Quantum Mechanics Cannot Produce Time: A Minimal Temporal-Field Requirement for a Structurally Complete Ontology

Quantum Mechanics Cannot Produce Time: A Minimal Temporal-Field Requirement for a Structurally Complete Ontology Matthew J. Hall* October 15, 2025 Abstract Claims that “everything is quantum” highlight the empirical reach of quantum mechanics (QM) from atoms to macroscopic circuits. Yet standard QM presupposes a continuous, ordered time parameter t while providing no mechanism that generates or explains it. We demonstrate, using textbook postulates and spectral constraints, that time cannot be produced internally by QM. Specifically: (i) the Schr ¨ odinger and Heisenberg pictures require an external ordering variable; (ii) Pauli-type obstructions forbid a selfadjoint time operator for semibounded Hamiltonians (spectra bounded from below); and (iii) relational reconstructions re-import temporal order via auxiliary clock systems. We then introduce the minimal fix: treat time as a weakly dynamical field τ(r, t) = t+ϕ(r, t) . This extension recovers all quantum predictions when ϕ→0 and yields testable deviations—quantised sidebands and tunnelling-rate shifts— arising from temporal modulation. For superconducting qubits under ∼1 kHz clock modulation with ϕ0∼10−4 , we estimate sidebands at 10−4 – 10−3 of the carrier, within current microwave-spectroscopy resolution. Consequently, quantum theory depends on time as a deeper substrate: without time, its equations cannot even be evaluated. Keywords: problem of time; quantum foundations; temporal field; quantum time; superconducting qubits; time crystals 1 Introduction Quantum mechanics has proven stunningly accurate across scales, explaining atomic spectra, superconducting qubits, and macroscopic quantum tunnelling. Such success inspires the slogan “everything is quantum.” Yet all these analyses rely on quantities like duration, frequency, and phase—concepts that presuppose a time parameter the theory itself cannot derive. Just as a map presupposes a territory it cannot draw itself, QM presupposes a time it cannot generate. Without an external clock, even a textbook qubit’s Rabi oscillations would lack a measurable frequency, rendering phrases like “ 5 GHz cycling” undefined (e.g., unmeasurable in a Ramsey experiment). This paper advances two points: 1. Non-derivability: Time is assumed, not derived, within standard QM. 2. Minimal sufficiency: A weakly dynamical temporal field restores structural completeness and yields falsifiable predictions. *Email: [email protected]. ORCID: 0009-0001-7066-2558. 1 2 Quantum dynamics presupposes external time 2.1 Schr¨ odinger and Heisenberg pictures State evolution follows iℏ∂ ∂t |ψ(t)⟩=ˆ H|ψ(t)⟩,(1) which only makes sense if texists as a smooth ordering parameter. In the Heisenberg picture, dˆ A dt =i ℏ[ˆ H, ˆ A] + ∂ˆ A ∂t explicit ,(2) again assumes d/dt is well defined. Laboratory observables—Rabi oscillations, coherence times—require a clock to measure frequency. Path integrals weight histories by ei ℏRL dt , undefined without dt . Measurement theory uses POVMs 1 whose relative frequencies are counted per unit time in repeated trials. Summary. QM describes change with respect to time; it does not describe how time itself comes to be. 3 Why time cannot be an internal quantum observable 3.1 Pauli-type obstruction Theorem 1 (Pauli-type).Let ˆ H be self-adjoint with spectrum bounded from below. Then no self-adjoint ˆ T satisfies [ˆ T, ˆ H]=iℏ I. Sketch. If such ˆ T existed, the Weyl relations imply e−iα ˆ T/ℏˆ H eiα ˆ T/ℏ=ˆ H+αI for all α∈R , shifting the spectrum arbitrarily and contradicting the lower bound. Remark 1.Time-of-arrival POVMs or maximally symmetric (non-self-adjoint) “time” operators define statistics given time, not time from subsystem dynamics; see e.g. [2]. 3.2 Relational and “timeless” reconstructions Page–Wootters and Wheeler–DeWitt approaches seek emergent time from correlations in a larger “clock + system” composite. However, relational models do not derive time internally for laboratory subsystems: • Lab systems are referenced to external clocks; internal clocks merely relocate the ordering, not remove it. • Universe-wide relational models require pre-ordered clock states for conditional probabilities, external to laboratory subsystems. Theorem 2 (Non-derivability of time for subsystem QM).Let (H,ˆ H) describe a laboratory subsystem with semibounded ˆ H (spectrum bounded from below). No construction within standard QM for that subsystem produces a continuous, ordered parameter functionally equivalent to physical time without importing additional (clock) degrees of freedom or external structure. Proof sketch. Pauli’s obstruction forbids a conjugate time operator; correlation-based clocks add external structure. Thus no operator or correlation within (H,ˆ H)generates the required ordered continuum. 1POVM = positive operator-valued measure (generalised quantum measurement). 2 Summary. Time enters every formulation as background structure, never as a product of the subsystem theory. 4 A minimal temporal-field extension 4.1 Ansatz, scope, and definition of ˆ H(τ) We posit a weakly dynamical temporal field τ(r, t) = t+ϕ(r, t),|ϕ|≪1,(3) reflecting that observed time is nearly uniform. Small ϕ ensures agreement with clock precision ( ∆t/t ≲ 10−18 ). We remain agnostic about the microphysics of ϕ (classical noise, quantum-gravitational fluctuations, or emergent), treating it phenomenologically. We replace ∂/∂t by ∂/∂τ in the dynamical law, iℏ∂ ∂τ |ψ(τ)⟩=ˆ H(τ)|ψ(τ)⟩2,(4) where ˆ H(τ) denotes the Hamiltonian with any explicit time dependence evaluated at τ . For time-independent ˆ H , we have ˆ H(τ)≡ˆ H ; for driven systems with ˆ H(t) = ˆ H0+Vcos(ωt) , the replacement t7→ τ shifts the explicit time dependence. By the chain rule, ∂ ∂τ =1 1+∂tϕ ∂ ∂t.(5) Combining Equations (4) and (5) yields iℏ1 1+∂tϕ ∂ ∂t |ψ(τ)⟩=ˆ H(τ)|ψ(τ)⟩,(6) reducing continuously to Equation (1) when ϕ→0. 4.2 Immediate consequences (Floquet picture) Floquet theory 3 is a natural lens here. If ϕ(t) = ϕ0cos(ωτt) , Floquet analysis yields quasi-energies En=E0+nℏωτ. A drive at frequency ωdevelops sidebands at ω±mωτwith amplitudes ∼Jm(ωτϕ0). Tunnelling rates. Instanton actions and escape exponents acquire a factor 1 + ∂tϕ , predicting small, phase-correlated deviations in measured lifetimes under controlled timing modulation. This connects with thermodynamic constraints on coherence and rates [8]. 4.3 Experimental illustration and estimates Superconducting qubit / Josephson device. Modulate the local reference (clock) at ωτ/2π∼1 kHz with depth ϕ0∼10−4 (chosen to stay below current clock precision yet inside high-sensitivity readout). Then the first sidebands scale like I±1 I0 ≈1 4(ωτϕ0)2∼10−4–10−3(0.01%–0.1%), detectable via high-resolution microwave spectroscopy as used in transmon devices [ 12 ]. Coherence-time shifts scale with ⟨∂tϕ⟩∼ωτϕ0(order 10−4), measurable by Ramsey/echo sequences. 2|ψ(τ)⟩denotes the state evolved with respect to the temporal field τ. 3Floquet theory is a framework for systems with periodic coefficients; quasi-energies label stroboscopic eigenstates. 3 Figure 1: Schematic spectrum: standard QM (single peak at ω ) versus temporal-field prediction (sidebands at ω±mωτ ). Sideband heights shown qualitatively following Jm(ωτϕ0) for ϕ0∼10−4 and ωτ/2π∼1 kHz . Summary. The temporal-field framework preserves unitarity, recovers standard QM as ϕ→0 , and predicts measurable, phase-linked deviations under controlled modulation. 5 Implications and broader context The slogan “everything is quantum” is valid only within time’s domain. QM describes dynamics inside time, not time itself. Treating time as a field: • clarifies connections to time-crystal research (Wilczek; Else et al.). Unlike time crystals (spontaneous temporal symmetry breaking), our ϕmodulates the external time parameter directly (akin to imposed clock jitter); • bridges to quantum-clock theory and limits [9, 10], informing feasible ϕ0, ωτregimes; • aligns with quantum-gravity programs where spacetime may emerge from correlations (e.g., group field theory) [ 11 ], suggesting ϕ could encode quantum-gravitational fluctuations testable via laboratory spectroscopy. Table 1: Standard QM vs. temporal-field predictions (qualitative). Estimates assume ϕ0∼10−4 , ωτ/2π∼ 1 kHz. Detection methods leverage standard circuit-QED techniques (e.g., Ramsey/echo) [12]. Observable Standard QM Temporal field (τ=t+ϕ) Experimental detection Spectral lines Single peak at ωSidebands at ω±mωτMicrowave spectroscopy Tunnelling rate Γ(fixed) Γ [1 + ∂tϕ](phase-sensitive) Time-resolved readout Coherence time T2T2×(1 + ϵ), ϵ ∼ωτϕ0∼10−4Ramsey/echo sequences 4 Summary. Recognising time as foundational re-orders the ontology: Temporal field →Quantum mechanics →Classical reality. 6 Conclusion Quantum mechanics, as verified by modern experiments, presupposes but cannot create time. Introducing a weak temporal field supplies the missing structural element, retaining empirical success while enabling new, falsifiable predictions (sidebands, tunnelling-rate shifts). Future clock-modulated spectroscopy in qubits or Josephson devices could test this framework, potentially revealing whether time’s structure bridges QM to quantum gravity and cosmology. Acknowledgements Thanks to colleagues in quantum foundations and condensed matter for discussions on Pauli’s theorem, time-crystal analogies, and clock-limited spectroscopy. References [1] W. Pauli, ¨ Uber das Verh ¨ altnis des Abschlusses der Elektronengruppen im Atom zum periodischen System, Z. Phys. 31, 765 (1925). [2] P. Busch, M. Grabowski, and P. Lahti, Operational Quantum Physics, 2nd ed., Springer (1997). [3] J. G. Muga, R. Sala Mayato, and I. Egusquiza (eds.), Time in Quantum Mechanics, Springer (2002, 2009). [4] D. N. Page and W. K. Wootters, Evolution without evolution: Dynamics described by stationary observables, Phys. Rev. D 27, 2885 (1983). [5] C. J. Isham, Canonical Quantum Gravity and the Problem of Time, in Integrable Systems, Quantum Groups, and Quantum Field Theories, Kluwer (1993). [6] F. Wilczek, Quantum Time Crystals, Phys. Rev. Lett. 109, 160401 (2012). [7] D. V. Else, B. Bauer, and C. Nayak, Floquet time crystals, Phys. Rev. Lett. 117, 090402 (2016). [8] S. Deffner and S. Campbell, Quantum Thermodynamics, Morgan & Claypool (2019). [9] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?, Nat. Commun. 8, 2269 (2017). [10] M. P. Woods, R. Silva, J. O. de Almeida, A. J. P. Garner, and M. Huber, Quantum Clocks and Their Synchronization, PRX Quantum 3, 023041 (2022). [11] D. Oriti, Emergent spacetime in quantum gravity, Front. Phys. 8, 294 (2020). [12] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A 76, 042319 (2007). 5 Appendix A: Chain rule and Floquet sketch Chain rule for τ=t+ϕ .For smooth ϕ(r, t) with |∂tϕ|≪1 , ∂τ= (∂t/∂τ)∂t=1 1+∂tϕ∂t. Substituting into ∂τψyields Equation (6). Floquet quasi-energies. For periodic coefficients with period Tτ= 2π/ωτ , solutions take the form ψ(t) = e−iεt/ℏu(t) with u(t+Tτ) = u(t) . Quasi-energies ε are defined modulo ℏωτ , giving observable sidebands at integer multiples of ωτ. 6