Integrating out the Dirac Sea

We introduce a new method for dealing with fermionic quantum field theories amenable to a mean-field-type approximation. In this book we focus on the relativistic Hartree approximation. Our aim is to integrate out the Dirac sea and derive a ‘no-sea effective theory‘ with positive energy single particle states only. As the derivation of the no-sea effective theory involves only standard Feynman diagrams, our approach is quite general and not restricted to particular space-time dimensions. We develop and illustrate the approach in the ‘large N‘ limit of the Gross-Neveu model family in 1+1 dimensions, which serves as an ideal testing ground for our no-sea effective theory approach. The chiral Gross-Neveu model, also referred to as Nambu-Jona-Lasinio model, turns out to be of particular interest. In this case, we explicitly derive a consistent effective theory featuring both elementary ‘Pi meson‘ fields and (positive energy) ‘quark‘ fields, starting from a purely fermionic quantum field theory. In the second part of this book, we apply our approach to the Walecka model in 1+1 and 3+1 dimensions.