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The Positive-Energy Principle for Quantum Dynamics

Pushpendra Singh

Published Oct 24, 2026• 0 citations
Publisher

Abstract

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 $\hat{G}$, the associated second-order parent wave equation naturally defines a nonnegative physical-energy operator through the spectral modulus $\hat{E}_{\mathrm{pe}} = |\hat{G}| = \sqrt{\hat{G}^2}$. Through the polar decomposition $\hat{G} = \hat{S}\hat{E}_{\mathrm{pe}}$, 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 $\hat{H}_D$ retains its conventional unbounded spectrum $(-\infty,-mc^2]\cup[mc^2,\infty)$, whereas the physical-energy spectrum is bounded below by the rest energy, $\sigma(\hat{E}_{\mathrm{pe}}) = [mc^2,\infty)$. 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 $L^2(\mathbb{R})$ and show that the temporal Hilbert transform $\mathcal{H}_t$ provides a natural real-linear operator realization of the modulus mapping via $\hbar\mathcal{H}_t\partial_t \leftrightarrow \hbar|\Omega|$. 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.

DOI
10.5281/zenodo.23614195

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