*6.1. Introduction*

In contrast to classical antennas, q-antenna systems are intrinsically stochastic. The ultimate goal of a candidate q-antenna theory is to supply rules and guidelines for estimating *probabilities* of potential detection (reception) processes ideally enacted somewhere within the near or far spacetime zones of the quantum sources. In conventional antenna theory, certain quantities are fundamental for analyzing and designing functioning actual devices able to radiate power in real-life settings. These often include directivity, gain (preferably with a measure of radiation efficiency), and the array factor [27,33,35]. As will be illustrated in this section, it turns out that in q-antennas, there also exist close counterparts for many of the conventional characterization measures/concepts already in use in -c-antennas. However, there are also fundamental differences between c- and q-antennas in terms of the physical interpretation and meaning of the results thus obtained since, clearly, electromagnetic and quantum radiation are very distinct (though related) physical processes. For the purpose of constructing working definitions for directivity and gain functions suitable for q-antennas, we explore next some possible antenna radiation pattern constructions made available to us by the relativistic QFT approach proposed above.

### *6.2. 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* ⊂ M<sup>4</sup> is given by the integral of the square of the probability amplitude *<sup>A</sup>*(*q*):

$$\Pr\{q \in Q\} = \frac{1}{\mathfrak{a}} \int\_{q \in Q'} \mathrm{d}^4 q \, |A(q)|^2 = \frac{1}{\mathfrak{a}} \int\_{\mathbf{q} \in Q} \frac{\mathrm{d}^3 q}{(2\pi)^3} \frac{|A(\mathbf{q})|^2}{2\omega\_\mathbf{q}},\tag{54}$$

where *Q* ⊂ R<sup>3</sup> is the projection of *Q* ⊂ M<sup>4</sup> onto R3, and we made use of (A34) to derive the second equality in (54). The number *α* ∈ R<sup>+</sup> 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.*

**Remark 17** (**Probabilistic normalization**)**.** *The correct relativistic normalization of the fourmomentum kets* |*p and* |*q, 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*

$$\Pr\{q \in \mathbb{M}^4\} = \frac{1}{\alpha} \int\_{q \in \mathbb{M}^4} \mathbf{d}^4 q |A(q)|^2 = 1. \tag{56}$$

*The latter is in fact a consequence of the normalization in* (54) *by*

$$\mathfrak{a} := \int\_{q \in \mathbb{M}^4} \mathbf{d}^4 q |A(q)|^2,\tag{57}$$

*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 convergen<sup>t</sup> in the usual sense since *exact* continuous momentum states such as |*q* or |**q** 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 *<sup>A</sup>*(*q*), for example the point source with *<sup>A</sup>*(*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 Coleman'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. Introduce the Euclidean ball

$$Q(\mathbf{q}\_0, \varepsilon) := \{ \mathbf{q} \in \mathbb{R}^3, |\mathbf{q} - \mathbf{q}\_0| < \varepsilon \},\tag{58}$$

where **q**0 ∈ R3, while *ε* ∈ R<sup>+</sup> is very small. (The norm |·| is that inherited from the standard Euclidean metric on R3). The probability of measuring a momentum **q** in the ball *Q*(**<sup>q</sup>**0,*<sup>ε</sup>*) is given by

$$\Pr\{q \in Q(\mathbf{q}\_0, \varepsilon)\} = \frac{1}{\alpha} \int\_{\mathbf{P} \in \mathcal{Q}(\mathbf{q}\_0, \varepsilon)} \frac{\mathbf{d}^3 q}{(2\pi)^3} \frac{|A(\mathbf{q})|^2}{2\omega\_\mathbf{q}} \simeq \frac{(4/3)\pi \varepsilon^3}{a(2\pi)^3} \frac{|A(\mathbf{q}\_0)|^2}{2\omega\_\mathbf{q}} = \frac{\varepsilon^3}{12\pi^2 \alpha} \frac{|A(\mathbf{q}\_0)|^2}{2\omega\_\mathbf{q}},\tag{59}$$

where (4/3)*πε*<sup>3</sup> is the volume of the ball *Q*(**<sup>q</sup>**0,*<sup>ε</sup>*) centered at **q**0 with radius *ε*. Therefore, the expression (59) gives the probability of measuring a momentum **p** = **q**0 after exciting the ground state |0 of the q-antenna to a higher-energy state |*q*<sup>0</sup>. 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 *<sup>x</sup>*, the probability amplitude is *<sup>A</sup>*(*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 **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 fourmomentum states |*q* 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.*

### *6.3. Constructing the Quantum Antenna Directivity Pattern*

Our goal here is to estimate the directive properties of a continuous source *J*(*x* ) located in the spacetime region *D*s. The probability amplitude of measuring the four-momentum *q* is denoted by *<sup>A</sup>*[*q*; *J*(*x* ), *<sup>D</sup>*s] but will be abbreviated to *<sup>A</sup>*(*q*) whenever there is no confusion about the source. Our main tool here is the following theorem.

**Theorem 5.** *Consider a q-antenna system J*(*x*), *x* ∈ *D*s ⊂ M4*. Let the corresponding probability amplitude be <sup>A</sup>*(*q*)*. Then, the following formula holds:*

$$|A(q)|^2 = |J(\mathbf{q}\_\prime \omega\_\mathbf{q})|^2,\tag{61}$$

*where J*(*q*) *is the spacetime Fourier transform of the radiating current source* (A45)*.*

**Proof.** See Appendix I.

**Remark 19.** *Since there is a one-to-one (injective) mapping between the Lorentz four-momentum states q* ∈ M<sup>4</sup> *and the conventional three-momentum vectors q* ∈ R<sup>3</sup> *via the dispersion relation* (A27)*, we may just write <sup>A</sup>*(*q*) *instead of <sup>A</sup>*(*q*, *<sup>ω</sup>q*)*.*

**Theorem 6.** *For the situation described in Theorem 5, the quantum radiation probability amplitude can be expressed as:*

$$\Pr\{k \in Q\} = \frac{1}{\alpha} \int\_{k \in Q} \frac{d^3k}{(2\pi)^3} \frac{|f(\mathbf{k})|^2}{2\omega\_\mathbf{k}},\tag{62}$$

*for a generic three-dimensional region Q* ⊂ R3*. Here, J*(*k*) := *J*(*k*, *ω*) *is the momentum space Fourier transform of the source defined by* (A45)*; see also Remark 19 on the reduced notation in momentum space used here.*

**Proof.** Use the probability law (54) in (61) and make the replacement **q** → **k**.

An immediate application of Theorem 6 is in expressing the total energy radiated by a quantum source directly in terms of the classical source function *J*(*x*), as will be illustrated through the next example.

**Example 4** (**Total radiated energy and momentum of a q-antenna in momentum space**)**.** The total energy radiated by the antenna in a momentum space region *Q* ⊂ R<sup>3</sup> can be obtained by multiplying the probability of particle production in the infinitesimal momentum space volume d3*k* by the particle's energy, which is ¯*h<sup>ω</sup>***k** (or just *ω***k** in natural units), and then integrating. With the help of (62), this procedure yields:

$$\mathcal{E}[Q] = \frac{1}{2} \int\_{\mathbf{k} \in Q} \frac{\mathbf{d}^3 k}{(2\pi)^3} |f(\mathbf{k})|^2,\tag{63}$$

where E is the total energy. Similarly, the expected value of the total radiated momentum can be estimated via the formula

$$\mathcal{P}[Q] = \frac{1}{2} \int\_{\mathbf{k} \in Q} \frac{\mathbf{d}^3 k}{(2\pi)^3} \frac{|J(\mathbf{k})|^2 \mathbf{k}}{\sqrt{|\mathbf{k}|^2 + m^2}},\tag{64}$$

where use was made of the fact that the momentum of the particle excitation associated with the **k**th state is ¯*h***k** (or **k** in natural units).

**Remark 20.** *It is interesting to note that the momentum space energy pattern of the q-antenna* (63) *is similar to the energy directivity of a classical source in generic medium. For example, see how general formulas of directivity for classical radiators were constructed recently in momentum space for generic homogeneous nonlocal domains [71–73].*

In order to put relation (62) into further practical use, we transform momentum representations into spherical coordinates so we may obtain some information about the directive properties of quantum radiating sources as illustrated by the following example.

**Example 5** (**Radiation angular directivity pattern of q-antennas in momentum space**)**.** Let us express the wavevector **k** in terms of the spherical angular coordinates *θ* and *ϕ* by using the standard transformation

$$\hat{k} = \hat{k}(\Omega) = \hat{x}\_1 \cos \varphi \sin \theta + \hat{x}\_2 \cos \varphi \sin \theta + \hat{x}\_3 \cos \theta,\tag{65}$$

where

$$
\Omega := (\theta, \varphi), \quad \hat{k}(\Omega) := \frac{\mathbf{k}}{|\mathbf{k}|}. \tag{66}
$$

Here, *<sup>x</sup>*<sup>ˆ</sup>*i*, *i* = 1, 2, 3, constitute a set of three mutually orthogonal Cartesian unit vectors. The angles 0 < *ϕ* < 2*π*, 0 < *θ* < *π* determine the direction of the unit vector ˆ *k*, or where the emitted quantum particle's momentum **p** = *h*¯**k** is pointing in three-dimensional space. Since physically and intuitively, one would expect that the directions in threedimensional position space along which momenta tend to maximally flow also correspond to the directions toward which most of the particle ensemble's energy and momentum are directed, we may use Theorem 6 in conjunction with (66) to estimate the quantum source's directive properties in generic three-dimensional scenarios. A simple way to achieve this is by re-expressing (62) using (65) and (66), resulting in

$$\Pr\{k \in [k\_{\min}, k\_{\max}], \,\,\Omega \in \Omega\_0\} = \frac{1}{a} \int\_{k \in [k\_{\min}, k\_{\max}]} \int\_{\Omega \in \Omega\_0} \text{d}k \,\text{d}\Omega \,\frac{k^2 \sin\theta}{(2\pi)^3} \frac{\left| \int [k, \hat{k}(\Omega)] \right|^2}{2\sqrt{k^2 + m^2}},\tag{67}$$

where

$$k := |\mathbf{k}|, \quad \mathbf{d}\Omega := \mathbf{d}\theta \mathbf{d}\varphi. \tag{68}$$

The angular set Ω0 in (67) is the three-dimensional (solid) angular sector inside where one is interested in characterizing the radiating quantum antenna system.

**Remark 21.** *It is important not to confuse k* ∈ R+*, defined by* (68)*, with the four-vector k* = *kμ* ∈ M<sup>4</sup> *(Appendix B). In this section, we use the shorthand notation* (68) *in order to simplify the presentation.*

Inspired by the expression (67), we may then introduce the following definition of momentum space directivity for quantum antennas:

**Definition 6** (**Quantum antenna directivity in momentum space**)**.** *The momentum space directivity of a q-antenna with source function J*(*x*), *x* ∈ *D*s *is defined as the following angular function*

$$\mathcal{D}(\boldsymbol{\varrho},\boldsymbol{\theta};k) := \frac{1}{\kappa\_k} \frac{k^2 \sin \theta}{(2\pi)^3} \frac{\left| \boldsymbol{J} \left[ \boldsymbol{k}, \hat{\boldsymbol{k}}(\boldsymbol{\varrho},\boldsymbol{\theta}) \right] \right|^2}{2\sqrt{k^2 + m^2}},\tag{69}$$

*where*

$$\alpha\_k := \frac{1}{4\pi} \int\_{4\pi} d\Omega \frac{k^2 \sin \theta}{(2\pi)^3} \frac{\left| \int \left[ k, \hat{k} (\varphi, \theta) \right] \right|^2}{2\sqrt{k^2 + m^2}} \tag{70}$$

*is a positive momentum-dependent probability noramalization constant.*

Expressed in terms of the momentum-space-type directivity (69), the total probability in (62) may be be put into the more compact form

$$\Pr\{k \in [k\_{\min}, k\_{\max}]\_\prime \: \, \Omega \in \Omega\_0\} = \int\_{k \in [k\_{\min}, k\_{\max}]} \mathrm{d}k \frac{a\_k}{\mathfrak{a}} \int\_{\Omega \in \Omega\_0} \mathrm{d}\Omega \,\mathcal{D}(\varphi, \theta; k). \tag{71}$$

The last expression explains the physical motivation behind Definition 6. Moreover, the following two remarks explain more about the engineering background to directivity concepts in antenna theory motivating the above definition itself.

**Remark 22.** *It is evident then from* (71) *that for a given* ˆ *k, the function* <sup>D</sup>(*k*, ˆ*k*) *is the angular probability density in the momentum space variable k. On the other hand, the mathematical dependence of <sup>D</sup>*(*k*, ˆ*k*) *on the angles ϕ and θ provides information on how the emitted quantum particles tend to flow along different directions in space. This is why D does indeed behave as a momentum-space radiation pattern (probability per unit momentum per solid angle).*

**Remark 23.** *The positive number αk is inserted into* (69) *in order to ensure that the total probability of radiation at all angles, evaluated at a single radial momentum value k, is equal to that of an isotropic source.<sup>23</sup> Intuitively, αk represents the total radiation angular density emitted by the q-antenna, so the ratio* D *given by* (69) *is the relative radiation intensity along one direction with respect to a standard isotropic source*

D0 := *P*0 4*π*, (72)

*where P*0 *is the constant radiation angular probability of such an isotropic reference antenna whose radiation angualr density is* D0*. That is, we have*

$$\int\_{\Omega \in 4\pi} d\Omega \,\mathcal{D}(\varphi, \theta; k) = 4\pi \mathcal{D}\_{0\prime} \tag{73}$$

*as expected from a typical directivity expression [35]. Physically, the ratio* 4*παk*/*<sup>α</sup> in* (71) *represents the fraction of the total quantum radiation contained in a sphere with radius k in momentum space* *relative to the total radiation obtained by including contributions coming from all values of the momentum magnitude k* := |*k*|*. The ability of a source to direct power along certain directions is measured by angle-dependent generalizations of such total ratios. Then, Definition 6 of the q-antenna directivity is a natural generalization of the corresponding definition in classical antenna theory as traditionally presented in texts such as [27,35].*

*6.4. The Probability Law of Receiving Radiated Quantum States: Source-Receiver Coupling Gain Estimation*

Consider the configuration shown in Figure 5. The quantum antenna communication system is characterized by the *gain functional*

$$\mathcal{G}: \mathcal{X} \times \mathcal{X} \to \mathbb{C}, \quad \mathcal{G} = \mathcal{G}[f\_{\mathbf{s} \boldsymbol{\omega}} f\_{\mathbf{r}}],\tag{74}$$

which is a bilinear functional in the source and receiver currents *J*s(*x* ) ∈ X , *x* ∈ *D*s, and *J*r(*x*) ∈ X , *x* ∈ *D*r, respectively. Here, X is the space of real-valued functions on the four-dimensional domains *D*s,r ⊂ M4. The complex number G gives the probability amplitude of information transmission (coupling amplitude) between source and receiver. This gain plays the role of "transmission coefficient" or the "coupling coefficient" often used in conventional electromagnetic communication systems, e.g., see [33,88]. From the fundamental theory of quantum antennas developed above, we may easily deduce the general expression of this functional by simply treating the receiver as a source with the "quantum reverse" process of that of the transmitter, i.e., the dual of the transmitter problem, e.g., by converting kets to bras, taking adjoints, complex conjugates, etc. Since for *real* sources and receiver currents *J*r,s(*x* ), the problem is fully reciprocal,<sup>24</sup> we can immediately write down the q-antenna system gain expression as follows:

$$\mathcal{G} = \langle 0 \vert \int\_{\mathbf{x} \in D\_{\mathbf{r}}} \mathbf{d}^{4} \mathbf{x} \int\_{\mathbf{x'} \in D\_{\mathbf{s}}} \mathbf{d}^{4} \mathbf{x'} \, l\_{\mathbf{r}}(\mathbf{x}) \mathcal{T} \phi(\mathbf{x}) \phi^{\dagger}(\mathbf{x'}) l\_{\mathbf{s}}(\mathbf{x'}) \vert 0 \rangle,\tag{75}$$

where (43) was used to realize the transmitter, and a similar form was adapted for the receiver.

**Figure 5.** Angular gain measurementconfiguration scenario for a generic quantum antenna source system *J*s(**<sup>x</sup>**, *t*), **x** ∈ *S*s, *t* ∈ *T*s.

**Example 6** (**Construction of practical angular gain pattern in q-antenna systems**)**.** We will use the general relativistic expression (75) in order to construct an angular expression characterizing quantum radiation for possible use in practical settings. To do that, we first need to break relativistic covariance in order to simplify the problem and facilitate calculations. The most natural choice for a preferred frame is to use a coordinate system at rest with either the lab frame of the transmitter or the receiver. We choose the source frame below. Consider a standard test receiver to be used for performing quantum measurement on the radiation emitted by the source *J*s(**x** , *t* ), where **x** ∈ *Ss*, and *Ss* is the spatial support of the source region. The receiver spatial region is *S*r, and its strength is given by *J*r(**<sup>x</sup>**, *t*). Both *t* and *t* will be measured with a single clock at rest in the source lab frame. Let the test (receiver) current be reduced to a concentrated delta source

$$J\_{\mathbf{f}}(\mathbf{x}) = J\_{\mathbf{f}}(\mathbf{x}, t) = \delta(\mathbf{x} - \mathbf{x}\_{\mathbf{f}}) J\_{\mathbf{f}}(t), \quad \forall \, t \in T\_{\mathbf{f}}.\tag{76}$$

That is, we measure the response at a sharp spatial location **x**r, while we impose the receiver function restriction *J*r(*t*) on the time measurement interval *T*r. We can use (66) to express the position vectors as

$$\mathbf{x}\_{\mathbf{r}} = |\mathbf{x}\_{\mathbf{r}}| \pounds\_{\mathbf{r}}(\Omega),\tag{77}$$

where the unit angle-only dependent unit vector **x**ˆr is defined by

$$\mathfrak{X}\_{\mathbf{r}}(\Omega) := \frac{\mathbf{x}\_{\mathbf{r}}}{|\mathbf{x}\_{\mathbf{r}}|}. \tag{78}$$

By substituting the relations (76), (77), and (78) into the gain Formula (75), the following angular expression is obtained:

$$\mathcal{G}(\Omega; |\mathbf{x}\_{\mathbf{r}}|) = \int\_{\mathbf{x'} \in S\_{\mathbf{x'}}, t' \in T\_{\mathbf{s}}} \mathrm{d}^3 \mathbf{x'} \mathrm{d}t' \, G\_q^a(\Omega; |\mathbf{x}\_{\mathbf{r}}|, \mathbf{x'}) I\_s(\mathbf{x'}), \tag{79}$$

where

$$\mathbf{G}\_{\boldsymbol{\eta}}^{\mathrm{d}}(\boldsymbol{\Omega};|\mathbf{x}\_{\mathrm{r}}|,\mathbf{x}'):=\int\_{t\in T\_{\mathrm{r}}}\mathrm{d}t\,\mathrm{G}\_{\boldsymbol{\eta}}(|\mathbf{x}\_{\mathrm{r}}|\hat{\mathbf{x}}\_{\mathrm{r}}(\boldsymbol{\Omega})-\mathbf{x}';t-t')f\_{\mathrm{r}}(t),\tag{80}$$

and the notation of (27) was used. The integral (80) can be computed numerically for known current source distributions *J*s(*x*) by using the special functions representation of the Feynman propagator *Gq*(*x* − *x* ) given in Appendix H.

**Remark 24.** *We note that the q-antenna system gain Function* (79) *depends on the distance between the source and the receiver in addition to the angles. In general, one studies the asymptotic behavior of Ga q* (<sup>Ω</sup>; |*<sup>x</sup>*r|, *x* ) *in the long distance limit* |*<sup>x</sup>*r| → <sup>∞</sup>*, with the hope that this behaves as* |*<sup>x</sup>*r| −*n*, *with n often a small integer (usually 1 or 2). From the asymptotic limits of the special functions used in Appendix H, some possible relations could be derived using the bessel function large argument approximation, which in turn then might be further utilized in order to eliminate the dependence on the radial distance in the gain pattern* (79)*. However, such detailed computational considerations are outside the scope of this paper, which is mainly focused on the general fundamental theory.*
