# Fundamental Spacetime Representations of Quantum Antenna Systems
## Appendix I Proof of Relation  
Proof of Relation (61), when there is a source \(J(x^{\prime}),x^{\prime}\in D_{\mathrm{s}}\), then the actual radiated (quantum) state of the q-antenna system is obtained by applying the operator  
\[\int_{x^{\prime}\in D_{\mathrm{s}}}\mathrm{d}^{4}x^{\prime}J(x^{\prime})\phi^ {\dagger}(x^{\prime})\]  
on the ground state \(|0\rangle\) in order to create the following one-particle excited state  
\[\int_{x^{\prime}\in D_{\mathrm{s}}}\mathrm{d}^{4}x^{\prime}J(x^{\prime})\phi^ {\dagger}(x^{\prime})|0\rangle.\]  
To measure the probability of having a four-momentum state \(|q\rangle\) in the previous excited q-antenna state, we compute the source probability amplitude \(A[q;J(x^{\prime}),D_{\mathrm{s}}]\) as follows:  
\[A[q;J(x^{\prime}),D_{\mathrm{s}}]\] \[:=\langle q|\int_{x^{\prime}\in D_{\mathrm{s}}}\mathrm{d}^{4}x^{ \prime}J(x^{\prime})\phi^{\dagger}(x^{\prime})|0\rangle=\int_{\mathbf{p}\in \mathbb{R}^{3}}\frac{\mathrm{d}^{3}p}{(2\pi)^{3/2}(2\omega_{\mathbf{p}})^{1/2 }}\int_{x^{\prime}\in D_{\mathrm{s}}}\mathrm{d}^{4}x^{\prime}J(x^{\prime})e^ {\mathrm{i}p_{x}x^{\prime\prime}}\langle q|\mathbf{p}\rangle\] \[=\int_{\mathbf{p}\in\mathbb{R}^{3}}\frac{\mathrm{d}^{3}p}{(2\pi) ^{3/2}(2\omega_{\mathbf{p}})^{1/2}}J(p)\delta(\mathbf{q}-\mathbf{p})(2\pi)^{ 3/2}(2\omega_{\mathbf{p}})^{1/2}=J(q)|_{q^{0}=\omega_{\mathbf{q}}}=J(\mathbf{ q},\omega_{\mathbf{q}}),\]  
where  
\[J(k):=\int_{x\in D_{\mathrm{s}}}\mathrm{d}^{4}x\,J(x)e^{\mathrm{i}k_{p}x^{ \prime\prime}}=\int_{x\in\mathbb{R}^{3}}\mathrm{d}^{3}x\int_{t\in\mathbb{R}} \mathrm{d}t\,J(\mathbf{x},t)e^{\mathrm{i}(\omega t-\mathbf{k}\cdot\mathbf{x})} \tag{107}\]  
is the four-dimensional (Minkowski) Fourier transform in spacetime. The second line in (107) follows from using the momentum-space expansion of the source field \(\phi(x^{\prime})\) given by (104), while in the third line, we used the normalized four-momentum state inner product formula  
\[\langle q|\mathbf{p}\rangle=(2\pi)^{3/2}(2\omega_{\mathbf{p}})^{1/2}\delta( \mathbf{q}-\mathbf{p}), \tag{108}\]  
then proceeded to evaluate the trivial resulting integral over a delta function. Notes1 Other possible long-term aims behind the spacetime theory of q-antennas 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 physical-layer applications, e.g., quantum communications, cryptography, computing, and so on. [PERSON]'s equations in vacuo are exactly linear [49, 51]; a photon does not self-interact 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 \(J(x)\) itself. Well-known 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 \(J(x)\) itself are outside the scope of the proposed theory. Otherwise, relativistic causality would preclude information transfer [51]. In general, there is an agreement in the literature that entanglement-based quantum communication links cannot transmit information at superluminal speeds even though the quantum correlations between entangled states persists at spacelike separated terminals [2, 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. The propagators coincide with well-known Green's functions in the case of free fields.