Fundamental Spacetime Representations of Quantum Antenna Systems
Abstract
:1. Introduction
 The source is a controllable current distribution function of both space and time.
 Ultimately, the source current is externally controlled.
 A spatiotemporal current distribution profile serving as a model for a quantum antenna source can control the radiation proprieties in both space and time, and usually for near and farfield scenarios as well.
 Understanding what is meant by quantum radiation.
 Understanding the role played by particle emission dynamics in light of the waveparticle duality characteristic of quantum phenomena.
 Understanding the complex role played by quantum fields, propagators, and Green’s functions in quantum radiation.
 Understanding the role played by manyparticle states/interactions in quantum radiation processes.
 Primary research objectives of the present article:
 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.
 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 (spin0) Klein–Gordon qantennas.
 Secondary (pedagogical) objectives of the present article:
 Introducing new applications of fundamental theory (here relativistic quantum mechanics) to different audiences, e.g., quantum engineering and quantum technology research.
 Introducing the subject of QFT in a selfsufficient manner by providing detailed appendices explaining how relativistic quantum mechanics is formulated for an audience familiar only with nonrelativistic quantum mechanics.1
2. Antenna Theory: Classical and Quantum Radiation Scenarios
 Classical electromagnetic radiation in free space or linear materials, when viewed from the perspective of its ultimate source (external field), is inherently linear.2
 On the other hand, quantum antennas involving higherorder processes (manyparticle interactions, npoint processes with $n>1$, etc.) are intrinsically nonlinear radiation problems (due to the manybody nature of interactions in quantum field theory).
3. The General Theory of Quantum Antenna Systems
3.1. Preliminary Considerations
3.2. A Generic Interaction Hamiltonian Description of Quantum Antenna Systems
 ${H}_{\mathrm{in}}$ captures intrinsic interaction in the fundamental quantum field of the qantenna system, e.g., self interactions such as polynomial Lagrangian terms containing powers of $\varphi \left(x\right)$ larger than three such as the mainstream interacting ${\varphi}^{4}$theory [54,56,57].
 The terms ${H}_{\mathrm{s}}$ and ${H}_{\mathrm{r}}$ describe, respectively, the interaction between the source and the receiver antennas on one side and the fundamental quantum field $\varphi \left(x\right)$ of the qantenna system on the other side. These interactions should be understood as processes localized within their respective spacetime domains ${D}_{\mathrm{s}}$ and ${D}_{\mathrm{r}}$.
 Finally, the term ${H}_{\mathrm{c}}$ corresponds to channel interactions and couplings, e.g., coupling of the excited quantum field $\varphi \left(x\right)$ with scattering objects located within the effective path of an excited quantum particle produced by the source and directed toward the receiver.6
3.3. The General Expansion Theorem of Quantum Radiation Fields
4. Linear Quantum Antenna Systems
4.1. Introduction
 Construct a quantum source model resembling the point (infinitesimal source) in classical antenna theory.
 Using the previous quantum point source model, construct the quantum state radiated by the qantenna due to arbitrary continuous or discrete source distribution (superposition principle).
 Construct Green’s function of the qantenna using the previous superposition integral.
 Introduce and evaluate the qantenna radiation pattern using Green’s function (mostly in the momentum space representation).
4.2. The Klein–Gordon Field Theory
4.3. An Elementary Model for Point Quantum Particle Excitation
 It emits quantum particles (massive particles when $m\ne 0$ and scalar Klein–Gordon particles when $m=0$) at particular spacetime positions.
 Once generated, the quantum field $\varphi \left(x\right)$ would somehow “propagate” the quantum particle in space and time.
4.4. The Feynman Propagator of Quantum Antennas
4.5. Generalization to Multiple Discrete and Continuous and Sources
 The classical source function$$J\left({x}^{\prime}\right):{D}_{\mathrm{s}}\subset {\mathbb{M}}^{4}\to \mathbb{R}.$$
 The quantum source operator$${J}_{q}\left({D}_{\mathrm{s}}\right):{\mathcal{D}}_{s}\to \mathcal{O},$$$${\mathcal{D}}_{s}:=\{D\subset {\mathbb{M}}^{4}\text{}D\text{}\mathrm{is}\text{}\mathrm{open},\text{}\mathrm{cl}\left(D\right)\text{}\mathrm{is}\text{}\mathrm{compact}\text{}\mathrm{in}\text{}{\mathbb{R}}^{4}\}$$
5. On the General Structure of Radiation Processes in Linear Quantum Antenna Systems
5.1. The General Structure of the Quantum Antenna Propagator Process
 We first must form the correct relativistic sum over all allowable momentum states. This is accomplished by the Lorentz invariant integral operator$${\int}_{\mathbf{p}\in {\mathbb{R}}^{3}}\frac{{\mathrm{d}}^{3}p}{{\left(2\pi \right)}^{3}}\frac{1}{2{\omega}_{\mathbf{p}}}.$$
 Each momentum state $\mathbf{p}\rangle $ will be summed over all possible source locations ${x}^{\prime}\in {D}_{\mathrm{s}}$ in the source region via the integral operator$${\int}_{{x}^{\prime}\in {D}_{\mathrm{s}}}{\mathrm{d}}^{4}{x}^{\prime}.$$This step is also relativistic, since ${D}_{\mathrm{s}}\subset {\mathbb{M}}^{4}$ and ${\mathrm{d}}^{4}{x}^{\prime}$ are Lorentz invariant.21
 The next crucial step is to multiply by the factor $\mathrm{exp}\left(\mathrm{i}{p}_{\mu}{x}^{\prime \mu}\right)$. This will trigger the production of a quantum wave (particle) emanating from ${x}^{\prime}$ and spreading radially away from the point source.
 Finally, in order to observe the radiation field at a distance, we multiply the wave produced at ${x}^{\prime}$ by a propagation factor $\mathrm{exp}(\mathrm{i}{p}_{\mu}{x}^{\mu})$. This will guarantee that the field has been effectively propagated and absorbed at the observation location x.
5.2. Comparative Analysis of the Three Fundamental Types of Antennas
5.3. On the Causal Spacetime Structure of Radiation Emitted by Quantum Antenna Systems
 Consider a point source case. Figure 4a indicates the future lightcone ${C}_{{x}^{\prime}}$ of the event located at ${x}^{\prime}$, i.e., the apex of the cone in the spacetime diagram given therein. Since we assume for simplicity that the operational principle behind our qantennabased communication link’s receiver is based on the process of annihilating the radiated particle at the observation point x, it follows that only receivers located inside the antenna causal lightcone ${C}_{{x}^{\prime}}$ can receive information from the point source.
 For potential receivers located outside the antenna causal cone, it is not possible to transmit information at all, unless one admits some superluminal mechanism to be used for sending information, which is currently not accepted by majority of scientists.
6. Quantum Antenna Radiation Patterns: Basic Constructions
6.1. Introduction
6.2. The Probability Law of Producing Radiated Quantum States
6.3. Constructing the Quantum Antenna Directivity Pattern
6.4. The Probability Law of Receiving Radiated Quantum States: SourceReceiver Coupling Gain Estimation
7. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
QFT  Quantum field theory 
QM  Quantum mechanics 
SR  Special relativity 
cantenna  Classical antenna 
qantenna  Quantum antenna 
qradiation  Quantum radiation 
qstate  Quantum state 
qsource  Quantum source 
csource  Classical source 
EM  Electromagnetic/Electromagnetics 
ACGF  Antenna current Green’s function 
Appendix A. Classical Antenna Theory
Appendix B. The Relativistic FourVector Formalism
Quantity  Expression 

$\mathbf{p}$ (particle momentum)  $\mathbf{p}=\mathbf{k}$ 
${E}_{\mathbf{p}}$ (particle energy)  ${E}_{\mathbf{p}}={\omega}_{\mathbf{p}}={\omega}_{\mathbf{k}}$ 
Relativistic dispersion relation  ${E}_{\mathbf{p}}^{2}={\mathbf{p}}^{2}+{m}^{2}$ 
${\partial}^{\mu}$ (fourvector differential operator)  ${\partial}^{\mu}=(\partial /\partial t,\nabla )$ 
${x}^{\mu}$ (position fourvector)  ${x}^{\mu}=(t,\mathbf{x})$ 
${k}^{\mu}$ (relativistic wavevector)  ${k}^{\mu}=(\omega ,\mathbf{k})$ 
${p}^{\mu}$ (photon fourmomentum vector)  ${p}^{\mu}=({E}_{\mathbf{p}},\mathbf{p})={k}^{\mu}$ 
${g}_{\mu \nu}$ (Lorentz metric tensor)  ${g}_{\mu \nu}=\mathrm{diag}(1,1,1,1)$ 
${p}^{\mu}{x}_{\mu}$ (fourvector inner product)  ${p}^{\mu}{x}_{\mu}={g}_{\mu \nu}{p}^{\mu}{x}^{\nu}=\omega t\mathbf{k}\text{}\xb7\text{}\mathbf{x}$ 
Appendix C. Natural Units
Appendix D. Dirac Interaction Picture
 In the Schrodinger picture, the state evolves in time according to the full Hamiltonian while the operators are constant.
 In the Heisenberg picture, the state is constant (timeindependent), but the operator evolves according to the full Hamiltonian.
 In the Dirac picture, both state and operators evolve with time. However, the time evolution is decoupled into two distinct and independent components. First, all interaction (Dirac) picture operators evolve according to the free Hamiltonian as per the corresponding Heisenberg Equation (A9). Second, the Dirac state ${\psi}_{\mathrm{I}}\left(t\right)\rangle $ evolves independently according to the dynamic Equation (A14).
Appendix E. On the Background to Theorem
Appendix F. The Neutral Klein–Gordon Field Theory
Appendix G. The Relativistic FieldTheoretic Canonical Quantization Algorithm
Algorithm A1 The Canonical Quantization Algorithm (General Formulation). 

Appendix H. On the Numerical Evaluation of the Propagator
Appendix I. Proof of Relation
Notes
1  Other possible longterm aims behind the spacetime theory of qantennas proposed below include the stimulation of fruitful collaboration between theoreticians, especially those working on problems related to foundations, and applied quantum physicists and engineers, whose attention is often more focused on algorithmic and physicallayer applications, e.g., quantum communications, cryptography, computing, and so on. 
2  Maxwell’s equations in vacuo are exactly linear [49,51]; a photon does not selfinteract with itself [52]. As in classical antenna theory, in the proposed quantum antenna theory given here, all nonlinearities are relegated to production of the source $J\left(x\right)$ itself. Wellknown examples include gun diodes (microwaves) and laser devices (optics), where the diode itself is nonlinear but the relation between the current or field as external source and the fields radiated into vacuum is linear. The physical nonlinear processes behind the source function $J\left(x\right)$ itself are outside the scope of the proposed theory. 
3  Otherwise, relativistic causality would preclude information transfer [51]. In general, there is an agreement in the literature that entanglementbased quantum communication links cannot transmit information at superluminal speeds even though the quantum correlations between entangled states persists at spacelike separated terminals [2,4]. 
4  This is a realistic assumption in our model, in conformity with the common practical situation where typical classical or quantum sources are supported by bounded spatial domains and radiate within a finite time interval while practical measurement times are also bounded. 
5  The propagators coincide with wellknown Green’s functions in the case of free fields. For interacting field theories, the propagators are not in general known, but viable approximations can be estimated using perturbation theory, in which the freefield Green’s function is used as a fundamental building block in order to compute more complex higherorder interaction processes [54,55,56,57,58,59,60]. 
6  By effective path, we just mean the spacetime trajectory in the Feynman’s path integral expansion of the propagator that contributes most to the total probability amplitude, e.g., see [57]. 
7  Cf. Appendix D. 
8  Interaction is a more general concept than quantum correlation, since two uncorrelated objects could interact, where in this case, the interaction terms are just the multiplication of the strengths of each process while the two remain, at least stochastically speaking, independent. An example illustrating this will be given shortly. 
9  
10  In QFT, integrals such as (20) are handled using fourpoint Green’s functions of the form ${G}^{\left(4\right)}({x}_{3},{x}_{4},{x}_{1},{x}_{2}$), where the latter is called the fourpoint correlation (Green’s) function [54,55,57]. In our case, we just choose ${x}_{3}={x}_{4}=x$, since x is the common observation point of the receiving qantenna system. 
11  The realization of the need to eventually differentiate a purely classical source function from within any stochastic system (including quantum systems) was originally proposed within the context of quantum optics in the 1960s [62]. 
12  For the purposes of illustrating the main ideas of qantenna systems, this assumption simplifies considerably the presentation, but the main ideas related to qantennas are unchanged when more complicated field theories are considered such as spin1 and spin$1/2$ theories. 
13  
14  
15  The generation of $\varphi \left(x\right)$ itself is not treated here for simplicity. However, note that computing the quantized fields of coupled matter–field systems is a fairly welldeveloped area in the physics and engineering literature, mostly using the methods of perturbation theory [38,54,55,67,77]. On the other hand, in this paper, our main focus is on how to deploy an already given or generated quantum field $\varphi \left(x\right)$ in order to construct the radiation pattern and the Green’s function of a qantenna system for use in applications in controlled radiation of quantum states. 
16  When reworked in the full momentum space $p\in {\mathbb{M}}^{4}$, the integral (21) becomes even more interesting, both computationally and conceptually, since one can show then that point source excitations lead to the production of virtual (offmassshell) particles [38,57]. However, we will not make use of these expansions in the present paper though they are expected to play an important role in developing the nearfield theory of radiating quantum source systems. 
17  The concept of particle localization in QFT is difficult both philosophically and mathematically, and several approaches have been proposed in the literature so far, apparently with no universal agreement on the ontological status of particles in field quantization. Such more advanced issues do not affect the practically oriented theory of quantum antennas developed in this paper. For some indepth discussion of localization in field theory, see [37,78]. 
18  
19  Momentum space means either $\mathbf{p}\in {\mathbb{R}}^{3}$ or $p\in {\mathbb{M}}^{4}$. In this paper, whenever the term momentum space is invoked, it is to be understood that we will mostly work with the former version, i.e., in three dimensions. 
20  In this paper, we do not consider the possible case of unbounded source domains such as infinite current sheets and lines. 
21  Cf. Appendix G. 
22  Cf. Appendix A. 
23  Indeed, this is how directivity is defined in classical antenna theory, e.g., see [35]. Note further that $\mathcal{D}$ in the LHS of (69) was directly expressed in terms of the spherical angles $\phi ,\theta ,$ in order to emphasize the spatial angular character of this momentum space function. See [71,72] for more details about the momentum space approach to directivity. 
24  See Coleman’s discussion of the generic detection process in highenergy physics as given in [38]. 
25  Indeed, negative energy/frequencies obtained as solutions to the massive particle dispersion equation ${E}^{2}={\mathbf{p}}^{2}+{m}^{2}$ are often reinterpreted as antiparticles and plane waves of the form $\mathrm{exp}(+\mathrm{i}{p}_{\mu}{x}^{\mu})$ are taken to represent outgoing antiparticles with momentum p and energy ${E}_{\mathbf{p}}$ [38]. 
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Mikki, S. Fundamental Spacetime Representations of Quantum Antenna Systems. Foundations 2022, 2, 251289. https://doi.org/10.3390/foundations2010019
Mikki S. Fundamental Spacetime Representations of Quantum Antenna Systems. Foundations. 2022; 2(1):251289. https://doi.org/10.3390/foundations2010019
Chicago/Turabian StyleMikki, Said. 2022. "Fundamental Spacetime Representations of Quantum Antenna Systems" Foundations 2, no. 1: 251289. https://doi.org/10.3390/foundations2010019