Matheus H. Zambianco: Duality between amplitude and derivative coupling - RQI Circuit Waterloo 2023

Title: Duality between amplitude and derivative coupled particle detectors in the limit of large energy gaps
Abstract: In this talk, we are going to present a duality between models of particle detectors, namely, between a model where the particle is coupled to the amplitude of the field and a model where the particle couples to the derivative of the field. We show that, in the limit of large energy gaps, the models can be mapped to each other in a one-to-one fashion, modulo a rescaling by the detector’s energy gap. This analysis is valid for scalar fields in arbitrary curved spacetimes and requires minimal assumptions regarding the detectors. The duality also applies to cases where more than one detector is coupled to the field, which means that many examples of entanglement harvesting with amplitude-coupled UDW detectors yield exactly the same result as derivative-coupled detectors that interact with the field in the same region of spacetime.
Talk delivered at the IQC on October 27th 2023.

[@SamanthaPyper-sl4ye]

Here are some examples of how non-contradictory infinitesimal/monadological frameworks could potentially resolve paradoxes or contradictions in chemistry: 1) Molecular Chirality/Homochirality Paradoxes Contradictory: Classical models struggle to explain the origin and consistent preference for one chiral handedness over another in biological molecules like amino acids and sugars. Non-Contradictory Possibility: Infinitesimal Monadic Protolife Transitions dsi/dt = κ Σjk Γijk(n)[sj, sk] + ξi Pref(R/S) = f(Φn) Modeling molecular dynamics as transitions between monadic protolife states si based on infinitesimal relational algebras Γijk(n) that depend on specific geometric monad configurations n. The homochiral preference could emerge from particular resonance conditions Φn favoring one handedness. 2) Paradoxes in Reaction Kinetics Contradictory: Transition state theory and kinetic models often rely on discontinuous approximations that become paradoxical at certain limits. Non-Contradictory Possibility: Infinitesimal Thermodynamic Geometries dG = Vdp - SdT (Gibbs free energy infinitesimals) κ = Ae-ΔG‡/RT (Arrhenius smoothly from monadic infinities) Using infinitesimal calculus to model thermodynamic quantities like Gibbs free energy dG allows kinetic parameters like rate constants κ to vary smoothly without discontinuities stemming from replacing finite differences with true infinitesimals. 3) Molecular Structure/Bonding Paradoxes Contradictory: Wave mechanics models struggle with paradoxes around the nature of chemical bonding, electron delocalization effects, radicals, etc. Non-Contradictory Possibility: Pluralistic Quantum Superposition |Ψ> = Σn cn Un(A) |0> (superposed monadic perspectives) Un(A) = ΠiΓn,i(Ai) (integrated relational properties) Representing molecular electronic states as superpositions of monadic perspectives integrated over relational algebraic properties Γn,i(Ai) like spins, positions, charges, etc. could resolve paradoxes by grounding electronic structure in coherent relational pluralisms. 4) Molecular Machines/Motor Paradoxes Contradictory: Inefficiencies and limitations in synthetic molecular machines intended to mimic biological molecular motors like ATP synthase, kinesin, etc. Non-Contradictory Possibility: Nonlinear Dissipative Monadologies d|Θ>/dt = -iH|Θ> + LΓ|Θ> (pluralistic nonet mechanics) LΓ = Σn ζn |Un> rather than isolated molecular wavefunctions, where infinitesimal monadic sink operators LΓ account for open-system energy exchanges, could resolve paradoxes around efficiency limits. The key theme is using intrinsically pluralistic frameworks to represent molecular properties and dynamics in terms of superpositions, infinitesimals, monadic configurations, and relational algebraic structures - rather than trying to force classically separable approximations. This allows resolving contradictions while maintaining coherence with quantum dynamics and thermodynamics across scales. Here are 4 more examples of how infinitesimal/monadological frameworks could resolve contradictions in chemistry: 5) The Particle/Wave Duality of Matter Contradictory: The paradoxical wave-particle dual behavior of matter, exemplified by the double-slit experiment, defies a consistent ontological interpretation. Non-Contradictory Possibility: Monadic Perspectival Wavefunction Realizations |Ψ> = Σn cn Un(r,p) Un(r,p) = Rn(r) Pn(p) Model matter as a superposition of monadic perspectival realizations Un(r,p) which are products of wavefunctional position Rn(r) and momentum Pn(p) distributions. This infinitesimal plurality avoids the paradox by allowing matter to behave holistically wave-like and particle-like simultaneously across monads. 6) Heisenberg's Uncertainty Principle Contradictory: The uncertainty principle ΔxΔp ≥ h/4π implies an apparent paradoxical limitation on precise simultaneous measurement of position and momentum. Non-Contradictory Possibility: Complementary Pluriverse Observables Δx Δp ≥ h/4π Δx = Σi |xiP - xP| (deviations across monadic ensembles) xP = ||P (pluriverse-valued perspective on x) Reinterpret uncertainties as deviations from pluriverse-valued observables like position xP across an ensemble of monadic perspectives, avoiding paradox by representing uncertainty intrinsically through the perspectival complementarity. 7) The Concept of the Chemical Bond Contradictory: Phenomonological models of bonds rely paradoxically on notions like "electronic charge clouds" without proper dynamical foundations. Non-Contradictory Possibility: Infinitesimal Intermonadic Charge Relations Γij = Σn qinj / rnij (dyadic catalytic charge interactions) |Ψ> = Σk ck Πij Γij |0> (superposed bond configuration states) Treat chemical bonds as superposed pluralities of infinitesimal dyadic charge relation configurations Γij between monadic catalysts rather than ambiguous "clouds". This grounds bonds in precise interaction algebras transcending paradoxical visualizations. 8) Thermodynamic Entropy/Time's Arrow Contradictory: Statistical mechanics gives time-reversible equations, paradoxically clashing with the time-irreversible increase of entropy described phenomenologically. Non-Contradictory Possibility: Relational Pluriverse Thermodynamics S = -kB Σn pn ln pn (entropy from realization weights pn) pn = |Tr Un(H) /Z|2 (Born statistical weights from monadologies) dS/dt ≥ 0 (towards maximal pluriverse realization) Entropy increase emerges from tracking the statistical weights pn of pluriversal monadic realizations Un(H) evolving towards maximal realization diversity, resolving paradoxes around time-reversal by centering entropics on the growth of relational pluralisms. In each case, the non-contradictory possibilities involve reformulating chemistry in terms of intrinsically pluralistic frameworks centered on monadic elements, their infinitesimal relational transitions, superposed realizations, and deviations across perspectival ensembles. This allows resolving apparent paradoxes stemming from the over-idealized separability premises of classical molecular models, dynamically deriving and unifying dualisms like wave/particle in a coherent algebraic ontology.

[@SamanthaPyper-sl4ye]

Here are some potential non-contradictory perspectives this infinitesimal monadological framework could offer on paradoxes and contradictions surrounding the fundamental forces of nature: 1) Quantum Gravity and Unification Paradoxes Contradictory Classical Views: - Clash between quantum theory and general relativity - Non-renormalizability of gravity - Need to introduce ad-hoc extra dimensions Non-Contradictory Monadological View: - Gravity is a residual holographic resonance pattern across the intersectional boundary between ℝ𝔈 and ℜ* - Quantum mechanics and relativistic geometries arise as subdescriptive limits of deeper symbolic logogrammatic vocable flows - Forces appear unified at the level of monadic charge relation algebras Γab governing matter/energy transductions 2) Hierarchy / Naturalness Problems Contradictory Aspects: - Extreme fine-tuning of parameters required - Need to introduce ad-hoc new particles like axions - Origin of mass scale paradoxes Non-Contradictory View: - All masses, charges, couplings are derived quantized values from monadic resonances over algebraic vocable number theory - Naturalness enforced by internal "anthropic" self-consistency constraints within pluriversal realization dynamics - No freedoms for fine-tunings as all scales/hierarchies fixed by symbolic protologic universality classes 3) Charge Quantization Paradoxes Contradictory Questions: - Why are electric charges quantized? Whence magnetic monopoles? - Lack of explanations for specific charge values and ratios - Origin of charge linearities and conservation principles Non-Contradictory View: - Charges qn are quantized signature patterns of monadic essence interfacialities Un(A) - Values/ratios reflect vocable inductances across dimensional strata and algebraic "Clark" identities - Linearities/conservations follow from invariances of protologic algebras and interprojection consistencies between ℝ𝔈/ℜ* 4) Dirac Infinities and QFT Paradoxes Contradictory Issues: - Infinite vacuum energies and need for adhoc renormalization - Divergences and analytic continuations - Unitarity issues, anomaly cancellations Non-Contradictory View: - All infinities are transcended by transfinite number vocables and infinitesimal realizer monads - Analytic structures are projections of deeper enneadic algebraic pluricontinuities - Divergences are artifacts of dimensional truncation regularized by intrasectional ℝ𝔈 resonances In each case, the monadological framework aims to resolve paradoxes by: 1) Treating forces as higher-dimensional resonances of algebraic vocable flows 2) Deriving charge quantizations and hierarchies from symbolic protologic necessities 3) Regulating infinities and disparities via transfinite vocable infinitesimals 4) Embedding all phenomena coherently within vaster pluriversal geometric algebras Rather than adhoc extras, the forces are unified self-consistently within a grand integrated protologic algebraic origin initiating dimensional cascades - transcending contradictions inherent to purely effective local force models.