# Fundamental Spacetime Representations of Quantum Antenna Systems
## 5 On the General Structure of Radiation Processes in Linear Quantum Antenna Systems
### On the Causal Spacetime Structure of Radiation Emitted by Quantum Antenna Systems  
In both special and general relativity, two events \(x\) and \(x^{\prime}\) with timelike distance, i.e., \(\left|x-x^{\prime}\right|^{2}>0\), can be causally connected [51, 53, 84]. This has a direct and obvious implication for the general spacetime theory of q-antennas developed here. 1. Consider a point source case. Figure 3(a) indicates the future lightcone \(C_{x^{\prime}}\) of the event located at \(x^{\prime}\), i.e., the apex of the cone in the spacetime diagram given therein. Since we assume for simplicity that the operational principle behind our q-antenna-based communication link's receiver is based on the process of annihilating the radiated particle at the observation point \(x\), it follows that _only receivers located inside the antenna causal lightcone \(C_{x^{\prime}}\) can receive information from the point source._
2. For potential receivers located _outside_ the antenna causal cone, it is not possible to transmit information at all, unless one admits some superluminal mechanism to be used for sending information, which is currently not accepted by majority of scientists. These two observations above are enough to determine the causal structure of q-antennas for the case of point sources. However, since arbitrary sources can be constructed from assembling clusters of point sources, the argument can be expanded, as will be shown next. Figure 3: The fundamental operational modes of c- and q- antennas. Both types requires a current Green’s function (ACGF) \(\mathbf{F}\)[28] to connect classical excitation field \(\mathbf{E}^{\mathrm{ex}}\) with a classical induced current \(\mathbf{J}\). Note that in both c- and q-antennas, the induced currents \(\mathbf{J}_{c}\) and \(\mathbf{J}_{q}\), respectively, are classical. However, the difference resides mainly in the “output” or the radiation state, which is classical (quantum) in c- (q-) antennas. For the case of continuous sources, the situation is qualitatively similar to the discrete scenario discussed above, but the detailed content of the antenna's causal domain becomes somehow more complicated, since the situation strongly depends on the geometry of the source region in the former case. In Figure 4b, we schematically illustrate the problem when a generic continuous source \(J(x^{\prime})\) is applied in the spacetime region \(D_{\mathrm{s}}\subset\mathbb{M}^{4}\). Since any current source function \(J(x^{\prime})\) can be expanded into a continuous sum of point sources, as per the sifting property of delta functions  
\[J(x^{\prime})=\int_{x\in\mathbb{M}^{4}}\mathrm{d}^{4}x\,J(x)\delta(x^{\prime} -x), \tag{50}\]  
one may then attempt to construct a causal lightcone for each point \(x^{\prime}\in D_{\mathrm{s}}\), which is again denoted by \(C_{x^{\prime}}\). A _continuum_ of causal cones is shown in Figure 4b where all cones appear to be aligned in parallel in spacetime because we ignore gravitational effects. We now introduce the following construction of the antenna's horizon causal structure, which is valid for both classical and quantum antennas:  
**Definition 5** (The antenna causal domain). : _The antenna causal domain associated with a given source region \(D_{\mathrm{s}}\), which we denote by \(C(D_{\mathrm{s}})\), is defined as simply the fusion (set-theoretic union) of all individual causal cones \(C_{x^{\prime}}\) based on events located inside the source region \(x^{\prime}\in D_{\mathrm{s}}\). That is, according to the recipe_  
\[C(D_{\mathrm{s}}):=\bigcup_{x^{\prime}\in D_{\mathrm{s}}}C_{x^{\prime}}. \tag{51}\]  
_The part of \(C(D_{\mathrm{s}})\) that is solely due to future lightcones \(C_{x^{\prime}}^{+}\) is called the antenna future causal domain. Similarly, the components of \(C(D_{\mathrm{s}})\) due to contributions emanating from past lightcones \(C_{x^{\prime}}^{-}\) lead to the antenna past causal domain.