# Fundamental Spacetime Representations of Quantum Antenna Systems
## 5 On the General Structure of Radiation Processes in Linear Quantum Antenna Systems
### The General Structure of the Quantum Antenna Propagator Process  
If we insert (29) into (43), interchanging the order of integrals, the following spectral (momentum-space) result is obtained.  
**Theorem 4**.: _The momentum space probability amplitude \(A(q)\) in Theorem 3 can be expanded in the spectral domain using the following integral formula:_  
\[A(x)=\int_{\mathbf{p}\in\mathbb{R}^{3}}\frac{d^{3}p}{(2\pi)^{3} }\frac{d^{4}p}{2\omega_{\mathbf{p}}}\int_{x^{\prime}\in D_{\mathrm{s}}}\mathrm{ d}^{4}x^{\prime}\,J(x^{\prime})e^{ip_{\mu}x^{\prime\mu}}e^{-ip_{\mu}x^{\mu}}. \tag{45}\]  
_Here, the spectral integration is performed with respect to all real momenta \(\mathbf{p}\) in \(\mathbb{R}^{3}\)._  
Proof.: In order to obtain (45), we assume that \(\mathrm{cl}(D_{\mathrm{s}})\) is compact in \(\mathbb{R}^{4}\) in order to make sure that a needed interchange of limiting operations can be justified. The rest of the proof is immediate.  
The anatomy of the quantum radiation's Green's function's spectral expansion (45) is illustrated in Figure 2. The various fundamental sub-processes composing the overall process of quantum radiation can be formally identified as follows:  
1. We first must form the correct relativistic sum over all allowable momentum states. This is accomplished by the Lorentz invariant integral operator \[\int_{\mathbf{p}\in\mathbb{R}^{3}}\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{1}{ 2\omega_{\mathbf{p}}}.\] (46)
2. Each momentum state \(|\mathbf{p}\rangle\) will be summed over all possible source locations \(x^{\prime}\in D_{\mathrm{s}}\) in the source region via the integral operator \[\int_{x^{\prime}\in D_{\mathrm{s}}}\mathrm{d}^{4}x^{\prime}.\] (47) This step is also relativistic, since \(D_{\mathrm{s}}\subset\mathbb{M}^{4}\) and \(\mathrm{d}^{4}x^{\prime}\) are Lorentz invariant.[21]
3. The next crucial step is to multiply by the factor \(\exp(\mathrm{i}p_{\mu}x^{\prime\mu})\). This will trigger the production of a quantum wave (particle) emanating from \(x^{\prime}\) and spreading radially away from the point source.
4. Finally, in order to observe the radiation field at a distance, we multiply the wave produced at \(x^{\prime}\) by a propagation factor \(\exp\left(-\mathrm{i}p_{\mu}x^{\mu}\right)\). This will guarantee that the field has been effectively propagated and absorbed at the observation location \(x\).  
**Remark 12**.: _It may be seen from the above algorithmic construction of the q-antenna propagator in the spectral domain that the problem of quantum radiation acquires a very intuitive and concrete structure when viewed from the spectral (momentum space) domain's perspective. This observation will be exploited in Section 6 when we explore candidate expressions for the q-antenna radiation pattern._  
**Remark 13**.: _The probability amplitude \(A(x)\) can be seen as a superposition of \"quantum plane waves\" each of the form \(\exp\left(ip_{\mathbf{\
u}}x^{\mathbf{\imath}}\right)\). This is in fact somehow similar to the plane-wave (Weyl) expansion of electromagnetic fields radiated by continuous sources [63; 64; 80; 81; 82]. As in the electromagnetic case, each point source will emit a continuum of plane waves (some of them are evanescent, and the others are pure propagating), with wavevector/momentum specified by \(\mathbf{p}\) and frequency \(\omega_{\mathbf{p}}\). The total sum of all these waves will produce an effective field moving gradually away from the source and approaching the far zone of the antenna by first going through the near zone. (The evanescent mode character of the momentum space representation becomes directly visible when we transform the above three-dimensional spectral integrals evaluated with respect to \(\mathbf{p}\in\mathbb{R}^{3}\) to equivalent four-dimensional integrals now performed with respect to \(p\in\mathbb{M}^{4}\), see for example [57])._