# Fundamental Spacetime Representations of Quantum Antenna Systems
## 6 Quantum Antenna Radiation Patterns: Basic Constructions
### The Probability Law of Producing Radiated Quantum States  
From the general rules of QFT [54, 56, 57], the probability of measuring a particle with four-momentum \(q\) within the range \(Q^{\prime}\subset\mathbb{M}^{4}\) is given by the integral of the square of the probability amplitude \(A(q)\):  
\[\Pr\{q\in Q\}=\frac{1}{\alpha}\int_{q\in Q^{\prime}}\mathrm{d}^{4}q\ |A(q)|^{2}=\frac{1}{\alpha}\int_{\mathbf{q}\in Q}\frac{\mathrm{d}^{3}q}{(2 \pi)^{3}}\frac{|A(\mathbf{q})|^{2}}{2\omega_{\mathbf{q}}}, \tag{54}\]  
where \(Q\subset\mathbb{R}^{3}\) is the projection of \(Q^{\prime}\subset\mathbb{M}^{4}\) onto \(\mathbb{R}^{3}\), and we made use of (A34) to derive the second equality in (54). The number \(\alpha\in\mathbb{R}^{+}\) is a normalization constant defined by (57) inserted in order to ensure that the probability of any event is between 0 and 1. **Remark 16** (Relativistic normalization): _In writing the second equality in (54), a delta function_  
\[\delta(p^{2}-m^{2}),\quad p^{0}>0, \tag{55}\]  
_is included in the \(d^{4}p\)-integral (54), i.e., the mass shell condition expressed by the dispersion relation (A27); see Appendix G for more details on this background. Figure 4: The fundamental causal structure of the q-antenna radiation process. For simplicity, we only show future lightcones. **Remark 17** (**Probabilistic normalization**). : _The correct relativistic normalization of the four-momentum kets \(|p\rangle\) and \(|q\rangle\), e.g., see relations such as (A38), which we have been already utilizing from the outset, does not automatically ensure the required probability normalization condition_  
\[\Pr\{q\in\mathbb{M}^{4}\}=\frac{1}{\alpha}\int_{q\in\mathbb{M}^{4}}\mathrm{d}^{ 4}q|A(q)|^{2}=1. \tag{56}\]  
_The latter is in fact a consequence of the normalization in (54) by_  
\[\alpha:=\int_{q\in\mathbb{M}^{4}}\mathrm{d}^{4}q|A(q)|^{2}, \tag{57}\]  
_whenever the integral converges._  
The normalization condition (56) is not a consequence of the completeness relation of the Lorentz momentum states (A38), but it should be enforced by hand in order to extract useful probability statements from the theory [85]. However, it should be stressed that the normalization integral (56) for the \(q\)-states is not convergent in the usual sense since _exact_ continuous momentum states such as \(|q\rangle\) or \(|\mathbf{q}\rangle\) are unormalizable [52, 54, 86]. For more detailed information about the proper rigorous mathematical theories that can deal with problems involving continuous states, see [37, 55, 67, 79]. In practice, we seldom need to directly compute continuous state representation, since one always works with \"discretized states\", where momentum or wavevectors are measured in a finite range [2, 87]. The set of all such finite-range states is normalizable [54]. Therefore, whenever expressions such as (56) are encountered, they should be understood in the above of sense of being approximated by discrete sum after which the normalization to unity becomes correct. Nevertheless, for some choices of \(A(q)\), for example the point source with \(A(q)=1\), the integral (56) diverges regardless of whether one works with continuous or discretized state representations. This is an example of the persistent and well-known problem of ultraviolet (UV) divergences in QFT [38]. However, this problem is not a major issue in q-antenna theory, since in practical settings, one often requires a smoothly switchable source. In such types of source systems, the Fourier amplitudes of the spacetime source function decay fast enough in momentum space in order to secure the convergence of infinite spectral integrals such as (56). For in-depth discussion of this scenario backed by several examples taken from the area of high-energy physics, see [PERSON]'s lectures [38]. We next give a few examples to illustrate this formulation. **Example 2** (**General quantum source's radiation probability law at a particular momentum**). : First, we construct a suitable probabilistic representation of a point source radiation function.