Light-Cone Expansion of the Dirac Sea in the Presence of Chiral and Scalar Potentials

We study the Dirac sea in the presence of external chiral and scalar/pseudoscalar potentials. In preparation, a method is developed for calculating the advanced and retarded Green's functions in an expansion around the light cone. For this, we first expand all Feynman diagrams and then explicitly sum up the perturbation series. The light-cone expansion expresses the Green's functions as an infinite sum of line integrals over the external potential and its partial derivatives. The Dirac sea is decomposed into a causal and a non-causal contribution. The causal contribution has a light-cone expansion which is closely related to the light-cone expansion of the Green's functions; it describes the singular behavior of the Dirac sea in terms of nested line integrals along the light cone. The non-causal contribution, on the other hand, is, to every order in perturbation theory, a smooth function in position space.Comment: 59 pages, LaTeX (published version

Fermion Systems in Discrete Space-Time

Fermion systems in discrete space-time are introduced as a model for physics on the Planck scale. We set up a variational principle which describes a non-local interaction of all fermions. This variational principle is symmetric under permutations of the discrete space-time points. We explain how for minimizers of the variational principle, the fermions spontaneously break this permutation symmetry and induce on space-time a discrete causal structure.Comment: 8 pages, LaTeX, few typos corrected (published version

The Chiral Index of the Fermionic Signature Operator

We define an index of the fermionic signature operator on even-dimensional globally hyperbolic spin manifolds of finite lifetime. The invariance of the index under homotopies is studied. The definition is generalized to causal fermion systems with a chiral grading. We give examples of space-times and Dirac operators thereon for which our index is non-trivial.Comment: 21 pages, LaTeX, 3 figures, minor corrections (published version

Fermion Systems in Discrete Space-Time - Outer Symmetries and Spontaneous Symmetry Breaking

A systematic procedure is developed for constructing fermion systems in discrete space-time which have a given outer symmetry. The construction is illustrated by simple examples. For the symmetric group, we derive constraints for the number of particles. In the physically interesting case of many particles and even more space-time points, this result shows that the permutation symmetry of discrete space-time is always spontaneously broken by the fermionic projector.Comment: 43 pages, LaTeX, few typos corrected (published version

The Principle of the Fermionic Projector: An Approach for Quantum Gravity?

In this short article we introduce the mathematical framework of the principle of the fermionic projector and set up a variational principle in discrete space-time. The underlying physical principles are discussed. We outline the connection to the continuum theory and state recent results. In the last two sections, we speculate on how it might be possible to describe quantum gravity within this framework.Comment: 18 pages, LaTeX, few typos corrected (published version