Quantum Field Theory with Noise, Filters, Scattering, and Curved Spacetime

Given a partition of space into two disjoint regions V 1 and V 2 with the separating surface being S , formulate the Einstein field equations for the metric on both these disjoint volumes and apply the boundary conditions namely continuity of the tangential and normal components of the velocity field and also continuity of the metric tensor and the matter density and pressure and on both sides of this interface S to solve for the metric and the density field, the pressure field, the velocity field on both these sides. A classical example is the collapse of a dust like sphere radially towards the centre. Let R ( t ) be the radius of an expanding sphere. For r ( t ) matter is present and we use the Robertson Walker metric in this region corresponding to a homogeneous and isotropic space-time while for r>R ( t ), we use the Schwarzchild solution corresponding to a spherically symmetric metric in the absence of matter. Design a neural network that will be able to estimate the metric tensor in both of these regions as functions of space-time from measurements carried out on electromagnetic fields in these two regions as well as on measurements carried out on particle geodesic trajectories. Assuming the metric tensor to be a quantum field, cause it to interact with a quantum electromagnetic field of photons and the quantum Dirac field of electrons and positrons and assuming the initial state of the gravitons, photons and electrons-positrons to be a tensor product of Bosonic and Fermionic coherent states, evaluate the quantum mechanical moments of these fields in this coherent state as it evolves with time by using perturbation theory applied to the free fields in the absence of interactions, ie, express the solution to the interacting fields using pertubation theory as polynomials in the free field creation and annihilation operators. Having done so, write a programme to estimate the coherent state parameters in terms of the quantum moments of the field. By setting up the Hamiltonian of the interacting fields in terms of the free field creation and annihilation operators after taking into account quantum nois into account, develop a quantum filtering theory assuming the Bosonic and Fermionic noise fields to be independent of each other. By quantizing small fluctuations in the metric tensor around the classical Robertson-Walker metric and analyzing the effects of these quantum metric fluctuations upon the Dirac and Maxwell fields, estimate the scale factor of the expanding universe as a function of time by analyzing the change in the scattering amplitudes of electrons, positrons and photons caused by gravitons in a coherent state. Train a neural network to output this expanding scale factor as well as the graviton coherent state by measuring scattering probabilities of electrons, positrons and photons. Develop the quantum filter when the noise comprises of both Bosonic and Fermionic noise present in the HPS qsde which are mutually independent, ie, they act in different tensor product components of a Boson Fock space which implies that non-demolition measurements can be achieved only by measuring the Bosonic noise components passed through the HP qsde.