The Schrödinger equation for a non-quantized matter field: a theoretical introduction for Physics Education researchers

Usual presentations of Quantum Field Theory (QFT) start by postulating the existence of a classical field that obeys to the wave equation, derived from a conveniently chosen Lagrangian density. But while electromagnetic fields are given a proper physical meaning even before quantization, matter fields acquire it mostly only when quantized. Therefore, in this paper, we aim to give a pedagogical construction that allows us to assign a physical meaning also to non-quantized matter fields. This operation is particularly important since we believe that Quantum Field Theory is more suited than Quantum Mechanics (QM) to introduce quantum physics in secondary schools. The purpose of this paper is to show how to attribute a physical meaning to classical (that means, not yet quantized) matter fields. It is not intended for a high school presentation, but, besides being addressed to those who teach QFT at university, and to teacher educators, it is primarily addressed to researchers in physics education and to all those who work for an educational presentation of quantum physics based on QFT.