Alternative Framework to Quantize Fermionic Fields

TL;DR

This work develops an information-theoretic extension of the stationary action principle to quantize fermionic fields. By incorporating a relative-entropy based information metric into the classical action and employing an extended canonical transformation, the authors derive a functional Schrödinger equation for Grassmann-valued fermions, obtaining the Floreanini-Jackiw representation from first principles. They demonstrate that the resulting Hamiltonian generates the Poincaré algebra and reproduce canonical results for particle creation in a constant external field, while also outlining applications to electromagnetic and non-Abelian interactions and to non-renormalizable fermionic couplings, which can lead to nonlinear Schrödinger-type dynamics. The framework provides a coherent, Lorentz-invariant alternative to standard second-quantization approaches and suggests new insights into field fluctuations, information metrics, and potential extensions to curved spacetimes and gravity.

Abstract

A variational framework is developed here to quantize fermionic fields based on the extended stationary action principle. From the first principle, we successfully derive the well-known Floreanini-Jackiw representation of the Schrödinger equation for the wave functional of fermionic fields - an equation typically introduced as a postulate in standard canonical quantization. The derivation is accomplished through three key contributions. At the conceptual level, the classical stationary action principle is augmented to include a correction term based on the relative entropy arising from field fluctuations. Then, an extended canonical transformation for fermionic fields is formulated that leads to the quantum version of the Hamilton-Jacobi equation in a form consistent with the Floreanini-Jackiw representation; Third, necessary functional calculus with Grassmann-valued field variables is developed for the variation procedure. The quantized Hamiltonian can generate the Poincaré algebra, thus satisfying the symmetry requirements of special relativity. Concrete calculation of the probability of particle creation for the fermionic field under the influence of constant external field confirms that the results agree with those using standard canonical quantization. We also show that the framework can be applied to develop theories of interaction between fermionic fields and other external fields such as electromagnetic fields, non-Abelian gauge fields, or another fermionic field. These results further establish that the present variational framework is a novel alternative to derive quantum field theories.