The Principle of the Fermionic Projector: A New Variational Principle in Space-Time

In this short review, we introduce the mathematical framework of the principle of the fermionic projector and set up a variational principle in discrete space-time. The connection to the continuum theory is outlined. Recent results and open problems are discussed. The principle of the fermionic projector [1] provides a new model of space-time together with the mathematical framework for the formulation of physical theories. It was proposed to formulate physics in this framework based on a particular variational principle. In this short review article we explain a few basic ideas of the approach and report on recent results and open problems. It is generally believed that the concept of a space-time continuum (like Minkowski space or a Lorentzian manifold) should be modified for distances as small as the Planck length. We here assume that space-time is discrete on the Planck scale. Our notion of “discrete space-time” differs from other discrete approaches (like for example lattice gauge theories or quantum foam models) in that we do not assume any structures or relations between the space-time points (like for example the nearest-neighbor relation on a space-time lattice). Instead, we set up a variational principle for an ensemble of quantum mechanical wave functions. The idea is that for mimimizers of our variational principle, these wave functions should induce relations between the discrete space-time points, which, in a suitable limit, should go over to the topological and causal structure of a Lorentzian manifold. More specifically, in this limit the wave functions should group to a configuration of Dirac seas. For clarity, we first introduce the mathematical framework (Section 1) and discuss it afterwards, working out the underlying physical principles (Section 2). Then we outline the connection to the continuum theory (Sections 3 and 4). We conclude with a statement of results and an outlook (Sections 5 and 6). 1 A Variational Principle in Discrete Space-Time We let (H,<.|.>) be a complex inner product space of signature (N,N). Thus <.|.> is linear in its second and antilinear in its first argument, and it is symmetric, <Ψ | Φ> = <Φ |Ψ> for all Ψ,Φ ∈ H , and non-degenerate, <Ψ | Φ> = 0 for all Φ ∈ H =⇒ Ψ = 0 .