The Yamagishi Time Operator and Clock Subsystem Theory ── POVM Circumvention of Pauli's Theorem and the Emergence of Time in the Static Wheeler--DeWitt Universe ── [Paper 54]
For the "non-existence of a time operator T" in quantum mechanics (Pauli's theorem) and the "disappearance of time" in quantum gravity (the Wheeler--DeWitt equation HΨ=0), we give a unified interpretation from the standpoint of computer architecture. First, we state Pauli's theorem in its precise form with premises made explicit --- when the spectrum of the Hamiltonian is bounded from below, no self-adjoint operator T satisfying the canonical relation [ H, T]=iħ exists --- and sketch its proof (Theorem 2.1). Second, we make explicit that this theorem does not deny the existence of time observables: time observables such as time of arrival genuinely exist as POVMs (positive operator valued measures) rather than self-adjoint operators (Section 3). In the Universe OS picture this reads naturally as time being readable but not writable --- a Read-Only attribute. Third, we cite the Page--Wootters mechanism (1983), showing that a rigorous mechanism for the internal emergence of time from a static global state is already established. The "static database plus execution loop" picture of this paper is positioned as a computational restatement of that mechanism. The first edition's formulation in terms of an "external rendering loop" is withdrawn, since it merely displaces the problem of time outside the system rather than resolving it, and is replaced by a formulation in terms of an internal clock subsystem (Correction record 5.1). Further corrections in the second edition: (i) the first edition claimed that the “Read-Only attribute” reading is consistent with the concept of Paper 29, but the second edition of Paper 29 withdraws that Read-Only attribute claim itself; the reference has been removed (Section 3.2). The reading of that section nevertheless rests on the distinction between POVMs and self-adjoint operators and does not depend on Paper 29. (ii) The claim that the correspondence table “guarantees” the picture is not mere metaphor has been weakened --- the existence of a correspondence does not imply explanatory power (Section 4.2). The contribution of this paper is confined to organising established content (Pauli's theorem, POVMs, the Page--Wootters mechanism) in the vocabulary of computer architecture. It claims no new physics. A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 量子力学における「時間演算子 T の非存在」(パウリの定理)および量子重力の「時間の消失」(ホイーラー=ドウィット方程式 HΨ=0)に対し、計算機アーキテクチャの観点から統一的解釈を与える。第一に、パウリの定理をその前提を明示した正確な形——ハミルトニアンのスペクトルが下に有界であるとき、正準共役関係 [ H, T]=iħ を満たす自己共役演算子 T は存在しない——で述べ、証明の骨子を示す(定理2.1)。第二に、この定理が時間観測量の存在を否定するものではないことを明示する。到着時刻等の時間観測量は自己共役演算子ではなくPOVM(正作用素値測度)として実在する(第3節)。これは宇宙OSの描像において「時間は読み出し可能だが書き込み不可」——すなわちRead-Only属性——として自然に読める。第三に、Page--Wootters機構(1983)を引用し、静的な大域状態から時間が内部的に創発する厳密な機構が既に確立していることを示す。本稿の「静的データベース+実行ループ」という描像は、この機構の計算機的言い換えとして位置づけられる。初版が採用していた「外部レンダリングループ」という定式化は、時間の問題を系外へ押し出すのみで解決にならないため撤回し、内部クロック部分系による定式化へ置き換える(訂正記録5.1)。 第2版で追加した訂正:(i) 初版は「Read-Only属性」の読みについて論文29の概念と整合すると述べていたが、論文29第2版は当該の Read-Only 属性の主張そのものを撤回している。参照を削除した(第3.2節)。ただし本節の読みはPOVMと自己共役演算子の区別に立脚しており、論文29に依存しない。 (ii) 対応表について「単なる比喩でないことを保証する」としていた表現を弱めた——対応がつくことは説明力を意味しない(第4.2節)。 本稿の寄与は、確立した内容(パウリの定理・POVM・Page--Wootters機構)を計算機アーキテクチャの語彙で整理した点に限られる。新たな物理を主張しない。 本稿の位置づけ。本稿は確立した物理法則に対する構造的解釈であり、新たな物理法則を提案するものではありません。示される関係はいずれも既知の式への代入、またはその幾何学的な読み替えであり、元の式は変わりません。目的はささやかで、繰り返し現れる幾何構造——とくに立体角 4π と指数 n = 2——を一つの描像の中で見やすくすることにあります。特定の単位系(例:G = 4π)を採る箇所は記法上の約束であって、自然についての主張ではありません。既存の結果と矛盾する意図はありません。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。
For the "non-existence of a time operator T" in quantum mechanics (Pauli's theorem) and the "disappearance of time" in quantum gravity (the Wheeler--DeWitt equation HΨ=0), we give a unified interpretation from the standpoint of computer architecture. First, we state Pauli's theorem in its precise form with premises made explicit --- when the spectrum of the Hamiltonian is bounded from below, no self-adjoint operator T satisfying the canonical relation [ H, T]=iħ exists --- and sketch its proof (Theorem 2.1). Second, we make explicit that this theorem does not deny the existence of time observables: time observables such as time of arrival genuinely exist as POVMs (positive operator valued measures) rather than self-adjoint operators (Section 3). In the Universe OS picture this reads naturally as time being readable but not writable --- a Read-Only attribute. Third, we cite the Page--Wootters mechanism (1983), showing that a rigorous mechanism for the internal emergence of time from a static global state is already established. The "static database plus execution loop" picture of this paper is positioned as a computational restatement of that mechanism. The first edition's formulation in terms of an "external rendering loop" is withdrawn, since it merely displaces the problem of time outside the system rather than resolving it, and is replaced by a formulation in terms of an internal clock subsystem (Correction record 5.1). Further corrections in the second edition: (i) the first edition claimed that the “Read-Only attribute” reading is consistent with the concept of Paper 29, but the second edition of Paper 29 withdraws that Read-Only attribute claim itself; the reference has been removed (Section 3.2). The reading of that section nevertheless rests on the distinction between POVMs and self-adjoint operators and does not depend on Paper 29. (ii) The claim that the correspondence table “guarantees” the picture is not mere metaphor has been weakened --- the existence of a correspondence does not imply explanatory power (Section 4.2). The contribution of this paper is confined to organising established content (Pauli's theorem, POVMs, the Page--Wootters mechanism) in the vocabulary of computer architecture. It claims no new physics. A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 量子力学における「時間演算子 T の非存在」(パウリの定理)および量子重力の「時間の消失」(ホイーラー=ドウィット方程式 HΨ=0)に対し、計算機アーキテクチャの観点から統一的解釈を与える。第一に、パウリの定理をその前提を明示した正確な形——ハミルトニアンのスペクトルが下に有界であるとき、正準共役関係 [ H, T]=iħ を満たす自己共役演算子 T は存在しない——で述べ、証明の骨子を示す(定理2.1)。第二に、この定理が時間観測量の存在を否定するものではないことを明示する。到着時刻等の時間観測量は自己共役演算子ではなくPOVM(正作用素値測度)として実在する(第3節)。これは宇宙OSの描像において「時間は読み出し可能だが書き込み不可」——すなわちRead-Only属性——として自然に読める。第三に、Page--Wootters機構(1983)を引用し、静的な大域状態から時間が内部的に創発する厳密な機構が既に確立していることを示す。本稿の「静的データベース+実行ループ」という描像は、この機構の計算機的言い換えとして位置づけられる。初版が採用していた「外部レンダリングループ」という定式化は、時間の問題を系外へ押し出すのみで解決にならないため撤回し、内部クロック部分系による定式化へ置き換える(訂正記録5.1)。 第2版で追加した訂正:(i) 初版は「Read-Only属性」の読みについて論文29の概念と整合すると述べていたが、論文29第2版は当該の Read-Only 属性の主張そのものを撤回している。参照を削除した(第3.2節)。ただし本節の読みはPOVMと自己共役演算子の区別に立脚しており、論文29に依存しない。 (ii) 対応表について「単なる比喩でないことを保証する」としていた表現を弱めた——対応がつくことは説明力を意味しない(第4.2節)。 本稿の寄与は、確立した内容(パウリの定理・POVM・Page--Wootters機構)を計算機アーキテクチャの語彙で整理した点に限られる。新たな物理を主張しない。 本稿の位置づけ。本稿は確立した物理法則に対する構造的解釈であり、新たな物理法則を提案するものではありません。示される関係はいずれも既知の式への代入、またはその幾何学的な読み替えであり、元の式は変わりません。目的はささやかで、繰り返し現れる幾何構造——とくに立体角 4π と指数 n = 2——を一つの描像の中で見やすくすることにあります。特定の単位系(例:G = 4π)を採る箇所は記法上の約束であって、自然についての主張ではありません。既存の結果と矛盾する意図はありません。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。
I. Introduction The problem of time in quantum gravity is the following. General relativity is a diffeomorphisminvariant theory: there is no preferred time variable, and the Hamiltonian constraint HΨ = 0 impliesthat all physical states are ‘frozen’ in the sense that they carry no manifest time dependence [Dirac1964; DeWitt 1967]. Yet quantum mechanics requires time as a background parameter againstwhich evolution is measured. Reconciling these two facts — the timelessness of diffeomorphisminvariant quantum gravity and the temporal structure required for quantum measurement theory—is the problem of time [Isham 1993; Kuchăř 1992]. The problem is not merely philosophical: ithas concrete consequences for the interpretation of quantum cosmological perturbations, includingthose that seed the CMB anisotropies we observe today.One of the most elegant proposed resolutions is the Page–Wootters (PW) mechanism [Page & Wootters 1983]: time is not a background parameter but a relational quantity, encoded in correlationsbetween a ‘clock’ subsystem and the rest of the universe. Physical time is the reading of the clock,and quantum evolution is the conditional state of the system given a particular clock reading. Thisidea has been placed on rigorous mathematical footing by Höhn, Smith & Lock (HSL) [2021], whoproved the equivalence of three apparently distinct quantization procedures — the Dirac constraintquantization, the Page–Wootters mechanism, and the deparametrized Heisenberg picture — undera precise set of conditions. The HSL result reduces the problem of time, for any system satisfyingthose conditions, to the problem of identifying a clock with the right algebraic properties.In Loop Quantum Cosmology (LQC), relational time has been implemented using material degreesof freedom: a minimally-coupled massless scalar field [Ashtekar, Pawlowski & Singh 2006], a GFTcondensate [Gielen & Sindoni 2016], or dust fields [Brown & Kuchăř 1995; Pawłowski 2014]. Allof these are material clocks: dynamical degrees of freedom that must be added to, or identified1within, the matter content of the model. No worked example exists in which the clock arises fromthe algebraic structure of the kinematic Hilbert space itself — as an invariant of the theory’s gaugegroup rather than an additional field. Such an algebraic clock would be conceptually preferable: itrequires no extension of the Hilbert space, its back-reaction on geometry is controlled, and it is notsubject to the multiple-choice ambiguity [Kuchăř 1992] that afflicts material clock choices when thesystem has multiple degrees of freedom.This paper provides the first such example. The LQC bounce, treated via the Pöschl–Teller (PT)effective potential established in the companion series [Hillard 2026a–d], admits a spectral threshold = √2 that is fixed by the zero of the SU(2) Casimir polynomial W() = ³(²−2)/[C·sinh(/2)],where C = 3.60329… is the Hillard constant [Hillard 2026a, Paper 8]. This zero is a representationtheoretic invariant of the kinematic Hilbert space: it exists prior to any dynamical input, beforeany clock is chosen. We show that instantiates the full HSL relational clock construction. TheClock Decomposition Theorem (Section III) establishes the tensor product factorization H̃_kin H H and constructs the covariant POVM on H. The Temporal Trinity Formula (Section IV)is identified as the HSL clock reading in all three quantization pictures, with explicit clock-changemap T (Appendix A). The result holds exactly: is not a perturbative approximation to a clockbut a genuine algebraic invariant.A relational clock that is an algebraic invariant makes a concrete, falsifiable prediction. The signchange multipole * — the CMB multipole at which the LQC bounce imprint changes sign in thepower spectrum [Hillard 2026a] — is precisely the PW temporal origin in Fourier space: * =/_B. When GFT interactions of strength are included, the PT potential is deformed to U_ =−(2−)sech²(), which remains within the exactly solvable PT family (Section VI). The thresholdshifts to () = √(2−), giving the exact non-perturbative prediction:()/ = √(1 − /2) − 1 −/4 + O(²).* (1)*The sign is negative: GFT interactions push * to lower multipoles. The detection threshold forLiteBIRD is _crit 4/*, placing the prediction within reach of the near-term CMB survey (SectionV). This is the first quantitative prediction from a relational clock interpretation of the LQC bounce.A word on the relationship between this paper and the companion series [Hillard 2026a–d]. ThePWinterpretation of the bounce established here depends on W() only through its functional form:W() ³(²−2). Any function with that polynomial zero structure has a scattering threshold at =√2, and any such function in the Hamiltonian constraint generates the relational clock constructionof Sections III–IV. The validity of the LQG derivation of W in [Hillard 2026a] is therefore not aprerequisite for the results of this paper. The companion series and the present paper are mutuallyreinforcing rather than circularly dependent: [Hillard 2026a] provides the physical motivation andprecise form of W; this paper provides the relational interpretation that explains why has thevalue it has.The paper is organised as follows. Section II reviews the relevant background on LQC, the PTeffective potential, and the HSL formalism. Section III establishes the Clock Decomposition Theorem: spectral factorization (Proposition 1), covariant POVM, and temporal origin. Section IVidentifies the Temporal Trinity Formula in all three HSL pictures (Lemma 1, Proposition 2). Section V derives the physical inner product by group averaging and the full observational prediction(Table 2). Section VI proves that the intertwiner deformation is shape-invariant and promotes theprediction to an exact formula (Proposition 3). Appendix A constructs the clock-change map Texplicitly (Proposition A.1, Corollary A.1).22Main results at a glanceClock Decomposition Theorem (Section III): H̃_kin H H + O(). POVM E(d) =|W()|²/‖W‖² d is covariant. = √2 is the temporal origin.Trinity Formula (Section IV): N_total = N + ln(··k_piv/) is a Dirac observable, clockchange invariant. T acts as identity on (Appendix A).Exact prediction (Sections V–VI): ()/ = √(1−/2) − 1 −/4. LiteBIRD detection threshold:_crit 4/*.First algebraic PW clock in LQC (Section V.4): is an SU(2) Casimir invariant, not amaterial d.o.f. Back-reaction is O(); no Hilbert space extension needed.
Part I — The Pauli Time-Operator Constraint: A Theorem About Regular Weyl Pairs, Not a Prohibition on Time Observables[PDF pages 1–33; internal pages i–28] Establishes that the standard Pauli prohibition rests on an unstated assumption — that the canonical commutation relation on a dense domain implies the Weyl relation as an equality of unitary operators on the full Hilbert space. This implication is shown to fail via Nelson’s analytic-vector theorem and the Galapon counterexample. Fifty explicit constructions of time observables for bounded-below Hamiltonians, spanning six categories, demonstrate that the Pauli constraint governs only regular Weyl pairs and imposes no restriction on time observables generally. Part II — Time as the Governing Operator in 3D+1 Spacetime[PDF pages 34–91; internal pages i–52] Part II opens with a Preface and About the Author section, reflecting continuity of argument from Part I. The mathematical liberation established in Part I — that time observables are not prohibited for bounded-below Hamiltonians — serves as the foundation for a complete physical framework. Time is promoted from a passive coordinate to an active operator possessing an elastic modulus (Et = c²), a dynamical equation derived from the geodesic equation, and an emergent origin recovered through the WKB approximation of the Wheeler–DeWitt equation. A modified Schrödinger equation incorporating an entropy-coupled decoherence kernel provides a continuous mechanism for wavefunction collapse without an external measurement postulate. An angular-bias projection formalism demonstrates how observers confined to 3D+1 spacetime perceive density distributions through a time-evolving angular window, yielding apparent dark matter profiles with NFW-like asymptotics — steep inner cusp transitioning to r⁻³ outer falloff — without invoking new particle species. Dark energy density is derived as a time-integrated kinetic accumulation, bounded at late times by the decoherence regulator. The Einstein field equations are extended with an entropy-time modulation weight Ξ, producing a closed system of field equations (B1–B7) that recovers standard general relativity when Ξ → 1, standard quantum mechanics when λ → 0, and ΛCDM cosmology when evaluated at the present epoch. Each core claim is supported by no fewer than five independent derivations with all variables defined at first use. Falsification criteria are provided, including SPARC rotation-curve consistency, altitude-dependent quantum decoherence time scaling, time-dependent dark energy equation of state, lensing surface density stability, and angular-bias dependence in quantum interference experiments. AuthorLance Thomas Davidson ORCID0009-0006-1245-1644 Year2026 Rights and LicensingCopyright 2026 Lance Thomas Davidson. All rights reserved.Subordinate license: Creative Commons Attribution–NonCommercial–NoDerivatives 4.0 International.12-month grace period declared for United States patent registration and filing intent.
Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially un-distinguishables.
Abstract Pauli's theorem asserts that no self-adjoint time operator can exist that is canonically conjugate to a Hamiltonian with a spectrum bounded from below, preventing time and energy from both being operators in standard quantum mechanics. In this paper, we propose a new reformulation that circumvents this constraint by inverting the roles of time and energy: time becomes a self-adjoint operator, while energy is treated as an external parameter. This dual perspective yields a mathematically equivalent yet physically distinct formulation, where quantum systems evolve according to internal clocks. The model accounts for the measurement problem without invoking collapse, offering an observer-relative interpretation tied to temporal misalignment. The proposed model also suggests a fresh perspective on the relationship between quantum systems and spacetime geometry.
This article generalizes the conditional probability interpretation of time in which time evolution is realized through entanglement between a clock and a system of interest. This formalism is based upon conditioning a solution to the Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock, being in a state corresponding to a time t . Doing so assigns a conditional state to the rest of the Universe | ψ S ( t ) ⟩ , referred to as the system. We demonstrate that when the total Hamiltonian appearing in the Wheeler-DeWitt equation contains an interaction term coupling the clock and system, the conditional state | ψ S ( t ) ⟩ satisfies a time-nonlocal Schrödinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator. This time-nonlocal Schrödinger equation is solved perturbatively and three examples of clock-system interactions are examined. One example considered supposes that the clock and system interact via Newtonian gravity, which leads to the system's Hamiltonian developing corrections on the order of G / c 4 and inversely proportional to the distance between the clock and system.
Canonical quantization applied to closed systems leads to static equations, the Wheeler-deWitt equation in Quantum Gravity and the time independent Schrödinger equation in Quantum Mechanics. How to restore time is the Problem of Time(s). Integrating developments are: a) entanglement of a microscopic system with its classical environment accords it a time evolution description, the time dependent Schrödinger equation, where t is the laboratory time measured by clocks; b) canonical quantization of Special Relativity yields both the Dirac Hamiltonian and a self adjoint "time" operator, restoring to position and time the equivalent footing accorded to energy and momentum in Relativistic Quantum Mechanics. It introduces an intrinsic time property τ associated with the mass of the system, and a basis additional to the usual configuration, momentum and energy basis. As a generator of momentum displacements and consequently of energy, it invalidates Pauli's objection to the existence of a time operator. It furthermore complies with the requirements to condition the other observables in the conditional interpretation of QG. As Pauli's objection figures explicit or implicitly in most current developments of QM and QG, its invalidation opens to research the effect of this new two times perspective on such developments.