# Fundamental Spacetime Representations of Quantum Antenna Systems
## 4 Linear Quantum Antenna Systems  
### Introduction  
The conceptual, physical, and mathematical structure of quantum antennas will now be further investigated in depth by focusing on the first term on the RHS of (16). This restriction is motivated by the desire to keep the presentation as simple as possible. In fact, most of the key concepts and structure to be developed in connection with the main component of the quantum source, the function \(J_{1}\), can be adapted and also applied to the higher-order components \(J_{n},n>1\), although the details are lengthy. For example, a radiation pattern could be constructed for each source mode function \(J_{n}\) similar to the one to be given below (see Section 6), after which the various contributions of all such terms are summed together in order to estimate the total radiation characteristics of the source system.  
Our approach is fundamentally based on the idea of propagators and Green's functions in field theory. Similar to the corresponding situation in QFT itself, the concept of Green's function is also fundamental in classical antenna theory [27; 28; 33; 35; 49; 63; 64]. There, one finds that Green's function connects the source (cause) with its produced radiation field (effect), which is a fact that has been frequently exploited as a useful computational tool in both classical electromagnetism [49; 63; 64; 65; 66] and quantum physics [54; 55; 67].  
However, for quantum radiating systems to behave as antennas, one must track down and separate from out of the total quantum system an ultimate terminal where a purely _classical_--i.e., deterministic--source function is identified. Through such a source function, the quantum source system may then allow for an _external_ user control for the purpose of sending information [10; 13].11 For the purpose of this paper, we will show that QFT allows the construction of a very natural and direct source model for q-antenna systems.  
Footnote 11: The idea of the quantum source is to define a quantum source system, which is a classical antenna.  
In order to simplify the presentation, we break down our method into several steps as follows:  
1. Construct a quantum source model resembling the point (infinitesimal source) in classical antenna theory.
2. Using the previous quantum point source model, construct the quantum state radiated by the q-antenna due to arbitrary continuous or discrete source distribution (superposition principle).
3. Construct Green's function of the q-antenna using the previous superposition integral.
4. Introduce and evaluate the q-antenna radiation pattern using Green's function (mostly in the momentum space representation).  
References on the [PERSON]'s function approach to classical antenna theory can be found in [28; 46; 47; 48]. The infinitesimal source approach is developed in [68]. The momentum space approach to electromagnetic theory and antennas is outlined in [69; 70; 71; 72; 73].  
For maximum clarity, this program will be carried out throughout the remainder of the present paper for the special case of a neutral Klein-Gordon field \(\phi(x)\) (the standard spin-0 scalar field theory, Section 4.2). In addition, and as already mentioned above, for emphasis on simplicity, we focus on the linear quantum antenna case. In fact, the key concepts introduced below through the Feynman propagator for the construction of a quantum radiation pattern are essentially the same whether the radiation regime under consideration is linear or nonlinear.