# Fundamental Spacetime Representations of Quantum Antenna Systems
## Appendix I Proof of Relation
For interacting field theories, the propagators are not in general known, but viable approximations can be estimated using perturbation theory, in which the free-field Green's function is used as a fundamental building block in order to compute more complex higher-order interaction processes [54, 55, 56, 57, 58, 59, 60]. By _effective path_, we just mean the spacetime trajectory in the [PERSON]'s path integral expansion of the propagator that contributes most to the total probability amplitude, e.g., see [57]. Cf. Appendix D. 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. Nowadays, such many-body effects are approached in the modern literature on condensed-matter physics, e.g., through the elaborate and elegant language of many-point correlation or Green's functions in QFT [59, 60, 61]. In QFT, integrals such as (20) are handled using four-point Green's functions of the form \(G^{(4)}(x_{3},x_{4},x_{1},x_{2})\), where the latter is called the four-point correlation ([PERSON]'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 q-antenna system. 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]. For the purposes of illustrating the main ideas of q-antenna systems, this assumption simplifies considerably the presentation, but the main ideas related to q-antennas are unchanged when more complicated field theories are considered such as spin-1 and spin-1/2 theories. For additional information on the physical processes modeled by a Klein-Gordon quantum field, see [74, 75]. For a more detailed discussion of relativistic QFT, see [38, 76]. The generation of \(\phi(x)\) itself is not treated here for simplicity. However, note that computing the quantized fields of coupled matter-field systems is a fairly well-developed 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 \(\phi(x)\) in order to construct the radiation pattern and the Green's function of a q-antenna system for use in applications in controlled radiation of quantum states. 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 that point source excitations lead to the production of virtual (off-mass-shell) 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 near-field theory of radiating quantum source systems. 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 in-depth discussion of localization in field theory, see [37, 78]. Recall that in QFT, antiparticles are interpreted as particles moving _backward_ in time [57, 58]. 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. In this paper, we do not consider the possible case of unbounded source domains such as infinite current sheets and lines.