# Fundamental Spacetime Representations of Quantum Antenna Systems
## 4 Linear Quantum Antenna Systems
### The Feynman Propagator of Quantum Antennas  
So far, we have succeeded in modeling the process in which a q-antenna emits a particle at a specific spacetime point \(x^{\prime}=(t^{\prime},\mathbf{x}^{\prime})\). We also explicated the momentum-space composition of the emitted one-particle radiation state and found that it is comprised of a superposition of multiple outgoing one-particle momentum states. Moreover, we estimated the probability amplitude of measuring a certain (generic) momentum in this radiated state. However, for practical applications, there is a need to actually compute the effective coupling between the source on one side and generic observation spacetime points located either in the near- or far-zone on the other side. To do so, we will make use of one of the most powerful methods in QFT, the _Feynman propagator_[38, 54, 55, 67]. Effectively, this will also directly provide us with the two-point Green's function \(G^{1}(x,x^{\prime})\) of the q-antenna system corresponding to the first term on the RHS of (15). The importance of moving to a mathematical description based on propagators stems from the fact that in QFT, one often finds that exciting the ground state at a specific spacetime point does _not_ automatically imply that the radiated particle will reach _every_ point in the far-zone with _significant_ probability. To ensure that the quantum wireless or molecular communication link's receiver has access to the radiated particles with significant probability of detection, we need to compute the probability amplitude of measuring a particle at a generic spacetime point \(x\) away from the source point \(x^{\prime}\). Since in QFT, there are no measurement operators as in nonrelativistic quantum mechanics, we may model the observer's interactions with the receiver by the process of particle _annihilation_[38]. This is one of the key ideas of quantum measurement in QFT to be adopted in this paper. In order to mathematically implement this idea, we compute the two-point Green's function of the q-antenna system, which is defined as follows:  
**Definition 4** (The two-point q-antenna Green's function). : _The q-antenna two-point (or one-particle) Green's function, denoted by \(G_{q}(x,x^{\prime})\), is defined as the probability amplitude of the process that a particle created at the point \(x^{\prime}\in D_{\rm s}\) in the source region, while the system was initially at the ground state \(|0\rangle\), will be annihilated at the later point \(x\in D_{\rm r}\) in the receiver (detection, observation) region, after which the q-antenna system will return again to the ground state._  
**Theorem 2**. : _The correct relativistic expression for \(G_{q}(x,x^{\prime})\) is given by the formula_  
\[G_{q}(x,x^{\prime})=\langle 0|\mathcal{T}\phi(x)\phi^{\dagger}(x^{\prime})|0\rangle, \tag{27}\]  
_where \(\mathcal{T}\) is the time-ordering operator defined by (A17)._  
Proof. : This follows directly from Definition 4, the quantum field expansion (A26) in terms of creation and annihilation operators, and the definition of the time-ordering symbol (A17). The time ordering operator is included in order to automatically deal with both particle and antiparticle emission.18  
**Remark 3**. : _Definition 4 is inspired by the propagator-based quantum field-theoretic formalism of condensed matter physics, see in particular [PERSON]'s elegant formulation in [60], which influenced our approach here. The same quantity \(G_{q}(x,x^{\prime})\) is constructed in perturbative QFT as the fundamental tool for computing scattering cross-sections in experiments involving fundamental particle interactions [38, 54, 67] and the ground state energy in many-body condensed matter physics [59, 60, 61]. In the expression (27), \(\mathcal{T}\), the time-ordering operator, is inserted in order to automatically ensure that fields with later time components \(x^{0}\) are always placed to the left of earlier ones._  
Theorem 2 provides a characterization of the q-antenna Green's function expressed directly in spacetime.