Emergent Time and Time Travel in Quantum Physics

As fascinating as the notion of time travel is, its long list of problematic issues-ranging from the classical [1,2] & [3, p.212ff] to the quantum theoretical [4][5][6][7]-presents a very solid case against it, and its absence is effectively a feature of the universe accessible to human experience.Yet if one wants to look closely at these arguments, cracks and loopholes start to appear-many of which shall be mentioned in the following.To an extent, it almost seems as if the best evidence against any notion of time travel is that of our daily experienced notion of time and causality, our intuition.Still, the known universe is bigger than our day-to-day experience, and particularly the field of high energy physics is ripe with examples experimentally, observationally, or theoretically open to inquiry-in which various concepts or preconceived notions of human-scale physics have to be let go.In no field of theoretical physics is this more apparent than in the quest for quantum gravity.The field of contenders is large, and still growing: String theory, loop quantum gravity, quantum geometrodynamics, asymptotic safety, causal dynamical triangulation, causal set theory, Hořava-Lifshitz gravity, . . .Each comes with its own set of new notions beyond what can be experienced directly by human senses.Yet, these proposals still have to address, in one way or another, an issue plaguing quantum gravity from its beginning: Time.This problem of time, and its eventual resolution, then naturally will also impact what these theories have to say about (or against) time travel. The 'problem of time' in classical canonical gravity is settled by a careful, gaugetheoretic analysis [8] that distinguishes the different rôles played by the Hamiltonian, either in the constraint/gauge fixing or in the definition of observables, respectively.In quantum gravity, however, the problem of time remains (at the very least) a topic of active discussion and research [9][10][11].The reason for this goes back even before a theory of quantum gravity was the focus: While there are ways to give position and momentum a meaning in terms of operators on (rigged) Hilbert spaces in quantum mechanics, this cannot be carried over to time and energy.This was concerning to researchers in quantum mechanics early on, as the time-energy uncertainty relations proved to be both reliable and instructive, while not fitting into the standard paradigm of the Robertson-Schrödinger uncertainty relations for (generic) Hermitian operators.Only the proof of Mandelstam and Tamm in 1945 improved this uncertainty relation's foundation, while begging the question of why a different approach was needed [12].Essentially, a series of impossibility results for time operators canonically conjugate to a Hamiltonian was presented: Schrödinger used an approach based on normalizability [13], similar to the argument for why plane waves require a rigged Hilbert space.Pauli, in a footnote, used a simpler argument juxtaposing discrete spectra and the continuous translation in energy such a time operator would induce [14,15].Finally (and in reply to what will be discussed shortly), Unruh and Wald gave a general argument for semi-bounded Hamiltonians linked to unitarity [16].Two at first glance quite distinct approaches were developed in reply to these no-go results. One approach was similar in spirit to the answer to early counterarguments to plane waves, e.g., eigenstates of definite position and momentum.While with position and momentum one relaxes the idea of being restricted to Hilbert space and instead works in a rigged Hilbert space [17][18][19], here, when faced with the above-cited no-go results concerning time operators, one relaxes the idea of Hermiticity.Instead, one introduces the notion of a positive, operator-valued measure (POVM).It can represent not just perfect measurements yielding a definite eigenstate, but also imprecise measurements [20].Besides allowing one to address the measurement problem in quantum field theory [21], phase observables [22], or the already mentioned, imprecise measurement processes [20,23]-and more important for the present context-this allows a notion of time measurement [24].Since the no-go theorems relate to Hermitian operators, the more general POVMs are not ruled out as time observables.As we will see, the context of time travel allows one to avoid these no-go results by other means, too.Still, the bigger picture of gauge theory provides the most pertinent background for extensions of the model to be described in this paper. Another approach that developed independently from POVMs was to carefully distinguish between a clock and time.Paraphrasing Einstein [25], time is meaningful only after specifying the clock measuring it.Early on, in developing canonical quantum gravity, DeWitt [26] pointed out how different subsystem's operators in a given Hamiltonian constraint could serve this purpose of an effective clock.Page and Wootters then took this idea and made it more general and workable, by phrasing time evolution in terms of conditional probabilities linking different subsystems [27,28]. 1 This Page-Wootters (PW) formalism was soon challenged on various grounds by Unruh, Wald [16] and Kuchař [29], leading to this approach lying dormant for some time. Recent developments then made the latter approach re-emerge [30][31][32][33][34][35], and both approaches converge [36,37].This also allowed one to address and re-contextualize the earlier criticisms of the PW formalism [37,38].In essence, the clocks chosen within the PW formalism become part of a gauge-theoretic picture of clocks; the 'physical Hilbert space' being a 'clock-neutral' picture, and different clocks representing different 'gauge conditions' [37].These results form part of a larger research programme that aims to ground the rulers and clocks at least implicitly employed in external symmetries with an operational underpinning [39][40][41][42][43][44].For our purposes below, the key takeaway is that the PW formalism still works, if its defects and the criticisms of it are understood as related to issues already familiar from electromagnetism: One has to choose whether a calculation should be gauge-fixed or gauge-independent.Depending on the question at hand, one or the other will be better suited to calculations or understanding.Our model below will be too simple to fully showcase these developments, but given the criticisms the PW formalism faced, it is important to keep the resolution in mind. Within this special issue's scope, many other pertinent links of time and time travel to quantum physics could be made.A selected, non-exhaustive list would include: Deutsch CTCs and (non/retro-)causal quantum processes [45][46][47][48][49], simulation of time (travel) in analogues [50][51][52], semi-classical/quantum stability [4][5][6][7]53], et cetera.For the purposes of this article, however, we would like to highlight only two further research avenues.One is that of different notions of time.Yet another, non-exhaustive list would include at least cosmological time, psychological time, parameter time, thermodynamic time, dynamical versus kinematical time, . . .While this distinction is studied mostly in the context of 'the arrow of time' [54], it is worth pointing out that many counterarguments to time travel have to rely at least on some conflation of these concepts.To give just one example, if thermodynamic arguments are brought forth, one needs to establish a more or less direct relation between the thermodynamic notion implied by the second law2 and the notion of time invoked in time travel.(This itself does not have to be the same notion as that of general relativity (GR), where time travel can be identified with closed, time-like curves (CTCs) or related concepts [56].In theories other than GR, a different notion of time might be relevant to describe time travel!) The other important point to make is that despite all the counterarguments, time travel is just one notion of many with questionable 'physicality' [57].Even within the context of GR, other notions of physicality vie for validity with the absence of time travel, for example, various kinds of completeness (geodesic, hole free, . . . ) or the validity of various kinds of energy conditions.These notions, however, are not all compatible with each other [58][59][60][61][62][63].One might find oneself in the uncomfortable situation that a dearly held and important property (stability, completeness, fulfilled energy conditions, . . . ) will require time travel to be permitted; and this just within GR.Other theories are unlikely to be free from such problems, especially as many of the comforting no-go theorems are specifically proven within the context of GR.It it therefore not surprising that in the context of classical field theories of gravity alone, there exist many different ways to realize time travel.This includes wormholes [3,64], warp drives [65], cosmologies [66][67][68][69], maximally extended space-times (such as Kerr near the central ring singularity) [70], and various more mathematical construction techniques [71][72][73].The latter can also be employed to distinguish between, for example, time travel (CTCs) and time machines (whatever structure creates or causes CTCs) [59,60,73]. This finally brings us to the goal of this paper: We want to study what can be said about the viability of time travel if time itself is only an emergent concept, as in the PW formalism.Concretely, we will be studying an example of two non-interacting harmonic oscillators similarly constrained to a fixed total energy of zero as in a Wheeler-DeWitt (WDW) equation, i.e., it will mimic a minisuperspace model of time travel.This toy model is by no means meant to exhibit an exhaustive description of whether and how time travel can arise in the PW formalism.Nor should this model be taken literally as a minisuperspace model, as no gravitational model is canonically quantized to arrive at it; rather it demonstrates possible phenomenology.We will see that the results in our toy model point to an extraordinarily bland version of Novikov's self-consistency conjecture [74][75][76].

The paper situates its work within: (i) the problem of time in quantum gravity and the Page–Wootters (PW) formalism, along with criticisms and modern gauge-theoretic reinterpretations [16,29–38]; (ii) approaches to time observables via POVMs and phase/time operators for harmonic oscillators [20–24,82–84]; (iii) broader contexts of time travel in physics, including Deutsch CTCs and non/retro-causal processes, analogue simulations, and semiclassical stability [4–7,45–52]; and (iv) GR-based mechanisms (wormholes, warp drives, cosmologies, chronology protection) and notions of physicality/energy conditions [3,56–73]. It highlights the distinction among different notions of time (cosmological, thermodynamic, parameter, psychological) and warns that many no-go arguments mix these notions. The work differentiates itself from quantum cosmology minisuperspace models by not restricting configuration space to a half-plane and by using ladder-operator methods throughout.

- Framework: Employ the Page–Wootters (PW) formalism to model emergent time by partitioning the total Hilbert space H = HC ⊗ HR with a stationary global state |Ψ⟩ satisfying a Hamiltonian constraint of Wheeler–DeWitt type. - Clock and time observable: Implement periodic time via a phase POVM for the harmonic oscillator. Construct time/phase states |θ⟩ as (overcomplete) eigenstates of the Susskind–Glogower operator W from the polar decomposition of the annihilation operator a = W|a|, leading to a normalized POVM B0(X) = (1/2π)∫X |θ⟩⟨θ| dθ on [0,2π). Time operators can be obtained from the POVM (Toeplitz operators), with covariant shifts Tθ* = e^{iĤ θ*} T0 e^{-iĤ θ*}. - Model: Two non-interacting harmonic oscillators serve as clock (C) and residual (R) subsystems with frequencies ωC and ωR. Impose a WDW-like Hamiltonian constraint Ĥ|Ψ⟩ = 0 with Ĥ = (ωC nC + ωC/2)1C ⊗ 1R − 1C ⊗ (ωR nR + ωR/2), effectively requiring total energy zero. - PW conditional dynamics: Define clock states |ψ(θ*)⟩C = e^{-iĤC θ*}|ψ(0*)⟩C and compute conditional expectations E(ĤR|θ*) = tr(ĤR Pθ* ρ)/tr(Pθ* ρ), where Pθ* = (|ψ(θ*)⟩⟨ψ(θ*)|) ⊗ 1R and ρ = |Ψ⟩⟨Ψ|. - State structure: Expand |Ψ⟩ = ∑_{n,n′} A_{n,n′} |n⟩C ⊗ |n′⟩R. Square-integrable solutions require a commensurability condition ωR/ωC = (2nC+1)/(2nR+1) for some nC,nR ∈ ℕ0. For diagonal A (n = n′) compute conditional expectations explicitly. For general A, show that A has the sparse factorization A = Δ1^T D Δ2 (selecting allowed (n,n′) pairs), and derive a general identity for the PW numerator/denominator with matrices Q = D1^† A^† D0 A D1, proving Q is diagonal, hence time-independence of conditional quantities.

- Existence/normalizability enforces a commensurability condition: square-integrable solutions to the constraint exist only if ωR/ωC = (2nC+1)/(2nR+1), nC,nR ∈ ℕ0 [Eq. (21)]. Deviations lead to non-normalizable states. - For diagonal universe states A_{n,n′} ∝ δ_{n,n′}, PW conditional expectations are time-independent: E(ĤR|θ*) = E(ĤR|0*). Thus, the residual system exhibits no evolution relative to the clock. - For general (non-diagonal) states, using the constraint-driven sparsity A = Δ1^T D Δ2 and the matrix identity Q = D1^† A^† D0 A D1, one proves Q is diagonal, making all e^{i(n−m)θ*} factors trivial (n=m). Therefore, both numerator and denominator in the PW conditional expression are θ*-independent: the evolution is again trivial. - The model hence realizes a ‘boring’ form of Novikov self-consistency: the system is self-consistent because nothing changes; periodic clock states do not induce nontrivial dynamics for the residual system under the given constraints. - Contrast to quantum cosmology: The result relies on having the full configuration space ℝ^2 and ladder-operator methods. Quantum cosmology often restricts one degree of freedom to a half-plane (e.g., scale factor a≥0), which alters the analysis and can lead to different, nontrivial dynamics. - Role of POVMs: While phase/time POVMs motivate the periodic time and provide a rigorous time observable, the final trivial-evolution result primarily follows from the constraint structure and commensurability, not from specific details of the time operator.

The study asks whether time travel is viable when time is emergent via the PW formalism. In the chosen toy model (two decoupled harmonic oscillators under a WDW-like constraint), emergent-time dynamics of the residual system relative to a periodic clock are found to be constant in time. This directly enforces a trivial form of Novikov self-consistency (each complete clock cycle returns the system to the same state)—a notion of ‘time travel’ without evolution or paradoxes. The work shows that conclusions about time travel in emergent-time frameworks are highly model-dependent: here, the Hamiltonian constraint and commensurability eliminate nontrivial dynamics, whereas modifications (interactions, additional subsystems, different state spaces or inner products) may yield different outcomes. The findings also highlight the subtle distinction between a periodic clock and actual time travel phenomena; mere periodicity does not guarantee nontrivial cycles. Compared to minisuperspace quantum cosmology, not restricting configuration domains and relying on ladder operators changes results substantially, underscoring sensitivity to modeling choices.

The paper presents a proof-of-concept study of time travel in a quantum system where time is emergent via the Page–Wootters formalism and implemented with phase/time POVMs. For two non-interacting harmonic oscillators subject to a WDW-like Hamiltonian constraint, only commensurate frequency ratios admit normalizable solutions, and the PW conditional dynamics are time-independent for both diagonal and non-diagonal global states, yielding a trivial, self-consistent ‘time travel’ scenario. The work demonstrates feasibility of using emergent-time tools to probe time-travel scenarios and clarifies that results are strongly model-dependent. Future directions include: adding more/different subsystems and ancillary/memory registers; introducing interactions; exploring entropic/thermodynamic considerations (e.g., observational entropy) for constrained systems; distinguishing periodic clocks from genuine time-travel behavior beyond binary self-consistency; analyzing quantum reference frame transformations and their impact on time-travel features; studying locality and ‘partial’ time travel within larger systems; and examining alternative inner products and other quantum gravity frameworks to test robustness of emergent-time conclusions.

- Toy model with two non-interacting harmonic oscillators; absence of interactions or ancillary systems likely suppresses nontrivial dynamics. - Strong reliance on the WDW-like constraint and a full ℝ^2 configuration space with ladder-operator methods; results can change drastically if one oscillator is restricted to a half-plane, as in quantum cosmology. - Existence of normalizable solutions hinges on an exact commensurability condition; slight deviations lead to non-normalizable states. - The emergent-time analysis uses stationary global states and specific PW conditional expectations; conclusions may not generalize to more sophisticated gauge-fixed/independent implementations with interactions. - The model is not derived from a concrete gravitational theory; it is theory-agnostic and may miss gravity-specific effects (e.g., singularity resolution).