# Fundamental Spacetime Representations of Quantum Antenna Systems
## 4 Linear Quantum Antenna Systems
### Generalization to Multiple Discrete and Continuous and Sources
: _In deriving (37), we implicitly assume that the order of the \(dx^{\prime}\)- and \(d^{3}p\)- integrals can be exchanged. This can be justified relatively easily when we impose the condition that \(D_{\rm s}\) is compact, which is an assumption we make throughout this paper that still covers most practical antenna systems. The details of the rigorous proof are lengthy and will not be given. For a comprehensive discussion of the rigorous mathematical theory behind representing integrals (continuum sums) over quantum kets in QFT, see [54, 55, 67, 79]._  
To summarize, we have managed so far to construct two fundamental types of quantities behaving as \"sources\" in q-antenna theory:  
1. The classical source function \[J(x^{\prime}):D_{\rm s}\subset\mathbb{M}^{4}\to\mathbb{R}.\] (39)
2. The quantum source operator \[J_{q}(D_{\rm s}):\mathcal{D}_{\rm s}\to\mathcal{O},\] (40) where \[[J(x_{1}^{\prime})\phi(x_{1}^{\prime})+J(x_{2}^{\prime})\phi(x_{2}^{\prime})] |0\rangle=\int_{\mathbf{p}\in\mathbb{R}^{3}}\frac{\mathrm{d}^{3}p}{(2\pi)^{3/ 2}}\frac{J(x_{1}^{\prime})e^{\mathrm{i}p_{x}x_{1}^{\prime\prime}}+J(x_{2}^{ \prime})e^{\mathrm{i}p_{y}x_{2}^{\prime\prime}}}{(2\omega_{\mathbf{p}})^{1/2} }|\mathbf{p}\rangle, \tag{32}\]0 (41) is the set of all open subsets in the Minkowski spacetime \(\mathbb{M}^{4}\) whose topological closure is compact in \(\mathbb{R}^{4}\). On the other hand, \(\mathcal{O}\) is defined as the space of all operators acting on elements of the Fock (occupation state Hilbert space representation [54, 55]) space of the q-antenna system.20  
Footnote 20: The source design problem in quantum antenna theory is a special case of discrete sources. **Remark 7** (**The source design problem in quantum antenna theory**). : _The classical source \(J(x^{\prime})\) is needed in order to delimit the actual \(q\)-antenna configuration in spacetime. As in classical antenna theory, an antenna source is considered known if its geometric support regions \(D_{\rm s}\) is known in addition to the value of the source \(J(x^{\prime})\) at each point \(x^{\prime}\in D_{\rm s}\). The situation is quite similar here. The \(q\)-antenna designer is interested in obtaining bounds or information on both \(D_{\rm s}\) and \(J(x^{\prime})\) in order to attain certain radiation states. This is the design problem that is usually solved by means of optimization methods._  
Once the classical source \(J(x^{\prime})\) is fixed, one can employ (38) in order to immediately construct the quantum source operator \(J_{q}\), which depends (among other things) on the geometrical and topological structure of the source support domain \(D_{\rm s}\). **Remark 8**. : _It should be noted that the operator \(J_{q}\) depends on the classical function \(J(x^{\prime})\) defined on the entire spacetime regions \(D_{\rm s}\). Thus, formally speaking, one should write that operator as \(J_{q}[D_{\rm s},J(x^{\prime})]\). However, to simplify the presentation, this is avoided, since it should always be clear from the context which specific classical source function \(J(x^{\prime})\) is involved with the quantum source operator \(J_{q}\)._  
The probability amplitude \(A(x)\) of receiving a particle/wave at the spacetime position \(x\) can now be readily computed in exactly the same way we did with the point quantum source at \(x^{\prime}\), i.e, the amplitude \(G_{q}(x-x^{\prime})\) in (27), simply by adding the various contributions coming from all source points \(x^{\prime}\in D_{\rm s}\). The calculations are  
\[ \begin{split} A(x)&=\langle 0|\mathcal{T}\phi(x)J_{q} (D_{\rm s})|0\rangle=\langle 0|\mathcal{T}\phi(x)\int_{x^{\prime}\in D_{\rm s}} \mathrm{d}^{4}x^{\prime}\ J(x^{\prime})\phi^{\dagger}(x^{\prime})|0\rangle\\ &=\int_{x^{\prime}\in D_{\rm s}}\mathrm{d}^{4}x^{\prime}\langle 0 |\mathcal{T}\phi(x)\phi^{\dagger}(x^{\prime})|0\rangle f(x^{\prime}),\end{split}  \tag{42}\]  
where (37) was employed in order to write down the second equality in (42).