DRSN VIII: Quantum Drift Geometry (De Rerum Spectrale Natura, Report VIII, Version 2.0)

DRSN VIII: QUANTUM DRIFT GEOMETRY De Rerum Spectrale Natura series REPORT VIII (Version 2.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 • Quantisation is formulated directly at the spectral level, promoting Dirac operators rather than background fields to quantum variables. • The drift parameter acts as a quantum deformation parameter, controlling fluctuations of spectral geometry. •One–loop and higher quantum corrections arise from spectral determinants and heat–kernel expansions. •Renormalisation is interpreted as spectral flow, yielding operatorial renormalisation group equations. • The resulting framework provides a background–independent formulation of quantum geometry and quantum gravity. Quantum Drift Geometry: Drift Quantisation, Spectral Determinants, and Renormalisation as Spectral Flow J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We develop a quantum formulation of drifted spectral geometry by promoting the drift parameter to a quantum variable and analysing fluctuations of the spectral action. Quantisation is implemented directly at the level of Dirac operators, avoiding background-dependent field decompositions. One–loop and higher quantum corrections arise from spectral determinants and functional traces, while renormalisation is interpreted as spectral flow. The resulting framework provides a background-independent approach to quantum geometry and quantum gravity within the drifted spectral paradigm. Keywords: Spectral Action; Quantum Geometry; Drift Quantisation; One–Loop Corrections; Renormalisation; Noncommutative Geometry. ∗jp[email protected] 3 CONTENTS I. Introduction 5 II. Quantum Spectral Framework 6 A. Drift as a Quantum Variable 6 B. Spectral Fluctuations 6 C. One–Loop Structure 6 D. Renormalisation as Spectral Flow 6 III. Quantum Corrections, Renormalisation, and Stability of the Drift Potential 7 A. One–Loop Corrections to the Spectral Potential 7 B. Renormalised Coefficients 8 C. Renormalisation Group Equations 8 D. Stability of the Spectral Condensate 8 E. Absence of Quantum Instabilities 9 F. Interpretation 9 IV. Non–Perturbative Effects, Path Integrals, and Spectral Measure 9 A. Spectral Path Integral 9 B. Integration over the Drift Sector 10 C. Spectral Measure and Operatorial Weights 10 D. Instanton–Like Spectral Configurations 10 E. Tunnelling and Vacuum Structure 10 F. Structural Summary 11 G. Remarks 11 V. Conclusions, Outlook, and Connections to Quantum Gravity 11 A. Summary of Results 11 B. Conceptual Implications 12 C. Relation to Quantum Gravity 12 D. Future Directions 12 Appendices 13 A. Spectral Determinants and Zeta Regularisation 13 4 B. Stability of the Quantum Spectral Measure 13 References 14 5 I. INTRODUCTION The previous reports of the DRSN series established drifted spectral geometry as a unifying framework for classical gravity, cosmology, branes, and holography. A natural next step is to address the quantum regime. Traditional approaches to quantum gravity rely on perturbative expansions around a fixed background geometry, leading to severe conceptual and technical difficulties. In contrast, spectral geometry encodes geometric information directly in the spectrum of Dirac operators. This suggests that quantisation should be implemented at the operatorial level, rather than through background-dependent field variables [1–3]. In the present report we pursue this idea by developing a quantum theory of drifted spectral geometry. The key principle is to treat the drift parameter as a quantum degree of freedom and to analyse quantum fluctuations of the spectral action. In this setting, loop corrections arise from spectral determinants, and renormalisation corresponds to controlled spectral flow. This approach preserves background independence and maintains analytic control over quantum corrections. The operator-theoretic framework of drifted spectral geometry developed in DRSN I–VII establishes the analytic and structural foundations on which the present quantum formulation is built, including bounded similarity deformations, spectral action dynamics, compactification control, brane localisation, and holographic correspondence [4–10]. About this report. This work constitutes Report VIII of the DRSN series (De Rerum Spectrale Natura), a sequence of independent but thematically unified studies on spectral drift geometry. The DRSN series is developed within an open research community on spectral geometry and fundamental physics; related materials, preprints and versioned updates are archived at https://zenodo.org/communities/dsrn/. The present report develops the quantum layer of drifted spectral geometry. Quantisation is implemented directly at the level of Dirac operators by promoting the drift parameter to a quantum degree of freedom and analysing spectral fluctuations of the spectral action. The emphasis is structural and analytic: quantum corrections arise from spectral determinants and heat-kernel expansions, renormalisation is interpreted as spectral flow, and non–perturbative effects are formulated operatorially. No new phenomenological claims are made. Subsequent reports will build on this quantum framework to address information-theoretic and arithmetic aspects of drifted spectral geometry. 6 II. QUANTUM SPECTRAL FRAMEWORK A. Drift as a Quantum Variable Classically, the drift parameter s labels a family of isospectral Dirac operators Ds . In the quantum theory, we promote s to a dynamical quantum variable and consider path integrals of the form Z=ZDsexp(−Tr(f(Ds/Λ))) .(II.1) This defines a quantum theory directly in terms of spectral data. B. Spectral Fluctuations Quantum fluctuations correspond to variations of the Dirac operator, Ds−→ Ds+δD, (II.2) where δD is a bounded operator encoding quantum corrections. The effective action at one loop is given by the spectral determinant Γ1-loop =1 2log detD2 s,(II.3) defined through zeta-function regularisation. C. One–Loop Structure The one–loop effective action can be expressed as Γ1-loop =−1 2Z∞ 0 dt tTre−tD2 s,(II.4) making explicit its dependence on heat-kernel coefficients [11,12]. Quantum corrections therefore renormalise the spectral coefficients already present at the classical level. D. Renormalisation as Spectral Flow Renormalisation is interpreted as the flow of spectral coefficients under changes of the cutoff Λand the drift parameter s . This yields renormalisation group equations directly at the spectral level, without reference to local counterterms [13]. 7 Classical Spectral Action Spectral Determinant Quantum Effective Action FIG. 1. Quantum drift geometry: loop corrections arise from spectral determinants. Remark 1. In this framework, renormalisation preserves the spectral form of the action. III. QUANTUM CORRECTIONS, RENORMALISATION, AND STABILITY OF THE DRIFT POTENTIAL In this section we analyse how quantum corrections modify the classical drift-induced spectral potential and assess the stability of its universal quartic structure. A key result is that the qualitative form of the potential is preserved under quantisation, ensuring robustness of the drift mechanism. A. One–Loop Corrections to the Spectral Potential At the classical level, the effective potential for the drift parameter takes the universal form Vcl(s)=α s2+β s4, β > 0.(III.1) Quantum fluctuations generate corrections through the one–loop effective action Γ1-loop(s) = 1 2log det(D2 s).(III.2) Using the heat kernel representation, these corrections can be written as Γ1-loop(s)=−1 2Z∞ 0 dt tTre−tD2 s.(III.3) The s –dependence of Γ 1-loop arises exclusively through the drifted heat-kernel coefficients, leading to quantum corrections of order s2and s4. 8 B. Renormalised Coefficients The quantum-corrected effective potential can be expressed as Veff (s)=αren s2+βren s4+O(s6),(III.4) with renormalised coefficients αren =α+δα, βren =β+δβ. (III.5) The corrections δα and δβ are determined by one–loop spectral integrals and depend logarithmically on the cutoff scale Λ. Crucially, the sign of βren remains positive, preserving stability at large |s|. C. Renormalisation Group Equations Differentiating the renormalised coefficients with respect to log Λyields spectral renormalisation group equations, dαren dlog Λ =βα(α, β),dβren dlog Λ =ββ(α, β),(III.6) where βα and ββ are spectral beta functions computed from heat-kernel coefficients. These equations govern the quantum flow of the drift potential and admit fixed points corresponding to scale-invariant spectral configurations. D. Stability of the Spectral Condensate The existence of a non-trivial condensate s∗ requires αren < 0and βren > 0. Quantum corrections may shift the location of the minimum, s2 ∗=−αren 2βren ,(III.7) but do not eliminate it. The second derivative of the effective potential at the minimum, V′′ eff (s∗)=−2αren >0,(III.8) ensures perturbative stability of the condensate against quantum fluctuations. 9 E. Absence of Quantum Instabilities A potential concern in quantum theories with scalar degrees of freedom is the generation of destabilising higher-order terms. In the spectral framework, such terms are suppressed by higherorder heat-kernel coefficients and remain subleading. Moreover, the operatorial origin of the drift ensures that no odd powers of sare generated, even at the quantum level. Quantum Effect Spectral Outcome One–loop corrections Renormalisation of α, β Higher loops Suppressed spectral terms UV behaviour Stable quartic potential IR behaviour Shifted condensate s∗ TABLE I. Quantum effects on the drift-induced spectral potential. F. Interpretation Quantum drift geometry thus exhibits a remarkable degree of stability. The universal quartic structure of the spectral potential is preserved under quantisation, and the geometric origin of the condensate remains intact. This robustness supports the interpretation of the drift condensate as a genuine quantum-geometric phenomenon rather than a classical artefact. IV. NON–PERTURBATIVE EFFECTS, PATH INTEGRALS, AND SPECTRAL MEASURE While the previous section addressed perturbative quantum corrections, a complete quantum formulation of drifted spectral geometry must also incorporate non–perturbative effects. In this section we outline a non–perturbative framework based on spectral path integrals and operatorial measures, remaining fully background independent [14]. A. Spectral Path Integral The quantum theory of drift geometry is defined by a path integral over spectral data rather than metric fields. Formally, one considers Z=ZDDexp(−Tr(f(D/Λ))) ,(IV.1)