditions T const, C 0, = H = the evolution of the remaining degrees of freedom relative to we remove the redundancy from among both the kinematical a T(q , pa). This gives rise to so-called relational Dirac observables and physical degrees of freedom. The surviving reduced phase (a.k.a. evolving constants of motion) which encode the answer space description, which no longer contains the clock degrees to the question “what is the value of the function f along the flow of freedom as dynamical variables, can be interpreted as the

Frontiers in Physics | www.frontiersin.org 4 June 2021 | Volume 9 | Article 587083 Höhn et al. Relational Dynamics in Relativistic Settings description of the dynamics relative to the temporal reference Hamiltonian constraint. This is the basis of the so-called system defined by the clock function T. But now we keep track problem of time in quantum gravity [1–3], and of statements of time evolution not in terms of the dynamical T, but in terms of that a quantum theory defined by a Hamiltonian constraint the parameter τ representing its “clock readings.” In particular, is timeless. However, such a theory is only “background- the temporal reference system is not described relative to itself, timeless,” i.e., physical states do not evolve with respect to e.g., one finds the tautology F (τ) τ. Accordingly, choosing the “external” gauge parameter u parametrizing the group T,T ≈ the “perspective” of a clock means choosing the corresponding generated by the Hamiltonian constraint. Instead, it is more clock degrees of freedom as the redundant ones and removing appropriate to regard the quantum theory on Hphys as a them. The theory is then deparametrized: it no longer contains clock-neutral quantum theory: it is a global description of a gauge-parameter u, nor a constraint, nor dynamical clock the physics prior to choosing an internal clock relative to variables—only true evolving degrees of freedom. which to describe the dynamics of the remaining degrees of freedom (as argued in [7, 30, 31]). Just as in the classical case, 2.2. Clock-Neutral Quantum Theory there will in general be many possible clock choices and the Following the Dirac prescription for quantizing constrained “quantum constraint surface” Hphys contains the information systems [10, 11, 101, 102], one first promotes the canonical about all these choices at once; it is thus by no means coordinates of Pkin to canonical position and momentum “internally timeless.” operators qa and p acting on a kinematical Hilbert space H . ˆ ˆa kin The goal is to suitably quantize the relational Dirac The Hamiltonian constraint in Equation (1) is then imposed observables in Equation (4), promoting them to families of by demanding that physical states of the quantum theory are operators F (τ) on H . This involves a quantization of ˆf ,T phys annihilated by the quantization of the constraint function the temporal reference system T and it is clear that in the quantum theory different choices of T will also lead to different C ψ 0. (5) ˆ H | phys = quantum relational Dirac observables. This will give rise to a multitude of gauge-invariant, relational quantum dynamics, Solutions to this Wheeler-DeWitt-like equation may be each expressed with respect to the evolution parameter τ, constructed from kinematical states ψ H via a group | kin ∈ kin which corresponds to the readings of the chosen quantum averaging operation [11, 103–107]2 clock (and is thus not a gauge parameter). The quantization of relational observables is non-trivial, especially because 1 iCH u ψphys δ(CH) ψkin du e− ˆ ψkin , (6) Equation (4) may not be globally defined on C, and depends | = ˆ | = 2π | ZG very much on the properties of the chosen clock. Steps toward systematically quantizing relational Dirac observables have been where G is the group generated by CH. Physical states are ˆ undertaken (e.g., in [7, 17, 20, 37, 38]) and part of this not normalizable in Hkin if they are improper eigenstates of C (i.e., if zero lies in the continuous part of its spectrum). article is devoted to further developing them for a class of ˆ H However, they are normalized with respect to the so-called relativistic models. physical inner product In analogy to the classical case, the clock-neutral description on the “quantum constraint surface” Hphys is redundant: since : the constraint is satisfied, not all the degrees of freedom ψphys φphys ψkin δ(CH) φkin kin (7) | phys = | ˆ | are independent. In particular, the sets of quantum relational Dirac observables relative to different clock choices—and thus where kin is the kinematical inner product and ψkin , φkin | | | ∈ different relational quantum dynamics—will be interdependent. Hkin reside in the equivalence class of states mapped to the same ψ , φ under the projection in Equation (6). The proposal is once more to associate the choice of clock with | phys | phys Equipped with this inner product, the space of solutions to the the choice of redundant degrees of freedom; moving to the Wheeler-DeWitt equation in Equation (5) can usually be Cauchy “perspective” of a given clock means considering the quantum relational observables relative to it as the independent ones, completed to form the so-called physical Hilbert space Hphys [11, 103–107]. and removing the (now redundant) dynamical clock degrees of A gauge-invariant (i.e., physical) observable F acting on H freedom altogether. This works through a quantum symmetry ˆ phys must satisfy the quantization of Equation (2) reduction procedure, i.e., the quantum analog of phase space reduction, which is tantamount to a quantum deparametrization and has been developed in Höhn and Vanrietvelde [30], Höhn C , F ψ 0. (8) ˆ H ˆ | phys = [31], and Höhn et al. [7] and will be further developed in h i section 5. In particular, this procedure is at the heart of changing Such an observable F is a quantum Dirac observable. ˆ from a description relative to one quantum clock to one relative Clearly, exp( iu C ) ψ ψ , i.e., physical states − ˆ H | phys=| phys to another clock, which we elaborate on in section 7. As such, do not evolve under the dynamical flow generated by the quantum symmetry reduction is the key element of a proposal for exploring a quantum version of general covariance [7, 30, 2In contrast to [11, 103–107] and for notational simplicity, we refrain from using the more rigorous formulation in terms of Gel’fand triples and algebraic duals of 31, 47, 70–72] and thereby also addressing the multiple choice (dense subsets of) Hilbert spaces. However, the remainder of this article could be problem in quantum gravity and cosmology [1, 2] (see also put into such a more precise formulation. [28, 32–34, 64, 65, 69, 77, 108]).

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3. QUADRATIC CLOCK HAMILTONIANS TABLE 1 | Examples of constraints of the form in Equation (10).

Building upon the clock-neutral discussion, we now assume that Non-relativistic particle and arbitrary system p2 CH HS the kinematical degrees of freedom described by Pkin and Hkin = 2m + split into a clock C and an “evolving” system S, which do not interact. This will permit us to choose a temporal reference Relativistic particle in inertial coordinates 2 2 2 system in the next section, and thence define a relational CH p p m = − t + + dynamics in both the classical and quantum theories. Homogeneous isotropic cosmology with massless scalar field 3.1. Classical Theory 2 2 CH p p 4k exp(4α) Suppose the classical theory describes a clock C associated with = φ − α − 2 the phase space PC T∗R R , and some system of interest ≃ ≃ Homogeneous cosmology (vacuum Bianchi models) S associated with a phase space PS, so that the kinematical 1 2 √ 0 1 2 √ k 2 : CH p0 k0 exp(2 2β ) p k exp( 4 3β+) − p phase space decomposes as P P P . We assume 2 2 + 2 kin = C × S = − + + + + − + − PC to be parametrized by the canonical pair (t, pt), but will Some examples of constraints of the form of Equation (10), i.e., with clock Hamiltonians not need to be specific about the structure of PS (other than quadratic in an appropriate canonical momentum. The last three (relativistic) examples each contain both cases s 1, depending on which degree of freedom is used to assuming it to be a finite dimensional symplectic manifold). = ± define the clock C. In the example of the Friedman-Lemaître-Robertson-Walker model Further suppose that the clock and system are not coupled, with homogeneous massless scalar field we have used α : ln a, where a is the scale = leading to a Hamiltonian constraint function that is a sum of their factor, and k is the spatial curvature constant [109–115] (here a choice of lapse function 3 3α N e has been made and included in the definition of CH). The shape of the Hamiltonian respective Hamiltonians = constraint for vacuum Bianchi models can be found (e.g., in [116]), and holds for types I, II, III, VIII, IX, and the Kantowski-Sachs models. Here β0, β , β are linear combinations CH HC HS 0, (9) + − = + ≈ of the Misner anisotropy parameters and k0, k , k are constants, each of which may be + − zero, depending on the model. where HC is a function on PC and HS is a function on PS. This article concerns clock Hamiltonians that are quadratic in the clock momentum, H s p2/2, where s 1, 1 , so that C = t ∈ {− + } the Hamiltonian constraint becomes 2 pt CH s HS 0. (10) = 2 + ≈ This class of clock Hamiltonians appears in a wide number of (special and general) relativistic and non-relativistic models—see Table 1 for examples. They are doubly degenerate; every value of HC has two solutions in terms of pt, except on the line defined by p 0. Note that p is a Dirac observable. t = t The constraint in Equation (10) can be factored into two constraints, each linear in pt [30, 31, 33]: FIGURE 1 | Depicted are the surfaces C (red) and C (green) defined by pt + − C sC C , for C : σ s H , (11) C 0 and C 0, respectively. The union of these surfaces is the constraint H σ S + = − = = + − = √2 + − surface C C C Pkin, while their intersection C C is characterized = + ∪ − ⊂ + ∩ − p by pt HS 0, and is depicted by the thick black line. We have assumed that where we have introduced the degeneracy label σ 1. Note = = = ± HS is not degenerate (see Figure 1 of [31] for a similar depiction when HS is that Equation (10) forces s HS to take non-positive values on C. doubly degenerate). In the s 1 case, σ 1 and σ 1 define the positive and = − = + = − negative frequency modes in the quantum theory, respectively. For simplicity, we shall henceforth use this terminology for both s 14. It follows that we can decompose the constraint surface = ± 3.2. Quantum Theory into a positive and a negative frequency sector [30, 31] The Dirac quantization of the kinematical phase space P kin = P P leads to the kinematical Hilbert space H H H C C C , (12) C × S kin ≃ C ⊗ S = + ∪ − describing the clock and system, where H L2(R) and H C ≃ S where Cσ is the set of solutions to Cσ 0 in Pkin. The is the Hilbert space associated with S. We assume the system = H intersection C C is defined by pt HS 0 (see Figure 1 Hamiltonian to be promoted to a self-adjoint operator HS on S. + ∩ − = = ˆ for an illustration). An element of Hkin may be expanded in the eigenstates of the clock and system Hamiltonians as 3The assumption that clock and system do not interact does not hold in generic general relativistic systems. However, it is satisfied in some commonly used examples (see [7] for a discussion and Table 1 for some examples). 4Hence, positive and negative frequency modes are defined by p < 0 and p > 0, t t ψkin dpt ψkin(pt, E) pt C E S , | = R | | respectively, which follows from setting Cσ 0. ZE Z = P

Frontiers in Physics | www.frontiersin.org 6 June 2021 | Volume 9 | Article 587083 Höhn et al. Relational Dynamics in Relativistic Settings where the integral-sum highlights that H may either have a Physical states are normalized with respect to the physical inner ˆ S continuous or discrete spectrum5. product introduced in Equation (7) Physical states of the theory satisfy Equation (5), which for the Hamiltonian constraint in Equation (10) becomes ψ φ : ψ δ(C ) φ (18) phys| physphys = kin| ˆ H | kinkin 2 pt ψσ∗(E) φσ (E), CH ψphys s ˆ IS IC HS ψphys 0. (13) = E σ ˆ | = 2 ⊗ + ⊗ ˆ | = σ Z ∈ SC   X P We assume here that this constraint has zero-eigenvalues, i.e., which takes the usual form of non-relativistic quantum that solutions to Equation (13) exist. Note that this requires the mechanics (σ -sector-wise), in line with the properties of Newton- spectrum of s H to contain non-positive eigenvalues, in analogy Wigner-type wave functions. This observation will be crucial ˆ S with the classical case. when discussing relativistic localization in section 6. Quantizing Cσ in Equation (11) yields [C , C ] 0, so that ˆ + ˆ − = the group averaging projector in Equation (6) can be expressed 4. COVARIANT CLOCKS as 4.1. Relational Dynamics With a Classical 1 δ(C ) δ(s C C ) δ(C ). (14) Covariant Clock ˆ H ˆ ˆ 1 ˆ σ = + − = 2( s H ) 2 Exploiting the splitting of the degrees of freedom into clock C ˆ S σ − X (our temporal reference system) and evolving system S, we now P The form of δ(C ) implies the decomposition of the physical choose a clock function T on C relative to which we describe the ˆ H Hilbert space into a direct sum of positive and negative frequency evolution of S in terms of relational observables, as discussed in section 2.1. sectors Hphys H H (see also [31, 104]). Acting with the ≃ + ⊕ − We could simply choose the phase space coordinate T t projector δ(C ) on an arbitrary kinematical state yields a physical = ˆ H as the clock function. It follows from Equation (2) that in the state s 1 case t runs “forward” on the positive frequency sector = − C and “backward” on the negative frequency sector C along ψ δ(C ) ψ + − phys ˆ H kin the flow generated by C ; for s 1 the converse holds. Note | = | H = + ψσ (E) that every point in C C corresponds to a static orbit of t 1/4 pt,σ (E) C E S , (15) + ∩ − = E σ (2 E ) | | (since pt 0 there), and t is therefore a maximally bad clock σ Z ∈ SC | | = X P function on C C . This leads to challenges in describing + ∩ − relational dynamics relative to t: inverse powers of p appear in where ψσ (E) are Newton-Wigner-type wave functions associated t to the positive and negative frequency modes [90]6: the construction of relational observables encoding the evolution of system degrees of freedom relative to t when canonical pairs 7 on PS are used [18, 23, 30, 31] . One can solve this problem ψkin pt,σ (E), E ψσ (E) : , (16) and obtain a well-defined relational dynamics by using affine = (2 E )1/4 | |
(rather than canonical) pairs of evolving phase space coordinates on PS in the construction of relational observables [31], or in and we have defined the function p σ (E) : σ √2 E and t, = − | | the quantum theory by carefully regularizing inverse powers of spectrum pt [30]. However, in this article we shall sidestep these challenges and σ : Spec(H ) Spec( H ) SC = ˆ S ∩ − ˆ C provide an arguably more elegant solution. We choose a different E Spec(H ) sE 0 . (17) clock function according to the classical covariance condition: ˆ S = ∈ ≤ that it be canonically conjugate to HC. This has the consequence  of incorporating the pathology at p 0 into the clock 5The way we have written physical states implicitly assumes the non-positive part t = of the spectrum of s H to be non-degenerate. Were this not the case, additional function (which will nevertheless be meaningfully quantized in ˆ S degeneracy labels would be necessary. However, this would not otherwise affect section 4.2), and leads to relational observables which work the subsequent analysis. See Höhn [31] for an explicit construction of the flat independently of the choice of phase space coordinates on PS. FLRW model with a massless scalar field, whose Hamiltonian constraint can also Solving T, HC 1, we find that a covariant clock function T be interpreted as a free relativistic particle, and thus features a 2-fold system { } = must be of the form T st/p g(p ), where g(p ) is an arbitrary energy degeneracy. = t + t t 6The fourth root comes about because the Newton-Wigner wave function is function. Henceforth, we choose g(pt) 0 for simplicity, so that = usually defined for Klein-Gordon systems where what we call E is in fact the we have square of the energy ω : p2 m2. Note also that for the Klein-Gordon case p = + one has a doubly degenerate system energy, which we are not considering here. p t In that case, it is more convenient to use a momentum, rather than an energy T s . (19) representation of physical states, and a distinct measure. Equation (16) can then = pt indeed be interpreted as the usual Newton-Wigner wave function, written in terms of kinematical states. This will be discussed in more detail in section 6 (see also 7These challenges are related to those facing the definition of time-of-arrival [31]). operators in quantum mechanics [18, 23, 30, 117–119].

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This clock function is well-defined everywhere, except on the ψ E (X) ψ that a measurement of E yields an outcome C| T | C T line p 0, where H is non-degenerate. It is clear that T runs t X R given that the clock was in the state ψ H . In t = C ∈ ⊂ | C ∈ C “forward” everywhere on C for both s 1, except on C C . order to be a good time observable, it should satisfy the so-called = ± + ∩ − The covariance condition, combined with our assumption covariance condition that the clock does not interact with the system, implies that T, C 1, which simplifies the form of the relational E (X t) U (t)E (X)U† (t), (23) { H} = T C T C Dirac observables in Equation (4). For example, the relational + = observable corresponding to the question “what is the value of iH t relative to the unitary group action U (t) : e ˆ C generated C = − the system observable fS when the clock T reads τ?” now takes by the clock Hamiltonian. As we shall see shortly, this will the simple form [7, 21] give rise to a generalization of canonical conjugacy of the time observable and the clock Hamiltonian, and permit us to n ∞ (τ T) extend the approach to relational quantum dynamics based on F (τ) − fS, HS n. (20) fS,T ≈ n! { } n 0 covariant clock POVMs [7] to relativistic models. In particular, X= we obtain a valid quantum time observable despite the classical 4.2. Covariant Quantum Time Observable clock pathologies. for Quadratic Hamiltonians In the present case, such an observable can be constructed purely from the self-adjoint quantization of the clock One might try to construct a time operator in the quantum Hamiltonian H and its eigenstates. The effect densities theory by directly quantizing the covariant clock function in ˆ C can be defined as a sum of “projections” Equation (19) on the clock Hilbert space H L2(R)[86, 120, C ≃ 121]. Choosing a symmetric ordering, this yields 1 E (dt) dt t, σ t, σ (24) T = 2π | | 1 1 1 σ T s t p− p− t . (21) X ˆ = 2 ˆ ˆt + ˆt ˆ onto the clock states corresponding to the clock reading t R in
∈ 1 the negative and positive frequency (i.e., positive and negative Here, pt− is defined in terms of a spectral decomposition such ˆ 10 that T p 0 is undefined, analogous to the classical case. While clock momentum) sector ˆ | t = the operator T is canonically conjugate to the clock Hamiltonian, ˆ 2 : itspt /2 [T, HC] i, it is a symmetric operator that does not admit a self- t, σ dpt pt θ( σ pt) e− pt . (25) ˆ ˆ = | = R | | − | adjoint extension [86, 88]. Since T is not self-adjoint, its status Z p ˆ 8 as an observable seems a priori unclear . This is a manifestation The covariance condition in Equation (23) is ensured by the fact of Pauli’s objection against the construction of time observables that the clock states transform as in quantum mechanics: For H bounded below, there does ˆ C not exist a self-adjoint operator satisfying [T, H ] i. Pauli’s ˆ ˆ C = t t′, σ UC(t) t′, σ . (26) conclusion was that we are forced to treat time as a classical | + = | parameter, different to the way other observables (e.g., position Note that the clock states are orthogonal to the pathological state and momentum) are treated [122]. p 0 , so that the covariant time observable does not have | t = However, it was later realized that by appealing to the support on it. The clock states are furthermore not mutually more general notion of an observable offered by a POVM, a 9 orthogonal: covariant time observable ET can be constructed whose first moment corresponds to the operator Tˆ [86, 87, 89]. Such a 1 σ σ δ πδ time observable is defined by a set of effect operator densities t′, ′ t, σσ ′ t t′ iP , (27) | = − − t t′ ET(dt) 0 normalized as R ET(dt) IC, and self-adjoint effect  −  ≥ : =
operators ET(X) ET(dt) associated with the probability = X R where P denotes the Cauchy principal value. Hence ET(dt) is not a true projector. Nevertheless, the following lemma demonstrates 8 R 1 2 Using the commutation relation [t, pt− ] ipt− , which follows from ˆ ˆ =1 − ˆ that the clock states t, σ form an over-complete basis for the multiplying [t, p ] i from both sides with p− , we can also write this operator ˆ ˆt = ˆt | as σ -frequency sector of HC, and in turn a properly normalized covariant time observable ET on Hkin. 1 i 1 T s p− t p− (22) ˆ = ˆt ˆ − 2 ˆt   10Compared to Braunstein et al. [89], we use a different definition of the i 1 We note in passing that the operator t p− is precisely the “complex time degeneracy label σ (here adapted to positive and negative frequency modes), ˆ − 2 ˆt operator” derived in Bojowald et al. [32] (see also [33, 34, 77]) when constructing a change the normalization slightly, fix the relative phase, keep the momentum relational Schrödinger picture for Wheeler-DeWitt type equations for constraints eigenstates as energy eigenstates and introduce s. For notational simplicity, we of the form Equation (13). also set an arbitrary function in Braunstein et al. [89] (accounting for a freedom 9We emphasize that the covariant time observable is a kinematical observable on in choosing the clock states) to zero. This is the quantum analog of the classical H and not a gauge-invariant Dirac observable on H , as by construction choice we made above, where we also set g(p ) in T t/p g(p ) to zero (see also kin phys t = t + t its moments will not commute with the constraint. In particular, it is a partial Supplementary Material of [7]). It would, however, be straightforward to reinsert observable in the sense of [123]. this g(pt) in each of the following expressions.

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Lemma 1. The clock states t, σ defined in Equation (25) integrate Lemma 4. The nth-moment operator defined in Equation (30) | (n) (n 1) n to projectors θ( σ pt) onto the positive/negative frequency sector satisfies [T , H ] i n T . Furthermore, ψ D(T ) we − ˆ ˆ ˆ C = ˆ − ∀| ∈ ˆ on HC have T(n) ψ Tn ψ . ˆ | = ˆ | 1 Proof: The proof is given in Supplementary Material. dt t, σ t, σ θ( σ pt) (28) 2π R | | = − ˆ Z We emphasize that the second statement of Lemma 4 does not and hence form a resolution of the identity as follows: hold on all of HC. 1 The effect density does not commute with the clock E(dt) dt t, σ t, σ IC . (29) Hamiltonian, [ET(dt), HC] 0, which implies the time R π R = 2 σ | | = ˆ = Z X Z indicated by the clock (i.e., a measurement outcome of ET) and the clock energy cannot be determined simultaneously. Supplementary Material Proof: The proof is given in . However, importantly, the following lemma shows that the clock th reading and the frequency sector, i.e., the value of σ can be The n -moment operator of the time observable ET is defined as simultaneously determined. Lemma 5. The effect density E (dt) of the covariant clock (n) : n 1 n T T ET(dt) t dt t t, σ t, σ . (30) σ ˆ = R =2π R | | POVM and the projectors onto the -sectors commute: Z σ Z [E (dt), θ( σ p )] 0. X T − ˆt = While the effect operators of the clock POVM are self-adjoint, the Proof: The proof is given in Supplementary Material. moment operators for n 1 are not due to the non-orthogonality ≥ in Equation (27). Nevertheless, the latter are viable quantum observables with a valid probability interpretation in terms of the Corollary 1. Since the effect density integrates to the effect and moment operators, this entails that [E (X), θ( σ p )] POVM; the only price to pay is that the different measurement T − ˆt = [T(n), θ( σ p )] 0, for all X R and n N. outcomes t are not perfectly distinguishable. ˆ − ˆt = ⊂ ∈ With this definition, we find that the first-moment operator (1) The significance of this lemma and corollary is that they Tˆ of ET is in fact equal to the operator Tˆ in Equation (21). This was previously noticed in Holevo [86] and Busch et al. [88] permit us to condition on the time indicated by the clock (for the s 1 case). This provides a concrete interpretation and the frequency sector simultaneously. This will become = + crucial when defining the quantum reduction maps below of the time observable ET in terms of the classical theory— that take us from the physical Hilbert space to the relational the time operator T(1), namely the first moment of the time ˆ Schrödinger and Heisenberg pictures which exist for each σ - observable E , is the quantization of the classical clock function T sector. This lemma is thus an important for extending the T in Equation (19). quantum reduction procedures of Höhn et al. [7] to the class of (1) models considered here. Lemma 2. The operator Tˆ and the first moment operator Tˆ of the covariant time observable E are equal, T T(1). T ˆ ≡ ˆ 5. THE TRINITY OF RELATIONAL Proof: The proof is given in Supplementary Material. QUANTUM DYNAMICS: QUADRATIC Equation (30) demonstrates that the time operator Tˆ CLOCK HAMILTONIANS automatically splits into a positive and negative frequency part, in contrast to t, the quantization of the phase space coordinate t. Having introduced the clock-neutral structure of the classical ˆ Next, we find that while the clock states are not orthogonal, and quantum theories in section 2, a natural partitioning of the kinematical degrees of freedom into a clock C and system S in they are “almost” eigenstates of the covariant time operator Tˆ on each σ -sector: section 3, and a covariant time observable ET in section 4, we are now able to construct a relational quantum dynamics, describing Lemma 3. The clock states t, σ defined in Equation (25) are not how S evolves relative to C. | (1) ψ D D As noted in the introduction, we showed in Höhn et al. eigenstates of Tˆ Tˆ . However, for all (Tˆ ), where (Tˆ ) = | ∈ [7] that three formulations of relational quantum dynamics, is the domain of T,ˆ they satisfy: namely (i) quantum relational Dirac observables, (ii) the ψ T t, σ t ψ t, σ , t R , σ 1. relational Schrödinger picture of the Page-Wootters formalism, | ˆ | = | ∀ ∈ = ± and (iii) the relational Heisenberg picture obtained through Proof: The proof is given in Supplementary Material. quantum deparametrization, are equivalent for models described by the Hamiltonian constraint in Equation (9) when the This leads to a another result, which underscores why the clock Hamiltonian has a continuous, non-degenerate spectrum; covariance condition Equation (23) can be regarded as yielding the three formulations form a trinity of relational quantum a generalization of canonical conjugacy: dynamics. Here we demonstrate that this equivalence extends to

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TABLE 2 | Summary of different Hilbert spaces. 5.1. The Three Faces of the Trinity 5.1.1. Dynamics (i): Quantum Relational Dirac Clock C and system S Hilbert spaces Observables H and H C S We now quantize the relational Dirac observables in Equation (20), substantiating the discussion of relational Kinematical Hilbert space quantum dynamics in the clock-neutral picture in section 2.2 Hkin HC HS ≃ ⊗ for Hamiltonian constraints of the form Equation (13). Quantization of relational Dirac observables has been studied Physical Hilbert space when the quantization of the classical time function T results in a

Hphys δ(CH)(Hkin) H H self-adjoint time operator T (see [4, 7, 11–20, 27–31, 35–38, 116] ≃ ˆ = + ⊕ − ˆ and references therein); however, when Tˆ fails to be self-adjoint, Physical system Hilbert space such as in Equation (21), a more general quantization procedure Hphys H H is needed. S σSC ( S) S = ⊆ Such a procedure was introduced in Höhn et al. [7] based The various Hilbert spaces appearing in the construction of the trinity. The physical system upon the quantization of Equation (20) using covariant time Hilbert space is the subspace of HS spanned by the energy eigenstates permitted upon solving the constraint. The σ -sector Hσ of Hphys is also defined through solutions to the observables. Applying this procedure to the present class of constraint Cσ . Finally, σ θ( s H ) is a projector onto the system subspace permitted ˆ SC = − ˆ S models described by quadratic clock Hamiltonians, we quantize upon solving the constraint in Equation (13). the relational Dirac observables in Equation (20) using the nth- moment operators defined in Equation (30):

n : ∞ i n constraints of the form in Equation (13), involving the doubly Ff ,T(τ) ET(dt) (t τ) fS, HS ˆ S = R ⊗ n! − ˆ ˆ n degenerate clock Hamiltonian11. Z n 0 X=   Thanks to the direct sum structure of the physical Hilbert dt † UCS(t) τ, σ τ, σ fS UCS(t) space Hphys H H and the separation of the clock = R 2π | | ⊗ ˆ = + ⊕ − σ moment operators (Equation 30), into non-degenerate positive X Z   and negative frequency sectors, all the technical results needed : G τ, σ τ, σ f , (31) = | | ⊗ ˆS σ σ for establishing the equivalence in Höhn et al. [7] will hold per - X   sector for the present class of models. We will thus state some : of the following results without proofs, referring the reader as where [fS, HS]n [[fS, HS]n 1, HS] is the nth-order nested ˆ ˆ = ˆ ˆ − ˆ approriate to the proofs of the corresponding results in Höhn commutator with the convention [f , H ] : f , U (t) : ˆS ˆ S 0 = ˆS CS = etal.[7], which apply here per σ -sector. In particular, Corollary 1 exp( it C ), and the second line follows upon a change of − ˆ H implies that we are permitted to simultaneously condition on the integration variable and invoking the covariance condition in clock reading and the frequency sector. Equation (26). The relational Dirac observable F (τ) is thus ˆfS,T Lastly, we also provide a discussion of the relational quantum revealed to be an incoherent average over the one-parameter dynamics obtained through reduced phase space quantization. non-compact gauge group G generated by the constraint operator In this case, one deparametrizes the model classically relative CH of the kinematical operator τ, σ τ, σ fS, which is the to the clock function T, which amounts to a classical symmetry ˆ | | ⊗ ˆ system observable of interest f paired with the projector onto reduction. While the relational quantum dynamics thus obtained ˆS the clock reading τ and the σ -frequency sector. Such a group yields a relational Heisenberg picture resembling dynamics (iii) averaging is known as the G-twirl operation and we denote it of the trinity, it is not always equivalent and thus not necessarily G as in the last line of Equation (31). G-twirl operations have part of the trinity. For this reason, we have moved the exposition previously been mostly studied in the context of spatial quantum of reduced phase space quantization to Supplementary Material. reference frames, e.g., see [130–132], but have also appeared in It is however useful for understanding why the quantum some constructions of quantum Dirac observables (e.g., see [7, symmetry reduction explained below is the quantum analog 11, 37, 38, 106])12. As discussed in Höhn et al. [7], this G-twirl of classical phase space reduction through deparametrization. constitutes the quantum analog of a gauge-invariant extension of We emphasize that symmetry reduction and quantization do a gauge-fixed quantity. not commute in general [7, 124–129]. A related discussion for relativistic constraints can also be found in Kaminski et al.[29]. 12The recent [37, 38] also develop a systematic quantization procedure for To aid the reader, we summarize the various Hilbert spaces relational Dirac observables, based on integral techniques rather than the sum appearing in the construction of the trinity in Table 2. techniques used here and in Höhn et al. [7], and which too yields an expression similar to the one in the second line of Equation (31). While the construction procedure in Chataignier [37, 38] encompasses a more general class of models (but implicitly assumes globally monotonic clocks too), it uses a more restrictive 11We also refer the reader to the recent work [38], which we became aware of choice of clock observables which, in contrast to the covariant clock POVMs while completing this manuscript. It extends some of the results of Höhn et al. [7] here and in Höhn et al. [7], are required to be self-adjoint. However, the two as well, though using the different formalism developed in Chataignier [37]. It also quantization procedures of relational observables are compatible and it will be does not employ covariant clocks in the case of quadratic clock Hamiltonians. fruitful to combine them.

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The relational Dirac observables F (τ) in Equation (31) For example, this superselection rule manifests as a ˆfS,T constitute a one-parameter family of strong Dirac observables superselection across positive and negative frequency modes on Hphys (Theorem 1 of [7] whose proof applies here in each in the case of the relativistic particle and across expanding σ -sector): and contracting solutions in the case of the FLRW model with a massless scalar field in Table 1 [31]. On account of the [Ff ,T(τ), CH] 0, τ R . (32) reducibility of the representation, one usually restricts to one ˆ S ˆ = ∀ ∈ frequency sector (e.g., see [113, 114, 116, 133]). One might We thus obtain a gauge-invariant relational quantum dynamics conjecture that the analogous superselection rule in a quantum by letting the evolution parameter τ in the physical expectation field theory would manifest as a superselection rule between values ψ F (τ) ψ run. phys|ˆfS,T | physphys matter and anti-matter sectors. The decomposition of F (τ) in Equation (31) into positive Superselection rules induced by the G-twirl are often ˆfS,T and negative frequency sectors gives rise to a reducible interpreted as arising from the lack of knowledge about a representation of the Dirac observable algebra on the physical reference frame, and that if an appropriate reference frame is used Hilbert space. More precisely, relational Dirac observables are the superselection rule can be lifted [130]. This interpretation superselected across the σ -frequency sectors, and the σ -sum in seems unsuitable here. Firstly, lifting the superselection rule Equation (31) should thus be understood as a direct sum. To would entail undoing the group averaging, in violation of gauge see this, consider the operator Q : θ( p ) θ(p ), where we invariance. Secondly, such an interpretation is usually tied ˆ = −ˆt − ˆt recall that θ( σ p ) is a projector onto the corresponding σ - to an average over a given group action which parametrizes − ˆt sector. By construction [Q, C ] 0, which means that Q is one’s ignorance about relative reference frame orientations. By ˆ ˆ H = ˆ a strong Dirac observable. Its eigenspaces, with eigenvalues 1 contrast, the origin of the superselection of Dirac observables + and 1, correspond to the positive and negative frequency sector here is not the group generated by the constraint, but is a − subspaces H and H . Furthermore, Q commutes with any consequence of a property of the constraint, i.e., the group ˆ + − generator. Indeed, the superselection rule above originates in the relational Dirac observable FfS,T(τ) in Equation (31) on account factorizability of the constraint and the ensuing decomposition of Lemma 5, which implies that Q and any self-adjoint FfS,T(τ) can be diagonalized in the same eigenbasis. This in turn implies of the projector onto the constraint (Equation 14). Both these properties rely on the absence of a t-dependent term in the the following superselection rule ˆ constraint Equation (13); if such a term is introduced, one Ff ,T(τ) F+ (τ) F− (τ), (33) generally finds [C , C ] 0, where the Cσ are the quantization ˆ S = ˆfS,T ⊕ ˆfS,T ˆ + ˆ − = ˆ of the classical factors, but we emphasize that CH s C C ˆ = ˆ + ˆ − σ : 13 in that case. While such a modified constraint may generate where F (τ) G τ, σ τ, σ fS) L(Hσ . ˆfS,T = | | ⊗ ˆ ∈ the same group14, no superselection rule across the σ -sectors While there do exist states in the physical Hilbert space
would arise. The above superselection rule can thus not be that exhibit coherence across the σ -frequency sectors, for associated with the lack of a shared physical reference frame. This example ψphys ψ ψ , where ψσ Hσ , such | ∼ | ++| − | ∈ resonates with the interpretation of the physical Hilbert space as coherence is not physically accessible because it does not affect a clock-neutral, i.e., temporal-reference-frame-neutral structure the expectation value of any relational Dirac observable on (see section 2.2). account of the decomposition in Equation (33). In other words, Consider now the projector σ θ( s H ) from H to superpositions and classical mixtures across the σ -frequency SC = − ˆ S S its subspace spanned by all system energy eigenstates E with sectors are indistinguishable. Hence, superpositions of physical | S E σ ; that is, those permitted upon solving the constraint states across σ -sectors are mixed states and the pure physical ∈ SC Equation (13). We shall henceforth denote this system Hilbert states are those of either H or H (see also [103, 105] for a + − phys subspace H : σ (H ) and call it the physical system discussion on superselection in group averaging). S = SC S Hilbert space. We will obtain two copies of the physical system 13In particular, when the spectrum of H does not contain zero, the G-twirl G can Hilbert space, one for each frequency sector. In analogy to the ˆ S on each σ -sector be weakly rewritten as a reduced G-twirl Gσ , i.e., one generated classical case, we introduce the quantum weak equality between by C , rather than C . Indeed, it is easy to see that the observables in Equation (31) ˆ σ ˆ H operators, signifying that they are equal on the “quantum satisfy constraint surface” Hphys: σ F (τ) G τ, σ τ, σ fS) δ(CH )( τ, σ τ, σ fS), ˆfS,T = | | ⊗ ˆ ≈ ˆ | | ⊗ ˆ where is the quantum weak equality introduced in Equation (34). Now use O O (34) ≈ ˆ 1 ≈ ˆ 2 Equation (14) and notice that δ(C σ ) τ, σ E S 0 when zero does not lie ˆ − | ⊗ | = H in the spectrum of H . This observation yields O1 ψphys O2 ψphys , ψphys phys . ˆ S ⇔ ˆ | = ˆ | ∀ | ∈ σ 1 F (τ) δ(Cσ )( τ, σ τ, σ fS), ˆfS,T 1 ˆ ˆ It follows from Lemma 1 of [7], whose proof applies here per ≈ 2( sH ) 2 | | ⊗ − ˆ S σ 1 -sector, that G ( τ, σ τ, σ f ) , 1 σ ˆS ≈ 2( sH ) 2 | | ⊗ − ˆ S FfS,T(τ) Fσ fS σ ,T(τ), (35) where the last weak equality is restricted to Hσ . When zero does lie in the ˆ ≈ ˆ SC SC spectrum of H , the decomposition Equation (14) is not well-defined and one ˆ S needs to regularize. 14For example, when C p2 H2(q , p , t) and H2 > 0 t R. H = t − i i ∀ ∈

Frontiers in Physics | www.frontiersin.org 11 June 2021 | Volume 9 | Article 587083 Höhn et al. Relational Dynamics in Relativistic Settings are weakly equal relational Dirac observables. Hence, the with the constraint. In this section we extend this result to relational Dirac observables in Equation (31) form weak relativistic settings. equivalence classes on H , where F (τ) F (τ) if Here we shall expand the Page-Wootters formalism to the phys ˆfS,T ∼ ˆgS,T σ f σ σ g σ . These weak equivalence classes are more general class of Hamiltonian constraints of the form SC ˆS SC = SC ˆS SC labeled by what we shall denote Equation (13) exploiting the covariant clock POVM ET of section 4.215. On the one hand, this will permit us to prove full phys phys equivalence of the so-obtained relational quantum dynamics with f : σ f σ L H , (36) ˆS = SC ˆS SC ∈ S the manifestly gauge-invariant formulation in terms of relational   Dirac observables on Hphys of Dynamics (i). As an aside, this will for arbitrary fS L (HS), where L denotes the set of linear also resolve the normalization issue of physical states appearing ˆ ∈ operators. For later use, we note that the algebras of the physical in Diaz et al. [50], where the kinematical rather than physical phys Hphys H system observables fS on S and the F phys (τ) on phys inner product was used to normalize physical states, thus yielding ˆ ˆfS ,T are weakly homomorphic with respect to addition, multiplication a divergence (when used for equal mass states). On the other and commutator relations. More precisely, hand, the covariant clock POVM will allow us, in section 6, to address the observation by Kuchaˇr[1] that using the Minkowski time observable leads to incorrect localization probabilities for F phys phys phys (τ) F phys (τ) ˆf g h ,T ˆf ,T S + S S ≈ S relativistic particles in the Page-Wootters formalism. F phys (τ) F phys (τ) The Page-Wootters formalism produces the system state ˆg ,T ˆh ,T + S S at clock time τ by conditioning physical states on the clock f , g , h L (H ). This is a consequence of Theorem 2 of reading τ [39, 40, 134, 135]. Henceforth focusing on the class ∀ S S S ∈ S Höhn et al. [7] (whose proof again applies here per σ -sector). of models defined by the constraint in Equation (13) and the Together with Höhn et al. [7], this translates the weak classical covariant clock POVM of section 4.2, and given the reducible H algebra homomorphism defined through relational observables representation of phys, we may additionally condition on the in Dittrich [21] into the quantum theory. frequency sector thanks to Lemma 5. In extension of Höhn et al. [7], we may use this conditioning to define two reduction maps 5.1.2. Dynamics (ii): The Page-Wootters Formalism Rσ : H Hphys S (τ) phys S,σ , one per σ -frequency sector, Suppose we are given a quantum Hamiltonian constraint → σ : Equation (5) which splits into a clock and system contribution RS (τ) τ, σ IS, (38) as in Equation (9), but for the moment not necessarily assuming = | ⊗ phys phys it to be of the quadratic form in Equation (13). Suppose further where H is a copy of H σ (H ), i.e., the subspace of S,σ S = SC S we are given some (kinematical) time observable on the clock the system Hilbert space permitted upon solving the constraint, Hilbert space, which need not necessarily be a clock POVM corresponding to the σ -frequency sector. As will become clear which is covariant with respect to the group generated by the shortly, the label S on the reduction map stands for “Schrödinger clock Hamiltonian, but is taken to define the clock reading. Page picture” and to distinguish it from the italic S which stands and Wootters [39, 40, 134, 135] proposed to extract a relational for the system, we henceforth write it in bold face. Due to the quantum dynamics between the clock and system from physical decomposition Hphys H H , we equip the two copies states in terms of conditional probabilities: what is the probability phys = + ⊕ − H with the frequency label σ in order to remind ourselves of an observable f associated with the system S giving a particular S,σ ˆS which reduced theory corresponds to which positive or negative outcome f , if the measurement of the clock’s time observable frequency mode. yields the time τ? If e (τ) and e (f ) are the projectors onto the C fS The reduced states (whose normalization factor 1/√2 will be clock reading τ and the system observable fˆS taking the value f , explained later), this conditional probability is postulated in the form 1 ψσ (τ) : Rσ (τ) ψ (39) Prob f and τ √ | S = S | phys Prob f when τ (37) 2 = Prob (τ)
ψ i τ E
ψ τ ψ σ (E) e− E S , phys eC( ) efS (f ) phys kin = E σ | | ⊗ | . Z ∈ SC ψ τ ψ P = phys eC( ) IS phys kin | ⊗ | where ψσ (E) is the Newton-Wigner type wave function defined This expression appears at first glimpse to be in violation of the in Equation (16), satisfy the Schrödinger equation with respect to H : constraints, as it acts with operators on physical states that are ˆ S not Dirac observables; this is the basis of Kuchaˇr’s criticism (b) d that the conditional probabilities of the Page-Wootters formalism i ψσ (τ) H ψσ (τ) . (40) S ˆ S S are incompatible with the constraints [1]. However, for a class dτ | = | of models we have shown in Höhn et al. [7] that the expression 15See also the recent [38] for a complementary approach. Note that it does not Equation (37) is a quantum analog of a gauge-fixed expression employ covariant POVMs for quadratic Hamiltonians, and is thus subject to of a manifestly gauge-invariant quantity and thus consistent Kuchaˇr’s criticism (a) described in section 1.

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We interpret this as the dynamics of S relative to the temporal reference frame C. In particular, this Schrödinger equation looks the same for both the positive and negative frequency sectors because the time defined by the covariant clock POVM ET runs forward in both sectors. This is clear from Equation (26) and is the quantum analog of the earlier classical observation that the clock function T runs “forward” on both frequency sectors C and C (in contrast to t)16. + Thanks− to Equation (29), the decomposition of the physical states into positive and negative frequency modes, Equation (15) can also be written as follows:

1 σ ψphys dt t, σ ψS (t) . (41) R | = 2√2 π σ | | X Z Together with Lemma 1, this implies that the σ -sector left inverse phys H Hσ of the reduction map defined in Equation (38) is S,σ → given by FIGURE 2 | A summary of the Page-Wootters reduction maps and their inverses. The analogous state of affairs holds for the quantum symmetry reduction maps and their inverses. σ 1 1 (RS (τ))− dt t, σ US(t τ) = 2π R | ⊗ − Z δ(C )( τ, σ I ), (42) = ˆ H | ⊗ S Using these reduction maps and their inverses, we can define where US(t) exp( i HS t), so that τ,σ : phys = − ˆ an encoding operation E L H L (Hσ ), mapping the S S,σ → (Rσ (τ)) 1 Rσ (τ) δ(C )( τ, σ τ, σ I ) observables in Equation (36), acting on the physical system S − S ˆ H S = | | ⊗ Hphys θ( σ p ) I , (43) Hilbert space S,σ , into Dirac observables on the σ -sector of ≈ − ˆt ⊗ S Hphys: where is the quantum weak equality, and thus ≈ Eτ,σ phys : Rσ τ 1 phys Rσ τ σ 1 σ S fˆS ( S ( ))− fˆS S ( ) (RS (τ))− RS (τ) Iphys . (44) = σ ≈   phys X δ(CH) τ, σ τ, σ f . (45) = ˆ | | ⊗ ˆS Hphys   Conversely, we can write the identity on S,σ in the form These encoded observables turn out to be the σ -sector part of the σ σ 1 relational Dirac observables in Equation (31), as articulated in the RS (τ)(RS (τ))− τ, σ δ(CH) τ, σ = | ˆ | following theorem. σ . = SC Theorem 1. Let f L (H ). The quantum relational Dirac The identity in the second line can be checked with the help of ˆS ∈ S observable F (τ) acting on H , Equation (31), reduces under Equation (27) (and also follows from the proof of Theorem 2 in ˆfS,T phys Rσ Hphys [7]). A summary of these maps can be found in Figure 2. S (τ) to the corresponding projected observable on S,σ ,

16Note that we could also define linear combinations of clock states τ : | = Rσ (τ) τ Rσ τ 1 phys cσ τ, σ . Clearly, then we would also find that S FfS,T( )( S ( ))− σSC fS σSC fS . σ | ˆ = ˆ ≡ ˆ : P ψS(τ) τ ψphys | = | phys phys Conversely, let f L H . The encoding operation in i τE ˆS S,σ cσ∗ ψσ (E) e− E S ∈ = E σ | Equation (45) of system observables coincides on the physical σ Z ∈ SC !   X P H satisfies the same Schrödinger equation (40). However, it is straightforward to Hilbert space phys with the quantum relational Dirac observables check, using Lemma 1, that these new clock states do not give rise to a resolution in Equation (31) projected into the σ -sector: of the identity, dτ τ τ I and so ψ dτ τ ψ (τ) . In fact, | | = C | phys = | | S if cσ 0 for σ , , then ψ (τ) will mix contributions from the positive = R= + − | S R Eτ,σ phys σ τ and negative frequency sectors such that it will become impossible to reconstruct S fˆS F phys ( ), (46) ≈ ˆfS ,T (either of the positive or negative frequency part of) the physical state from it.   That is, a reduction map τ IS, which only conditions on the clock time, | ⊗ σ would not be invertible. This is another consequence of the superselection rule where F (τ) F phys (τ)(θ( σ p ) I )—c.f. Equation (33). ˆ phys ˆf ,T t S discussed above which entails that superpositions and mixtures across σ -sectors fS ,T = S − ˆ ⊗ are indistinguishable through Dirac observables. It is also another reason why we condition also on the frequency sector when defining the reduction map in Proof: The proof of Theorem 3 in Höhn et al. [7] applies here per Equation (38). σ -sector.

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Note that the relational Dirac observables F (τ) commute (1) Applying the Page-Wootters reduction map Rσ (τ) to the ˆfS,T S with the projectors θ( σ p ) due to the reducible representation physical Hilbert space H yields a relational Schrödinger − ˆt phys in Equation (31). picture with respect to the clock C on the physical system Apart from providing the σ -sector-wise dictionary between Hphys Hilbert space S,σ corresponding to the σ -frequency the observables on the physical Hilbert space and the physical sector. system Hilbert space, Theorem 1, in conjunction with the (2) σ -sector wise, the relational quantum dynamics encoded in weak equivalence in Equation (35), also implies an equivalence the relational Dirac observables on the physical Hilbert space between the full sets of relational Dirac observables Fσ (τ) on ˆ phys is equivalent to the dynamics in the relational Schrödinger fS ,T phys picture on the physical system Hilbert space of the Page- H and system observables on H . σ S,σ Wootters formalism. Crucially, the expectation values of the relational Dirac (3) Given the invertibility of the reduction map, Theorem 2 observables Equation (31) in the physical inner product phys Hphys Equation (18) coincide, σ -sector-wise, with the expectation formally shows that if fˆS is self-adjoint on S,σ , then so σ phys is F (τ) on Hσ . values of the physically permitted system observables fˆS in the ˆfS,T Hphys states solving the Schrödinger equation Equation (40) on S,σ . We note that the expression in the second line of Equation (47) also defines an inner product on the Theorem 2. Let f L (H ), and denote its associated operator space of solutions to the Wheeler-DeWitt-type constraint ˆS ∈ S Hphys phys Equation (13), which is equivalent to the physical inner on S by fˆS σSC fˆS σSC . Then = product in Equation (18) obtained through group averaging. σ 1 σ phys σ These two inner products thus define the same physical ψphys F (τ) φphys ψS (τ) f φS (τ) , |ˆfS,T | phys = 2 | ˆS | Hilbert space Hphys. The expression in the second line of Equation (47) is the adaptation of the Page-Wootters σ Rσ where ψS (τ) √2 S (τ) ψphys as in Equation (39). Hence, inner product introduced in Smith and Ahmadi [44] | = | to the reducible representation of the physical Hilbert 1 phys space associated to Hamiltonian constraints with quadratic ψ F (τ) φ ψσ (τ) f φσ (τ) . phys ˆfS,T phys phys S ˆS S clock Hamiltonians. | | = 2 σ | | X Proof: The proof of Theorem 4 in Höhn et al. [7] applies here per σ -sector. 5.1.3. Dynamics (iii): Quantum Deparametrization Classically, one can perform a symmetry reduction of the Hence, the expectation values in the relational Schrödinger clock-neutral constraint surface by gauge-fixing the flow of picture (i.e., the Page-Wootters formalism) are equivalent the constraint. In this case, this yields two copies of a gauge- to the gauge-invariant ones of the corresponding relational fixed reduced phase space, one for each frequency sector, Dirac observables on Hphys. Accordingly, equations, such as each equipped with a standard Hamiltonian dynamics, hence Equation (37) (adapted to σ -sectors) are not in violation of the yielding a deparametrized theory (see Supplementary Material). constraint as claimed by Kuchaˇr[1]. In contrast to the classical constraint surface, the “quantum σ constraint surface” H is automatically gauge-invariant This immediately implies that the reduction maps RS (τ) phys preserve inner products per σ -sector as follows. since the exponentiation of the symmetry generator C acts ˆ H trivially on all physical states and Dirac observables. Hence, Corollary 2. Setting f σ in Theorem 2 yields ˆS = SC there is no more gauge-fixing in the quantum theory after solving the constraint. Nevertheless, following Höhn and 1 σ σ Vanrietvelde [30], Höhn [31], and Höhn et al. [7], we now ψphys θ( σ pt) IS φphys ψS (τ) φS (τ) , | − ˆ ⊗ | phys = 2 | demonstrate the quantum analog of the classical symmetry reduction procedure for the class of models considered in this where ψσ (τ) √2 Rσ (τ) ψ . Hence, | S = S | phys article. As such it is the quantum analog of deparametrization, which we henceforth refer to as quantum deparametrization. 1 ψ φ ψσ τ φσ τ This quantum symmetry reduction maps the clock-neutral phys phys phys S ( ) S ( ) (47) | = 2 σ | Dirac quantization to a relational Heisenberg picture X ψ τ σ τ σ φ relative to clock observable ET, and involves the following phys ( , , IS) phys kin . = σ | | | ⊗ | two steps. X 1. Constraint trivialization: A transformation of the constraint The reason for introducing the normalization factor 1/√2 in such that it only acts on the chosen reference system (here Equation (39) is now clear: it permits us to work with normalized clock C), fixing its degrees of freedom. The classical analog is states ψσ (τ) ψσ (τ) 1 ψ ψ in each reduced S | S = = phys| physphys a canonical transformation which turns the constraint into a σ -sector and in the physical Hilbert space simultaneously. momentum variable and separates gauge-variant from gauge- The above results show: invariant degrees of freedom [7, 71].

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2. Condition on classical gauge fixing conditions: A “projection” Hence, per σ -frequency sector, the trivialized physical which removes the now redundant reference frame degrees of states are product states with respect to the tensor product freedom17. decomposition of the kinematical Hilbert space. Recalling the discussion of the superselection rule across σ -sectors, the We begin by defining the constraint trivialization map physical state in Equation (50) is indistinguishable from a T ǫ : H T ǫ(H ) relative to the covariant time T, phys → T, phys separable mixed state when it contains both positive and negative observable E . This map will transform the physical Hilbert T frequency modes. One can therefore also view the trivialization space into a new Hilbert space, while preserving inner products as a σ -sector-wise disentangling operation, given that physical and algebraic properties of observables states in Equation (15) appear to be entangled. However, we emphasize that this notion of entanglement is kinematical and n 2 n : ∞ i (n) ǫ not gauge-invariant (see [7] for a detailed discussion of this and T ǫ T HS s T, = n! ˆ ⊗ ˆ + 2 n 0   how the trivialization can also be used to clarify the role of X= entanglement in the Page-Wootters formalism). 1 i t (H s ǫ2/2) dt t, σ t, σ e ˆ S+ . (48) The clock factors in Equation (50) have become redundant, π R = 2 σ | | ⊗ X Z apart from distinguishing between the positive and negative frequency sector. Indeed, if we had ǫ 0, then (disregarding = In analogy to Höhn and Vanrietvelde [30], Höhn [31], and Höhn the diverging prefactor) both the negative and positive frequency ǫ > et al. [7], we introduce an arbitrary positive parameter 0 so terms in Equation (50) would have a common redundant factor that the map becomes invertible. Note that s ǫ2/2 Spec(H ). ˆ C p 0 , so that one could no longer distinguish between them ∈ | t = C T at the level of the eigenbases of pt and HS in which the states have Lemma 6. The left inverse of the trivialization T,ǫ is given by ˆ ˆ been expanded. That illustrates why TT,ǫ 0 is not invertible when 1 = 1 2 acting on physical states. Indeed, T − is undefined on states of T 1 i t (HS s ǫ /2) T,ǫ 0 −ǫ dt t, σ t, σ e− ˆ + , = T, = 2π R | | ⊗ the form p 0 ψ , since T(n) is not defined on p 0 . σ Z | t = C | S ˆ | t = C X This is similar to the construction of the trivialization maps in 1 : Höhn and Vanrietvelde [30] and Höhn [31], except that here so that, for any ǫ > 0, T − TT,ǫ(Hphys) Hphys and T,ǫ → the decomposition into positive and negative frequency sectors 1 proceeds somewhat differently. This concludes step 1. above. T − T ǫ I . T,ǫ ◦ T, ≈ phys In order to complete the reduction to the states of the relational Heisenberg picture, and thus also complete step 2. Proof: The proof of Lemma 2 in Höhn et al. [7] applies here above, we “project” out the redundant clock factor of the σ -sector wise. trivialized states by projecting onto the classical gauge-fixing condition T τ (see Supplementary Material for a discussion The main property of the trivialization map is summarized in = the following lemma. of the classical gauge-fixing). That is, we now proceed as in the Page-Wootters reduction and condition states in the trivialized T H Lemma 7. The map TT,ǫ trivializes the constraint in Equation (13) physical Hilbert space T,ǫ( phys) on the clock reading τ, to the clock degrees of freedom separating positive and negative frequency modes. Altogether, the quantum symmetry reduction map takes the form s T C T 1 ∗ (p2 ǫ2) I , (49) T,ǫ ˆ H T−,ǫ t S σ i τ s ǫ2/2 ≈ 2 ˆ − ⊗ R : e− ( τ, σ I ) T ǫ. (51) H = | ⊗ S T, where ∗ is the quantum weak equality on the trivialized physical Using that ≈ Hilbert space TT,ǫ(Hphys), and transforms physical states into a i τ s ǫ2/2 sum of two product states with a fixed and redundant clock factor τ, σ p σ ′ǫ δσσ √ǫ e , (52) | t = − = ′ 1 which is another reason why ǫ > 0 is chosen, in Equation (50) TT,ǫ ψphys pt σǫ C (50) | = √ǫ σ | = − one finds τ-independent system states X ψ (E) E . σ S σ ⊗ E σSC | R ψ ψσ (E) E Z ∈ H | phys = | S P ZE σSC P∈ Proof: The proof of Lemma 2 in Höhn et al. [7] applies here : 1 σ ψS , (53) σ -sector wise. = √2 |

17 We put projection in quotation marks because it is not a true projection as appropriate for a Heisenberg picture (compare with when applied to the physical Hilbert space as it only removes redundancy in the description, i.e., degrees of freedom which are fixed through the constraint. No Equation 39). This is also the reason why, in contrast to physical information is lost, which is why this operation will be invertible. It would the Page-Wootters reduction in Equation (38), we do not label however be a projection on Hkin. the quantum deparametrization map with a τ, despite the latter

Frontiers in Physics | www.frontiersin.org 15 June 2021 | Volume 9 | Article 587083 Höhn et al. Relational Dynamics in Relativistic Settings a priori appearing on the right hand side of Equation (51) and Proof: The proof of Lemma 3 in Höhn et al. [7] applies here why we equip it with an H label. The factor 1/√2 has again been σ -sector wise. introduced for normalization purposes. Since the wave function Given the Heisenberg-type states in Equation (53), we may σ phys ψ (E) √2 ψσ (E), (54) consider evolving Heisenberg observables on H S ≡ S,σ σ is square-integrable/summable, it is clear that ψ is an element phys phys | S f (τ) U†(τ) f U (τ). (56) Hphys ˆS S ˆS S of the physical system Hilbert space S,σ , corresponding to the = phys σ -sector. We therefore also have Rσ (τ) : H H , just as Indeed, the following theorem shows that these Heisenberg H phys → S,σ in Page-Wootters reduction. Using Lemmas 6 and 7, it is now also observables are equivalent to the relational Dirac observables clear how to invert the quantum symmetry reduction—at least on the σ -sector of the physical Hilbert space Hσ , thereby per σ -sector: demonstrating that the quantum symmetry reduction map yields a relational Heisenberg picture. To this end, we employ these 1 Rσ 1 : T 1 reduction maps and their inverses to define another encoding ( H)− −ǫ pt σǫ C IS = T, √ǫ | = − ⊗ σ : phys  operation E L H L (Hσ ),  H S,σ → Hphys H 18   defines a map S,σ σ , so that → Eσ phys τ : Rσ 1 phys τ Rσ H fˆS ( ) ( H)− fˆS ( ) H. (57) σ 1 σ = (R )− ψ √2 θ( σ p ) ψ . (55) H | S = − ˆt | phys   Theorem 3. Let f L (H ). The quantum relational Dirac Hence, from the physical system Hilbert space of the ˆS ∈ S observable F (τ) acting on H , Equation (31), reduces under positive/negative frequency modes one can only recover ˆfS,T phys Rσ to the corresponding projected evolving observable in the the positive/negative frequency sector of the physical Hilbert H Hphys space. Note that the inverse map is independent of τ in contrast Heisenberg picture on S,σ , Equation (56), i.e., to the Page-Wottters case. σ σ 1 R F (τ)(R )− σ f (τ) σ More precisely, the following holds. H ˆfS,T H = SC ˆS SC phys Lemma 8. The quantum symmetry reduction map is weakly equal f (τ). ≡ ˆS to the Page-Wootters reduction map and an (inverse) system time phys phys evolution Conversely, let f (τ) L H be any evolving Heisenberg ˆS ∈ S,σ Rσ † observable. The encoding operation  in Equation (57) of system H τ, σ US (τ), ≈ | ⊗ observables coincides on the physical Hilbert space Hphys with the U†(τ) Rσ (τ). = S S quantum relational Dirac observables in Equation (31) projected into the σ -sector: Similarly, the inverse of the quantum symmetry reduction is equal to a system time evolution and the inverse of the Page-Wootters Eσ phys τ σ τ H fˆS ( ) F phys ( ). (58) reduction: ≈ ˆfS ,T   σ 1 (R )− δ(C ) τ, σ U (τ) Proof: The proof of Theorem 5 in Höhn et al. [7] applies here per H = ˆ H | ⊗ S σ 1 σ -sector. (R (τ))− U (τ). = S S
Once more, the theorem establishes an equivalence between Hence the full sets of relational Dirac observables relative to clock ET on σ 1 σ Hσ and evolving Heisenberg observables on the physical system (RH)− RH θ( σ pt) IS ≈ − ˆ ⊗ Hphys Hilbert space of the σ -modes S,σ . Hence, one can recover the and action of the relational Dirac observables only σ -sector wise from σ σ 1 σ σ the Heisenberg observables. R (R )− ψ ψ . H H | S =| S Lemma 8 and Theorem 2 directly imply that we again have preservation of expectation values per σ -sector, as the following 18This is understood as appending the new clock tensor factor p σǫ to the | t =− C reduced system state ψσ and then applying the inverse of the trivialization (recall theorem shows. | S that the conditioning of physical states on clock readings is not a true projection phys † Theorem 4. Let f L (H ) and f (τ) U (τ) σ f and thus invertible, cf. previous footnote). Note that embedding the reduced ˆS ∈ S ˆS = S SC ˆS system states back into the physical Hilbert space is a priori highly ambiguous σSC US(τ) be its associated evolving Heisenberg operator on phys since the system state alone no longer carries any information about the clock H . Then state which had been projected out. However, here it is the physical interpretation S of the reduced state as being the description of the system S relative to the 1 phys temporal reference system C that singles out the embedding into the clock-neutral ψ Fσ (τ) φ ψσ f (τ) φσ , phys ˆfS,T phys phys S ˆS S (i.e., temporal-reference-system-neutral) Hphys. This physical interpretation is, of | | = 2 | | course, added information, but it is crucial. For a more detailed discussion of this σ σ topic, see Vanrietvelde et al. [71]. where ψ √2 R ψ . | S = H | phys

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Proof: The proof of Theorem 6 in Höhn et al. [7] applies here per Supplementary Material for the class of models of this article. σ -sector. In particular, we may think of the physical system Hilbert phys space H for the σ -sector as the quantum analog of the Therefore, the quantum symmetry reduction map R is an S,σ H gauge-fixed reduced phase space obtained by imposing for isometry, as we state in the following corollary. example the gauge T 0 on the classical σ -frequency 19 = sector Cσ . Also classically, one obtains two identically looking Corollary 3. Setting f σ in Theorem 4 yields ˆS = SC gauge-fixed reduced phase spaces, one for each frequency sector. Consequently, the relational Schrödinger and Heisenberg 1 σ σ ψphys θ( σ pt) IS φphys ψ φ , pictures can both be understood as the quantum analog of a | − ˆ ⊗ | phys = 2 S | S gauge-fixed formulation of a manifestly gauge-invariant theory. σ Rσ In this light, Theorems 1 and 3 imply that the encoding where ψS √2 H ψphys . Hence, | = | operations of system observables in Equations (45) and (57) 1 can be understood as the quantum analog of gauge-invariantly ψ φ ψσ φσ phys phys phys S S . extending a gauge-fixed quantity (see also [7]). Similarly, | = 2 σ | X the alternative physical inner product in the second line of Equation (47) is the quantum analog of a gauge-fixed version Accordingly, we can work with normalized states in each reduced of the manifestly gauge-invariant physical inner product σ -sector and in the physical Hilbert space simultaneously. obtained through group averaging in Equation (18). Indeed, In conclusion: ψ ( τ, σ τ, σ I ) φ is the (kinematical) σ phys| | | ⊗ S | physkin (1) Applying the quantum symmetry reduction map Rσ to the expectation value of the ‘projector’ onto clock time τ in H P clock-neutral picture on the physical Hilbert space Hphys physical states. However, it is clear from the unitarity of the yields a relational Heisenberg picture with respect to the clock Hphys Schrödinger dynamics on S,σ and Equation (47) that this C on the physical system Hilbert space of the σ -modes, inner product does not depend on τ (the “gauge”), in line with Hphys S,σ . the interpretation of it being the quantum analog of a gauge-fixed (2) σ -sector wise, the relational quantum dynamics encoded in version of a manifestly gauge-invariant quantity. the relational Dirac observables on the physical Hilbert space is equivalent to the dynamics in the relational Heisenberg 5.3. Interlude: Alternative Route picture on the physical system Hilbert space. As an aside, we mention that there exists an alternative route (3) Given the invertibility of the reduction map, Theorem 4 to establishing a trinity for clock Hamiltonians quadratic in phys τ Hphys formally shows that if fˆS ( ) is self-adjoint on S,σ , then momenta. This again exploits the reducible representation on σ so is F (τ) on Hσ . the physical Hilbert space. The σ -sector of H is defined ˆfS,T phys pt by the constraint Cσ H H , where H : ˆ and 5.1.4. Equivalence of Dynamics (ii) and (iii) ˆ = ˆ C′ + ˆ S′ ˆ C′ = √2 The previous subsections establish a σ -sector wise equivalence H : σ s H . Clearly, H is now a non-degenerate clock ˆ S′ = − ˆ S ˆ C′ between the relational dynamics, on the one hand, in the Hamiltonian.q In Höhn et al. [7] we established the trinity for clock-neutral picture of Dirac quantization and, on the non-degenerate clock Hamiltonians and the σ -sector defined by other, the relational Schrödinger and Heisenberg pictures, Cσ H H yields a special case of that. This immediately ˆ = ˆ C′ + ˆ S′ obtained through Page-Wootters reduction and quantum implies a trinity per σ -sector, however, now relative to a clock deparametrization, respectively. It is thus evident that also POVM which is covariant with respect to H . It is evident that ˆ C′ the relational Schrödinger and Heisenberg pictures are indeed the covariant clock POVM is in this case defined through the equivalent up to the unitary US(τ) as they should. This is directly eigenstates of t which (up to a factor of √2) is also the first ˆ implied by Lemma 8 which shows that the Page-Wotters and moment of the POVM. Indeed, the equivalence between the quantum symmetry reduction maps are (weakly) related by clock-neutral Dirac quantization and the relational Heisenberg US(τ). picture has previously been established for models with quadratic clock Hamiltonians precisely in this manner in Höhn and 5.2. Quantum Analogs of Gauge-Fixing and Vanrietvelde [30]andHöhn[31] (for a related discussion see also Gauge-Invariant Extensions [29, 38]). However, as mentioned in section 4.1, one either has to In contrast to the classical constraint surface, the “quantum regularize the relational observables or write them as functions of constraint surface” Hphys is automatically gauge-invariant since affine, rather than canonical pairs of evolving degrees of freedom. the exponentiation of the symmetry generator C acts trivially This is a consequence of the square root nature of H . None of ˆ H ˆ S′ on all physical states and Dirac observables. Nevertheless, these extra steps were needed in the trinity construction of this extending the interpretation established in Höhn et al. [7], we can understand the quantum symmetry reduction map R H 19However, note that the quantization of this classical reduced phase space will Rσ [and given their unitary relation, also S (τ)] as the quantum Hphys in some cases, but not in general, coincide with the quantum theory on S,σ analog of a classical phase space reduction through gauge- due to the generic inequivalence between Dirac and reduced quantization (see fixing. For completeness, the latter procedure is explained in Supplementary Material).

Frontiers in Physics | www.frontiersin.org 17 June 2021 | Volume 9 | Article 587083 Höhn et al. Relational Dynamics in Relativistic Settings article, which is based on a clock POVM which is covariant with one may further check that [31, 104] p2 respect to s ˆt , rather than p /√2. 2 ˆt φphys ψphys φ+ , ψ+ φ− , ψ− , | phys = phys phys KG − phys phys KG 6. RELATIVISTIC LOCALIZATION:     (60) ADDRESSING KUCHAR’Sˇ CRITICISM where In his seminal review on the problem of time, Kuchaˇr raised three criticisms against the Page-Wootters formalism [1]: the σ σ 3 σ ∗ σ φphys, ψphys i d x φphys(t, x) ∂t ψphys(t, x) Page-Wootters conditional probability in Equation (37) (a) yields KG = R3   Z h   the wrong localization probabilities for a relativistic particle, (b) σ ∗ σ ∂t φ (t, x) ψ (t, x) , (61) violates the Hamiltonian constraint, and (c) produces incorrect − phys phys   i transition probabilities. As mentioned in the introduction, is the Klein-Gordon inner product in which positive frequency criticisms (b) and (c) have been resolved in Höhn et al. [7]—see modes are positive semi-definite, negative frequency modes Theorem 2 which extends the resolution of (b) to the present class are negative semi-definite and positive and negative frequency of models. modes are mutually orthogonal. The physical inner product ˇ Here, we shall now also address Kuchar’s first criticism (a) is thus equivalent to the Klein-Gordon inner product (with on relativistic localization, which is more subtle to resolve. The correctly inverted sign for the negative frequency modes), which main reason, as is well-known from the theorems of Perez- provides the correct and conserved normalization for the free Wilde [92] and Malament [93] (see also the discussion in relativistic particle21. This raises hopes that the conditional [94, 95]), is that there is no relativistically covariant position- probabilities of the Page-Wootters formalism may yield the operator-based notion of localization which is compatible with correct localization probability for the relativistic particle. Note relativistic causality and positivity of energy. This is a key that so far we have not yet made a choice of time operator. motivation for quantum field theory [90, 94]—and here a Suppose now that the Minkowski time operator t, quantized challenge for specifying what the “right” localization probability ˆ as a self-adjoint operator on Hkin, is used to define the projector for a relativistic particle should be. Instead, one may resort onto clock time t as e (t) t t and e x x is the projector C =| | x =| | to an approximate and relativistically non-covariant notion onto position x. This time operator is not covariant with respect of localization proposed by Newton and Wigner [90, 91]. to the quadratic clock Hamiltonian. The conditional probability We will address criticism (a) by demonstrating that our Equation (37) then becomes formulation of the Page-Wootters formalism, based on covariant clocks for relativistic models, yields a localization in such an 2 ψphys(t, x) approximate sense. Prob x when t | ′ | ′ , (62) = d3x ψ (t, x ) 2 For the sake of an explicit argument, we shall, just like R3 phys
| | ˇ Kuchar [1], focus solely on the free relativistic particle, whose where ψ (t, x) ( t xR) ψ is a general solution to Table 1 phys = |⊗ | | phys Hamiltonian constraint reads (cf. ) the Klein-Gordon equation. As Kuchaˇr pointed out [1], while this 2 2 2 would be the correct localization probability for a non-relativistic CH pt p m , ˆ = −ˆ + ˆ + particle, it is the wrong result for a relativistic particle. Indeed, where p denotes the spatial momentum vector. However, the apart from not separating positive and negative frequency ˆ argument could be extended to the entire class of models modes, which is necessary for a probabilistic interpretation considered in this manuscript. It is straightforward to check e.g., if ψphys contains both positive and negative frequency that the physical inner product Equation (18) reads in this case modes then the denominator in Equation 62 is not conserved), [31, 104]20 Equation (62) neither coincides with the charge density of the Klein-Gordon current in Equation (61), nor with the d3p φ ψ φ ε ψ ε Newton-Wigner approximate localization probability [90, 91]. phys phys phys kin∗ ( p, p) kin( p, p) | = R3 2 εp In particular, one can not interpret a solution ψ (t, x) to Z h phys the Klein-Gordon equation as a probability amplitude to find φ∗ ( εp, p) ψ ( εp, p) , (59) + kin − kin − the relativistic particle at position x at time t. The reason, as i 2 2 explained in Haag [90], is that the conserved density is the one where εp p m is the relativistic energy of the particle and = + in Equation (61) and ψphys and ∂tψphys inside it are not only the first and second term in the integrand correspond to negative p dependent, but related by a non-local convolution and positive frequency modes, respectively. Fourier transforming to solutions to the Klein-Gordon equation in Minkowski space 3 ′ ′ ′ ∂tψphys(t, x) d x ε(x x ) ψphys(t, x ), 1 d3p = R3 − ψσ i(x p σ t εp) ψ σ ε Z phys(t, x) 3/2 e − kin( p, p), = (2π) R3 2 ε − 21 Z p This also resolves the normalization issue appearing in Diaz et al. [50] where physical states were normalized using the kinematical, rather than physical inner 20Note that here we have a doubly degenerate system energy H p2 m2 in product, thus yielding a divergent normalization (for equal mass states) in contrast ˆ S = ˆ + contrast to the expression in Equation (18) where we ignored degeneracy. to here.

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′ where ε(x x ) is the Fourier transform of i ε . Equation (64) as a Newton-Wigner wave function as well, but − − p By contrast, let us now exhibit what form of conditional expressed relative to a different time coordinate τ. Indeed, in probabilities the covariant clock POVM ET of section 4.2 gives line with this interpretation, we find that in this case too the rise to. We now insert e (τ) τ, σ τ, σ and, as before, physical inner product, Equation (60), for the positive frequency C = σ | | ex into the conditional probability Equation (37). The crucial modes assumes the form of the standard Schrödinger theory P difference between the covariant clock POVM ET and the clock inner product operator t (which is covariant with respect to C , but not C ) ˆ ˆ σ ˆ H is that the denominator of Equation (37) is equal to the physical 3 ∗ inner product in the former case (see Corollary 2) but not in the φphys+ , ψphys+ d x φS+(τ, x) ψS+(τ, x). KG = R3 22 Z latter . Supposing that we work with normalized physical system  
states ψσ (τ) ψσ (τ) 1, Theorem 2 implies S | S = The analogous statement is true for the negative frequency modes. In that sense, Equation (63), in contrast to Equation (62), 1 σ 2 Prob x when τ ψS (τ, x) (63) does admit the interpretation as a valid, yet approximate = 2 σ | |
X localization probability for the relativistic particle per frequency ψ F (τ) ψ , sector, just like in the standard Newton-Wigner case23. = phys|ˆex,T | physphys Accordingly, computing the conditional probabilities of the σ : Page-Wootters formalism relative to the covariant clock POVM, where ψS (τ, x) √2( τ, σ x ) ψphys and τ is now not = |⊗ | | rather than the non-covariant Minkowski time operator t, Minkowski time. For concreteness, let us now focus on positive ˆ frequency modes. Using Equations (15) and (25), one obtains leads to an acceptable localization probability for a relativistic particle, thereby addressing also Kuchaˇr’s first criticism (a). 3 1 d p 2 Given the equivalence of the Page-Wootters formalism with the ψ τ x i(x p τ εp) ψ ε p S+( , ) 3/2 e − kin( p, ). clock-neutral and the relational Heisenberg pictures, established = (2π) R3 2 εp − Z through the trinity in section 5, this result also equips (64) p the quantum relational Dirac observables F (τ) and the ˆx,T evolving Heisenberg observables x(τ) with the interpretation of This is almost the Newton-Wigner position space wave function providing an approximate, Newton-Wigner type localization in for positive frequency modes, which relative to Minkowski time Minkowski space. reads [90]

3 ′ ′ ψNW+ (t, x) d x K(x x ) ψphys+ (t, x ) (65) 7. CHANGING QUANTUM CLOCKS = R3 − Z 1 d3p So far we have worked with a single choice of clock. Let us i(x p t εp) ψ ε 3/2 e − kin( p, p), = (2π) R3 2 εp − now showcase how to change from the evolution relative to Z one choice of clock to that relative to another. Our discussion p where K(x) is the Fourier transform of 2εp. The key property will apply to both the relational Schrödinger picture of the 2 Page-Wootters formalism and the relational Heisenberg picture of ψ+ (t, x) is that, while not relativistically covariant, it | NW | p does admit the interpretation of an approximate localization obtained through quantum deparametrization. probability, with accuracy of the order of the Compton wave For concreteness, suppose we are given a Hamiltonian length, for finding the particle at position x at Minkowski time constraint of the form t [90, 91]. In particular, p2 p2 C s ˆ1 s ˆ2 H , (66) ˆ H 1 2 ˆ S 3 ∗ = 2 + 2 + φphys+ , ψphys+ d x φNW+ (t, x) ψNW+ (t, x), KG = R3   Z
where s 1 and p denotes the momentum of clock subsystem i.e., the Klein-Gordon inner product assumes the usual i = ± ˆi C , i 1, 2 and we have suppressed tensor products with identity Schrödinger form for the Newton-Wigner wave function. i = operators. In particular, suppose H does not depend on either Noting that due to Equation (19) we can heuristically view ˆ S of the clock degrees of freedom. We will work with the covariant τ as t/ε , and comparing with Equation (65) we can interpret p clock POVM of section 4.2 for both clock choices. For example, the constraints of the relativistic particle, the flat (k 0) FLRW 22It is instructive to see how this is linked to the (non-)covariance of the clock = observable with respect to C . Let e be either the covariant e (τ) or non- model with a massless scalar field and the Bianchi I and II models ˆ H C C covariant eC(t). The denominator of Equation (37) reads 23 The physical inner product for the positive frequency solutions ψ+ (t, x) to the ψphys eC IS ψphys ψkin δ(CH ) (eC IS) ψphys . phys | ⊗ | kin = | ˆ ⊗ | kin Klein-Gordon equation takes, of course, the standard Klein-Gordon (and not the Equation (44) implies that δ(C )(e (τ) I ) I . This exploits the covariance Schrödinger) form Equation (61). Nevertheless, the kernel K(x x′) in the non- ˆ H C ⊗ S ≈ phys − and immediately shows that the denominator coincides with the physical inner local convolution in the first line in Equation (65) decreases quickly as a function product Equation (18). By contrast, the non-covariance entails δ(C )(e (t) I ) of m x x′ [90]. Hence, for massive particles, we can interpret even Equation (62) ˆ H C ⊗ S = | − | Iphys, so that in this case the denominator differs from the physical inner product. as providing an approximate localization.

Frontiers in Physics | www.frontiersin.org 19 June 2021 | Volume 9 | Article 587083 Höhn et al. Relational Dynamics in Relativistic Settings from Table 1 are of the above form24. Our subsequent discussion In particular, owing to our focus on covariant clock POVMs, will thus directly apply to these models. all the results and proofs [7] apply σ -sector-wise to the present Since we will exploit the Page-Wootters and symmetry case. However, we will also study novel effects, such as the reduction maps as “quantum coordinate maps” from the clock- temporal frame dependence of comparing clock readings. neutral picture to the given “clock perspective,” we will be able to change from the description of the dynamics relative to one clock 7.1. State Transformations to that relative to another in close analogy to coordinate changes Denote by Rσi , where I S, H , the Page-Wootters or I ∈ { } on a manifold. Due to the shape of the constraint in Equation (66) quantum symmetry reduction map to the σi-sector of clock we now have superselection of Dirac observables and the physical Ci. For notational simplicity, we drop the dependence of the Hilbert space across both the σ1-frequency sectors of clock C1 Page-Wootters reduction on the clock reading τi of clock and the σ2-frequency sectors of clock C2. The physical Hilbert Ci for the moment. The temporal frame change (TFC) map σi σj phys phys phys phys space takes the form → : H H H H I J C ,σ S,σ C ,σ S,σ from clock Ci’s σi-sector → j i ⊗ i → i j ⊗ j to clock Cj’s σj-sector then reads H Hσ σ , (67) phys = 1, 2 σ1,σ2 σi σj σj σ 1 M → : R R i I J J I − . (68) : → = ◦ where Hσ ,σ Hσ Hσ is the overlap of the σ1-sector of 1 2 = 1 ∩ 2
clock C1 and the σ2-sector of clock C2. As we have seen the phys Here H denotes the physical clock Cj Hilbert space reduction maps are only invertible per frequency sector. Hence, Cj,σi corresponding to the σ -sector of clock C , i.e., the subspace of we will only be able to change from a given σ1-sector to the part of i i H compatible with solutions to the constraint Equation (66) the σ2-frequency sector which is contained in it. In other words, Cj and similarly for the other Hilbert spaces. When I J in the “quantum coordinate changes” are restricted to each overlap = H Equation (68), then the TFC map changes not only the temporal σ1,σ2 . reference frame, but also between the corresponding relational The method of changing temporal reference frames exhibited σ σ i→ j : below is a direct extension of several previous works: Höhn Heisenberg and Schrödinger pictures. Let us write I σ σ = i→ j and Vanrietvelde [30] and Höhn [31] developed the method σ - I I when no relational picture change takes place. → sector-wise for states and observables in the relational Heisenberg More explicitly, the TFC map from the σ1-frequency sector of picture for Hamiltonians of the type in Equation (66) for two clock C1 in the relational Schrödinger picture to the σ2-frequency example models, but used clock operators canonically conjugate sector of clock C2 in the relational Schrödinger picture takes the to the clock momenta p (and thus not a clock POVM covariant form ˆi with respect to the full clock Hamiltonian). The method of σ σ transforming relational observables from one clock description 1→ 2 τ , σ I δ(C ) τ , σ I . S 2 2 C1S ˆ H 1 1 C2S to another was demonstrated in Höhn and Vanrietvelde [30] and = | ⊗ | ⊗

Höhn [31] for a subset of relational Dirac observables, paying, Here we have made use of Equations (38) and (42) and the however, detailed attention to regularization necessities arising covariant clock states Equation (25) for both clocks. The reduced from time-of-arrival observables [18, 23, 117–119]. Our previous states transform under this map as follows: article [7] developed the method comprehensively for both states and observables for clock Hamiltonians with non-degenerate σ2 σ1 σ2 σ1 θ( σ1 p1) IS ψC S C (τ2) S → ψC S C (τ1) , and continuous spectrum in both the relational Schrödinger and − ˆ ⊗ | 1 | 2 = | 2 | 1 Heisenberg pictures; specifically, the transformation of arbitrary
(69) relational observables corresponding to relations between S and σi the clocks was developed for the corresponding class of models. where we made use of Equation (43) and ψC S C (τi) is the | j | i In Castro-Ruiz et al. [47] the clock change method was exhibited relational Schrödinger picture state of clock Cj and system S for state transformations in the relational Schrödinger picture relative to clock Ci, which is chosen as the temporal reference for ideal clocks whose Hamiltonian coincides with the clock frame, in its σi-sector. In other words, the Heaviside-function momentum itself. Our discussion can also be viewed as a full on the l.h.s. highlights that we can only map from the σ1-sector quantum extension of the semiclassical method in Bojowald et al. of clock C1 to that part of the σ2-sector of clock C2 which is [32, 33] and Höhn et al. [34] which is equivalent at semiclassical contained in the σ1-sector of clock C1. This is clear, because any order, however, also applies to clock functions which are non- reduction map is only invertible on its associated σ -sector: from monotonic, i.e., have turning points, in contrast to the other the σ1 relational Schrödinger picture one can only recover the works mentioned (See [70–72, 74] for related spatial quantum σ1-sector of the physical Hilbert space. Hence, the subsequent frame changes). Page-Wootters reduction map to the σ2-sector of clock C2 in Equation (68) can then only yield information in the overlap 24Indeed, the Hamiltonian constraint of the vacuum Bianchi I and II models can of the σ1- and σ2-sectors in the physical Hilbert space (see also be written in the form [116] [30, 31] for explicit examples of this situation in the relational 2 2 2 p0 p p 4√3β Heisenberg picture). This is a manifestation of the superselection C ˆ ˆ− ˆ+ k e ˆ+ . ˆ H − =− 2 + 2 + 2 + + across both the σ1- and the σ2-sectors.

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clock-neutral structure, providing a description of the dynamics prior to having chosen a temporal reference frame relative to which the other degrees of freedom evolve. In line with this, in terms of different one-parameter families of relational Dirac observables, the physical Hilbert space contains the complete information about the dynamics relative to all the different possible clock choices at once. 7.2. Observable Transformations Just as we transformed reduced states from the perspective of one clock to the perspective of another by passing through the gauge-invariant physical Hilbert space (see Figure 3), we now FIGURE 3 | Schematic representation of a temporal frame change, as defined transform the description of observables relative to one clock through Equation (68). The figure encompasses both the relational to that relative to the other by passing through the gauge- Schrödinger picture of the Page-Wotters formalism and the relational Heisenberg picture of the quantum deparametrization (as well as their invariant algebra of Dirac observables on the physical Hilbert i mixtures). Viewing the reduction maps RI+ as quantum coordinate maps, any space. The observable transformations will thus be dual to such temporal frame change takes the form of a quantum coordinate the state transformations. The idea is always that we describe transformation from the description relative to clock C1 to the one relative to the same physics, encoded in the gauge-invariant states and clock C2. Just as coordinate transformation pass through the observables of the clock-neutral physical Hilbert space, but reference-frame-neutral manifold, the quantum coordinate transformations pass through the clock-neutral physical Hilbert space in line with the general relative to different temporal frames. Again, we have to pay discussion of the clock-neutral structure and quantum general covariance in attention to the two superselection rules on the clock-neutral section 2. Due to the double superselection rule, the quantum coordinate physical Hilbert space and we will demonstrate the observable transformations have to preserve the overlaps of the frequency sectors of C1 transformations separately for the relational Schrödinger and and C2. Here we illustrate the example of the overlap of the positive frequency sectors of both clocks, so that the corresponding frame transformation passes Heisenberg pictures.

through H , (cf. Equation 67). +1 +2 7.2.1. Observable Transformations in the Relational Schrödinger Picture phys Suppose we are given an observable OC S C describing σ ˆ 2 | 1 Similarly, the TFC map from the 1-frequency sector of clock certain properties of the composite system C2S in the σ C1 in the relational Heisenberg picture to the 2-frequency sector relational Schrödinger picture of clock C1 in either frequency of clock C2 in the relational Heisenberg picture reads sector of the latter25. Owing to Theorem 1, we can write this as a reduction of a corresponding relational Dirac σ1 σ2 † → τ2, σ2 U (τ2) (70) observable on H : H = | ⊗ C1S phys  δ(C ) τ, σ U (τ ) , H 1 1 C2S 1 σ1 σ1 1 phys × ˆ | ⊗ R (τ1) FO ,T (τ1)(R (τ1))− O . S ˆ C2S C1 1 S ˆ C2S C1
| = | where we have made use of Lemma 8. Using the same lemma, the reduced states transform under this map in complete analogy to We can now also map the same relational Dirac observable into Equation (69) the σ2-sector of the relational Schrödinger picture of clock C2:

phys σ2 σ2 1 σ2 σ1 σ2 σ1 O (τ , τ ) : R (τ ) F (τ )(R (τ )) . θ( σ p ) I ψ → ψ , C1S C2 1 2 S 2 OC2S C1 ,T1 1 S 2 − 1 1 S C1S C2 S C2S C1 ˆ = ˆ | − ˆ ⊗ | | = | | | h i (71) p2 : ˆj phys with UCjS(τi) exp i τi (sj HS) the evolution operator = − 2 + ˆ The result will be the image of the original observable OC S C ,   ˆ 2 | 1 of the composite system of clock Cj and system S relative to describing properties of C2S relative to C1, in the “perspective” phys clock Ci. of clock C2. Hence, if O depends non-trivially on C2, ˆ C2S C1 Note that, interpreting the reduction maps as the “quantum an indirect self-reference effect| occurs in the last equation [7]. coordinate maps” taking one from the clock-neutral physical Notice that, while the original observable in the Schrödinger Hilbert space to a specific “clock perspective,” any such TFC picture of C1 is independent of the evolution parameter τ1, the map in Equation (68) takes the same compositional form as coordinate changes on a manifold. In particular, any such 25Recall that the label “phys” highlights that the observable acts on the physical C2S Hilbert space, i.e., on σ HC HS , where σ temporal frame change proceeds by mapping via the clock- C2SC1 2 C2SC1 2 ⊗ = p2 θ s1 s2 ˆ HS is the projector onto the subspace
corresponding to the neutral physical Hilbert space in analogy to how coordinate − 2 + ˆ p2 p2 changes always proceed via the manifold (see Figure 3). This spectrumh  σ : iSpec s ˆ2 H Spec s ˆ1 permitted by the constraint C2SC1 = 2 2 + ˆ S ∩ − 1 2 observation lies at the heart of the perspective-neutral approach Equation (66) (cf. Equation 17). We can thus also understand this observable as phys : to quantum reference frame changes [7, 30, 31, 71, 72]. It is also a projection O σ OC S C σ of some observable OC S C ˆ C2S C1 = C2SC1 ˆ 2 | 1 C2SC1 ˆ 2 | 1 ∈ the reason why we may interpret the physical Hilbert space as a L(H H ); cf.| Equation (36). C2 ⊗ S

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description of that same observable in the Schrödinger picture the description relative to clock C1 to one relative to clock C2, relative to clock C1 will generally depend on both evolution or vice versa. Once more, this is a direct consequence of the parameters τ1, τ2. The dependence on τ1 is a consequence of double superselection rule induced by the shape of the constraint it being the reduction of a relational Dirac observable with Equation (66). evolution parameter τ1, but into the “perspective” of C2. The possible τ2 dependence may arise as a consequence of said 7.2.2. Observable Transformations in the Relational phys indirect self-reference. For example, suppose O T2 Heisenberg Picture ˆ C2S C1 = ˆ ⊗ | The argumentation for the relational Heisenberg picture IS so that the relational Dirac observable is FOC S C ,T1 (τ1) ˆ 2 | 1 = proceeds in complete analogy. We thus just quote the results, F (τ ). The observable on the l.h.s. in Equation (71) then ˆT2,T1 1 which immediately follow from those of the previous subsection describes how the first moment operator T associated with C ˆ 2 2 through use of Lemma 8. Of course, in this case, the reduced evolves relative to C from the “perspective” of C ; this certainly 1 2 observables have an explicit dependence on the evolution should yield a τ dependence. We will explain this in more 1 parameter (Equation 56). detail shortly. The observable transformations from the relational Taking into account the two superselection rules across the Heisenberg picture of the σ -sector of clock C into the σ - and σ -sectors, these observations imply that observable 1 1 1 2 relational Heisenberg picture of the σ -sector of clock C are transformations from the relational Schrödinger picture of the 2 2 given by σ1-sector of clock C1 into the relational Schrödinger picture of the σ2-sector of clock C2 read σ1 σ2 phys σ1 σ2 1 → O (τ1) → − σ σ phys σ σ 1 σ σ 1 phys σ σ H ˆ C2S C1 H 1→ 2 O 1→ 2 − 2→ 1 − O 2→ 1 | S ˆ C2S C1 S S ˆ C2S C1 S σ2 σ2 1 | = | R FO ,T (τ1) θ( σ1
p1) IC S R − σ σ 1 phys σ σ 1 H ˆ C2S C1 1 2 H R 2 R 1 −
R 1
R 2 − ≈ | − ˆ ⊗ S (τ2) S (τ1) OC S C S (τ1) S (τ2) † phys = ◦ ˆ 2 | 1 ◦ θ( σ p ) I U (τ ) O (τ
, τ ) U
(τ ) (75) 1 1 S C1S 2 C1S C2 1 2 C1S 2 σ2 τ1,σ1 phys σ2 1 = − ˆ ⊗ ˆ R (τ ) E O
R (τ ) −
(72) | S 2 S C2S C1 S 2 : H = ˆ | θ( σ1p1) IS
OC S C (τ1, τ2), = − ˆ ⊗ ˆ 1 | 2 Rσ2  
Rσ2 1 S (τ2) FOC S C ,T1 (τ1) θ( σ1p1) IC2S S (τ2) − ≈ ˆ 2 | 1 − ˆ ⊗
phys phys θ( σ1p1) IS OC S C (τ1, τ2).

where O (τ1, τ2) is given by Equations (71) and (73). Thanks = − ˆ ⊗ ˆ 1 2 ˆ C1S C2 | to the double| superselection rule, this transformation preserves
In the second line we made use of Equation (45), in the third of expectation values again per overlap of a σ1- with a σ2-sector, in Theorem 1, and in the fourth of Equation (71) and the fact that obvious analogy to Equation (74). θ( σ p ) commutes with the reduction map of the C clock and − 1 ˆ1 2 with FO ,T (τ1) (see Lemma 5). ˆ C2S C1 1 Observe| that the structure of this transformation shows that 7.3. Occurrence of Indirect Clock reduced observables relative to one clock will transform always Self-Reference via the gauge-invariant Dirac observable algebra to reduced Finally, let us now come back to the indirect self-reference observables relative to another clock. effect of clock C2 alluded to above. The following theorem, Using Equations (38), (42), and (45), we can write this which is adapted from Höhn et al. [7] and whose proof transformation as applies here per pair of σ1- and σ2-sector, reveals the necessary and sufficient conditions for this indirect self-reference phys θ( σ1p1) IS OC S C (τ1, τ2) τ2, σ2 δ(CH) to occur: − ˆ ⊗ ˆ 1 | 2 = | ˆ phys τ1, σ1
τ1, σ1 O δ(CH) τ2, σ2 . (73) phys phys phys ˆ C2S C1 ˆ Theorem 5. Consider an operator O L(H H ) | | ⊗ | | C S C C2 S ˆ 2 | 1 ∈ ⊗   of the composite system C2S described from the perspective of This transformation reveals that expectation values are preserved clock C1. From the perspective of clock C2, this operator is in the following manner: phys phys independent of τ2, so that O (τ1, τ2) O (τ1) ˆ C1S C2 = ˆ C1S C2 ∈ σ phys σ phys phys | | ψ 2 (τ ) θ( σ p ) I O (τ , τ ) φ 2 (τ ) L(H H ) if and only if C S C 2 1 1 S C S C 1 2 C S C 2 C1 S 1 | 2 | − ˆ ⊗ ˆ 1 | 2 | 1 | 2 ⊗ σ1
phys σ1 ψ (τ1) θ( σ2p2) IS O θ( σ2p2) IS φ (τ1) . = C2S C1 | − ˆ ⊗ ˆ C2S C1 − ˆ ⊗ | C2S C1 | | | phys phys phys

(74) OC S C OC C fS C , ˆ 2 | 1 = ˆ 2| 1 i ⊗ ˆ | 1 i i     The projectors onto the σ2-sector on the r.h.s. appears because X the C2 reduction map in Equation (72) induces such a projection (compare this with the state transformations Equation (69) which phys Hphys phys where (fS C )i is an operator on S and (OC C )i is a constant ˆ | 1 ˆ 2| 1 are dual). In other words, only the physical information in the phys p2 of motion, (O ) , s ˆ2 0. Furthermore, in this case the σ σ C2 C1 i 2 2 overlap of the 1- and 2-sector is preserved when changing from ˆ | = h i

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H transformed observable reads operator transforms to an operator OC S C (τ1, τ2) that satisfies the ˆ 1 | 2 Heisenberg equation of motion in clock C2 time τ2 without an phys explicitly τ2 dependent term, θ( σ1p1) IS OC S C (τ1) − ˆ ⊗ ˆ 1 | 2   phys 2 G τ , σ τ , σ f (76) d H p2 H σC1SC2 C1S 1 1 1 1 S C1 O (τ , τ ) i s ˆ H , O (τ , τ ) , = | | ⊗ ˆ i C1S C2 1 2 2 S C1S C2 1 2 " | dτ ˆ | = 2 + ˆ ˆ | Xi     2   phys t , σ O δ(C ) t , σ , if and only if 2 2 C C H 2 2 σC1SC2 | ˆ 2| 1 i ˆ | #   phys phys phys O (τ ) O f (τ ) , 2 C2S C1 1 C2 C1 ˆS C1 1 p1 ˆ | = ˆ | i ⊗ | i where σ θ s2 s1 ˆ HS is the projector onto i C1SC2 = − 2 + ˆ X     H H the physical subspace ofh C1 S, t2i, σ2 is an arbitrary σ2- ⊗ | phys p2 phys sector clock state of C , and G is the G-twirl over the group ˆ1 2 C1S and OC C is a constant of motion, [s1 2 , OC C ] 0. 2 ˆ 2 1 i ˆ 2 1 = p1 | | generated by the evolution generator s1 ˆ HS of the composite   2 + ˆ system C1S. The interpretation of the transformations is of course completely analogous to the relational Schrödinger picture. That is to say, the indirect self-reference effect and thus τ2- dependence of Equation (72) is absent if and only if the relational Dirac observable encoding how C2S properties evolve relative 7.4. Application: Comparing Clock to C1 does not contain any degrees of freedom of clock C2 Readings that evolve. phys phys One application of the temporal frame change method developed When O I f , i.e., only the evolution of system C2S C1 C2 S C1 above is comparing readings of different clocks. This is also a ˆ | = ⊗ ˆ | degrees of freedom relative to C1 is described, Theorem 5 entails prerequisite for developing a notion of clock synchronization. that the transformation to the description relative to C2 simplifies For example, we may wish to compare the evolution of some as follows: system property fˆS relative to clock C1 with fˆS relative to clock phys C2. These two relational evolutions will be encoded in two one- θ( σ1p1) IS O (τ1) ˆ C1S C2 parameter families of Dirac observable of the form F (τ ) − ˆ ⊗ | = IC2 fS,T1 1 phys ˆ ⊗
G τ , σ τ , σ f . and F (τ ). In order to relate these two dynamics, we σC1SC2 C1S 1 1 1 1 S C σC1SC2 IC fS,T2 2 | | ⊗ ˆ | 1 ˆ 1 ⊗   need a consistent method for relating the different clock readings In particular, the transformed system observable is perspective τ1, τ2. While classically, there is an unambiguous way to answer τ independent, i.e., its description relative to C1 and C2 coincide the question “what is the value of the reading 2 of clock C2, when clock C reads τ ?” namely by setting τ (τ ) : F (τ ), if and only if it is a constant of motion (see [7] for the proof 1 1 2 1 = T2,T1 1 of this statement, which again applies here per pair of σ1- this is not so in the quantum theory because both clocks are and σ2-sector): now described in terms of quantum operators and their relation depends on the quantum state. In fact, we shall argue shortly that Corollary 4. An operator of C2S relative to C1 comparing clock readings is generally dependent on the choice of temporal frame (here either C1 or C2) in the quantum theory. phys phys OC S C IC2 fS C . ˆ 2 | 1 = ⊗ ˆ | 1 7.4.1. Three Ways of Comparing Clock Readings transforms under a temporal frame change map to the perspective To address this conundrum in the quantum theory, let us recall the conditional probabilities in Equation (37) and ask for the of C2 as follows probability that C2 reads τ2 when C1 reads τ1 (ignoring frequency phys phys sectors for simplicity for the moment): OC S C IC1 fS C , ˆ 1 | 2 = ⊗ ˆ | 2 P(T τ T τ ) phys phys phys 2 = 2| 1 = 1 where fS C fS C if and only if fS C is a constant of motion, ˆ 1 = ˆ 2 ˆ 1 ψ eC (τ1) eC (τ2) IS ψ phys | | | = phys| 1 ⊗ 2 ⊗ | physkin [fS C , HS] 0. ˆ 1 ˆ = P(T1 τ1 T2 τ2). (77) | = = | = Theorem 5 translates as follows into the relational Heisenberg picture (see [7] for the proof which applies here per pair of σ1- Here we have assumed that the physical state is normalized such and σ2-sector): that by Corollary 2 also the reduced states in the Schrödinger picture of either clock are normalized. phys phys phys Corollary 5. Let O (τ1) L(H H ) be an ˆ C2S C1 ∈ C2 ⊗ S operator describing the dynamics| of properties of the composite Comparing clock readings. Given the conditional probabilities system C2S relative to C1 in the Heisenberg picture. Under a Equation (77), we may consider the following three generally temporal frame change Equation (75) to the perspective of C2, this distinct options for comparing clock readings.

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(A) The clock reading of C2 when C1 reads τ1 is defined to be the is the σ2-sector nth-moment of the covariant clock POVM value of τ that maximizes the conditional probability P(T corresponding to C . In the second line of Equation (81) we have 2 2 = 2 τ T τ ). This assumes the distribution to have a unique made use of Equations (38) and (80), while in the third line we 2| 1 = 1 maximal peak. invoked Theorem 2. Note that by Equation (74), the expression (B) The clock reading of C2 when C1 reads τ1 is defined to be the in Equation (81) defines an expectation value which is preserved expectation value during a temporal frame change between C1 and C2. Thanks to Lemmas 3 and 4 we can write the nth-moment : τ2(τ1) dτ ′ τ ′ P(T2 τ ′ T1 τ1). (78) in Equation (81) also in the form = R = | = Z (n) σ1 n σ1 (C) The clock reading of C2 when C1 reads τ1 is defined to be τ2 (τ1) ψC S C (τ1) T2,σ IS ψC S C (τ1) / = 2 1 | ˆ 2 ⊗ | 2 1 (n) 1 n | | τ (τ ) for n > 1, where n 2 1 ψσ1,σ2 FT I ,T (τ1)) ψσ1,σ2 , = | ˆ 2 ⊗ S 1 | phys   (n) : n τ (τ1) dτ ′ (τ ′) P(T2 τ ′ T1 τ1) (79) σ1 n phys phys 2 as long as ψ (τ1) D(T ) H H . Since = R = | = | C2S C1 ∈ ˆ 2 ∩ C2 ⊗ S Z / | (n) 1 n   is the nth-moment of the conditional probability distribution in τ (τ ) τ (τ ) for n > 1 for general states, 2 1 = 2 1 Equation (77). definitions  (B) and (C) will generically not be equivalent. In Relating different clock readings in terms of expectation values,as the sequel, we shall mostly consider definition (B) in extension in (B), is arguably the most natural choice and has originally been of Höhn and Vanrietvelde [30], Höhn [31], Bojowald et al. discussed in Höhn and Vanrietvelde [30], Höhn [31], Bojowald [32, 33], Höhn et al. [34], and Smith and Ahmadi [45]. This et al. [32, 33], Höhn et al. [34], and Smith and Ahmadi [45]; we seems to be the physically most appealing one, especially if an expand on this here. ensemble interpretation could be developed for the models under Clearly, the two definitions (A) and (B) only agree when consideration. Definition (A) is only unambiguous when the the conditional probability distribution is peaked on the conditional probability distribution has a single maximal peak expectation value. Furthermore, all three definitions (A)–(C) and definition (C) is operationally unnatural and convoluted. agree in the special case that P(T τ T τ ) δ(τ τ ), That is, we set for the value of the reading of clock C2 when C1 2 = ′| 1 = 1 = ′ − 1 i.e., when there are no fluctuations in the conditional probability reads τ1: distribution. : (1) τ2(τ1) τ2 (τ1). (82) 7.4.2. Comparing Clock Readings for Quadratic = Clock Hamiltonians The following discussion, however, qualitatively also applies to Let us now explore these definitions in our present class of models definition (C). defined by Equation (66), taking into account the different frequency sectors again. Minding the double superselection rule, 7.4.3. Comparing Clock Readings Is Temporal Frame we replace Equation (77) by Dependent Notice that definitions (A)–(C) treat C2 as the fluctuating Pσ σ (T τ T τ ) 1, 2 2 = 2| 1 = 1 subsystem. We can thus interpret them as providing a definition σ1 σ2 ψ e (τ1) e (τ2) IS ψ of the clock reading of C2 relative to the temporal reference frame = phys| C1 ⊗ C2 ⊗ | physkin C1. Conversely, we can of course switch the roles of C1 and C2 ψσ1,σ2 eC1 (τ1) eC2 (τ2) IS ψσ1,σ2 kin , (80) = | ⊗ ⊗ | above and ask for the clock reading of C1 relative to C2. Resorting H where ψσ1,σ2 σ1,σ2 lies in the overlap of the σ1- and to definition (B), this would yield | ∈ σi : 1 σ2-sectors (see Equation 67) and e (τi) τi, σi τi, σi , Ci = 2π | | i 1,2. We can then write the nth-moment of the conditional τ (τ ) dτ τ P (T τ T τ ). (83) = 1 2 ′ ′ σ1,σ2 2 2 1 ′ probability distribution in Equations (78) and (79) for n N, = R = | = ∈ Z thus considering both definitions (B) and (C), as follows: Dropping the labels of the arguments in Equations (81) and (83), (n) n both of which run over all of R, it is important to note that τ1(τ) τ2 (τ1) dτ ′ (τ ′) Pσ1,σ2 (T2 τ ′ T1 τ1) = R = | = and τ (τ) will generally not be the same functions of τ. This is Z 2 σ (n) σ because generally P (T τ T τ) P (T τ T ψ 1 (τ ) T I ψ 1 (τ ) (81) σ1,σ2 2 ′ 1 σ1,σ2 2 1 C2S C1 1 2,σ2 S C2S C1 1 = | = = = | = = | | ˆ ⊗ | | τ ′) in Equation (80). Said another way, the evolution of C2 from ψσ ,σ F (n) (τ1)) ψσ ,σ , the perspective of C according to definition (B) may differ from 1 2 ˆT IS,T1 1 2 1 = | 2 ⊗ | phys the evolution of C1 relative to C2 (for the same physical state). where by Equation (30) One might wonder whether the function τ1(τ2) in Equation (83) is the inversion of τ2(τ1) in Equation (81), (n) 1 n T2,σ dt t t, σ2 t, σ2 i.e., obtained by solving τ2(τ1) for τ1. Classically, this is certainly ˆ 2 = 2π R | | Z the case and it would entail that for a fixed clock reading τ1∗ θ( σ p ) T(n) θ( σ p ) of C one finds τ (τ (τ )) τ . Physically this would mean = − 2 ˆ2 ˆ 2 − 2 ˆ2 1 1 2 1∗ = 1∗

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that both temporal reference frames C1 and C2 agree that when from the perspective of C1, then according to Equation (69) the C1 reads τ1∗, C2 reads the value τ2(τ1∗). This does occur in a corresponding initial state of C1S from the perspective of C2 is special case when definitions (A)–(C) all coincide, namely when θ( σ p ) I ψσ2 (τ (τ )) Pσ ,σ (T2 τ2(τ ∗) T1 τ ′) δ(τ ′ τ ∗) in Equation (83) 1 1 S C1S C2 2 1∗ 1 2 = 1 | = = − 1 − ˆ ⊗ | | in which case expectation value, most probable value and the Rσ2 Rσ1 1 σ1 S (τ2(τ1∗
)) S (τ1∗) − ψC S C (τ1∗) . (84) value defined through the nth-moment all agree. While this = ◦ | 2 | 1 does happen in simple models with a high degree of symmetry We can then evaluate the ‘same’
reduced system observable between C and C [31], this will in more interesting cases not phys 1 2 I f in the two states, where i 1 when evaluated be the case because the physical state will generically have a Ci ⊗ ˆS = relative to C2 and vice versa (cf. Corollary 4), in order to different spread along the τ1 and τ2 axes [30, 32–34]. In our case phys this means that the wave function compare the evolution of property fˆS relative to the two clocks in different quantum states (which amount also to quantum σ1,σ2 : states of the clocks). To avoid confusion, we emphasize, that ψC S C (τ1, τ2) ( τ1, σ1 τ2, σ2 φS ) ψphys , 2 | 1 = |⊗ |⊗ | | phys ICi fˆS , i 1, 2, correspond to two different relational phys ⊗ = φ H Dirac observables FIC fS,T1 (τ1) and FIC fS,T2 (τ2) on the clock- for some physical system state S S , which can be viewed ˆ 2 ⊗ ˆ 1 ⊗ | ∈ neutral physical Hilbert space Hphys; in particular, the two are not as either a wave function in the C1 or C2 relational Schrödinger σ σ related by the TFC map 1→ 2 . Hence, by evaluating these two picture, may have a different spread in τ1 than in τ2.Insuchacase S reduced observables in the relational Schrödinger states related we will generally find τ1(τ2(τ ∗)) τ ∗. This effect will occur in 1 = 1 σ1 σ2 the class of models considered here because physical states need via the TFC map S → by Equation (84), we can compare two genuinely distinct relational dynamics. The construction in the not have the same momentum distribution in p1 and p2 (and thus relational Heisenberg picture is of course completely analogous. neither in τ1 or τ2) due to the presence of the system S. This effect has also been demonstrated in a semiclassical approach in In Höhn et al. [7] and Castro-Ruiz [47] a frame dependent various models in Bojowald et al. [32, 33] and Höhn et al. [34] temporal non-locality effect was exhibited for idealized clocks whose Hamiltonian is the unbounded momentum operator. For where one finds discrepancies of the order of h between τ1∗ and τ (τ τ (τ )). example, when clock C2 is seen to be in a superposition of 1 2∗ = 2 1∗ In conclusion, this effect can be interpreted as a temporal two peaked states and in a product relation with S from the frame dependence of comparing clock readings according to perspective of C1, then C1S will generally be entangled as seen definition (B) [or (C)]: if from the perspective of the temporal from the perspective of C2 and undergo a superposition of time evolutions. This effect applies here per overlap of the different reference frame defined by C1 the clock C2 reads τ2(τ1∗) σ1- and σ2-sectors. It will be interesting to study how such a (computed according to Equation 81) when C1 reads τ1∗, then conversely from the perspective of the temporal reference frame frame dependent temporal locality affects the (potentially frame dependent) comparison and synchronization of the clocks and defined by C2 the clock C1 will not in general read τ1∗ when C2 the comparison of the evolutions of S relative to C1 and C2 reads the value τ2(τ1∗). That is, C1 and C2 will generally disagree about the pairings of their clock readings. in different quantum states, corresponding to different choices Let us now also briefly comment on the notion of quantum of the clock-neutral physical states. Such an exploration will clock synchronization. Using the state dependent relation appear elsewhere. Finally, these temporal frame changes and clock Equation (82), we could ask for which state would yield τ2(τ1∗) = synchronizations will be relevant in quantum cosmology. τ1∗ so that C1 and C2 read the same value when C1 reads the value For example, recently it was pointed out that singularity τ1∗. Even stronger, we could ask whether there are states for which resolution in quantum cosmology depends on the choice of clock τ2(τ1) τ1 const, for all τ1 R, so that, up to a constant = + ∈ which one uses to define a relational dynamics [77]. The different offset, C1 and C2 are always synchronized. Equation (81) tells us that this is the case if Pσ σ (T τ T τ ) δ(τ τ relational dynamics employed in Gielen and Menéndez-Pidal 1, 2 2 = ′| 1 = 1 = ′ − 1 − const). Again, while this happens in simple models [31], this will [77] can be interpreted as different choices of reduced dynamics generically not happen for the models in the class which we are in the sense of our relational Schrödinger/Heisenberg picture. studying on account of the above observations concerning the Temporal frame changes as developed here can in principle frame dependence of comparing clock readings. Such a notion of be used to study the temporal frame dependence of the fate of synchronization is therefore too strong and can generally not be cosmological singularities more systematically. implemented. It will furthermore generally be frame dependent too. 8. CONCLUSIONS

7.4.4. Comparing a System’s Evolution Relative to In this work we demonstrated the equivalence of three Two Clocks distinct approaches to relational quantum dynamics—relational Returning to our original ambition, it is thus more useful to Dirac observables, the Page-Wootters formalism, and quantum employ the more general (frame dependent) clock comparison, deparametrizations—for models described by a Hamiltonian according to definition (B), in order to compare the evolutions constraint in which the momentum of the system being employed of S with respect to C1 and C2. Working in the relational as a clock appears quadratically. Since this class of models σ1 Schrödinger picture, if ψC S C (τ1∗) is the initial state of C2S encompasses many relativistic settings, we have thereby extended | 2 | 1

Frontiers in Physics | www.frontiersin.org 25 June 2021 | Volume 9 | Article 587083 Höhn et al. Relational Dynamics in Relativistic Settings our previous results of Höhn et al. [7] into a relativistic context. In conjunction with our previous article [7], we have A crucial ingredient in this extension has been a clock POVM thus resolved all three criticisms (a)–(c) (see Introduction) which is covariant with respect to the group generated by the that Kuchaˇr raised against the Page-Wootters formalism in Hamiltonian constraint and is used to describe the temporal Kuchaˇr [1]. The Page-Wootters formalism is therefore a reference frame defined by the clock. This choice differs from the viable approach to relational quantum dynamics. Through the usual resort to self-adjoint clock operators in relativistic settings. equivalence established by the trinity, it also equips the relational Owing to a superselection rule induced by the shape observable formulation and deparametrizations with a consistent of the Hamiltonian constraint across positive and negative conditional probability interpretation. In particular, relational frequency modes, this equivalence, which we refer to as the observables describing the evolution of a position operator trinity of relational quantum dynamics, holds frequency sector relative to a covariant clock POVM yield a Newton-Wigner type wise. Moreover, we further develop the method of temporal localization in relativistic settings. quantum frame changes [7, 30–34, 47, 69] in this setting to address the multiple choice problem. This method is DATA AVAILABILITY STATEMENT then used to explore an indirect self-reference phenomenon that arises when transforming between clock perspectives The original contributions presented in the study are included and to reveal the temporal frame and state dependence of in the article/Supplementary Material, further inquiries can be comparing or even synchronizing the readings of different directed to the corresponding author/s. quantum clocks. This result adds to the growing list of quantum reference frame dependent physical properties, such AUTHOR CONTRIBUTIONS as entanglement [70, 72, 74], spin [73], classicality [72] or objectivity [79, 80] of a subsystem, superpositions [70, 72], All authors have contributed extensively to the research and certain quantum resources [78], measurements [70, 76], causal writing of the article with the lead input from PH. relations [47, 83], temporal locality [7, 47], and even spacetime singularity resolution [77]. The temporal frame changes may FUNDING also be employed to extend recent proposals for studying time dilation effects of quantum clocks [45, 136, 137] (see also PH was grateful for support from the Simons Foundation [137–140]). Furthermore, it would be interesting to expand through an It-from-Qubit Fellowship and the Foundational the temporal frame changes to cosmological perturbation Questions Institute through Grant number FQXi-RFP-1801A. theory to study the temporal frame dependence of power AS acknowledges support from the Natural Sciences and spectra [64, 65]. Engineering Research Council of Canada and the Dartmouth Importantly, the covariant clock POVM permitted us to Society of Fellows. ML acknowledges support from the ESQ resolve Kuchaˇr’s criticism that the Page-Wootters formalism Discovery Grant of the Austrian Academy of Sciences (ÖAW), does not produce the correct localization probability for a as well as from the Austrian Science Fund (FWF) through the relativistic particle in Minkowski space [1]. Indeed, such START project Y879-N27. This work was supported in part incorrect localization probabilities arise when conditioning on by funding from Okinawa Institute of Science and Technology times defined by the quantization of an inertial Minkwoski Graduate University. The initial stages of this project were time coordinate. We showed that conditioning instead on the made possible through the support of a grant from the John covariant clock POVM surprisingly produces a Newton-Wigner Templeton Foundation. type localization probability, which, while approximate and not fully covariant, is usually regarded as the best possible SUPPLEMENTARY MATERIAL notion of localization in relativistic quantum mechanics [90, 94]. This result underscores the benefits of covariant clock The Supplementary Material for this article can be found POVMs in defining a consistent relational quantum dynamics online at: https://www.frontiersin.org/articles/10.3389/fphy. [7, 44, 45, 96, 97]. 2021.587083/full#supplementary-material

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