# Fundamental Spacetime Representations of Quantum Antenna Systems
## 4 Linear Quantum Antenna Systems
### An Elementary Model for Point Quantum Particle Excitation  
In our model, a fundamental spacetime quantum field \(\phi(x)\) is associated with the q-antenna. [15] We start by provisionally identifying \(\phi(x)\) as the \"quantum source field\" of the q-antenna system, i.e., the fundamental quantum field of the system directly produced by the source \(J(x),x\in D_{\mathrm{s}}\). Roughly speaking, this terminology indicates that this quantum field plays a double role:  
1. It emits quantum particles (massive particles when \(m\
eq 0\) and scalar Klein-Gordon particles when \(m=0\)) at particular spacetime positions. 2. Once generated, the quantum field \(\phi(x)\) would somehow \"propagate\" the quantum particle in space and time. Equivalently, we say that the quantum field \(\phi(x)\) enjoys the double role of being both the producer and propagator of the quantum radiation particle. Next, let us hit the vacuum state \(|0\rangle\) with the q-antenna source field \(\phi(x^{\prime})\) operator; i.e., we wish to excite a quantum particle at the position \(x^{\prime}=(t^{\prime},\mathbf{x}^{\prime})\). Using (A26) and the standard facts \(a_{\mathbf{p}}|0\rangle=0\) and \(a_{\mathbf{p}}^{\dagger}|0\rangle=|\mathbf{p}\rangle\), we compute  
\[\phi^{\dagger}(x^{\prime})|0\rangle=\int_{\mathbf{p}\in\mathbb{R}^{3}}\frac{ \mathrm{d}^{3}p}{(2\pi)^{3/2}}\frac{e^{\mathrm{i}p_{\mathbf{p}}x^{\prime \prime}}}{(2\omega_{\mathbf{p}})^{1/2}}|\mathbf{p}\rangle. \tag{21}\]  
Therefore, we managed to set up a superposition of _outgoing particles_, i.e., radiation quanta _leaving_ the position \(\mathbf{x}^{\prime}\) at time \(t^{\prime}\). This may explain why \(\phi(x)\) was duped the quantum source field of the q-antenna: The field literally \"creates\" many quantum particles radiating away from the initial source location. (The latter source plays the role of the \"initial state\" in quantum mechanics, though it should be noted that QFT does not use the same concept of the state). Furthermore, with the help of the expansion (21), we are now in possession of a satisfactory understanding of the composition of the quantum states radiated by this point-like quantum source: they are essentially superpositions of pure momentum states \(|\mathbf{p}\rangle\), with scaling factors (momentum state excitation strength) given by the total factor of the integrand of (21) multiplying each such momentum ket. [16]  
Let us now compute how much probability amplitude there is in the new excited q-antenna state (21) when an observer tries to measure the q-antenna's radiation field's momentum. If the observation momentum eigenvalue is \(\mathbf{q}\), then we form the relativistic bra  
\[\langle q|=(2\pi)^{3/2}(2\omega_{\mathbf{p}})^{1/2}\langle\mathbf{q}|, \tag{22}\]through which one may estimate the amplitude \(A(q)\) by calculating the matrix element  
\[A(q):=\langle q|\phi^{\dagger}(x^{\prime})|0\rangle=\int_{\mathbf{p}\in\mathbb{R} ^{3}}\frac{\mathrm{d}^{3}p}{(2\pi)^{3/2}}\frac{e^{\mathrm{i}p_{x}x^{\prime y}}} {(2\omega_{\mathbf{p}})^{1/2}}\langle q|\mathbf{p}\rangle. \tag{23}\]  
Using the basic orthogonality relation between momentum eigenstate [57]  
\[\langle\mathbf{q}|\mathbf{p}\rangle=\delta(\mathbf{q}-\mathbf{p}), \tag{24}\]  
the integral (23) can be readily evaluated, yielding  
\[A(q)=e^{\mathrm{i}q_{x}x^{\prime y}}=e^{\mathrm{i}(\omega_{\mathbf{q}}- \mathbf{q}\cdot\mathbf{x}^{\prime})}. \tag{25}\]  
This is a deceptively simple-looking relation, but it underlies a powerful structure that enjoys considerable importance in the theory of q-antennas. From the engineering point of view, the expression (25) may be shown to lead to the emergence of the classical antenna array factor when moving into the quantum context, hence the ability to shape the quantum radiation emitted by q-antennas using techniques borrowed from what is essentially classical antenna theory.