**1. Introduction**

The main objective of this paper is to develop conceptual, physical, and mathematical foundations for quantum antenna theory based on a very broad approach to generic quantum fields produced and consumed by source and sink systems separated in spacetime. Quantum antenna technology is a recent emerging subfield within the larger and more fluid research area often referred to as quantum engineering, quantum technologies, or just quantum information processing. In particular, and within this subfield, we find that the main intention behind the desire for developing a new "quantum antenna technology" is to serve the needs of current and future quantum *communication* systems [1–4], where information is transmitted using quantum states [5,6], regardless of whether digital data are encoded as classical bits or qbits [1,7–10], with obvious applications to physical-layer security [11–14]. However, the peculiar system known as "the quantum antenna" may

**Citation:** Mikki, S. FundamentalSpacetime Representations of Quantum Antenna Systems. *Foundations* **2022**, *2*, 251–289. https://doi.org/10.3390/ foundations2010019

Academic Editor: Michal Hnatiˇc

Received: 15 November 2021 Accepted: 3 February 2022 Published: 2 March 2022

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also perform functions other than mere information transmission in wireless quantum links, for example, quantum tomography, quantum state estimation, biophotonics, sensing, molecular communications, space exploration, and other applications [13,15–26].

We believe that quantum antenna theory and technology may be viewed as an attempt to synthesize three already established subfields: (1) classical antennas [27,28], (2) optical nanoantennas [29,30], and (3) quantum emitters [29,31]. As such, a quantum antenna is often confused with one of these three topics. For example, occasionally, laser sources or photodiodes are treated as quantum antennas by some authors working in free space optical communications. This is *not* what we understand by the term here. A laser source could constitute a part of the q-antenna system, while a larger part, for example a terahertz or an optical antenna [29,32], may serve as a secondary reflector as in nanoantennas [30]. In addition, an atom emitting a photon after undergoing a transition from excited to ground states is not considered an "antenna" in this paper. Instead, we propose the following general definition:

**Definition 1** (**Antennas**)**.** *By the term antenna, we follow classical antenna theory where the antenna system is defined as an externally controlled spatiotemporal current distribution capable of controlling the spatiotemporal properties of the radiation emitted by the antenna system.*

It will be seen that the most salient point in this definition is the emphasis on the following three features:


Hence, the existence of *spatiotemporal*, *externally* controlled source current distributions capable of modifying its produced *radiation* in both space *and* time is the main content of antenna theory.

Clearly enough, mainstream examples in quantum optics such as a laser source by itself, or an isolated atom undergoing a spontaneous emission of light, do not fall under a conceptual umbrella such as the one supplied by Definition 1, at least not naturally. However, such conventional quantum optical sources may still serve as essential *sub*components of such systems. For example, probably, the most natural method to inject a controlled *time* signal into a radiating nanostructure is through a modulated laser beam. However, controlling the *spatial* distribution of the radiating *current* excited by this time signal would require the use of *additional* methods and objects; for instance, an optical antenna, or an engineered metamaterial, or an optimized array of point sources. Even though the data to be transmitted are encoded into time signals, in general, the quantum antenna system is much more complex than its time excitation method. There is a need, then, to understand the complex and multifaceted nature of the spatiotemporal structure of the radiation field emitted by a generic quantum system at a very broad level. This paper is a contribution toward this goal.

The classical theory of controlled radiation has been extensively studied and developed in applied electromagnetics [27,28,33–35]. On the other hand, the general theory of quantum antennas has not been investigated in depth so far, in spite of the publication in recent years of a number of reports and proposals about the subject, not restricted only to quantum wireless communications but also involving other topics as well, e.g., see [15–21]. Some of the main conceptual and philosophical hurdles that a viable theory of quantum antennas need to overcome include, though by no means are restricted to, the following issues:


It should be noted that these conceptual and technical issues are still considered difficult open research fields in both foundational and applied research as they have not been resolved even within standard quantum field theory itself. For instance, it is still not clear what the ultimate meaning of "particle" in quantum fields is [36,37]; questions about the nature of quantum excitations in interacting field theories have been asked in the past [38] and are still being investigated up to date [39]. Therefore, there is a need to re-examine the subject of quantum antennas at a very general and fundamental level, that of developing possible foundations for the topic that may help illuminate current and future open theoretical problems on one hand, and to help devise and evolve new genera of quantum systems and applications on the other hand. A viable candidate for such foundational approach, we believe, is to formulate the entire problem of quantum antennas and quantum radiation using a relativistic spacetime formalism (QFT in this case).

Our intended goals in the present article include a wide spectrum. Most of these objectives are research related and can be classified as part of the emerging area of quantum antenna theory within both quantum physics and quantum engineering. Yet, additionally, other aspects in our work are pedagogical, relating to the wish to reach a wide multidisciplinary audience. We summarize these two flavors of our main goals as follows:

	- 1. Generalizing the concept of antennas beyond acoustic and electromagnetic antennas, the two concepts that have dominated the field so far, by demonstrating how relativistic QFT can be used to formulate a single and unified concept of "quantum radiators" valid for a large number of possible radiation processes in nature.
	- 2. Providing a concrete illustration of some of the potential algorithmic capabilities of the spacetime formalism of quantum antennas by constructing various possible candidates for radiation pattern functions and gains in the case of the quantum (spin-0) Klein–Gordon q-antennas.
	- 1. Introducing new applications of fundamental theory (here relativistic quantum mechanics) to different audiences, e.g., quantum engineering and quantum technology research.
	- 2. Introducing the subject of QFT in a self-sufficient manner by providing detailed appendices explaining how relativistic quantum mechanics is formulated for an audience familiar only with nonrelativistic quantum mechanics.<sup>1</sup>

This paper is organized as follows. In Section 2, the classical theory of antennas is re-examined, and the comparison with the new, more general concept of quantum source radiators is explored at the thematic level in order to prep the reader for the subsequent, more technical quantum-field theoretic treatment. This is followed in Section 3 by a very broad view on the theory of quantum radiators developed based on interacting quantum mechanics, without much emphasis at this stage on the relativistic quantum field theoretic scenario. The purpose is to map out the generic structure of the problem and to highlight the distinction between linear and nonlinear quantum antennas. Starting from Section 4, we narrow down our focus to the special but rich enough special case of the neutral Klein– Gordon field linear quantum antenna, which appears to exhibit many of the salient features of the general quantum antenna system. Then, the abstract formal and physical structure of such a system is explicated in Section 5. To provide more concrete applications of the theory, Section 6 presents a series of examples and constructions aiming at illustrating how one may define radiation pattern measures such as directivity and gain (transmission) coefficients in quantum antenna systems. Finally, we end up with conclusions.

### **2. Antenna Theory: Classical and Quantum Radiation Scenarios**

In analogy to classical antenna theory [27,35,40], we formally define a *quantum antenna* in terms of the mathematical representation of a radiating source term as follows:

**Definition 2** (**Quantum Antennas**)**.** *Within the context of relativistic quantum field theory, a quantum antenna (q-antenna)* J *is an operator map of the form*

$$\mathcal{J}: I \to \Psi,\tag{1}$$

*where* Ψ *is a quantum field system produced by the abstract source function*

$$J: \mathbb{M}^4 \to \mathbb{K},\tag{2}$$

*which maps the Minkowski spacetime* M<sup>4</sup> *of special relativity to the set* K = R *(real numbers) or* K = C *(complex numbers). We assume that the function J*(*x*), *x* ∈ *D*s ⊂ M<sup>4</sup> *is integrable over any spacetime region D*s *of interest. We further assume that the source J*(*x*) *is compactly supported; i.e., cl*{*<sup>D</sup>*s} *is compact where cl is the closure operator in the standard Euclidean topology on* R4*.*

Physically, the main property of the q-antenna (source) function *J*(*x*) is that it controls the quantum radiation <sup>Ψ</sup>(*x*), *x* ∈ M4, emitted by the source system *J*(*x*), *x* ∈ *D*s ⊂ M4. The generic configuration itself is shown in Figure 1. In fact, much of this paper will concentrate on understanding the precise nature of how a generic source function *J*(*x*) can control the spacetime structure of the quantum field emission <sup>Ψ</sup>(*x*) in applications characteristic of the special context of an arbitrary quantum *communication* system (source-channel-detector system). Thus, the article's main scope is the theoretical foundations of the physical layer of the quantum communication link structure, e.g., how a quantum state can be directed to a given spacetime region through the choice of a proper abstract source function. The particular discipline devoted to this problem is the area we dub *quantum antenna theory*, which is a relatively new interdisciplinary research area.

**Figure 1.** The fundamental configuration of a quantum antenna communication system in spacetime.

**Remark 1.** *It may be noted here that there is a similarity between the quantum communication problem [7] and the quantum measurement problem in quantum physics [41–43]. The analogy is manifested by the common underlying abstract structure shared by the two distinct processes of communications and generalized quantum measurement [1,8]. Both involve a source of information (often called quantum state preparation in QM), which is localized within a given spacetime region; and a process of destructive consumption of the transmitted information, this time localized at a receiver or detector located at a timelike-distant spacetime region [43,44]. The quantum receiver system's ultimate objective is to extract the maximum amount of information embodied in the received quantum state in QM, or quantum field in QFT, through the judicious utilization of optimized combinations of measurement operations and signal processing algorithms.*

Quantum antenna systems turn out to be considerably more complex to understand and analyze than conventional antennas, where the latter antenna type is often approached within the framework based on *classical* electromagnetism (and sometimes acoustics). A reason behind this noticeable difference in complexity between the two theories might be related to the fact that a proper and completely satisfactory understanding of quantum interactions is ultimately based on relativistic QFT, which is inherently a *many-particle* world picture. That is, in QFT, it is inconsistent to assume the possibility of a singleparticle-interaction configuration due to the inherent tension that this assumption will bring with special relativity (SR) and other fundamental principles [38]. On the other hand, a traditional classical antenna is essentially a "one-point physical process" in which a generic source point *x* = (**x** , *t* ) radiates into another arbitrary observation point *x* = (**x**, *t*), where the total radiation is just the linear superposition of all contributions emanating from all relevant individual non-interacting spacetime points [45].

The most obvious and rigorous way to understand the classical process of radiation and reception of electromagnetic signals is through the two fundamental Green's functions of the antenna system, the source or current Green's function **<sup>F</sup>**(**<sup>x</sup>**, **x** , *t* − *t* ) [28,46–48], and the radiation Green's function **<sup>G</sup>**(**x** − **x** , *t* − *t* ) [49–51]. These two are essentially defined via their crucial superposition-like integral summation rules (A1) and (A2); see Appendix A, where the Green's function approach to classical antenna theory is briefly reviewed and additional references are given.

However, when we next move toward examining the emerging theory of quantum antenna systems, the natural question is whether this generalized structure of classical antennas persists. It turns out that radiation formulas similar to the Green's function-based expressions (A1)–(A3), Appendix A, can *not* in general be maintained in the case of *quantum* radiation. However, in this paper, we focus on ways to retain as much classical structure as possible in the new quantum antenna theory. This turns out to be possible for the case of *linear* quantum antenna systems, which will be investigated in details below.

To summarize, we find that the following general situation holds in antenna theory:


To be more precise, it can be shown that only *first*-order quantum radiation processes may lead to linear operator radiation relations similar to (A1)–(A3). This is the process that corresponds to what we call *linear quantum antennas*, which will be studied in more details in this paper, while a more complete theory of relativistic nonlinear quantum antennas is relegated to future publications. All other higher-order processes, which are ultimately due to the many-body quantum interactions between source points, can be shown to lead to *nonlinear* contributions to the total probability amplitude of the quantum radiation emitted by the source system.

### **3. The General Theory of Quantum Antenna Systems**

### *3.1. Preliminary Considerations*

According to Definition 2, a q-antenna system consists of a *spacetime source region D*s together with a current source function defined on this region. The *receiving* q-antenna is located in the compact spacetime region *D*r. The overall spacetime configuration is illustrated in Figure 1. To allow information transmission from the source to the receiver regions, the two spacetime domains *D*s and *D*r must have timelike separation [53].<sup>3</sup> For simplicity, *D*s and *D*r as spacetime regions are assumed to be factorizable into the forms

$$D\_{\mathbf{s}} = \mathbf{S}\_{\mathbf{s}} \times T\_{\mathbf{s}\prime} \qquad D\_{\mathbf{r}} = \mathbf{S}\_{\mathbf{r}} \times T\_{\mathbf{r}\prime} \tag{3}$$

where *S*s and *S*r are the three-dimensional spatial subdomain components of the original four-dimensional regions *D*s,r satisfying

$$\mathbb{M}^4 \supset D\_{\mathbf{s},\mathbf{r}} \supset S\_{\mathbf{s},\mathbf{r}} \subset \mathbb{R}^3,\tag{4}$$

while *T*s,r ⊂ R is the "temporal component" (timelike slice) of the source/receive regions *D*s,r, respectively. Here, both *D*s,r and *T*s,r are assumed to be *compact*, i.e., *D*s,r and *T*s,r are closed and bounded sets in the standard Euclidean metric space R3. 4

Throughout this section, the quantum field *φ*(*x*) is allowed to refer to a general field with possible spins 0, 1/2, 1, 2, etc., without worrying much about indices (in the next sections, we work only with scalar fields for simplicity). Moreover, any particular QFT under consideration may itself be of the free (non-interacting) or interacting types. Interacting field theories often require the use of perturbation theory in order to obtain practical results. However, all information about the interacting system is conveniently encoded in the *propagator*, which will be used extensively in our theory below.<sup>5</sup> Therefore, for generality, in this paper, we utilize the propagator concept as the fundamental mathematical carrier of physical information about the generic quantum antenna system in spacetime.

#### *3.2. A Generic Interaction Hamiltonian Description of Quantum Antenna Systems*

Here, a high-level view on interacting field theory is provided. Note that a full treatment of interacting field theories is outside the scope of the present paper, whose main goal is to supply the reader with a minimal picture of how eventually quantum antenna theory should be fully formulated when specific physical layouts, often described with their own Hamiltonian, are introduced. On the other hand, starting from Section 4, we work mainly with the scalar field theory (the Klein–Gordon theory) in order to illustrate the general structure of quantum radiation with minimum knowledge of the full details of matter–field interaction mechanisms. Readers interested in more information related to specialized physical layouts may consult numerous other publications, for example those quoted in the Introduction section of this paper.

We assume that all relevant quantum field operators can be described within one and the same large enough combined Hilbert space H, which in the case of QFT is a Fock space F. The total Hamiltonian operator of the system is written as

$$H = H\_0 + H\_{\text{int}\prime} \tag{5}$$

where *H*0 is the *free* Hamiltonian (non-interacting part that is usually solvable), while *H*int is the *interacting* Hamiltonian, which is a time-dependent operator that in turn can be expanded into four basic component as follows

$$H\_{\rm int} = H\_{\rm in} + H\_{\rm 5} + H\_{\rm I} + H\_{\rm 6}.\tag{6}$$

Here, we have the following three categories of interaction Hamiltonian terms:


A couple of important observations on this decomposition of the interacting Hamiltonian are in order. First, to keep the discussion at the most general level, we assume that the

intrinsic interacting Hamiltonian, if nonzero, is present all the time. That is, it becomes an essential ingredient of the fundamental field *φ*(*x*) of the q-antenna system. In this sense, the latter field is defined as precisely that quantum field corresponding to the Hamiltonian

$$H\_{\Phi} = H\_0 + H\_{\text{in}}.\tag{7}$$

On the other hand, the source, receiver, and channel Hamiltonian terms are treated in this theory as *extrinsic* interactions, i.e., external disturbances coupled to the fundamental quantum field *φ*(*x*) associated with the Hamiltonian *<sup>H</sup>φ*. As expected in quantum physics, coupling often leads to nonlinear equations of motion [54]. In this sense, the uncoupled fundamental field *φ*(*x*) is perturbatively transformed into a *new* coupled field *φ*(*x*) with a full Hamiltonian

$$H\_{\Phi'} = H\_{\Phi} + H\_{\\$} + H\_{\Gamma} + H\_{\mathbb{C}} \tag{8}$$

solved using perturbation Dyson or path integral expansions [55]. Following standard conventions in QFT, we do not change the notation of the field but always use *φ*(*x*) while clearly stating which Hamiltonian is being used (if needed). In addition, we will not label Hamiltonians by their fields or spacetime arguments unless this is needed.

**Remark 2.** *Sometimes, it is more convenient to group channel interactions H*c *and intrinsic field self-interactions H*in *into one term*

$$H'\_{\rm in} = H\_{\rm in} + H\_{\rm c}.\tag{9}$$

*The reason is that there is indeed some similarity between the two types of interactions above. They are both unrelated to the transmitter (source) and receiver terminals and can be considered then as indigenous components of the q-antenna system field itself. However, there are also some differences, since channel couplings are not the same everywhere but are localized at the scattering objects themselves. On the other hand, self-interactions and also intrinsic mutual interactions captured by the term H*in *are generally "turned on" most of the time. Nevertheless, in some applications, it might be useful to group H*in *and H*c *with each other under the rubric of scattering-based interaction processes in order to distinguish them from transmitter and receiver types of interactions (information source and sink).*

Let us examine now how the source and the receiver terminals of the system may interact. Working in the Schrodinger picture,<sup>7</sup> Let the state of the source at the end of the interaction interval *T*s = [*t*s, *t*s] be denoted by |*J*, *<sup>D</sup>*<sup>s</sup>*<sup>t</sup>*s . Here, *J* is a generic symbol for the overall set of (classical) disturbances supported on *D*s ultimately constituting our proposed formal Definition 3 of the q-antenna current source system. Then, we may write

$$\left| \left| f, D\_{\mathfrak{k}} \right\rangle\_{t\_{\mathfrak{k}}} = P(t\_{\mathfrak{k}}', t\_{\mathfrak{k}}; f, D\_{\mathfrak{k}}) \left| 0\_{\text{in}} \right\rangle,\tag{10}$$

where *<sup>P</sup>*(*t*s, *t*s; *J*, *<sup>D</sup>*s) is the interaction picture (Dirac) propagator (A18), which is reviewed in Appendix D for more details. That is, the computation of this propagator is based on the substitution *H*I = *H*s into (A19). The special state labeled |0in is the ground state of the *intrinsically interacting* system, i.e., the state satisfying

$$(H\_0 + \underbrace{H\_{\text{in}} + H\_{\text{c}}}\_{H\_{\text{in}}'})|0\_{\text{in}}\rangle = 0,\tag{11}$$

where, as discussed above, we have here opted for generality by including all possible wireless channel scattering effects captured by *H*c with the field intrinsic self-interaction Hamiltonian *H*in. Then, the ground state |0in represents both the (i) true or actual physical initial state of the q-antenna system *before* interacting with the source *J* localized in the spacetime domain *D*s; and (ii) the final state reattained *after* finishing interaction with the receiver localized in the disjoint domain *D*r. On the other hand, the *bare* ground state |0 is the ground state of the *free* Hamiltonian *H*0 satisfying

$$H\_0|0\rangle = 0.\tag{12}$$

An exactly analogous general analysis can be conducted in order to understand the receiving q-antenna interaction problem. The latter is that concerned with what happens after interaction with the detector during the time interval *T*r = [*t* r, *<sup>t</sup>*r], change in fields, states, observation outcomes, etc.

### *3.3. 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*s ⊂ M4.

**Definition 3** (**Generalized quantum antenna source**)**.** *A generalized quantum source is defined as a countable set of real-valued functions Jn of the form*

$$J\_n: \underbrace{D\_s \times D\_s \cdot \cdot \cdot \times D\_s}\_{n \text{ times}} \to \mathbb{R}, \quad n \in \mathbb{N}, \tag{13}$$

*where Ds is compact. In other words, a generalized quantum antenna source is defined as the set*

$$\mathcal{J} = \{f\_n, n \in \mathbb{N}\} \tag{14}$$

*of all real-valued functions on all product spaces of D*<sup>s</sup>*.*

The motivations behind the apparently abstract definition are in fact the actual physical relevance of all the functions *Jn* mentioned there. It turns out that *J*1 represents nothing but a direct linear current source, while all higher-order sources *Jn*, *n* > 1, can be interpreted as mutual interaction<sup>8</sup> strength factors. To see this, we now present the following fundamental theorem about quantum sources:

**Theorem 1** (**The q-antenna decomposition theorem**)**.** *Let <sup>A</sup>*(*x*) ∈ C *be the probability amplitude of the observation (particle annihilation) of the quantum antenna field at a location x* ∈ M4*. Then, it follows that when expressed in terms of the generalized q-antenna source of Definition 3, the amplitude <sup>A</sup>*(*x*) *can be expanded as*

$$A(\mathbf{x}) = \sum\_{n=1}^{\infty} \int \prod\_{l=1}^{n} \mathbf{d}^{4} \mathbf{x}\_{l} \, G^{n}(\mathbf{x}, \mathbf{x}\_{1}, \dots, \mathbf{x}\_{n}) f(\mathbf{x}\_{1}, \dots, \mathbf{x}\_{n}), \tag{15}$$

*where all integrals are performed within the source region D*<sup>s</sup>*. Here, J*(*<sup>x</sup>*1, ... , *xn*) *is identified with Jn in Definition 3. The set of functions Gn are* (*n* + <sup>1</sup>)*-point Green's functions where n is a labeling superscript, not a power.*

See Appendix E for some background to the decomposition theorem, where a general discussion and additional references are given. A complete and rigorous treatment of Theorem 1 is outside the scope of the present paper, since the proof is fairly lengthy though relatively straightforward, and the details are not needed in order to understand linear quantum antennas (the case mostly discussed in the remaining parts of this paper). A general idea of the scope of the proof is briefly outlined in Appendix E.

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:

$$A(\mathbf{x}) = \underbrace{\int \mathbf{d}^4 \mathbf{x}\_1 \mathbf{G}^1(\mathbf{x}, \mathbf{x}\_1) I(\mathbf{x}\_1)}\_{\text{Linear quantum radiation integral}} + \underbrace{\sum\_{n=2}^{\infty} \int \prod\_{l=1}^n \mathbf{d}^4 \mathbf{x}\_l \ G^n(\mathbf{x}, \mathbf{x}\_1, \dots, \mathbf{x}\_n) I(\mathbf{x}\_1, \dots, \mathbf{x}\_n)}\_{\text{Nonlinear quantum radiation higher-order integrals}}.\tag{16}$$

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 Green's function *<sup>G</sup>*<sup>1</sup>(*<sup>x</sup>*, *x*) 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 *<sup>A</sup>*(*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*(*<sup>x</sup>*1, ... , *xN*). These are *joint interaction* terms describing *coupling phenomena* among the generic points *x*1, ... , *xn* ∈ *D*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<sup>9</sup> 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:

$$J\_n = f(\mathbf{x}\_1, \cdot, \cdot, \mathbf{x}\_n) = \Pi\_{l=1}^n f(\mathbf{x}\_l) = f(\mathbf{x}\_1) f(\mathbf{x}\_2) \times \dots \times f(\mathbf{x}\_n). \tag{17}$$

Therefore, in such theory, the second term on the RHS of (16) looks like

$$\int \mathrm{d}^4 \mathbf{x}\_1 \mathrm{d}^4 \mathbf{x}\_2 \, G^2(\mathbf{x}, \mathbf{x}\_1, \mathbf{x}\_2) f(\mathbf{x}\_1) f(\mathbf{x}\_2) \, \tag{18}$$

which is essentially nonlinear in the current source. In the more general case, when there is a correlation between *x*1 and *x*2, the following condition holds:

$$J(\mathbf{x}\_1, \mathbf{x}\_2) \neq J(\mathbf{x}\_1)J(\mathbf{x}\_2). \tag{19}$$

However, still even in this correlated latter case, one may expand the function *J*(*<sup>x</sup>*1, *<sup>x</sup>*2) in Taylor series around the uncorrelated case *J*(*<sup>x</sup>*1, *<sup>x</sup>*2) = *J*(*<sup>x</sup>*1)*J*(*<sup>x</sup>*2) in order to understand the general structure of the problem whose general form is now given by the second-order process<sup>10</sup>

$$
\int \mathrm{d}^4 \mathbf{x}\_1 \mathrm{d}^4 \mathbf{x}\_2 \, G^2(\mathbf{x}, \mathbf{x}\_1, \mathbf{x}\_2) f(\mathbf{x}\_1, \mathbf{x}\_2) . \tag{20}
$$

Nevertheless, the message of this example is clear: A higher-order process, i.e., *Jn* with *n* > 1, introduces nonlinear contributions to the total probability amplitude *<sup>A</sup>*(*x*) as per (16).

### **4. Linear Quantum Antenna Systems**
