The Causal Perturbation Expansion Revisited: Rescaling the Interacting Dirac Sea

Details The Causal Perturbation Expansion Revisited: Rescaling the Interacting Dirac Sea The Causal Perturbation Expansion Revisited: Rescaling the Interacting Dirac Sea

2009-01-03

arxiv.org/abs/0901.0334v3

J. Math. Phys. 51 (2010) 072301

Physics

Mathematical Physics

34 pages, LaTeX, introduction expanded

Scientific paper

10.1063/1.3449058

The causal perturbation expansion defines the Dirac sea in the presence of a time-dependent external field. It yields an operator whose image generalizes the vacuum solutions of negative energy and thus gives a canonical splitting of the solution space into two subspaces. After giving a self-contained introduction to the ideas and techniques, we show that this operator is in general not idempotent. We modify the standard construction by a rescaling procedure giving a projector on the generalized negative-energy subspace. The resulting rescaled causal perturbation expansion uniquely defines the fermionic projector in terms of a series of distributional solutions of the Dirac equation. The technical core of the paper is to work out the combinatorics of the expansion in detail. It is also shown that the fermionic projector with interaction can be obtained from the free projector by a unitary transformation. We finally analyze the consequences of the rescaling procedure on the light-cone expansion.

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