Canonical quantization of the complex scalar field without making use of its real and imaginary parts

Abstract

We proceed to the canonical quantization of the complex scalar field without making use of its real and imaginary parts. Our motivation is to formally connect, as tightly as possible, the quantum-field notions of particle and antiparticle$\unicode{x2014}$most prominently represented, formally, by creation and annihilation operators$\unicode{x2014}$to the initial classical field theory$\unicode{x2014}$whose main formal object is the field amplitude at a given spacetime point. Our point of view is that doing this via the use of the real and imaginary parts of the field is not satisfying. The derivation demands to consider, just before quantization, the field and its complex conjugate as independent fields, which yields a system of two copies of independent complex scalar fields. One then proceeds to quantization with these two copies, which leads to the introduction of two families of creation and annihilation operators, corresponding to particles for the creation operators, and to antiparticles for the annihilation operators. One realizes that having two such families is the only hope for being able to "invert" the definitions of the creation and annihilation in terms of the Fourier quantized fields, so as to finally obtain an expression of the direct-space fields in terms of these creation and annihilation operators, because the real-field condition used in the case of a real scalar field does not hold for a complex scalar field. This final goal is then met by introducing the complex-conjugate constraint at the quantum level, that is, that the second independent field copy is actually the complex conjugate of the first. All derivations are led in a rigorous, and in particular purely deductive way, and all standard results are recovered that way. While we reckon our derivation exists in the literature, we have not found it.