The Principle of the Fermionic Projector, Appendices

TL;DR

This paper introduces the principle of the fermionic projector as a new mathematical framework for physics beyond the standard model, employing variational principles and hyperbolic differential equations to analyze fundamental interactions.

Contribution

It provides a rigorous definition of the fermionic projector and develops a variational principle that models interactions similar to the standard model.

Findings

Resolution of Klein's paradox via the fermionic projector

Formulation of a new variational principle in space-time

Derivation of effective interactions resembling the standard model

Abstract

The "principle of the fermionic projector" provides a new mathematical framework for the formulation of physical theories and is a promising approach for physics beyond the standard model. The book begins with a brief review of relativity, relativistic quantum mechanics and classical gauge theories, with the emphasis on the basic physical concepts and the mathematical foundations. The external field problem and Klein's paradox are discussed and then resolved by introducing the so-called fermionic projector, a global object in space-time which generalizes the notion of the Dirac sea. The mathematical core of the book is to give a precise definition of the fermionic projector and to employ methods of hyperbolic differential equations for its detailed analysis. The fermionic projector makes it possible to formulate a new type of variational principles in space-time. The mathematical tools for the analysis of the corresponding Euler-Lagrange equations are developed. A particular variational principle is proposed which gives rise to an effective interaction showing many similarities to the interactions of the standard model. The main chapters of the book are easily accessible for beginning graduate students in mathematics or physics. Several appendices provide supplementary material which will be useful to the experienced researcher.