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This book develops a new quantum Field Theory of Particle Spaces and shows that it leads directly to Octonion Cosmology. It begins by defining fermion and boson space fields for an arbitrary space-time dimension. It then proceeds to develop perturbation theory and to calculate the S-matrix element for the annihilation of a fermion-antifermion pair with the production of a scalar boson. The scalar boson has an internal array that maps to the dimension array of a space using a unitary group. Subsequently the dimension array of the boson determines the pattern of internal symmetries of each space of Octonion Cosmology including internal symmetry splitting. This process leads to the author's NEWQUeST, NEWUTMOST, and NEWUST. It provides a complete derivation of Octonion Cosmology with the resulting features of elementary particles. The author's Theory of Everything is thereby reduced to the Quantum Field Theory of Spaces. It has a clear, direct line of development based on the author's papers in the 1970's, and books in the Twenty-First Century. The author suggests that Quantum Field Theory is the lingua franca of Nature.