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An Exactly Solvable Model of Local Energy Oscillations in Everettian Branching

Muhammad Ghazi Al Gifari

Published Sep 11, 2026• 0 citations
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Abstract

Abstract The apparent local nonconservation of energy within individual branches during Everettian measurement remains a subtle foundational question in quantum mechanics. While the global expectation value of energy is strictly invariant under unitary time evolution, localized observer branches generically experience dynamic energy fluctuations. In this work, we present an exactly solvable two qubit model governed by an excitation exchange Hamiltonian structurally analogous to the Jaynes Cummings interaction. By decomposing the universalwavefunction |Ψ⟩ through structural projection operators, we derive closed form analytical expressions for the energetic trajectories of the emergent branches. We show that the dynamic asymmetry between a static branch and an oscillatory branch arises directly from total excitation number conservation rather than an eigenstate condition. Furthermore, we prove that local energy oscillations are precisely counterbalanced by time dependent shifts in branch probability weights via the resolution of identity, ensuring absolute global energy conservation. A parametric sweep confirms that this exact compensation mechanism persists across all interaction regimes. These results establish a rigorous mathematical demonstration that local energy oscillations in Everettian branching do not signal genuine thermodynamic violations, but emerge naturally as an observational artifact of subsystem confinement within a globally unitary framework.

DOI
10.21203/rs.3.rs-10990617/v1

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Abstract The apparent local nonconservation of energy within individual branches during Everettian measurement remains a subtle foundational question in quantum mechanics. While the global expectation value of energy is strictly invariant under unitary time evolution, localized observer branches generically experience dynamic energy fluctuations. In this work, we present an exactly solvable two qubit model governed by an excitation exchange Hamiltonian structurally analogous to the Jaynes Cummings interaction. By decomposing the universalwavefunction |Ψ⟩ through structural projection operators, we derive closed form analytical expressions for the energetic trajectories of the emergent branches. We show that the dynamic asymmetry between a static branch and an oscillatory branch arises directly from total excitation number conservation rather than an eigenstate condition. Furthermore, we prove that local energy oscillations are precisely counterbalanced by time dependent shifts in branch probability weights via the resolution of identity, ensuring absolute global energy conservation. A parametric sweep confirms that this exact compensation mechanism persists across all interaction regimes. These results establish a rigorous mathematical demonstration that local energy oscillations in Everettian branching do not signal genuine thermodynamic violations, but emerge naturally as an observational artifact of subsystem confinement within a globally unitary framework.
Starting from the unitary evolution of a closed quantum system, we rigorously demonstrate that the projection of the global wavefunction onto an arbitrary local subsystem is governed by an exact, time-dependent non-Hermitian Schrödinger equation. Crucially, the derivation does not rely on conventional approximations such as the Born approximation, the Markov approximation, or the wide-band limit. The central quantity is the \textit{instantaneous self-energy}, a time-dependent and generally non-Hermitian operator that encodes environmental backaction and, together with the subsystem Hamiltonian, forms the exact time-local generator of the projected dynamics. By benchmarking the conventional non-Hermitian approximation against this exact framework, we systematically expose its limitations. These results provide a rigorous microscopic foundation for the emergence of effective non-Hermitian dynamics in quantum systems.
We investigate the anomalous energy change of the measurement apparatus when a qubit is measured in bases that do not commute with energy. We model two possible measurement implementations: one is a quantum clock model with a completely time-independent Hamiltonian, while the other is a Jaynes-Cummings model that is time-dependent but conserves the total excitation number. We look at the mean energy change of the measurement apparatus in both models, conditioned on the qubit postselection, and we find that this change can be much greater than the level spacing of the qubit, like an anomalous weak value. In the clock model, the expression for the apparatus energy shift explicitly contains the weak value of the qubit Hamiltonian. However, in our case, no explicit weak measurements are carried out. Our two models give different results, which we explain to be a consequence of the nondegenerate spectrum of the Jaynes-Cummings model. We compare our calculations in the Jaynes-Cummings model with the experimental data of J. Stevens et al., [Phys. Rev. Lett. 129, 110601 (2022)], and we find good agreement when the conditions of our derivation are valid.
A recently proposed local measurement axiom (Postulate M) replaces the instantaneous global projection of textbook quantum mechanics with branch selection triggered by completed local decoherence, while a companion ψ-ontic analysis identifies continuous spontaneous localization (CSL) as a concrete local mechanism for the selection itself. Here we test this two-stage account quantitatively in the Einstein–Podolsky–Rosen (EPR) scenario. We integrate the nonlinear CSL stochastic Schrödinger equation for an entangled spin pair in which one particle couples to a local spin bath with random couplings, bath self-dynamics, and intra-bath interactions, while the noise field and collapse operator act only at the measurement site. Analytically, we derive an exact law for the maximal Clauser–Horne–Shimony–Holt (CHSH) value under local decoherence, Smax⁡(t)=21+sin⁡2(2θ) ∣z(t)∣2 S_{\max}(t)=2\sqrt{1+\sin^{2}(2\theta)\,|z(t)|^{2}} Smax​(t)=21+sin2(2θ)∣z(t)∣2​, with z(t) the decoherence factor and θ the entanglement angle: the CHSH excess over the classical bound is consumed, never the measurement-basis probabilities, which are exactly conserved. We further show that the selection stage is governed by an exact martingale: the branch weight obeys a scalar Itô equation whose derivation requires only the orthogonality of the branches—supplied by the spin structure, not by the environment—so the Born rule emerges as an absorption probability irrespective of the residual bath coherence. Numerically, the law holds to machine precision (∼ 10−15) for all three bath classes, including interacting environments, and through finite-size recurrences; stochastic trajectories confirm Born statistics to within sampling error, with all realizations fully resolved. The ensemble-averaged reduced state of the distant particle is invariant under the local collapse-operator axis and rate—parameter independence as a structural theorem, verified to machine precision in the unitary sector—and branch amplitudes are invariant across all environmental initial conditions and dynamics (∼ 10−11), confirming that outcome randomness is not of environmental origin. In the symmetric configuration, with independent noises at both wings, the two local noises provably drive a single global branch martingale at the summed rate: selection is one joint event whose absorbing states are the correlated branches—mismatched outcomes are dynamically excluded (zero events in 300 realizations, against ∼ 25% each for a product-state control)—the joint statistics are invariant under swapping which wing completes first (93% vs 9% orderings, distributions equal within sampling error), and sampled outcomes reproduce the singlet’s measurement correlation E(β) = − cos β—with β the angle between Alice’s and Bob’s measurement axes—including a CHSH value |S| = 2.710±0.073, consistent with the Tsirelson bound 2√2 within sampling errors. Decoherence and local stochastic selection jointly reproduce the empirical content of quantum measurement in the EPR scenario with no nonlocal dynamical element: the only nonclassical resource is the nonseparability of the initial state. Nonlocal projection is thus dispensable not only axiomatically and ontologically but operationally.
The Jaynes-Cummings (JC) Hamiltonian is a paradigmatic model of light-matter interaction and, more generally, qubit-boson interactions, widely used across atomic, optical, and superconducting qubit platforms. In the multi-qubit setting, where n qubits are identically coupled to a single boson mode, this interaction is known as the Tavis-Cummings (TC) Hamiltonian. The structure of the TC model is usually understood in terms of two standard symmetries: permutation invariance of the qubits and a U(1) symmetry associated with conservation of the total excitation number. Here we identify an additional, independent "accidental" symmetry of the TC Hamiltonian and construct the corresponding conserved observable. We show that, for n>2 qubits, this symmetry imposes strong constraints on the realizable unitary transformations. These constraints persist in the presence of the global JzJ_zJz​ Hamiltonian, but are removed by adding Jz2J_z^2Jz2​, even though Jz2J_z^2Jz2​ preserves both permutation invariance and the U(1) symmetry. Finally, we explain the origin of this previously unnoticed symmetry using Schwinger's boson representation of angular momentum. These restrictions have important implications for controllability of the TC system and for its applications to quantum computing, which are investigated further in a companion paper.
A multiscale thermodynamic model is considered, in which cosmological dynamics enforce persistent non-equilibrium conditions through recursive energy exchange across hierarchically ordered subsystems. The internal energy of each subsystem is recursively determined by energetic interactions with its subcomponents, forming a nested hierarchy extending up to cosmological scales. The total energy of the universe is assumed to be constant, imposing global consistency conditions on local dynamics. On the quantum scale, subsystems remain thermodynamically constrained in their accessible state space due to the unresolved energetic embedding imposed by higher-order couplings. As a result, quantum behavior is interpreted as an effective projection of unresolved thermodynamic interactions. In this view, the wave function serves as a mathematical representation of a subsystem's thermodynamic embedding, summarizing the unresolved energetic couplings with its environment, as shaped by recursive interactions across cosmological and microscopic scales. Phenomena such as zero-point energy and vacuum fluctuations are thereby understood as residual effects of structural energy constraints. Classical mechanics arises as a limiting case under full energetic resolution, while the quantum formalism reflects thermodynamic incompleteness. This formulation bridges statistical mechanics and quantum theory without metaphysical assumptions. It remains fully compatible with standard formalism, offering a thermodynamic interpretation based solely on energy conservation and hierarchical organization. All effects arise from scale-dependent resolution, not from violations of established physics.