# nLab causal structure

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

Given a Lorentzian manifold, then its causal structure is the information at each tangent space of the subspace of lightlike tangent vectors – the “light cones”. This information distinguishes timelike from the spacelike vectors. Hence if one thinks of the Lorentzian manifold as modelling a spacetime in physics, then this information encodes, at each point, the directions along which causal influences may propagate in this spacetime.

This causal stucture is closely related to the underlying conformal structure. One may also define a concept of a manifold with causal structure or causal manifold without reference to a concept of Lorentzian pseudo-Riemannian structure (Bannier 88, Rainer 99, Khavkine 12).

## In algebraic quantum field theory

In the formalization of quantum field theory in terms of locally covariant AQFT a QFT over a background field of classical gravity is axiomatized as a causally local net of observables on a category of Lorentzian manifolds. Making more of the degrees of freedom of gravity become quantized themselves would mean to replace the latter with a category of manifolds with less structure than Lorentzian manifold structure. Since the causal structure is necessary to express “local net” at all, one idea is to consider local nets on a category of causal manifolds.

This idea is mentioned as motivation for developing concepts of causal manifolds for instance in Rainer 99, and in a more sophisticated version in Khavkine 12.

• Ulrich Bannier, On generally covariant quantum field theory and generalized causal and dynamical structures, Communications in Mathematical Physics, 118(1):163–170, March 1988

• Martin Rainer, Cones and causal structures on topological and differentiable manifolds (arXiv:gr-qc/9905106)

• Renee Hoekzema, On the Topology of Lorentzian manifolds, Essay as a part of the 2010-2011 lecture on “Quantum Fields in curved spacetimes” by W.G. Unruh 2011 (pdf)

• Igor Khavkine, Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory (arXiv:1211.1914)