Causal Fermion Systems: A Quantum Space-Time Emerging From an Action Principle

Abstract

Abstract. Causal fermion systems are introduced as a general mathematical framework for formulating relativistic quantum theory. By specializing, we recover earlier notions like fermion systems in discrete space-time, the fermionic projector and causal variational principles. We review how an effect of spontaneous structure formation gives rise to a topology and a causal structure in space-time. Moreover, we outline how to construct a spin connection and curvature, leading to a proposal for a "quantum geometry" in the Lorentzian setting. We review recent numerical and analytical results on the support of minimizers of causal variational principles which reveal a "quantization effect" resulting in a discreteness of space-time. A brief survey is given on the correspondence to quantum field theory and gauge theories.

The chiral index of the fermionic signature operator

“…The advantage of working with a causal fermion system is that the underlying space-time does not need to be a Lorentzian manifold, but it can be a more general "quantum space-time" (for more details see [2]). We now recall a few basic notions from [3]. On F we consider the topology induced by the operator norm A := sup{ Au H with u H = 1}.…”

“…After providing the necessary preliminaries (Section 2), this definition will be given in Section 3 in space-times of finite lifetime. In order to work out the mathematical essence of our index, in Section 4 we also give its definition in the general setting of causal fermion systems (for an introduction to causal fermion systems see [3] or [4]). Section 5 is devoted to a variant of the chiral index which applies in the special case of the massless Dirac equation and a Dirac operator which is odd with respect to the chiral grading.…”

Lorentzian spectral geometry for globally hyperbolic surfaces

“…We now estimate the eigenvalues of S from above and below using the min-max principle. Since the spectrum is symmetric (see Proposition 2.7), we know that 17) giving the upper bound…”

Causal Fermion Systems: An Overview

“…Combining these transformations should give rise to an approximate symmetry of the wave evaluation operator (1.18) in the sense that if we compare the transformation of the space-time point with the unitary transformation by setting 17) then the operator E τ : H → C 0 (M, SM ) should be sufficiently small. Here "small" means for example that E vanishes on the orthogonal complement of a finite-dimensional subspace of H; for details see [21,Section 6].…”

“…To avoid confusion, we remark that in earlier papers (see [15], [17]) a slightly different definition of the causal structure was used. But the modified definition used here seems preferable.…”