*4.1. Introduction*

The conceptual, physical, and mathematical structure of quantum antennas will now be further investigated in depth by focusing on the first term on the RHS of (16). This restriction is motivated by the desire to keep the presentation as simple as possible. In fact, most of the key concepts and structure to be developed in connection with the main component of the quantum source, the function *J*1, can be adapted and also applied to the higher-order components *Jn*, *n* > 1, although the details are lengthy. For example, a radiation pattern could be constructed for each source mode function *Jn* similar to the one to be given below (see Section 6), after which the various contributions of all such terms are summed together in order to estimate the total radiation characteristics of the source system.

Our approach is fundamentally based on the idea of propagators and Green's functions in field theory. Similar to the corresponding situation in QFT itself, the concept of Green's function is also fundamental in classical antenna theory [27,28,33,35,49,63,64]. There, one finds that Green's function connects the source (cause) with its produced radiation field (effect), which is a fact that has been frequently exploited as a useful computational tool in both classical electromagnetism [49,63–66] and quantum physics [54,55,67].

However, for quantum radiating systems to behave as antennas, one must track down and separate from out of the total quantum system an ultimate terminal where a purely *classical*—i.e., deterministic—source function is identified. Through such a source function, the quantum source system may then allow for an *external* user control for the purpose of sending information [10,13].<sup>11</sup> For the purpose of this paper, we will show that QFT allows the construction of a very natural and direct source model for q-antenna systems.

In order to simplify the presentation, we break down our method into several steps as follows:


References on the Green's function approach to classical antenna theory can be found in [28,46–48]. The infinitesimal source approach is developed in [68]. The momentum space approach to electromagnetic theory and antennas is outlined in [69–73].

For maximum clarity, this program will be carried out throughout the remainder of the present paper for the special case of a neutral Klein–Gordon field *φ*(*x*) (the standard spin-0 scalar field theory, Section 4.2). In addition, and as already mentioned above, for emphasis on simplicity, we focus on the linear quantum antenna case. In fact, the key concepts introduced below through the Feynman propagator for the construction of a quantum radiation pattern are essentially the same whether the radiation regime under consideration is linear or nonlinear.

### *4.2. The Klein–Gordon Field Theory*

In relativistic quantum field theory [38,54,55], everything takes place in the four-dimensional Minkowski spacetime [51,53], denoted by M4, which is a linear vector space endowed with a special metric, the Lorentz metric *gμν*, see Table A1 in Appendix B, where the relativistic notation and key quantum formulas in the natural unit systems are reviewed. (In what follows, we work in the natural unit systems where *c*, the speed of light, and *h*¯, the Planck constant, are both reduced to unity (*c* = *h*¯ = 1); see Appendix C). Without loss of generality, and as mentioned earlier, the q-antenna quantum field *φ*(*x*) is assumed to

be that of scalar massive theories.<sup>12</sup> To further simplify the mathematical manipulations, we focus on the neutral massive Klein–Gordon field theory, which is outlined in Appendix F. For massive particles (spin-0 particles in our scalar theory example), the dispersion relation (A27) governs the behavior of particles emitted when the q-antenna system is used in applications such as quantum molecular communications. On the other hand, massless (*m* = 0) particles models "scalar photons" or photons with polarization ignored, since in this case, the Klein–Gordon equation reduces to the (scalar) wave equations.<sup>13</sup> In general, adding spin to the theory does not involve any major changes in the main conceptual ideas related to q-antennas and hence will be left for future work.

We also further note that the field modal expansion (A26) is intrinsically Lorentz invariant, even though the spectral integration performed there is carried with respect to the non-Lorentz invariant volume measure d3*x* because the standard method of *normalizing factor* has been already employed in our formulation.<sup>14</sup> In other words, the spacetime q-antenna theory developed here is fully relativistic. A breakdown of relativistic invariance, as exemplified by choices of spatiotemporal decomposition (slicing) of the source or receiver regions *D*s,r, e.g., as in (3), can be introduced later as an external restriction enforced by hand to simplify the calculations and the presentation.

### *4.3. An Elementary Model for Point Quantum Particle Excitation*

In our model, a fundamental spacetime quantum field *φ*(*x*) is associated with the q-antenna.<sup>15</sup> We start by provisionally identifying *φ*(*x*) as the "quantum source field" of the q-antenna system, i.e., the fundamental quantum field of the system directly produced by the source *J*(*x*), *x* ∈ *D*s. Roughly speaking, this terminology indicates that this quantum field plays a double role:


Equivalently, we say that the quantum field *φ*(*x*) enjoys the double role of being both the producer and propagator of the quantum radiation particle.

Next, let us hit the vacuum state |0 with the q-antenna source field *φ*(*x*) operator; i.e., we wish to excite a quantum particle at the position *x* = (*t*, **<sup>x</sup>**). Using (A26) and the standard facts *<sup>a</sup>***p**|<sup>0</sup> = 0 and *<sup>a</sup>*†**p**|<sup>0</sup> = |**<sup>p</sup>**, we compute

$$\phi^\dagger(\mathbf{x'})|0\rangle = \int\_{\mathbf{p}\in\mathbb{R}^3} \frac{\mathbf{d}^3 p}{(2\pi)^{3/2}} \frac{e^{i p\_\mu \mathbf{x'}^\mu}}{(2\omega\_\mathbf{p})^{1/2}} |\mathbf{p}\rangle. \tag{21}$$

Therefore, we managed to set up a superposition of *outgoing particles*, i.e., radiation quanta *leaving* the position **x** at time *t*. This may explain why *φ*(*x*) was duped the quantum source field of the q-antenna: The field literally "creates" many quantum particles radiating away from the initial source location. (The latter source plays the role of the "initial state" in quantum mechanics, though it should be noted that QFT does not use the same concept of the state).

Furthermore, with the help of the expansion (21), we are now in possession of a satisfactory understanding of the composition of the quantum states radiated by this pointlike quantum source: they are essentially superpositions of pure momentum states |**<sup>p</sup>**, with scaling factors (momentum state excitation strength) given by the total factor of the integrand of (21) multiplying each such momentum ket.<sup>16</sup>

Let us now compute how much probability amplitude there is in the new excited q-antenna state (21) when an observer tries to measure the q-antenna's radiation field's momentum. If the observation momentum eigenvalue is **q**, then we form the relativistic bra

$$
\langle q \vert = (2\pi)^{3/2} (2\omega\_\mathbf{p})^{1/2} \langle \mathbf{q} \vert\_\prime \tag{22}
$$

through which one may estimate the amplitude *<sup>A</sup>*(*q*) by calculating the matrix element

$$A(q) := \langle q \vert \phi^\dagger(\mathbf{x'}) \vert 0 \rangle = \int\_{\mathbf{p} \in \mathbb{R}^3} \frac{\mathbf{d}^3 p}{(2\pi)^{3/2}} \frac{e^{i p\_\mu \mathbf{x'}^\mu}}{(2\omega\_\mathbf{p})^{1/2}} \langle q \vert \mathbf{p} \rangle. \tag{23}$$

Using the basic orthogonality relation between momentum eigenstate [57]

$$
\langle \mathbf{q} | \mathbf{p} \rangle = \delta(\mathbf{q} - \mathbf{p}),
\tag{24}
$$

the integral (23) can be readily evaluated, yielding

$$A(q) = \varepsilon^{\mathrm{i}q\_{\mu}\mathbf{x}^{\prime\mu}} = \varepsilon^{\mathrm{i}(\omega\_{\mathbf{q}} - \mathbf{q} \cdot \mathbf{x}^{\prime})}.\tag{25}$$

This is a deceptively simple-looking relation, but it underlies a powerful structure that enjoys considerable importance in the theory of q-antennas. From the engineering point of view, the expression (25) may be shown to lead to the emergence of the classical antenna array factor when moving into the quantum context, hence the ability to shape the quantum radiation emitted by q-antennas using techniques borrowed from what is essentially classical antenna theory.

From the physics point of view, (25) says that the complex probability amplitude of finding the radiated particle emitted at the spacetime point *x* at the four-momentum state |*q* is simply exp <sup>i</sup>*qμx<sup>μ</sup>*. Furthermore, the relation (25) also confirms the provisional interpretation proposed above that the state *φ*†(*x* )|0 may be viewed as a one-particle quantum field excitation state "localized" at the spacetime point *<sup>x</sup>*. The reason is that such interpretations is reminiscent of the standard relation

$$
\langle \mathbf{p} | \mathbf{x} \rangle = \exp(i \mathbf{p} \cdot \mathbf{x}) \tag{26}
$$

in nonrelativistic quantum mechanics, suggesting a "quasi-localization" of the particle's momentum state at *x*. From (25), we obtain Pr{*q* = *q* } = 1 for any four-momentum range *q* ∈ M4. This is completely natural since, as in nonrelativistic quantum mechanics, momentum eigenkets are maximally nonlocalized. Since we have just established that the state *φ*†(*x*)|<sup>0</sup> represents what, within QFT, corresponds to a pure one-particle momentum state, total nonlocalizablity of the conjugate position parameter is expected. However, note that in standard perturbative QFT, it is very difficult to mathematically describe the complete localization of particles. For an in-depth discussion of this problem, see the footnotes.<sup>17</sup>

### *4.4. The Feynman Propagator of Quantum Antennas*

So far, we have succeeded in modeling the process in which a q-antenna emits a particle at a specific spacetime point *x* = (*t* , **x** ). We also explicated the momentum-space composition of the emitted one-particle radiation state and found that it is comprised of a superposition of multiple outgoing one-particle momentum states. Moreover, we estimated the probability amplitude of measuring a certain (generic) momentum in this radiated state. However, for practical applications, there is a need to actually compute the effective coupling between the source on one side and generic observation spacetime points located either in the near- or far-zone on the other side. To do so, we will make use of one of the most powerful methods in QFT, the *Feynman propagator* [38,54,55,67]. Effectively, this will also directly provide us with the two-point Green's function *<sup>G</sup>*<sup>1</sup>(*<sup>x</sup>*, *x* ) of the q-antenna system corresponding to the first term on the RHS of (15).

The importance of moving to a mathematical description based on propagators stems from the fact that in QFT, one often finds that exciting the ground state at a specific spacetime point does *not* automatically imply that the radiated particle will reach *every* point in the far-zone with *significant* probability. To ensure that the quantum wireless or molecular communication link's receiver has access to the radiated particles with significant probability of detection, we need to compute the probability amplitude of measuring a particle at a generic spacetime point *x* away from the source point *<sup>x</sup>*. Since in QFT, there are no measurement operators as in nonrelativistic quantum mechanics, we may model the observer's interactions with the receiver by the process of particle *annihilation* [38]. This is one of the key ideas of quantum measurement in QFT to be adopted in this paper.

In order to mathematically implement this idea, we compute the two-point Green's function of the q-antenna system, which is defined as follows:

**Definition 4** (**The two-point q-antenna Green's function**)**.** *The q-antenna two-point (or oneparticle) Green's function, denoted by Gq*(*<sup>x</sup>*, *x* )*, is defined as the probability amplitude of the process that a particle created at the point x* ∈ *D*s *in the source region, while the system was initially at the ground state* |<sup>0</sup>*, will be annihilated at the later point x* ∈ *D*r *in the receiver (detection, observation) region, after which the q-antenna system will return again to the ground state.*

**Theorem 2.** *The correct relativistic expression for Gq*(*<sup>x</sup>*, *x* ) *is given by the formula*

$$G\_q(\mathbf{x}, \mathbf{x}') = \langle 0 | \mathcal{T} \phi(\mathbf{x}) \phi^\dagger(\mathbf{x}') | 0 \rangle,\tag{27}$$

*where* T *is the time-ordering operator defined by* (A17)*.*

**Proof.** This follows directly from Definition 4, the quantum field expansion (A26) in terms of creation and annihilation operators, and the definition of the time-ordering symbol (A17). The time ordering operator is included in order to automatically deal with both particle and antiparticle emission.<sup>18</sup>

**Remark 3.** *Definition 4 is inspired by the propagator-based quantum field-theoretic formalism of condensed matter physics, see in particular Mattuck's elegant formulation in [60], which influenced our approach here. The same quantity Gq*(*<sup>x</sup>*, *x* ) *is constructed in perturbative QFT as the fundamental tool for computing scattering cross-sections in experiments involving fundamental particle interactions [38,54,67] and the ground state energy in many-body condensed matter physics [59–61]. In the expression* (27)*,* T *, the time-ordering operator, is inserted in order to automatically ensure that fields with later time components x*0 *are always placed to the left of earlier ones.*

Theorem 2 provides a characterization of the q-antenna Green's function expressed directly in spacetime. However, in most applications of QFT, it is the *momentum* space representation that often proves to be *the* most useful space to do calculations in.<sup>19</sup> As will become more evident below, this is also the case in q-antenna theory. We would like then to derive an expression for the propagator in momentum space. In fact, this is a quite straightforward process: taking the conjugate of (21), multiplying the result with (21), then making use of the basic position-momentum eigenket orthogonality relation

$$
\langle \mathbf{q} | \mathbf{p} \rangle = \delta(\mathbf{q} - \mathbf{p}),
\tag{28}
$$

the q-antenna Green's function readily evaluates to

$$G\_{\boldsymbol{\eta}}(\mathbf{x}, \mathbf{x}') = \int\_{\mathbf{p} \in \mathbb{R}^3} \frac{\mathbf{d}^3 p}{(2\pi)^3} \frac{e^{-i p\_{\boldsymbol{\mu}} (\mathbf{x}^{\boldsymbol{\mu}} - \mathbf{x}'^{\boldsymbol{\mu}})}}{2\omega\_{\mathbf{p}}}.\tag{29}$$

This spectral expansion of the q-antenna Green's function is very important and will be illustrated in several examples, while its structure is investigated in depth below.

**Remark 4.** *It should be noted that in all momentum space integrals of the form* (29)*, a rigorous treatment would require that we place a step function of the form* <sup>Θ</sup>(*x*<sup>0</sup> − *x*<sup>0</sup>) *before the integral (or integrand) in order to transition from the time-ordered form* (27) *to the final expression* (29)*. Here, we are effectively focusing on the causal or retarded radiation problem where it is understood that*

*particles observed at x possess a clock time x*0 *that is later with respect to the spacetime creation point <sup>x</sup>, whose internal clock starts ticking at x*0*; i.e., x*0 > *x*<sup>0</sup> *and* <sup>Θ</sup>(*x*<sup>0</sup> − *x*<sup>0</sup>) = 1*. If antiparticles are to be included, this restriction is not needed. The advantage of using the Feynman propagator (among many other things) is that it naturally and economically leads to a direct and efficient computational formulation of the problem of quantum radiation, where no distinction between particles and antiparticles is required, since the propagator* (27) *can handle both at the same time.*

**Remark 5.** *The integral representation of Green's Function* (29) *can also be put in the following* (*x*, *t*)*-form more suitable for future use in antenna theory:*

$$\mathbf{G}\_{\mathbf{q}}(\mathbf{x} - \mathbf{x}', t - t') = \int\_{\mathbf{k} \in \mathbb{R}^3} \frac{d^3 k}{(2\pi)^3} \frac{e^{-i|\mathbf{k}|(t - t')}}{2\sqrt{|\mathbf{k}|^2 + m^2}} e^{i\mathbf{k} \cdot (\mathbf{x} - \mathbf{x}')},\tag{30}$$

*where the dispersion relation* (A27) *was utilized. Moreover, the natural-unit replacement p* → *k from Table A1 was used. For the computational evaluations of Green's functions in terms of special functions, see Appendix H.*

### *4.5. Generalization to Multiple Discrete and Continuous and Sources*

Suppose that now, we apply the quantum source field again but at a *different* spacetime point, say *<sup>x</sup>*2, while we rename *x* in (21) as *x*1. Furthermore, we note that each excitation of the vacuum by a field *φ*(*x*) localized at *x* can be controlled by some position-dependent "scaling factor", say *J*(*x*). This is again the abstract source representation of the q-antenna in the spirit of Definition 2. Then, the total quantum source field is

$$J(\mathbf{x}\_1')\phi(\mathbf{x}\_1') + J(\mathbf{x}\_1')\phi(\mathbf{x}\_2'). \tag{31}$$

Again, if we hit the vacuum state by this new operator, the radiated quantum state can be written as

$$[f(\mathbf{x}\_1')\phi(\mathbf{x}\_1') + f(\mathbf{x}\_2')\phi(\mathbf{x}\_2')]|0\rangle = \int\_{\mathbf{p}\in\mathbb{R}^3} \frac{\mathbf{d}^3 p}{(2\pi)^{3/2}} \frac{f(\mathbf{x}\_1')e^{ip\_\mu \mathbf{x}\_1'^\mu} + f(\mathbf{x}\_2')e^{ip\_\mu \mathbf{x}\_2'^\mu}}{(2\omega\_\mathbf{p})^{1/2}} |\mathbf{p}\rangle\_\prime \tag{32}$$

where we used (21). Physically, the expression (32) means that two clusters of outgoing waves (quantum particles) are emitted, one emanating from the spacetime point *x μ* 1 = (*t*1, **<sup>x</sup>**1), while the other cluster is directly radiated from *xμ*2 = (*t*2, **<sup>x</sup>**2). In all cases, note that the linearity of the quantum source field operator *φ*(*x*), as manifested by the expression *φ*(*x*)|<sup>0</sup> in (21), is the ultimate basis behind the applicability of the principle of superposition in q-antenna systems.

It is straightforward to generalize the q-antenna's radiation Formula (32) to the generic scenario of an arbitrary number of *N* discrete point sources located at the spacetime points *<sup>x</sup>n*, each with its own excitation strength specified by *J*(*xn*), *n* = 1, 2, ... , *N*. This leads to the expression

$$J\_q(\mathbf{x'})|0\rangle = \int\_{\mathbf{p} \in \mathbb{R}^3} \frac{\mathbf{d}^3 p}{(2\pi)^{3/2}} \frac{\sum\_{n=1}^N J(\mathbf{x'\_n}) e^{i p\_\mu \mathbf{x'\_n^\mu}}}{(2\omega\_\mathbf{p})^{1/2}} |\mathbf{p}\rangle\_\prime \tag{33}$$

where

$$J\_{\mathfrak{f}}(\mathbf{x}') := \sum\_{n=1}^{N} J(\mathbf{x}'\_n) \phi(\mathbf{x}'\_n) \tag{34}$$

is the effective (discrete) quantum source distribution operator.

It should become clear now how to generalize from discrete to continuous sources. Let us expand the continuous classical current source function *J*(*x*) in terms of a finite number of spacetime (four-dimensional) Dirac delta functions *<sup>δ</sup>*(*x*) as follows:

$$J(\mathbf{x}') = \sum\_{n=1}^{N} J(\mathbf{x}'\_n) \delta(\mathbf{x}' - \mathbf{x}'\_n),\tag{35}$$

where the source domain *D*s, the spacetime region upon which the source is supported, is in this case the discrete set

$$D\_{\sf s} := \{ \mathbf{x}\_1', \mathbf{x}\_2', \dots, \mathbf{x}\_N' \}. \tag{36}$$

Using the sifting property of the Dirac delta function, it is evident that (33) can be rewritten as

$$J\_q(D\_s)|0\rangle = \int\_{\mathbf{p} \in \mathbb{R}^3} \frac{\mathbf{d}^3 p}{(2\pi)^{\frac{3}{2}} (2\omega\_\mathbf{p})^{\frac{1}{2}}} \int\_{\mathbf{x'} \in D\_s} \mathbf{d}^4 \mathbf{x'} f(\mathbf{x'}) e^{i p\_\mathbf{p} \cdot \mathbf{x''\_n}} |\mathbf{p}\rangle,\tag{37}$$

where the quantum source operator is given by the integral

$$J\_q(D\_s) := \int\_{\mathbf{x'} \in D\_s} \mathbf{d}^4 \mathbf{x'} f(\mathbf{x'}) \phi^\dagger(\mathbf{x'}).\tag{38}$$

While (33) and (37) are formally equivalent in the special case of discrete sources, it is the *second* form (37) that is needed for writing down the correct expression corresponding to the continuous source case. Indeed, when the set *D*s becomes dense (which implies *N* → ∞), the relation (37) continues to hold.

**Remark 6.** *In deriving* (37)*, we implicitly assume that the order of the dx- and d*3 *p- integrals can be exchanged. This can be justified relatively easily when we impose the condition that D*s *is compact, which is an assumption we make throughout this paper that still covers most practical antenna systems. The details of the rigorous proof are lengthy and will not be given. For a comprehensive discussion of the rigorous mathematical theory behind representing integrals (continuum sums) over quantum kets in QFT, see [54,55,67,79].*

To summarize, we have managed so far to construct two fundamental types of quantities behaving as "sources" in q-antenna theory:

1. The classical source function

$$J(\mathbf{x'}): D\_{\mathbf{s}} \subset \mathbb{M}^4 \to \mathbb{R}.\tag{39}$$

2. The quantum source operator

$$J\_q(D\_s): \mathcal{D}\_s \to \mathcal{O}\_\prime \tag{40}$$

where

$$\mathcal{D}\_s := \{ D \subset \mathbb{M}^4 \mid D \text{ is open, } \text{cl}(D) \text{ is compact in } \mathbb{R}^4 \}\tag{41}$$

is the set of all open subsets in the Minkowski spacetime M<sup>4</sup> whose topological closure is compact in R4. On the other hand, O is defined as the space of all operators acting on elements of the Fock (occupation state Hilbert space representation [54,55]) space of the q-antenna system.<sup>20</sup>

**Remark 7** (**The source design problem in quantum antenna theory**)**.** *The classical source J*(*x*) *is needed in order to delimit the actual q-antenna configuration in spacetime. As in classical antenna theory, an antenna source is considered known if its geometric support regions D*s *is known in addition to the value of the source J*(*x*) *at each point x* ∈ *D*<sup>s</sup>*. The situation is quite similar here. The q-antenna designer is interested in obtaining bounds or information on both D*s *and J*(*x*) *in order to attain certain radiation states. This is the design problem that is usually solved by means of optimization methods.*

Once the classical source *J*(*x*) is fixed, one can employ (38) in order to immediately construct the quantum source operator *Jq*, which depends (among other things) on the geometrical and topological structure of the source support domain *D*s.

**Remark 8.** *It should be noted that the operator Jq depends on the classical function J*(*x*) *defined on the entire spacetime regions D*<sup>s</sup>*. Thus, formally speaking, one should write that operator as Jq*[*<sup>D</sup>*s, *J*(*x*)]*. However, to simplify the presentation, this is avoided, since it should always be clear from the context which specific classical source function J*(*x*) *is involved with the quantum source operator Jq.*

The probability amplitude *<sup>A</sup>*(*x*) of receiving a particle/wave at the spacetime position *x* can now be readily computed in exactly the same way we did with the point quantum source at *<sup>x</sup>*, i.e, the amplitude *Gq*(*x* − *x*) in (27), simply by adding the various contributions coming from all source points *x* ∈ *D*s. The calculations are

$$\begin{split} A(\mathbf{x}) &= \langle 0|\mathcal{T}\boldsymbol{\phi}(\mathbf{x})I\_{\boldsymbol{\theta}}(D\_{\mathbf{s}})|0\rangle = \langle 0|\mathcal{T}\boldsymbol{\phi}(\mathbf{x}) \int\_{\mathbf{x}' \in D\_{\mathbf{s}}} \mathbf{d}^{4}\mathbf{x}' \, I(\mathbf{x}')\boldsymbol{\phi}^{\dagger}(\mathbf{x}')|0\rangle \\ &= \int\_{\mathbf{x}' \in D\_{\mathbf{s}}} \mathbf{d}^{4}\mathbf{x}' \langle 0|\mathcal{T}\boldsymbol{\phi}(\mathbf{x})\boldsymbol{\phi}^{\dagger}(\mathbf{x}')|0\rangle I(\mathbf{x}'), \end{split} \tag{42}$$

where (37) was employed in order to write down the second equality in (42). Note that we assume that *x* and *x* are well separated from each other such that there is no overlap between the observation spacetime region *D*r and points *x* ∈ *D*s in the source domain. This allows as to freely move the time-ordering operator T so to obtain (with negligible error) the third equality in (42) above, making our radiation formulas ultimately valid in the *exterior* region of the q-antenna system.

Finally, with the help of (27), we arrive at our main q-antenna radiation amplitude formula summarized by the following theorem:

**Theorem 3.** *The probability amplitude of a continuous source q-antenna system J*(*x*), *x* ∈ *D*s ⊂ <sup>M</sup>4*, where the closure of D*s *is compact in* <sup>R</sup>4*, is given by the following superposition integral*

$$A(\mathbf{x}) = \int\_{\mathbf{x}' \in D\_{\mathbf{s}}} \mathbf{d}^4 \mathbf{x}' G\_{\mathbf{q}}(\mathbf{x} - \mathbf{x}') f(\mathbf{x}'),\tag{43}$$

*where Gq*(*x* − *x*) *is the q-antenna Green's function in Definition 4.*

**Remark 9.** *We may rewrite* (43) *in a form more familiar to antenna engineers by recruiting the notation x* = (*t*, *x*) *and x* = (*t*, *<sup>x</sup>*)*, viz., in the "*(*<sup>x</sup>*, *t*)*-format", allowing us to restate* (43) *as*

$$A(\mathbf{r},t) = \int\_{\mathbf{x}' \in S\_{\mathbf{s}}} \int\_{t' \in T\_{\mathbf{s}}} \mathbf{d}t' \, \mathbf{d}^3 \mathbf{x}' \, \mathbf{G}\_{\mathbf{q}}(\mathbf{x} - \mathbf{x}', t - t') f(\mathbf{x}', t'), \tag{44}$$

*where S*s *and T*s *are the spatial and temporal components of the four-dimensional spacetime source region D*s*; see* (3)*. Thereby, we find that the spatial support domain of the q-antenna's source system, i.e., its spatial extension properties, is captured by S*<sup>s</sup>*, while T*s *is the time interval of the application of the source.*

**Remark 10.** *The relation D*s = *S*s × *T*s *holds locally in D*<sup>s</sup>*. The reason is that for attaining wide generality, we are already implicitly permitting D*s *to posses a manifold structure in which, locally speaking, each point x* ∈ *D*s *is homeomorphic (topologically isomorphic) to* <sup>R</sup>*d*, *d* = 2, 3, 4*. That is, from the topological viewpoint, the source region D*s ⊂ M<sup>4</sup> *can be said to be of type (0 + 1) for q-antenna point sources; type (1 + 1) for one-dimensional sources such as wire or loop antenna; type (2 + 1) for surface radiators; or full (3 + 1) type for volumetric sources; and so on.*

**Remark 11.** *Either one of the two expressions* (43) *or* (44) *may be used to express the probability amplitude of receiving a particle at a specific location x* = (*t*, *x*) *when radiated by a source distribution written as either J*(*x*) *or J*(*x*, *<sup>t</sup>*)*. Both are Lorentz invariant, but the form* (43) *expresses that more clearly. On the other hand, the form* (44) *may be utilized when a concrete frame of reference (lab frame) is used such that an unambiguous decomposition of the source region D*s *into spatial*

*and temporal part can be made (this is in fact the case in most practical scenarios). Physically, the radiation formulas* (43) *or* (44) *express quantum radiation by a source J*(*x*) *in terms of the q-antenna Green's function, which happens in this particular case to coincide with the Feynman propagator of QFT.*

### **5. On the General Structure of Radiation Processes in Linear Quantum Antenna Systems**

Before moving next to the construction of practical definitions for the q-antenna directivity and gain patterns, we pause for a moment in order to provide a deeper insight into the nature of quantum radiation in q-antenna systems based on the new q-antenna Green's function *Gq*(*x* − *x* ) whose main concept was introduced earlier via Definition 4.

#### *5.1. The General Structure of the Quantum Antenna Propagator Process*

If we insert (29) into (43), interchanging the order of integrals, the following spectral (momentum-space) result is obtained.

**Theorem 4.** *The momentum space probability amplitude <sup>A</sup>*(*q*) *in Theorem 3 can be expanded in the spectral domain using the following integral formula:*

$$A(\mathbf{x}) = \int\_{\mathbf{P} \in \mathbb{R}^3} \frac{d^3 p}{(2\pi)^3 \, 2\omega\_\mathbf{P}} \int\_{\mathbf{x'} \in D\_\mathbf{s}} \mathbf{d}^4 \mathbf{x'} J(\mathbf{x'}) e^{i p\_\mathbf{p} \mathbf{x'}^\mu} e^{-i p\_\mathbf{p} \mathbf{x''}}.\tag{45}$$

*Here, the spectral integration is performed with respect to all real momenta p in* R3*.*

**Proof.** In order to obtain (45), we assume that cl(*<sup>D</sup>*s) is compact in R<sup>4</sup> in order to make sure that a needed interchange of limiting operations can be justified. The rest of the proof is immediate.

The anatomy of the quantum radiation's Green's function's spectral expansion (45) is illustrated in Figure 2. The various fundamental sub-processes composing the overall process of quantum radiation can be formally identified as follows:

1. We first must form the correct relativistic sum over all allowable momentum states. This is accomplished by the Lorentz invariant integral operator

$$\int\_{\mathbf{p}\in\mathbb{R}^3} \frac{\mathbf{d}^3 p}{(2\pi)^3} \frac{1}{2\omega\_\mathbf{p}}.\tag{46}$$

2. Each momentum state |**p** will be summed over all possible source locations *x* ∈ *D*s in the source region via the integral operator

$$\int\_{\mathbf{x}' \in D\_{\mathbf{s}}} \mathbf{d}^4 \mathbf{x}'.\tag{47}$$

This step is also relativistic, since *D*s ⊂ M<sup>4</sup> and d4*x* are Lorentz invariant.<sup>21</sup>


**Figure 2.** The anatomy of the q-antenna's propagator-based quantum radiation process.

**Remark 12.** *It may be seen from the above algorithmic construction of the q-antenna propagator in the spectral domain that the problem of quantum radiation acquires a very intuitive and concrete structure when viewed from the spectral (momentum space) domain's perspective. This observation will be exploited in Section 6 when we explore candidate expressions for the q-antenna radiation pattern.*

**Remark 13.** *The probability amplitude <sup>A</sup>*(*x*) *can be seen as a superposition of "quantum plane waves" each of the form* exp *ipμx<sup>μ</sup>. This is in fact somehow similar to the plane-wave (Weyl) expansion of electromagnetic fields radiated by continuous sources [63,64,80–82]. As in the electromagnetic case, each point source will emit a continuum of plane waves (some of them are evanescent, and the others are pure propagating), with wavevector/momentum specified by p and frequency <sup>ω</sup>p. The total sum of all these waves will produce an effective field moving gradually away from the source and approaching the far zone of the antenna by first going through the near zone. (The evanescent mode character of the momentum space representation becomes directly visible when we transform the above three-dimensional spectral integrals evaluated with respect to p* ∈ R<sup>3</sup> *to equivalent four-dimensional integrals now performed with respect to p* ∈ <sup>M</sup>4*, see for example [57]).*

#### *5.2. Comparative Analysis of the Three Fundamental Types of Antennas*

At this point, it is instructive to give a bird's eye view on the various genera of antenna theories available to us so far. *Classical antennas* (c-antennas) involve excitation with an external electric field **E**ex, which in turn induces a current on the antenna via the current Green's function [28,46–48].<sup>22</sup> This is illustrated in Figure 3a. Here, it is essential to note that the "system input" is a classical field, while the "system output" is *also* the classical radiated fields **E** and **H**. On the other hand, the *quantum-optics* approach to defining *quantum antennas*, which is the one currently most often mentioned in connection with applications to quantum communications [1,2,13], treats the input as a classical source while the output is a quantum state |*α*(*t*) [10,13,62,83]. This is illustrated in Figure 3b, where the q-antenna current Green's function **F***q* is *still* a *classical* function, just like the induced radiating current **J***q*(**x**).

$$\xrightarrow{\mathbf{E}^{\mathrm{ex}}(\mathbf{x},t)} \xrightarrow{\mathbf{\upint}} \underbrace{\left[\mathbf{\upint}\_{\mathbf{c}}(\mathbf{x},\mathbf{x}';t,t')\right]}\_{\mathbf{\upint}} \xrightarrow{\mathbf{J}\_{c}(\mathbf{x},t)} \underbrace{\left[\mathbf{c}\text{-channel}\mathbf{rel}\right]}\_{\mathbf{\upint}} \xrightarrow{\mathbf{E}(\mathbf{x},t)} \mathbf{\upint}(\mathbf{x},t)$$

(**a**) Electromagnetics-based c-antennas.

$$\xrightarrow{\mathbf{F}^{\infty}(\mathbf{x},t)} \xrightarrow{\mathbf{F}\_{q}(\mathbf{x},\mathbf{x}';t,t')} \xrightarrow{\mathbf{J}\_{q}(\mathbf{x},t)} \xrightarrow{\mathbf{J}\_{q}(\mathbf{x},t)} \xrightarrow{|\mathbf{a}\times\mathbf{t}|} \xrightarrow{|\mathbf{a}\times(t)|}$$

(**b**) Quantum-optics approach to q-antennas.

$$\xrightarrow{J(\mathbf{x'})} \xrightarrow{} \boxed{F\_q(D\_{\mathbf{s}\mathbf{s'}}\mathbf{x'})} \xrightarrow{J\_q(D\_{\mathbf{s}})} \boxed{\mathbf{q-channel}} \xrightarrow{} \xrightarrow{} \mathbf{A^{(\mathbf{x})}}$$

(**c**) Relativistic-QFT approach to q-antennas.

**Figure 3.** The fundamental operational modes of c- and q- antennas. Both types requires a current Green's function (ACGF) **F ¯** [28] to connect classical excitation field **E**ex with a classical induced current **J**. Note that in both c- and q-antennas, the induced currents **J***c* and **J***q*, respectively, are classical. However, the difference resides mainly in the "output" or the radiation state, which is classical (quantum) in c- (q-) antennas.

Consider now the *third* type of antennas shown in Figure 3c. The input to the system *J*(*x*) (exactly like the previous two antenna types) is still a classical source, but now the radiating source *Jq*(*<sup>D</sup>*s) is an *operator* that can describe quantum particles emission (quantum radiation) from within the q-antenna's source spacetime region *D*s. In fact, we may rewrite (38) in the revealing integral form

$$J\_q(D\_\mathbf{s}) := \int\_{\mathbf{x'} \in \mathbb{M}^4} \mathbf{d}^4 \mathbf{x'} F\_q(D\_\mathbf{s'} \mathbf{x'}) f(\mathbf{x'}),\tag{48}$$

where the q-antenna current Green's function in this case is simply given by

$$F\_{\emptyset}(D\_{\mathbf{s}}, \mathbf{x}') := \begin{cases} \boldsymbol{\Phi}^{\dagger}(\mathbf{x}'), & \mathbf{x}' \in D\_{\mathbf{s}'} \\ 0, & \mathbf{x}' \notin D\_{\mathbf{s}}. \end{cases} \tag{49}$$

Clearly, this is quite different from the two cases depicted in Figure 3a,b. In the case Figure 3c, captured by the expression (49), the current Green's function itself is a *quantum field*, i.e., an operator-valued function on spacetime. The source *Jq* reproduced by the Green's function superposition integral is also an operator, and the ultimate "output" coming out from the relativistic QFT-based antenna system is the probability amplitude *<sup>A</sup>*(*q*) of annihilating a particle at some generic observation point *x* ∈ *D*r ⊂ M4.

#### *5.3. On the Causal Spacetime Structure of Radiation Emitted by Quantum Antenna Systems*

In both special and general relativity, two events *x* and *x* with timelike distance, i.e., |*x* − *x*|<sup>2</sup> > 0, can be causally connected [51,53,84]. This has a direct and obvious implication for the general spacetime theory of q-antennas developed here.


These two observations above are enough to determine the causal structure of qantennas for the case of point sources. However, since arbitrary sources can be constructed from assembling clusters of point sources, the argumen<sup>t</sup> can be expanded, as will be shown next.

For the case of continuous sources, the situation is qualitatively similar to the discrete scenario discussed above, but the detailed content of the antenna's causal domain becomes somehow more complicated, since the situation strongly depends on the geometry of the source region in the former case. In Figure 4b, we schematically illustrate the problem when a generic continuous source *J*(*x* ) is applied in the spacetime region *D*s ⊂ M4. Since any current source function *J*(*x* ) can be expanded into a continuous sum of point sources, as per the sifting property of delta functions

$$J(\mathbf{x'}) = \int\_{\mathbf{x} \in \mathbb{M}^4} \mathbf{d}^4 \mathbf{x} \, J(\mathbf{x}) \delta(\mathbf{x'} - \mathbf{x}),\tag{50}$$

one may then attempt to construct a causal lightcone for each point *x* ∈ *D*s, which is again denoted by *Cx* . A *continuum* of causal cones is shown in Figure 4b where all cones appear to be aligned in parallel in spacetime because we ignore gravitational effects.

We now introduce the following construction of the antenna's horizon causal structure, which is valid for both classical and quantum antennas:

**Definition 5** (**The antenna causal domain**)**.** *The antenna causal domain associated with a given source region D*<sup>s</sup>*, which we denote by <sup>C</sup>*(*<sup>D</sup>*s)*, is defined as simply the fusion (set-theoretic union) of all individual causal cones Cx based on events located inside the source region x* ∈ *D*<sup>s</sup>*. That is, according to the recipe*

$$\mathbb{C}(D\_{\mathfrak{s}}) := \bigcup\_{\mathfrak{x}' \in D\_{\mathfrak{s}}} \mathbb{C}\_{\mathfrak{x}'}.\tag{51}$$

*The part of <sup>C</sup>*(*<sup>D</sup>*s) *that is solely due to future lightcones C*<sup>+</sup> *x is called the antenna future causal domain. Similarly, the components of <sup>C</sup>*(*<sup>D</sup>*s) *due to contributions emanating from past lightcones C*− *x lead to the antenna past causal domain. Then, we have*

$$\mathbb{C}^{+}(D\_{\mathbf{s}}) := \bigcup\_{\mathbf{x'} \in D\_{\mathbf{s}}} \mathbb{C}^{+}\_{\mathbf{x'}} \quad \mathbb{C}^{-}(D\_{\mathbf{s}}) := \bigcup\_{\mathbf{x'} \in D\_{\mathbf{s}}} \mathbb{C}^{-}\_{\mathbf{x'}} \tag{52}$$

*and*

$$
\mathbb{C}(D\_{\mathfrak{s}}) = \mathbb{C}^+(D\_{\mathfrak{s}}) \cup \mathbb{C}^-(D\_{\mathfrak{s}}).\tag{53}
$$

*Note that the same constructions can be laid out for the receiving q-antenna case when the detector is in the receive domain D*<sup>r</sup>*.*

**Remark 14.** *Note that it is still possible to incorporate gravitation into our model, since the effect of gravitational fields is mainly to tilt the lightcones locally, where the tilting is directly determined by the gravitational potential/metric tensor gμν [41]. For applications of q-antennas in deep space communications, these gravitational effects may have to be taken into account. Consequently, the constructions provided by Definition 5 may become useful in astronomical and cosmological applications of either classical or quantum antennas. That is especially true when the large-scale structural impact of the gravitational field on a planned solar or even future interstellar communication links is important.*

**Remark 15.** *The causal domain <sup>C</sup>*(*<sup>D</sup>*s) *is a function of the source region D*<sup>s</sup>*. Moreover, it should be emphasized that in general, the antenna causal domain, as defined above, need not constitute a simple cone in itself. In addition, in principle, the receivers located inside <sup>C</sup>*(*<sup>D</sup>*s) *can receive information transmitted by radiation emitted from inside D*<sup>s</sup>*. On the other hand, noncausal receivers, e.g., see some cases schematically depicted in Figure 4b, can never receive information from an antenna system whose source is supported by the region D*<sup>s</sup>*.*

Overall, the analysis above has highlighted an organic interlinking of spacetime with causality within the context of generic quantum communication systems utilizing quantum antennas. This suggests the need to pay closer attention to the *global* (i.e., topological) structure of spacetime domains when such future quantum technologies are incorporated

in the analysis, design, and construction of long-distance space communication systems at the solar, extra-solar, astronomical, or even cosmological scales.

**Figure 4.** The fundamental causal structure of the q-antenna radiation process. For simplicity, we only show future lightcones.

### **6. Quantum Antenna Radiation Patterns: Basic Constructions**
