# Fundamental Spacetime Representations of Quantum Antenna Systems
## 3 The General Theory of Quantum Antenna Systems
### The General Expansion Theorem of Quantum Radiation Fields  
We now generalize the concept of quantum source in order to take into account many-point interactions. Let us consider a generic quantum source located in the region \(D_{\mathrm{s}}\subset\mathbb{M}^{4}\). **Definition 3** (**Generalized quantum antenna source**). : _A generalized quantum source is defined as a countable set of real-valued functions \(J_{n}\) of the form_  
\[J_{n}:\underbrace{D_{\mathrm{s}}\times D_{\mathrm{s}}\cdots\times D_{\mathrm{ s}}}_{n\ \mathrm{times}}\rightarrow\mathbb{R},\ \ \ n\in\mathbb{N}, \tag{13}\]  
_where \(D_{\mathrm{s}}\) is compact. In other words, a generalized quantum antenna  
We discuss the physical meaning of Theorem 1. For clarity, let us rewrite the general expression (15) as a sum of first-order term (linear radiation) and all higher-order terms:  
\[A(x)=\underbrace{\int\mathrm{d}^{4}x_{1}G^{1}(x,x_{1})J(x_{1})}_{\text{Linear quantum radiation integral}}+\underbrace{\sum_{n=2}^{\infty}\int\prod_{l=1}^{n}\mathrm{d}^{4}x_{l}\ G^{n}(x,x_{1},\dots,x_{n})J(x_{1},\dots,x_{n})}_{\text{ Nonlinear quantum radiation higher-order integrals}}. \tag{16}\]  
The first term on the RHS represents conventional linear quantum radiation processes and will be studied in detail in the remaining parts of this paper. It involves a usual quantum source function \(J(x)\) (see Definition 2), with a two-point [PERSON]'s function \(G^{1}(x,x^{\prime})\) serving as a \"system spacetime transfer function\" of the antenna system (or \"generalized impulse response\" using the terminology of signal processing and the engineering sciences). Note that while \(A(x)\) is a complex probability amplitude the like of which is completely absent in the classical world, radiation expressions analogous to the first term on the RHS of (16) do possess some--at least formal--structural similarity to the classical radiation formula (see Section 5.2 for an in-depth comparative analysis between different antenna types). On the other hand, the remaining terms in (16) involve _mutual_ source functions of the form \(J(x_{1},\dots,x_{N})\). These are _joint interaction_ terms describing _coupling phenomena_ among the generic points \(x_{1},\dots,x_{n}\in D_{\text{s}},n>1\), which may introduce mutual correlation between some or all points of the source systems, which in turn are ultimately explainable as many-body effects [9] as illustrated by the following example. **Example 1** (**Classical current source**). : In order to appreciate why the presence of joint source functions signifies interactions, let us consider the well-known case of a classical current source (i.e., the backreaction of the quantum field on the source is ignored) when such source function \(J(x)\) is inserted into the Lagrangian of quantum field theory [14, 62]. In that case, one may obtain an exact solution of the interaction problem. It turns out that higher-order processes in this solution can be all expressed as simple direct multiplications of the same source function \(J(x)\), i.e., we have:  
\[J_{n}=J(x_{1},\cdots,x_{n})=\Pi_{l=1}^{n}J(x_{l})=J(x_{1})J(x_{2})\times\cdots J (x_{n}). \tag{17}\]  
Therefore, in such theory, the second term on the RHS of (16) looks like  
\[\int\mathrm{d}^{4}x_{1}\mathrm{d}^{4}x_{2}\ G^{2}(x,x_{1},x_{2})J(x_{1})J(x_{2 }), \tag{18}\]  
which is essentially nonlinear in the current source.