The Hill-Wheeler Equation as a Quantum Mechanical Fermi Distribution: A New Statistical Framework for Elementary Particles

This research proposes a new statistical mechanical approach in quantum field theory: the concept of fermion-boson duality and transition functions. In conventional quantum field theory, fermions (such as electrons) and bosons (such as photons) are distinguished as particles with fundamentally different statistical properties. However, this study examines the possibility that the statistical properties of particles change depending on energy scales, constructing a mathematical framework to describe a dual transition where electrons, which are fermionic at low energies, show bosonic properties at high energies, and conversely, photons, which are bosonic at low energies, show fermionic properties at high energies. To describe this transition, we introduce energy-dependent transition functions and apply them to quantum electrodynamics calculations, demonstrating that ultraviolet divergences in conventional theory are naturally suppressed. As a specific numerical example, we perform calculations of electron self-energy and demonstrate that with the introduction of transition functions, divergent integrals converge to finite values. This statistical mechanical approach suggests the possibility of regularizing quantum field theory calculations in a physically meaningful way without introducing artificial cutoffs or renormalization.