# Fundamental Spacetime Representations of Quantum Antenna Systems
## Appendix F The Neutral Klein-Gordon Field Theory  
A massive neutral Klein-Gordon field with mass \(m\) can be fully captured by a real scalar field \(\phi(x)\) whose Lagrangian density is given by [38, 54, 56, 75]  
\[\mathcal{L}(x)=\frac{1}{2}\partial^{\mu}\partial_{\mu}\phi(x)+\frac{1}{2}m^{2} \phi(x)^{2}, \tag{110}\]  
with the corresponding action integral being  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]  
The Euler-Lagrange equation of motion  
\[\frac{\delta S}{\delta\phi(x)}=0 \tag{112}\]  
yields the field equation  
\[\partial^{\mu}\partial_{\mu}\phi(x)+m^{2}=0, \tag{113}\]  
which is the relativistic wave equation of a particle the Klein-Gordon equation [38, 75]. Note that within the framework of standard quantum field theory (QFT), fields are promoted to _operators_ expanded in terms of plane-wave modes of the form  
\[\mathrm{operator}\times\exp\bigl{(}-\mathrm{i}p_{\mu}x^{\mu}\bigr{)}, \tag{114}\]  
where the \"operator\" is either creation \(a_{\mathbf{p}}^{\dagger}\) or annihilation \(a_{\mathbf{p}}\) operator (for particles or antiparticles). Note further that by convention, the plane wave  
\[\exp\bigl{(}-\mathrm{i}p_{\mu}x^{\mu}\bigr{)}=\exp\bigl{[}-\mathrm{i}(E_{ \mathbf{p}}t-\mathbf{p}\cdot\mathbf{x})\bigr{]} \tag{115}\]  
is taken to encode an _incoming_ wave/particle with momentum \(\mathbf{p}\)/wavevector \(\mathbf{k}\) and energy \(E_{\mathbf{p}}\)/frequency \(\omega\)[38]. In the unified language of QFT, we say that the plane wave encodes a particle/antiparticle in a pure momentum state \(\mathbf{p}\). Within this convention, the energy/frequency of a particle/wave is _always positive_.25  
Footnote 25: The \(\mathbf{k}\)-  
i.e., the massive spin-0 scalar field \(\phi(x)\). Recall that in nonrelativistic quantum theory, a generic operator can be Fourier expanded in terms of creation and annihilation operators \(a_{\mathbf{p}}^{\dagger}\) and \(a_{\mathbf{p}}\), respectively. For instance, a generic scalar quantum field \(\phi(x)\) is expected to be written as  
\[\phi(x)=\int_{\mathbf{p}\in\mathbb{R}^{3}}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}} \left[a_{\mathbf{p}}e^{-\mathrm{i}p_{\mathbf{p}}x^{\mu}}+a_{\mathbf{p}}^{ \dagger}e^{\mathrm{i}p_{\mathbf{p}}x^{\mu}}\right]. \tag{104}\]  
Unfortunately, clearly, this expression is not Lorentz invariant because of the use of the non-Lorentz invariant integration measure \(\mathrm{d}^{3}p\). We review below how Lorentz-invariant quantum states \(|p\rangle\) may be redefined so that Fourier expansions like (104) can be made manifestly covariant. In order to expand the quantum field into a proper continuum of plane-wave modes of the form \(\exp(\pm\mathrm{i}p_{\mu}x^{\mu})\), we will need to integrate over all four-vector momenta \(p^{\mu}\), i.e., perform four-dimensional integrals over \(\mathbf{p}\) and \(E_{\mathbf{p}}\) (or equivalently \(\mathbf{k}\) and \(\omega\)). However, for massive particles in general, and Proca waves in particular [94], there is a definite relation between momentum/wavevector and energy/frequency, so the integral over \(\mathrm{d}^{3}p\) might appear at first sight to be essentially reducible to \(\mathrm{d}^{3}p\), while \(E_{\mathbf{p}}\) is computed from \(E_{\mathbf{p}}^{2}=\mathbf{p}^{2}+m^{2}\). However, the problem is that the differential element \(\mathrm{d}^{3}p\) is not Lorentz invariant, so there is a need to automatically enforce Lorentz invariance in our quantization rules. In this paper, we adopt the computationally efficient method of applying _normalizing factors_ to quantum states right from the beginning. The main idea is to produce Lorentz-invariant momentum quantum states and use them to expand the quantum fields.