# Fundamental Spacetime Representations of Quantum Antenna Systems
## 6 Quantum Antenna Radiation Patterns: Basic Constructions
### Constructing the Quantum Antenna Directivity Pattern
** Let us express the wavevector \(\mathbf{k}\) in terms of the spherical angular coordinates \(\theta\) and \(\varphi\) by using the standard transformation  
\[\hat{k}=\hat{k}(\Omega)=\hat{x}_{1}\cos\varphi\sin\theta+\hat{x}_{2}\cos \varphi\sin\theta+\hat{x}_{3}\cos\theta, \tag{65}\]  
where  
\[\Omega:=(\theta,\varphi),\quad\hat{k}(\Omega):=\frac{\mathbf{k}}{\left| \mathbf{k}\right|}. \tag{66}\]  
Here, \(\hat{x}_{i},i=1,2,3\), constitute a set of three mutually orthogonal Cartesian unit vectors. The angles \(0<\varphi<2\pi,0<\theta<\pi\) determine the direction of the unit vector \(\hat{k}\), or where the emitted quantum particle's momentum \(\mathbf{p}=\hbar\mathbf{k}\) is pointing in three-dimensional space. Since physically and intuitively, one would expect that the directions in three-dimensional position space along which momenta tend to maximally flow also correspond to the directions toward which most of the particle ensemble's energy and momentum are directed, we may use Theorem 6 in conjunction with (66) to estimate the quantum source's directive properties in generic three-dimensional scenarios. A simple way to achieve this is by re-expressing (62) using (65) and (66), resulting in  
\[\Pr\{k\in[k_{\min},k_{\max}],\ \Omega\in\Omega_{0}\}=\frac{1}{\alpha}\int \limits_{k\in[k_{\min},k_{\max}]}\int\limits_{\Omega\in\Omega_{0}}\mathrm{d}k \,\mathrm{d}\Omega\,\frac{k^{2}\sin\theta}{(2\pi)^{3}}\frac{\left|J\right[k, \hat{k}(\Omega)]\right|^{2}}{2\sqrt{k^{2}+m^{2}}}, \tag{67}\]where  
\[k:=|\mathbf{k}|,\ \ \ \mathrm{d}\Omega:=\mathrm{d}\theta\mathrm{d}\varphi. \tag{68}\]  
The angular set \(\Omega_{0}\) in (67) is the three-dimensional (solid) angular sector inside where one is interested in characterizing the radiating quantum antenna system. **Remark 21**. : _It is important not to confuse \(k\in\mathbb{R}^{+}\), defined by (68), with the four-vector \(k=k^{\mu}\in\mathbb{M}^{4}\) (Appendix B). In this section, we use the shorthand notation (68) in order to simplify the presentation._  
Inspired by the expression (67), we may then introduce the following definition of momentum space directivity for quantum antennas:  
**Definition 6** (**Quantum antenna directivity in momentum space**). : _The momentum space directivity of a \(q\)-antenna with source function \(J(x),x\in D_{\mathrm{s}}\) is defined as the following angular function_  
\[\mathcal{D}(\varphi,\theta;k):=\frac{1}{\alpha_{k}}\frac{k^{2}\sin\theta}{(2 \pi)^{3}}\frac{\left|J\left[k,\hat{k}(\varphi,\theta)\right]\right|^{2}}{2 \sqrt{k^{2}+m^{2}}}, \tag{69}\]  
_where_  
\[\alpha_{k}:=\frac{1}{4\pi}\int_{4\pi}d\Omega\frac{k^{2}\sin\theta}{(2\pi)^{3} }\frac{\left|J\left[k,\hat{k}(\varphi,\theta)\right]\right|^{2}}{2\sqrt{k^{2} +m^{2}}} \tag{70}\]  
_is a positive momentum-dependent probability normalization constant._  
Expressed in terms of the momentum-space-type directivity (69), the total probability in (62) may be be put into the more compact form  
\[\Pr\{\mathbf{k}\in Q\}=\frac{1}{\alpha}\int_{\mathbf{k}\in Q}\frac{d^{3}k}{(2\pi)^{3} }\frac{|J(\mathbf{k})|^{2}}{2\omega_{\mathbf{k}}}, \tag{62}\]0  
The last expression explains the physical motivation behind Definition 6. Moreover, the following two remarks explain more about the engineering background to directivity concepts in antenna theory motivating the above definition itself. **Remark 22**. : _It is evident then from (71) that for a given \(\hat{k}\), the function \(\mathcal{D}(k,\hat{k})\) is the angular probability density in the momentum space variable \(k\). On the other hand, the mathematical dependence of \(D(k,\hat{k})\) on the angles \(\varphi\) and \(\theta\) provides information on how the emitted quantum particles tend to flow along different directions in space. This is why \(D\) does indeed behave as a momentum-space radiation pattern (probability per unit momentum per solid angle)._  
**Remark 23**.