# Fundamental Spacetime Representations of Quantum Antenna Systems
## 4 Linear Quantum Antenna Systems
### Generalization to Multiple Discrete and Continuous and Sources  
Suppose that now, we apply the quantum source field again but at a _different_ spacetime point, say \(x_{2}^{\prime}\), while we rename \(x^{\prime}\) in (21) as \(x_{1}^{\prime}\). Furthermore, we note that each excitation of the vacuum by a field \(\phi(x^{\prime})\) localized at \(x^{\prime}\) can be controlled by some position-dependent \"scaling factor\", say \(J(x^{\prime})\). This is again the abstract source representation of the q-antenna in the spirit of Definition 2. Then, the total quantum source field is  
\[J(x_{1}^{\prime})\phi(x_{1}^{\prime})+J(x_{1}^{\prime})\phi(x_{2}^{\prime}). \tag{31}\]  
Again, if we hit the vacuum state by this new operator, the radiated quantum state can be written as  
\[[J(x_{1}^{\prime})\phi(x_{1}^{\prime})+J(x_{2}^{\prime})\phi(x_{2}^{\prime})] |0\rangle=\int_{\mathbf{p}\in\mathbb{R}^{3}}\frac{\mathrm{d}^{3}p}{(2\pi)^{3/ 2}}\frac{J(x_{1}^{\prime})e^{\mathrm{i}p_{x}x_{1}^{\prime\prime}}+J(x_{2}^{ \prime})e^{\mathrm{i}p_{y}x_{2}^{\prime\prime}}}{(2\omega_{\mathbf{p}})^{1/2} }|\mathbf{p}\rangle, \tag{32}\]  
where we used (21). Physically, the expression (32) means that two clusters of outgoing waves (quantum particles) are emitted, one emanating from the spacetime point \(x_{1}^{\prime\prime\prime}=(t_{1}^{\prime},\mathbf{x}_{1}^{\prime})\), while the other cluster is directly radiated from \(x_{2}^{\prime\prime\prime}=(t_{2}^{\prime},\mathbf{x}_{2}^{\prime})\). In all cases, note that the linearity of the quantum source field operator \(\phi(x^{\prime})\), as manifested by the expression \(\phi(x^{\prime})|0\rangle\) in (21), is the ultimate basis behind the applicability of the principle of superposition in q-antenna systems. It is straightforward to generalize the q-antenna's radiation Formula (32) to the generic scenario of an arbitrary number of \(N\) discrete point sources located at the spacetime points \(x_{n}^{\prime}\), each with its own excitation strength specified by \(J(x_{n}^{\prime})\), \(n=1,2,\ldots,N\). This leads to the expression  
\[J_{q}(x^{\prime})|0\rangle=\int_{\mathbf{p}\in\mathbb{R}^{3}}\frac{\mathrm{d} ^{3}p}{(2\pi)^{3/2}}\frac{\sum_{n=1}^{N}J(x_{n}^{\prime})e^{\mathrm{i}p_{x}x _{n}^{\prime\prime}}}{(2\omega_{\mathbf{p}})^{1/2}}|\mathbf{p}\rangle, \tag{33}\]  
where  
\[J_{q}(x^{\prime}):=\sum_{n=1}^{N}J(x_{n}^{\prime})\phi(x_{n}^{\prime}) \tag{34}\]  
is the effective (discrete) quantum source distribution operator. It should become clear now how to generalize from discrete to continuous sources. Let us expand the continuous classical current source function \(J(x^{\prime})\) in terms of a finite number of spacetime (four-dimensional) Dirac delta functions \(\delta(x^{\prime})\) as follows:  
\[J(x^{\prime})=\sum_{n=1}^{N}J(x_{n}^{\prime})\delta(x^{\prime}-x_{n}^{\prime}), \tag{35}\]where the source domain \(D_{\rm s}\), the spacetime region upon which the source is supported, is in this case the discrete set  
\[D_{\rm s}:=\{x^{\prime}_{1},x^{\prime}_{2},\ldots,x^{\prime}_{N}\}. \tag{36}\]  
Using the sifting property of the Dirac delta function, it is evident that (33) can be rewritten as  
\[J_{q}(D_{\rm s})|0\rangle=\int_{\mathbb{P}\in\mathbb{R}^{3}}\frac{\mathrm{d}^{ 3}p}{(2\pi)^{\frac{3}{2}}(2\omega_{\mathbb{P}})^{\frac{1}{2}}}\int_{x^{\prime }\in D_{\rm s}}\mathrm{d}^{4}x^{\prime}J(x^{\prime})e^{\mathrm{i}p_{\rm s}x^{ \prime\prime}_{n}}|\mathbb{P}\rangle, \tag{37}\]  
where the quantum source operator is given by the integral  
\[J_{q}(D_{\rm s}):=\int_{x^{\prime}\in D_{\rm s}}\mathrm{d}^{4}x^{\prime}J(x^{ \prime})\phi^{\dagger}(x^{\prime}). \tag{38}\]  
While (33) and (37) are formally equivalent in the special case of discrete sources, it is the _second_ form (37) that is needed for writing down the correct expression corresponding to the continuous source case. Indeed, when the set \(D_{\rm s}\) becomes dense (which implies \(N\to\infty\)), the relation (37) continues to hold. **Remark 6**.