# Fundamental Spacetime Representations of Quantum Antenna Systems
## 3 The General Theory of Quantum Antenna Systems
### A Generic Interaction Hamiltonian Description of Quantum Antenna Systems  
Here, a high-level view on interacting field theory is provided. Note that a full treatment of interacting field theories is outside the scope of the present paper, whose main goal is to supply the reader with a minimal picture of how eventually quantum antenna theory should be fully formulated when specific physical layouts, often described with their own Hamiltonian, are introduced. On the other hand, starting from Section 4, we work mainly with the scalar field theory (the Klein-Gordon theory) in order to illustrate the general structure of quantum radiation with minimum knowledge of the full details of matter-field interaction mechanisms. Readers interested in more information related to specialized physical layouts may consult numerous other publications, for example those quoted in the Introduction section of this paper. We assume that all relevant quantum field operators can be described within one and the same large enough combined Hilbert space \(\mathcal{H}\), which in the case of QFT is a Fock space \(\mathcal{F}\). The total Hamiltonian operator of the system is written as  
\[H=H_{0}+H_{\rm int}, \tag{5}\]  
where \(H_{0}\) is the _free_ Hamiltonian (non-interacting part that is usually solvable), while \(H_{\rm int}\) is the _interacting_ Hamiltonian, which is a time-dependent operator that in turn can be expanded into four basic component as follows  
\[H_{\rm int}=H_{\rm in}+H_{\rm s}+H_{\rm r}+H_{\rm c}. \tag{6}\]  
Here, we have the following three categories of interaction Hamiltonian terms:  
1. \(H_{\rm in}\) captures _intrinsic_ interaction in the fundamental quantum field of the q-antenna system, e.g., self interactions such as polynomial Lagrangian terms containing powers of \(\phi(x)\) larger than three such as the mainstream interacting \(\phi^{4}\)-theory [54; 56; 57]. 2. The terms \(H_{\rm s}\) and \(H_{\rm r}\) describe, respectively, the interaction between the source and the receiver antennas on one side and the fundamental quantum field \(\phi(x)\) of the q-antenna system on the other side. These interactions should be understood as processes localized within their respective spacetime domains \(D_{\rm s}\) and \(D_{\rm r}\). 3. Finally, the term \(H_{\rm c}\) corresponds to _channel_ interactions and couplings, e.g., coupling of the excited quantum field \(\phi(x)\) with scattering objects located within the effective path of an excited quantum particle produced by the source and directed toward the receiver.6  
Footnote 6: The term \(H_{\rm c}\) corresponds to _channel_ interactions and couplings, e.g., coupling of the excited quantum field \(\phi(x)\) with scattering objects located within the effective path of an excited quantum particle produced by the source and directed toward the receiver.7  
Footnote 7: The term \(H_{\rm c}\) corresponds to _channel_ interactions and couplings, e.g., coupling of the excited quantum field \(\phi(x)\) with scattering objects located within the effective path of an excited quantum particle produced by the source and directed toward the receiver.8  
Footnote 8: The term \(H_{\rm c}\) corresponds to _channel_ interactions and couplings, e.g., coupling of the excited quantum field \(\phi(x)\) with scattering objects located within the effective path of an excited quantum particle produced by the source and directed toward the receiver.8  
A couple of important observations on this decomposition of the interacting Hamiltonian are in order. First, to keep the discussion at the most general level, we assume that the intrinsic interacting Hamiltonian, if nonzero, is present all the time. That is, it becomes an essential ingredient of the fundamental field \(\phi(x)\) of the q-antenna system.