# Fundamental Spacetime Representations of Quantum Antenna Systems
## 6 Quantum Antenna Radiation Patterns: Basic Constructions
### The Probability Law of Producing Radiated Quantum States
Introduce the Euclidean ball  
\[Q(\mathbf{q}_{0},\varepsilon):=\{\mathbf{q}\in\mathbb{R}^{3},|\mathbf{q}- \mathbf{q}_{0}|<\varepsilon\}, \tag{58}\]  
where \(\mathbf{q}_{0}\in\mathbb{R}^{3}\), while \(\varepsilon\in\mathbb{R}^{+}\) is very small. (The norm \(|\cdot|\) is that inherited from the standard Euclidean metric on \(\mathbb{R}^{3}\)). The probability of measuring a momentum \(\mathbf{q}\) in the ball \(Q(\mathbf{q}_{0},\varepsilon)\) is given by  
\[\Pr\{q\in Q(\mathbf{q}_{0},\varepsilon)\}=\frac{1}{\alpha}\int_{\mathbf{p}\in Q (\mathbf{q}_{0},\varepsilon)}\frac{\mathrm{d}^{3}q}{(2\pi)^{3}}\frac{|A( \mathbf{q})|^{2}}{2\omega_{\mathbf{q}}}\simeq\frac{(4/3)\pi\varepsilon^{3}}{ \alpha(2\pi)^{3}}\frac{|A(\mathbf{q}_{0})|^{2}}{2\omega_{\mathbf{q}_{0}}}= \frac{\varepsilon^{3}}{12\pi^{2}\alpha}\frac{|A(\mathbf{q}_{0})|^{2}}{2\omega _{\mathbf{q}_{0}}}, \tag{59}\]  
where \((4/3)\pi\varepsilon^{3}\) is the volume of the ball \(Q(\mathbf{q}_{0},\varepsilon)\) centered at \(\mathbf{q}_{0}\) with radius \(\varepsilon\). Therefore, the expression (59) gives the probability of measuring a momentum \(\mathbf{p}=\mathbf{q}_{0}\) after exciting the ground state \(|0\rangle\) of the q-antenna to a higher-energy state \(|q_{0}\rangle\). It is completely general, regardless of how the vacuum state was excited. **Example 3** (**Point source quantum radiation probability model**). : Let us apply the rule (54) to the single quantum source probability amplitude (25). By treating (25) as a special case of (54), we find that for a point quantum source firing at the spacetime point \(x^{\prime}\), the probability amplitude is \(A(q)=1\); hence, the total radiation probability is given by  
\[\Pr\{\mathbf{q}=\mathbf{q}_{0}\}\simeq\frac{1}{\alpha}\frac{\varepsilon^{3}}{12 \pi^{2}\omega_{\mathbf{q}_{0}}}=\frac{\varepsilon^{3}}{12\pi^{2}\alpha\sqrt{| \mathbf{q}_{0}|^{2}+m^{2}}}, \tag{60}\]  
after using the dispersion relation (A27). Since the expression (60) depends only on the amplitude of the momentum \(\mathbf{q}_{0}\), not its direction, we conclude that the q-antenna with a single scalar quantum source is an _isotropic_ radiator since it radiates its various four-momentum states \(|q\rangle\) in the same manner in all directions. This is not very surprising, since the quantum field of the q-antenna is scalar, and in classical antenna theory, a scalar point source is also isotropic [27]. **Remark 18**. : _From the antenna viewpoint, the examples above suggest that the quantum state of the q-antenna is inherently global and hence extendable everywhere. Therefore, we can not guarantee that the q-antenna has effectively radiated into every spacetime point until a concrete measurement process is performed at some position \(x\), after which one may determine the probability of actual detection there using the receive antenna model with the help of the q-antenna Green's function in Theorem 2. Such a process requires introducing a more sophisticated approach to measure probability amplitudes when the observation point is included as a parameter in the system. This subject will be taken up again in Section 6.4._