# Fundamental Spacetime Representations of Quantum Antenna Systems
## 2 Antenna Theory: Classical and Quantum Radiation Scenarios  
In analogy to classical antenna theory [27; 35; 40], we formally define a _quantum antenna_ in terms of the mathematical representation of a radiating source term as follows:  
**Definition 2** (**Quantum Antennas**). : _Within the context of relativistic quantum field theory, a quantum antenna (\(q\)-antenna) \(\mathcal{J}\) is an operator map of the form_  
\[\mathcal{J}:J\to\Psi, \tag{1}\]  
_where \(\Psi\) is a quantum field system produced by the abstract source function_  
\[J:\mathbb{M}^{4}\to\mathbb{K}, \tag{2}\]  
_which maps the Minkowski spacetime \(\mathbb{M}^{4}\) of special relativity to the set \(\mathbb{K}=\mathbb{R}\) (real numbers) or \(\mathbb{K}=\mathbb{C}\) (complex numbers). We assume that the function \(J(x),x\in D_{\rm s}\subset\mathbb{M}^{4}\) is integrable over any spacetime region \(D_{\rm s}\) of interest. We further assume that the source \(J(x)\) is compactly supported; i.e., \(cl\{D_{\rm s}\}\) is compact where \(cl\) is the closure operator in the standard Euclidean topology on \(\mathbb{R}^{4}\)._  
Physically, the main property of the q-antenna (source) function \(J(x)\) is that it controls the quantum radiation \(\Psi(x),x\in\mathbb{M}^{4}\), emitted by the source system \(J(x),x\in D_{\rm s}\subset\mathbb{M}^{4}\). The generic configuration itself is shown in Figure 1. In fact, much of this paper will concentrate on understanding the precise nature of how a generic source function \(J(x)\) can control the spacetime structure of the quantum field emission \(\Psi(x)\) in applications characteristic of the special context of an arbitrary quantum _communication_ system (source-channel-detector system). Thus, the article's main scope is the theoretical foundations of the physical layer of the quantum communication link structure, e.g., how a quantum state can be directed to a given spacetime region through the choice of a proper abstract source function. The particular discipline devoted to this problem is the area we dub _quantum antenna theory_, which is a relatively new interdisciplinary research area. **Remark 1**. : _It may be noted here that there is a similarity between the quantum communication problem [7] and the quantum measurement problem in quantum physics [41; 42; 43]. The analogy is manifested by the common underlying abstract structure shared by the two distinct processes of communications and generalized quantum measurement [1; 8]. Both involve a source of information (often called quantum state preparation in QM), which is localized within a given spacetime region; and a process of destructive consumption of the transmitted information, this time localized at a receiver or detector located at a timelike-distant spacetime region [43; 44]. The quantum receiver system's ultimate objective is to extract the maximum amount of information embodied in the received quantum state in QM, or quantum field in QFT, through the judicious utilization of optimized combinations of measurement operations and signal processing algorithms._  
Figure 1: The fundamental configuration of a quantum antenna communication system in spacetime. Quantum antenna systems turn out to be considerably more complex to understand and analyze than conventional antennas, where the latter antenna type is often approached within the framework based on _classical_ electromagnetism (and sometimes acoustics). A reason behind this noticeable difference in complexity between the two theories might be related to the fact that a proper and completely satisfactory understanding of quantum interactions is ultimately based on relativistic QFT, which is inherently a _many-particle_ world picture. That is, in QFT, it is inconsistent to assume the possibility of a single-particle-interaction configuration due to the inherent tension that this assumption will bring with special relativity (SR) and other fundamental principles [38].