We prove an exact quantum conservation law for a harmonic oscillator coupled to a ghost degree of freedom: a second classical conserved quantity lifts to a quantum operator that commutes with the Hamiltonian with no hbar corrections, yielding a rigorous, state-independent upper bound on the mean squared phase-space radius for all time and every quantum state with finite initial second moments. The proof uses only canonical commutation relations and the Leibniz rule; it requires no confining potential, no spectral assumptions, and no perturbative expansion. The interaction studied here is bounded and vanishes at large separations, the generic situation in effective field theory, yet this suffices to guarantee quantum stability in the sense of bounded second moments. Three independent numerical frameworks (Heisenberg picture, Schrodinger picture, and Fock-space diagonalization) confirm wavepacket confinement below the analytic bound, a real energy spectrum, and Poisson level statistics numerically consistent with an integrable structure. The absence of a confining potential means the proof is silent on spectral discreteness and the existence of a ground state; those questions, addressed for polynomial confining interactions in concurrent work, remain open for the interaction class studied here and represent the sharpest targets for future work. Ghost quantum instability is therefore not an inevitable consequence of a wrong-sign kinetic term but depends critically on the interaction structure.
We prove an exact quantum conservation law for a harmonic oscillator coupled to a ghost degree of freedom: a second classical conserved quantity lifts to a quantum operator that commutes with the Hamiltonian with no hbar corrections, yielding a rigorous, state-independent upper bound on the mean squared phase-space radius for all time and every quantum state with finite initial second moments. The proof uses only canonical commutation relations and the Leibniz rule; it requires no confining potential, no spectral assumptions, and no perturbative expansion. The interaction studied here is bounded and vanishes at large separations, the generic situation in effective field theory, yet this suffices to guarantee quantum stability in the sense of bounded second moments. Three independent numerical frameworks (Heisenberg picture, Schrodinger picture, and Fock-space diagonalization) confirm wavepacket confinement below the analytic bound, a real energy spectrum, and Poisson level statistics numerically consistent with an integrable structure. The absence of a confining potential means the proof is silent on spectral discreteness and the existence of a ground state; those questions, addressed for polynomial confining interactions in concurrent work, remain open for the interaction class studied here and represent the sharpest targets for future work. Ghost quantum instability is therefore not an inevitable consequence of a wrong-sign kinetic term but depends critically on the interaction structure.
We quantize a classically stable system of a harmonic oscillator polynomially coupled to a ghost with negative kinetic energy. We prove that due to an integral of motion with a positive discrete spectrum: i) the Hamiltonian has a pure point spectrum unbounded in both directions, ii) the evolution is manifestly unitary, iii) the vacuum is well-defined, iv) expectation values for squares of canonical variables are bounded. Numerical solutions of the Schrödinger equation confirm these results. We argue that the discrete spectrum of the integral of motion enforces stability for extended interactions.
We formulate a second-order organization of quantum dynamics in which physical energy is identified with the magnitude of the temporal oscillation frequency, whereas the algebraic sign of the frequency specifies temporal phase orientation rather than the sign of energy. For any time-independent self-adjoint first-order evolution generator G^, the associated second-order parent wave equation naturally defines a nonnegative physical-energy operator through the spectral modulus E^pe=∣G^∣=G^2. Through the polar decomposition G^=S^E^pe, this construction decouples the nonnegative physical-energy magnitude from the signed spectrum of the first-order generator. Applied to the free Dirac equation, the Dirac Hamiltonian H^D retains its conventional unbounded spectrum (−∞,−mc2]∪[mc2,∞), whereas the physical-energy spectrum is bounded below by the rest energy, σ(E^pe)=[mc2,∞). For generators with a spectral gap around zero, the second-order dynamics admits a positive-definite conserved wave norm and an equivalent two-component unitary evolution. We further prove a positive-frequency spectral completeness theorem in L2(R) and show that the temporal Hilbert transform Ht provides a natural real-linear operator realization of the modulus mapping via ℏHt∂t↔ℏ∣Ω∣. Direct numerical diagonalization and high-precision time-domain integration of the one-dimensional Dirac equation confirm both the spectral modulus mapping and the exact conservation of quadratic wave energy across opposite temporal orientations. While this kinematic framework places quantum energy on the same nonnegative footing as classical field and signal energies, promoting the two temporal orientations to full particle and antiparticle sectors requires a second-quantized field theory incorporating charge conjugation, spin-statistics, locality, and interactions.
Arbitrarily small changes in the commutation relations suffice to transform the usual singular quantum theories into regular quantum theories. This process is an extension of canonical quantization that we call general quantization. Here we apply general quantization to the time-independent linear harmonic oscillator. The unstable Heisenberg group becomes the stable group SO(3). This freezes out the zero-point energy of very soft or very hard oscillators, like those responsible for the infrared or ultraviolet divergencies of usual field theories, without much changing the medium oscillators. It produces pronounced violations of equipartition and of the usual uncertainty relations for soft or hard oscillators, and interactions between the previously uncoupled excitation quanta of the oscillator, weakly attractive for medium quanta, strongly repulsive for soft or hard quanta.
The quantum mechanics of an oscillator interacting with a field in the limit of zero damping (\ensuremath{\lambda}+0) and infinite field extension (\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}) depends on the order in which the limits are taken. If the limit \ensuremath{\lambda}\ensuremath{\rightarrow}0 is taken after the limit \ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}, one obtains Gaussian wave functions for all stationary states (m=0,1,2,. . .) of the particle. These Gaussians do have the correct standard values for the variances 〈${\ensuremath{\zeta}}^{2}$〉n but otherwise obviously differ from the standard harmonic-oscillator result. The latter arises only if the zero-damping limit is taken before the infinite-system limit.
Within a strong coupling expansion, we construct local quasi-conserved operators for a class of Hamiltonians that includes both integrable and non-integrable models. We explicitly show that at the lowest orders of perturbation theory the structure of the operators is independent of the system details. Higher order contributions are investigated numerically by means of an ab initio method for computing the time evolution of local operators in the Heisenberg picture. The numerical analysis suggests that the quasi-conserved operators could be approximations of a quasi-local conservation law, even if the model is non-integrable.
The Born rule — that the probability of obtaining outcome |phi> in a measurement of a quantum system in state |psi> is P = |<phi|psi>|^2 — is one of the most empirically well-confirmed and conceptually contested statements in physics. Existing derivation attempts (Gleason’s theorem; Deutsch–Wallace decision theory; Zurek envariance; many-worlds branch counting) each establish it under their own assumptions, but none has achieved universal acceptance. We show that in the Three Time Dimensions (3+3) spacetime framework (de Haan 2026, book DOI 10.5281/zenodo.19633127), in which the third time dimension t_3 is compactified as a discrete two-sphere S^2 with 2^152 Planck-area cells, the Born rule emerges through three independent geometric routes that all converge on the same result P = |<phi|psi>|^2. Route 1 (Malus’s Law on S^2): the wavefunction is literally a direction on the t_3 S^2; the Bloch sphere IS this S^2; Malus’s 1809 law P = cos^2(theta) is identical to the Born rule for bosons, with SU(2) double cover giving cos^2(theta/2) for fermions. Route 2 (Gleason + S^2 non-contextuality): Gleason’s 1957 theorem uniquely selects |<phi|psi>|^2 given non-contextuality, which in (3+3) is a consequence of the S^2 Riemannian metric rather than an independent postulate. Route 3 (Lévy measure + t_2 isotropy): the t_2 expansion perturbs all t_3 directions isotropically; the unique continuous binary-outcome probability measure is the Lévy / Haar-invariant spherical measure, which gives P(+n_A) = cos^2(theta/2). The three routes use distinct mathematical tools (projection, Hilbert-space theorem, stochastic analysis) and distinct physical inputs (geometry, Riemannian metric, dynamical isotropy). They must converge if the geometric identification is correct; they would diverge if it were wrong. Malus’s 1809 law is the Born rule avant la lettre. The empirical optical law taught to undergraduates as a fact is, under this reading, a direct geometric consequence of photon polarisations being directions on S^2 — a statement about projection geometry that predates quantum mechanics by a century. The same geometric identification simultaneously resolves four foundational puzzles: (1) Schrödinger’s cat is never in superposition (the cat’s ~10^29 t_3 modes are entrained by environmental t_2 perturbation within femtoseconds of the first atomic interaction); (2) the three-polariser paradox dissolves (non-commuting measurements are non-commuting rotations on S^2); (3) the Bell/Tsirelson bound 2sqrt(2) is a geometric consequence of S^2 having dimension exactly 2 — empirical confirmation by loophole-free Bell tests (Hensen, Giustina, Shalm 2015) confirms the S^2 identification; (4) the cosmological decoherence floor tau_cosm = 1/H_0 ~ 14.5 Gyr is a structural upper bound on quantum coherence set by the t_2 cosmic expansion — distinguishing the framework from many-worlds interpretations. Three supporting elegance claims establish the framework’s quantum-foundational coherence: (A) the complex-valuedness of psi in C follows from t_2 being a rotation (the imaginary unit i is the generator of the t_2 precession rotation); (B) the Schrödinger equation i hbar dpsi/dt = H psi is the infinitesimal form of t_2 precession rotating the phase of the t_3 configuration; (C) the canonical commutation [x, p] = i hbar follows from the foam lattice structure. The paper is complementary to the companion Quantum Computing preprint [DOI 10.5281/zenodo.19651560] which uses the Bloch-sphere-as-S^2 identification for applied hardware-level proposals. The Born Rule paper provides the foundational underpinning for why that identification is correct at the level of quantum-mechanical first principles. The paper is honest about what remains open. Four specific items are explicitly acknowledged (§11): (1) the framework does not yet have a 6D Lagrangian formulation from which the Born rule would fall out of path-integral quantisation; (2) the uniqueness of the S^2 identification is only partially addressed — higher-dimensional spheres are ruled out by the Tsirelson bound, but tori and Calabi–Yau alternatives are not examined; (3) the path-integral reformulation of quantum mechanics within (3+3) has not been constructed; (4) detailed comparison with QBism, relational quantum mechanics, and consistent histories is outside the paper’s scope. Falsifiability. Six routes: Tsirelson-bound overshoot/undershoot beyond 2sqrt(2) + measurement precision; indefinite quantum coherence beyond cosmological timescales; real-valued QM experimental confirmation (already ruled out by Renou et al. 2021, reconfirmed by independent tests); detection of genuine contextuality; detection of discrete S^2 cell structure at near-Planck scales. The Born rule, in (3+3), is not a mysterious postulate; it is Malus’s Law on a compact sphere that nobody knew was there.