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

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.Comment: 23 pages, LaTeX, 2 figures, footnote added on page

Gravity and Matter in Causal Set Theory

The goal of this paper is to propose an approach to the formulation of dynamics for causal sets and coupled matter fields. We start from the continuum version of the action for a Klein-Gordon field coupled to gravity, and rewrite it first using quantities that have a direct correspondent in the case of a causal set, namely volumes, causal relations, and timelike lengths, as variables to describe the geometry. In this step, the local Lagrangian density $L(f;x)$ for a set of fields $f$ is recast into a quasilocal expression $L_0(f;p,q)$ that depends on pairs of causally related points $p \prec q$ and is a function of the values of $f$ in the Alexandrov set defined by those points, and whose limit as $p$ and $q$ approach a common point $x$ is $L(f;x)$. We then describe how to discretize $L_0(f;p,q)$, and use it to define a discrete action.Comment: 13 pages, no figures; In version 2, friendlier results than in version 1 are obtained following much shorter derivation

From Discrete Space-Time to Minkowski Space: Basic Mechanisms, Methods and Perspectives

This survey article reviews recent results on fermion system in discrete space-time and corresponding systems in Minkowski space. After a basic introduction to the discrete setting, we explain a mechanism of spontaneous symmetry breaking which leads to the emergence of a discrete causal structure. As methods to study the transition between discrete space-time and Minkowski space, we describe a lattice model for a static and isotropic space-time, outline the analysis of regularization tails of vacuum Dirac sea configurations, and introduce a Lorentz invariant action for the masses of the Dirac seas. We mention the method of the continuum limit, which allows to analyze interacting systems. Open problems are discussed.Comment: 25 pages, LaTeX, 6 figures, minor improvements (published version

Network Cosmology

Prediction and control of the dynamics of complex networks is a central problem in network science. Structural and dynamical similarities of different real networks suggest that some universal laws might accurately describe the dynamics of these networks, albeit the nature and common origin of such laws remain elusive. Here we show that the causal network representing the large-scale structure of spacetime in our accelerating universe is a power-law graph with strong clustering, similar to many complex networks such as the Internet, social, or biological networks. We prove that this structural similarity is a consequence of the asymptotic equivalence between the large-scale growth dynamics of complex networks and causal networks. This equivalence suggests that unexpectedly similar laws govern the dynamics of complex networks and spacetime in the universe, with implications to network science and cosmology

The moduli space of isometry classes of globally hyperbolic spacetimes

This is the last article in a series of three initiated by the second author. We elaborate on the concepts and theorems constructed in the previous articles. In particular, we prove that the GH and the GGH uniformities previously introduced on the moduli space of isometry classes of globally hyperbolic spacetimes are different, but the Cauchy sequences which give rise to well-defined limit spaces coincide. We then examine properties of the strong metric introduced earlier on each spacetime, and answer some questions concerning causality of limit spaces. Progress is made towards a general definition of causality, and it is proven that the GGH limit of a Cauchy sequence of $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz spaces is again a $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz space. Finally, we give a necessary and sufficient condition, similar to the one of Gromov for the Riemannian case, for a class of Lorentz spaces to be precompact.Comment: 29 pages, 9 figures, submitted to Class. Quant. Gra

Extended cardiolipin anchorage to cytochrome c: model for protein-mitochondrial membrane binding.

Two models have been proposed to explain the interaction of cytochrome c with cardiolipin (CL) vesicles. In one case, an acyl chain of the phospholipid accommodates into a hydrophobic channel of the protein located close the Asn52 residue, whereas the alternative model considers the insertion of the acyl chain in the region of the Met80-containing loop. In an attempt to clarify which proposal offers a more appropriate explanation of cytochrome c-CL binding, we have undertaken a spectroscopic and kinetic study of the wild type and the Asn52Ile mutant of iso-1-cytochrome c from yeast to investigate the interaction of cytochrome c with CL vesicles, considered here a model for the CL-containing mitochondrial membrane. Replacement of Asn52, an invariant residue located in a small helix segment of the protein, may provide data useful to gain novel information on which region of cytochrome c is involved in the binding reaction with CL vesicles. In agreement with our recent results revealing that two distinct transitions take place in the cytochrome c-CL binding reaction, data obtained here support a model in which two (instead of one, as considered so far) adjacent acyl chains of the liposome are inserted, one at each of the hydrophobic sites, into the same cytochrome c molecule to form the cytochrome c-CL complex

A numerical study of the correspondence between paths in a causal set and geodesics in the continuum

This paper presents the results of a computational study related to the path-geodesic correspondence in causal sets. For intervals in flat spacetimes, and in selected curved spacetimes, we present evidence that the longest maximal chains (the longest paths) in the corresponding causal set intervals statistically approach the geodesic for that interval in the appropriate continuum limit.Comment: To the celebration of the 60th birthday of Rafael D. Sorki

Nonperturbative dynamics for abstract (p,q) string networks

We describe abstract (p,q) string networks which are the string networks of Sen without the information about their embedding in a background spacetime. The non-perturbative dynamical formulation invented for spin networks, in terms of causal evolution of dual triangulations, is applied to them. The formal transition amplitudes are sums over discrete causal histories that evolve (p,q) string networks. The dynamics depend on two free SL(2,Z) invariant functions which describe the amplitudes for the local evolution moves.Comment: Latex, 12 pages, epsfig, 7 figures, minor change

Relationships between various characterisations of wave tails

One can define several properties of wave equations that correspond to the absence of tails in their solutions, the most common one by far being Huygens' principle. Not all of these definitions are equivalent, although they are sometimes assumed to be. We analyse this issue in detail for linear scalar waves, establishing some relationships between the various properties. Huygens' principle is almost always equivalent to the characteristic propagation property, and in two spacetime dimensions the latter is equivalent to the zeroth order progressing wave propagation property. Higher order progressing waves in general do have tails, and do not seem to admit a simple physical characterisation, but they are nevertheless useful because of their close association with exactly solvable two-dimensional equations.Comment: Plain TeX, 26 page