# Fundamental Spacetime Representations of Quantum Antenna Systems
## 3 The General Theory of Quantum Antenna Systems
### A Generic Interaction Hamiltonian Description of Quantum Antenna Systems
In this sense, the latter field is defined as precisely that quantum field corresponding to the Hamiltonian  
\[H_{\phi}=H_{0}+H_{\text{in}}. \tag{7}\]  
On the other hand, the source, receiver, and channel Hamiltonian terms are treated in this theory as _extrinsic_ interactions, i.e., external disturbances coupled to the fundamental quantum field \(\phi(x)\) associated with the Hamiltonian \(H_{\phi}\). As expected in quantum physics, coupling often leads to nonlinear equations of motion [54]. In this sense, the uncoupled fundamental field \(\phi(x)\) is perturbatively transformed into a _new_ coupled field \(\phi^{\prime}(x)\) with a full Hamiltonian  
\[H_{\phi^{\prime}}=H_{\phi}+H_{\text{s}}+H_{\text{r}}+H_{\text{c}} \tag{8}\]  
solved using perturbation Dyson or path integral expansions [55]. Following standard conventions in QFT, we do not change the notation of the field but always use \(\phi(x)\) while clearly stating which Hamiltonian is being used (if needed). In addition, we will not label Hamiltonians by their fields or spacetime arguments unless this is needed. **Remark 2**. : _Sometimes, it is more convenient to group channel interactions \(H_{\text{c}}\) and intrinsic field self-interactions \(H_{\text{in}}\) into one term_  
\[H_{\text{in}}^{\prime}=H_{\text{in}}+H_{\text{c}}. \tag{9}\]  
_The reason is that there is indeed some similarity between the two types of interactions above. They are both unrelated to the transmitter (source) and receiver terminals and can be considered then as indigenous components of the q-antenna system field itself. However, there are also some differences, since channel couplings are not the same everywhere but are localized at the scattering objects themselves. On the other hand, self-interactions and also intrinsic mutual interactions captured by the term \(H_{\text{in}}\) are generally \"turned on\" most of the time. Nevertheless, in some applications, it might be useful to group \(H_{\text{in}}\) and \(H_{\text{c}}\) with each other under the rubric of scattering-based interaction processes in order to distinguish them from transmitter and receiver types of interactions (information source and sink)._  
Let us examine now how the source and the receiver terminals of the system may interact. Working in the Schrodinger picture,7 Let the state of the source at the end of the interaction interval \(T_{\text{s}}=[t_{s}^{\prime},t_{\text{s}}]\) be denoted by \(\ket{J,D_{\text{s}}}_{t_{\text{s}}}\). Here, \(J\) is a generic symbol for the overall set of (classical) disturbances supported on \(D_{\text{s}}\) ultimately constituting our proposed formal Definition 3 of the q-antenna current source system. Then, we may write  
Footnote 7: The source is a source of the source, and the receiver is a source of the source. \[\ket{J,D_{\text{s}}}_{t_{\text{s}}}=P(t_{\text{s}}^{\prime},t_{\text{s}};J,D_ {\text{s}})\ket{0_{\text{in}}}, \tag{10}\]  
where \(P(t_{s}^{\prime},t_{\text{s}};J,D_{\text{s}})\) is the interaction picture (Dirac) propagator (A18), which is reviewed in Appendix D for more details. That is, the computation of this propagator is based on the substitution \(H_{\text{I}}=H_{\text{s}}\) into (A19). The special state labeled \(\ket{0_{\text{in}}}\) is the ground state of the _intrinsically interacting_ system, i.e., the state satisfying  
\[(H_{0}+\underbrace{H_{\text{in}}+H_{\text{c}}}_{H_{\text{in}}^{\prime}})\ket{ 0_{\text{in}}}=0, \tag{11}\]  
where, as discussed above, we have here opted for generality by including all possible wireless channel scattering effects captured by \(H_{\text{c}}\) with the field intrinsic self-interaction Hamiltonian \(H_{\text{in}}\). Then, the ground state \(\ket{0_{\text{in}}}\) represents both the (i) true or actual physical initial state of the q-antenna system _before_ interacting with the source \(J\) localized in the spacetime domain \(D_{\text{s}}\); and (ii) the final state reattained _after_ finishing interaction with the receiver localized in the disjoint domain \(D_{\mathbf{r}}\).