# Fundamental Spacetime Representations of Quantum Antenna Systems
## Appendix F The Neutral Klein-Gordon Field Theory
The essence of the method of normalizing factors is to perform a transformation of the form  
\[\int_{\mathbf{p}\in\mathbb{R}^{3}}\mathrm{d}^{3}p\ \longrightarrow\int_{p\in \mathbb{M}^{4}}\mathrm{d}^{4}p \tag{105}\]  
taking us from the non-Lorentz invariant space \(\mathbb{R}^{3}\) to Minkowski space \(\mathbb{M}^{4}\), where \(\mathrm{d}^{4}p\) is clearly invariant since \(p:=p^{\mu}=(p^{0},\mathbf{p})\). However, note that the mass shell condition  
\[p^{2}:=p^{\mu}p_{\mu}=m^{2} \tag{106}\]  
forces this integration to remain on the cone defined by equations of this type, i.e., a four-submanifold embedded into \(\mathbb{M}^{4}\) whose equation is (106). Then, we may write  
\[\mathrm{d}^{4}p=\mathrm{d}^{3}p\delta\Big{(}p^{2}-m^{2}\Big{)}\Theta\Big{(}p^ {0}\Big{)}, \tag{107}\]  
where \(\Theta(\cdot)\) is the Heaviside unit step function, which is inserted by hand in order to prevent the appearance of negative energies. It can be shown that [38, 76]  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]0  
and hence, we conclude  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]1  
where the factor \(1/(2\pi)^{3}\) was intentionally inserted in order to make the final integral looks like an inverse Fourier transform (this inserted factor will be compensated for shortly when we define the normalizing factors of the Lorentz-invariant state \(|p\rangle\)). Therefore, we have managed then to reduce a four-dimensional integration in \(\mathbb{M}^{4}\) into a regular volume integral in \(\mathbb{R}^{3}\). Note that by writing the RHS of in (109) without \(\delta(E-E_{\mathbf{p}})\Theta(p^{0})\), it is to be implicitly understood that - algorithmically speaking - every appearance of \(E\) or \(\omega\) in the various possible expressions placed to the right of the integral operator sign \(\int\mathrm{d}^{3}p\) should be automatically replaced by \(+E_{\mathbf{p}}\) or \(+\omega_{\mathbf{k}}\). A Lorentz-invariant momentum state \(|p\rangle\) should be constructed from the standard three-vector momentum states \(|\mathbf{p}\rangle\), which obey the normalization rule  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]2  
The most obvious construction of the Lorentz-invariant momentum state then is  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]3  
For example, the inner product of such states is  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]4  
The definition (114) can be readily used to yield the desired Lorentz-invariant completeness relation  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]5  
where \(\mathds{1}\) is the unity operator. The computational utility of the method of normalizing factors stems from the fact that a completeness relation originally holding for all states \(\{|p\rangle,p\in\mathbb{M}^{4}\}\) in the full four-dimensional Minkowski space is now reduced to carrying out integration in the regular Euclidean space \(\mathbb{R}^{3}\). However, the price to be paid for such simplification is that a momentum-dependent factor \(1/2E_{\mathbf{p}}\) must be inserted into the integrand of the reduced integral, which slightly complicates the evaluation of various related expressions such as probability amplitudes. Finally, we can create the Lorentz state by applying a new normalized creation operator \(b_{\mathbf{p}}^{\dagger}\) to the ground or vacuum state \(|0\rangle\), i.e., \(|p\rangle=b_{\mathbf{p}}^{\dagger}|0\rangle\) and \(b_{\mathbf{p}}|0\rangle=0\). Based on (114), we expect  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]6  
The field expansion now, based on (116), becomes  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]7  
which after using (117) leads to  
\[S=\int\mathrm{d}^{4}x\,\mathcal{L}(x). \tag{111}\]8  
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1: Start with a Lagrangian density \(\mathcal{L}\). 2: Find the momentum density. From this knowledge, construct the Hamiltonian. 3: Promote the field and the corresponding momentum variables to operator-valued fields. Apply the standard commutation relations.