Complex scalar relativistic field as a probability amplitude

TL;DR

The work addresses interpreting a relativistic complex scalar field as a probability amplitude while avoiding negative-energy solutions of the Klein–Gordon–Fock equation. It develops a first-order-in-time relativistic equation by a phase transformation $\varphi(x)=\psi(x)e^{-i\mu c t}$, derives a continuity equation for probability density, and shows the general solution splits into two positive-energy excitations with distinct dispersions $\omega_k^{(\pm)}=c\sqrt{\mu^{2}+k^{2}} \mp c\mu$. Through a Lagrangian formulation and Noether analysis, it constructs a consistent second-quantized framework with two bosonic sectors and a normally ordered Hamiltonian $H=\int d^3k\big[ \hbar\omega^{(+)}_k a^{(+)dagger}(\mathbf{k})a^{(+)}(\mathbf{k}) + \hbar\omega^{(-)}_k b^{(-)dagger}(\mathbf{k})b^{(-)}(\mathbf{k}) \big]$, together with vacuum structure and one-particle states. The approach yields a probabilistic interpretation for $\psi$, reveals a nonlocal high-order-derivative structure, and provides a multiparticle QFT description that could correspond to light and heavier scalar meson pairs in a particle-physics context.

Abstract

A relativistic equation for a neutral complex field as a probability amplitude is proposed. The continuity equation for the probability density is obtained. It is shown that there are two types of excitations of this field, which describe particles with positive energy and different dispersion laws. Based on the Lagrangian formalism, conservation laws are obtained. The transition to secondary quantization is considered.