QNNs and EKF for Transmission Line Control and Field Estimation

In this paper, we formulate the basic Maxwell-Dirac equations for the electromagnetic field of photons interacting with the Dirac field of electrons and positrons taking into account noisy components of the electromagnetic field modeled in the context of the Hudson-Parthasarathy quantum stochastic calculus. These equations are formulated within a waveguide and result in Helmholtz differential equations for the z component of the em fields with Dirac current source. We demonstrate how quantum noise perturbs these fields and how one can calculate the moments of the resulting perturbed quantum fields due to interaction and noise in a coherent state of the photons and Fermions. We also discuss perturbations to these fields produced by quantum gravitational fields using the covariant Maxwell equations of general relativity. Specifically, using elementary perturbation theory, we derive Helmholtz equations for the z components of the perturbed electromagnetic fields with current sources being functions of the unperturbed electromagnetic and Dirac fields and also the gravitational metric perturbations. We indicate how these calculations can be used in quantum guide parameter estimation problems. We also indicate how by expressing the Hamiltonian of the Maxwell-Dirac field in terms of creation and annihilation operators and creation and annihilation quantum stochastic processes for the noise, we can derive the Hudson-Parthasarathy noisy Schrodinger equation for the unitary evolution in second quantized field space and how by measuring non-demolition noise, we can estimate the evolving state of the Maxwell-Dirac field on a real time basis. The paper provides a self-contained discussion of quantum noise in field theory. Some aspects of supersymmetry, string gauge field theory and non-Abelian gauge field theory in the presence of noise are also discussed here. The paper provides a self-contained account of how to treat quantum Bosonic and Fermionic noisy fields in the context of quantum field theory by highlighting the fact that quantum noisy fields have an additional time-component of quantum randomness as compared with that of conventional quantum fields.