# Fundamental Spacetime Representations of Quantum Antenna Systems
## 4 Linear Quantum Antenna Systems
### Generalization to Multiple Discrete and Continuous and Sources
Note that we assume that \(x\) and \(x^{\prime}\) are well separated from each other such that there is no overlap between the observation spacetime region \(D_{\rm r}\) and points \(x^{\prime}\in D_{\rm s}\) in the source domain. This allows as to freely move the time-ordering operator \(\mathcal{T}\) so to obtain (with negligible error) the third equality in (42) above, making our radiation formulas ultimately valid in the _exterior_ region of the q-antenna system. Finally, with the help of (27), we arrive at our main q-antenna radiation amplitude formula summarized by the following theorem:  
**Theorem 3**. : _The probability amplitude of a continuous source q-antenna system \(J(x),x\in D_{\rm s}\subset\mathbb{M}^{4}\), where the closure of \(D_{\rm s}\) is compact in \(\mathbb{R}^{4}\), is given by the following superposition integral_  
\[A(x)=\int_{x^{\prime}\in D_{\rm s}}\mathrm{d}^{4}x^{\prime}G_{q}(x-x^{\prime} )J(x^{\prime}), \tag{43}\]  
_where \(G_{q}(x-x^{\prime})\) is the \(q\)-antenna Green's function in Definition 4._  
**Remark 9**. : _We may rewrite (43) in a form more familiar to antenna engineers by recruiting the notation \(x=(t,\mathbf{x})\) and \(x^{\prime}=(t^{\prime},\mathbf{x}^{\prime})\), viz., in the \"\((\mathbf{x},t)\)-format\", allowing us to restate (43) as_  
\[A(\mathbf{r},t)=\int_{\mathbf{x}^{\prime}\in S_{\rm s}}\int_{t^{\prime}\in T_{\rm s}} \mathrm{d}t^{\prime}\ \mathrm{d}^{3}\mathbf{x}^{\prime}\,G_{q}(\mathbf{x}-\mathbf{x}^{\prime},t-t^{\prime})J( \mathbf{x}^{\prime},t^{\prime}), \tag{44}\]  
_where \(S_{\rm s}\) and \(T_{\rm s}\) are the spatial and temporal components of the four-dimensional spacetime source region \(D_{\rm s}\); see (3). Thereby, we find that the spatial support domain of the q-antenna's source system, i.e., its spatial extension properties, is captured by \(S_{\rm s}\), while \(T_{\rm s}\) is the time interval of the application of the source._  
**Remark 10**. : _The relation \(D_{\rm s}=S_{\rm s}\times T_{\rm s}\) holds locally in \(D_{\rm s}\). The reason is that for attaining wide generality, we are already implicitly permitting \(D_{\rm s}\) to posses a manifold structure in which, locally speaking, each point \(x^{\prime}\in D_{\rm s}\) is homeomorphic (topologically isomorphic) to \(\mathbb{R}^{d},d=2,3,4\). That is, from the topological viewpoint, the source region \(D_{\rm s}\subset\mathbb{M}^{4}\) can be said to be of type (\(0+1\)) for \(q\)-antenna point sources; type (\(1+1\)) for one-dimensional sources such as wire or loop antenna; type (\(2+1\)) for surface radiators; or full (\(3+1\)) type for volumetric sources; and so on._  
**Remark 11**. : _Either one of the two expressions (43) or (44) may be used to express the probability amplitude of receiving a particle at a specific location \(x=(t,\mathbf{x})\) when radiated by a source distribution written as either \(J(x^{\prime})\) or \(J(x^{\prime},t^{\prime})\). Both are Lorentz invariant, but the form (43) expresses that more clearly. On the other hand, the form (44) may be utilized when a concrete frame of reference (lab frame) is used such that an unambiguous decomposition of the source region \(D_{\rm s}\) into spatialand temporal part can be made (this is in fact the case in most practical scenarios). Physically, the radiation formulas (43) or (44) express quantum radiation by a source \(J(x)\) in terms of the \(q\)-antenna Green's function, which happens in this particular case to coincide with the Feynman propagator of QFT._