# Fundamental Spacetime Representations of Quantum Antenna Systems
## 6 Quantum Antenna Radiation Patterns: Basic Constructions
### Constructing the Quantum Antenna Directivity Pattern  
Our goal here is to estimate the directive properties of a continuous source \(J(x^{\prime})\) located in the spacetime region \(D_{\mathrm{s}}\). The probability amplitude of measuring the four-momentum \(q\) is denoted by \(A[q;J(x^{\prime}),D_{\mathrm{s}}]\) but will be abbreviated to \(A(q)\) whenever there is no confusion about the source. Our main tool here is the following theorem. **Theorem 5**. : _Consider a q-antenna system \(J(x),x\in D_{\mathrm{s}}\subset\mathbb{M}^{4}\). Let the corresponding probability amplitude be \(A(q)\). Then, the following formula holds:_  
\[|A(q)|^{2}=|J(\mathbf{q},\omega_{\mathbf{q}})|^{2}, \tag{61}\]  
_where \(J(q)\) is the spacetime Fourier transform of the radiating current source (A45)._  
Proof. : See Appendix I. **Remark 19**. : _Since there is a one-to-one (injective) mapping between the Lorentz four-momentum states \(q\in\mathbb{M}^{4}\) and the conventional three-momentum vectors \(\mathbf{q}\in\mathbb{R}^{3}\) via the dispersion relation (A27), we may just write \(A(\mathbf{q})\) instead of \(A(\mathbf{q},\omega_{\mathbf{q}})\)._  
**Theorem 6**. : _For the situation described in Theorem 5, the quantum radiation probability amplitude can be expressed as:_  
\[\Pr\{\mathbf{k}\in Q\}=\frac{1}{\alpha}\int_{\mathbf{k}\in Q}\frac{d^{3}k}{(2\pi)^{3} }\frac{|J(\mathbf{k})|^{2}}{2\omega_{\mathbf{k}}}, \tag{62}\]  
_for a generic three-dimensional region \(Q\subset\mathbb{R}^{3}\). Here, \(J(\mathbf{k}):=J(\mathbf{k},\omega)\) is the momentum space Fourier transform of the source defined by (A45); see also Remark 19 on the reduced notation in momentum space used here._  
Proof. : Use the probability law (54) in (61) and make the replacement \(\mathbf{q}\to\mathbf{k}\). An immediate application of Theorem 6 is in expressing the total energy radiated by a quantum source directly in terms of the classical source function \(J(x)\), as will be illustrated through the next example. **Example 4 (Total radiated energy and momentum of a q-antenna in momentum space). ** The total energy radiated by the antenna in a momentum space region \(Q\subset\mathbb{R}^{3}\) can be obtained by multiplying the probability of particle production in the infinitesimal momentum space volume \(\mathrm{d}^{3}k\) by the particle's energy, which is \(\hbar\omega_{\mathbf{k}}\) (or just \(\omega_{\mathbf{k}}\) in natural units), and then integrating. With the help of (62), this procedure yields:  
\[\mathcal{E}[Q]=\frac{1}{2}\int_{\mathbf{k}\in Q}\frac{\mathrm{d}^{3}k}{(2\pi)^ {3}}|J(\mathbf{k})|^{2}, \tag{63}\]  
where \(\mathcal{E}\) is the total energy. Similarly, the expected value of the total radiated momentum can be estimated via the formula  
\[\mathcal{P}[Q]=\frac{1}{2}\int_{\mathbf{k}\in Q}\frac{\mathrm{d}^{3}k}{(2\pi) ^{3}}\frac{|J(\mathbf{k})|^{2}\mathbf{k}}{\sqrt{\left|\mathbf{k}\right|^{2}+ m^{2}}}, \tag{64}\]  
where use was made of the fact that the momentum of the particle excitation associated with the \(\mathbf{k}\)th state is \(\hbar\mathbf{k}\) (or \(\mathbf{k}\) in natural units). **Remark 20**. : _It is interesting to note that the momentum space energy pattern of the q-antenna (63) is similar to the energy directivity of a classical source in generic medium. For example, see how general formulas of directivity for classical radiators were constructed recently in momentum space for generic homogeneous nonlocal domains [71, 72, 73]._  
In order to put relation (62) into further practical use, we transform momentum representations into spherical coordinates so we may obtain some information about the directive properties of quantum radiating sources as illustrated by the following example. **Example 5 (Radiation angular directivity pattern of q-antennas in momentum space).