Causality, Order, Information and Topology Prakash Panangaden1 1School of Computer Science McGill University 5th June 2013 / Causal structure in quantum theory. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 1 / 37 2 Causal Structure 3 Domain Theory 4 Domains and causal structure 5 Interval Domains 6 Reconstructing spacetime Outline 1 Introduction Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 2 / 37 3 Domain Theory 4 Domains and causal structure 5 Interval Domains 6 Reconstructing spacetime Outline 1 Introduction 2 Causal Structure Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 2 / 37 4 Domains and causal structure 5 Interval Domains 6 Reconstructing spacetime Outline 1 Introduction 2 Causal Structure 3 Domain Theory Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 2 / 37 5 Interval Domains 6 Reconstructing spacetime Outline 1 Introduction 2 Causal Structure 3 Domain Theory 4 Domains and causal structure Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 2 / 37 6 Reconstructing spacetime Outline 1 Introduction 2 Causal Structure 3 Domain Theory 4 Domains and causal structure 5 Interval Domains Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 2 / 37 Outline 1 Introduction 2 Causal Structure 3 Domain Theory 4 Domains and causal structure 5 Interval Domains 6 Reconstructing spacetime Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 2 / 37 The topology can be derived from this. Ordered topological spaces (domains) were used by Dana Scott to model computation as information processing. Spacetime carries a natural domain structure. Introduction Overview Causal structure - mathematically modelled as a partial order - can be taken to be the fundamental structure of spacetime. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 3 / 37 Ordered topological spaces (domains) were used by Dana Scott to model computation as information processing. Spacetime carries a natural domain structure. Introduction Overview Causal structure - mathematically modelled as a partial order - can be taken to be the fundamental structure of spacetime. The topology can be derived from this. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 3 / 37 Spacetime carries a natural domain structure. Introduction Overview Causal structure - mathematically modelled as a partial order - can be taken to be the fundamental structure of spacetime. The topology can be derived from this. Ordered topological spaces (domains) were used by Dana Scott to model computation as information processing. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 3 / 37 Introduction Overview Causal structure - mathematically modelled as a partial order - can be taken to be the fundamental structure of spacetime. The topology can be derived from this. Ordered topological spaces (domains) were used by Dana Scott to model computation as information processing. Spacetime carries a natural domain structure. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 3 / 37 A finite piece of information about the output should only require a finite piece of information about the input. This is just what the − δ definition says. Data types are domains (ordered topological spaces) and computable functions are continuous. Introduction Background Scott’s vision: computability should be continuity in some topology. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 4 / 37 This is just what the − δ definition says. Data types are domains (ordered topological spaces) and computable functions are continuous. Introduction Background Scott’s vision: computability should be continuity in some topology. A finite piece of information about the output should only require a finite piece of information about the input. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 4 / 37 Data types are domains (ordered topological spaces) and computable functions are continuous. Introduction Background Scott’s vision: computability should be continuity in some topology. A finite piece of information about the output should only require a finite piece of information about the input. This is just what the − δ definition says. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 4 / 37 Introduction Background Scott’s vision: computability should be continuity in some topology. A finite piece of information about the output should only require a finite piece of information about the input. This is just what the − δ definition says. Data types are domains (ordered topological spaces) and computable functions are continuous. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 4 / 37 A (globally hyperbolic) spacetime can be given domain structure: approximate points. [CMP Nov’06] The space of causal curves in the Vietoris topology is compact (cf. Sorkin-Woolgar) [GRG ’06] The geometry can be captured by a Martin “measurement.” [AMS Symposia in Pure and Appliedd Math 2012] Introduction Summary of Results The causal order alone determines the topology of globally hyperbolic spacetimes. [CMP Nov’06] Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 5 / 37 The space of causal curves in the Vietoris topology is compact (cf. Sorkin-Woolgar) [GRG ’06] The geometry can be captured by a Martin “measurement.” [AMS Symposia in Pure and Appliedd Math 2012] Introduction Summary of Results The causal order alone determines the topology of globally hyperbolic spacetimes. [CMP Nov’06] A (globally hyperbolic) spacetime can be given domain structure: approximate points. [CMP Nov’06] Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 5 / 37 The geometry can be captured by a Martin “measurement.” [AMS Symposia in Pure and Appliedd Math 2012] Introduction Summary of Results The causal order alone determines the topology of globally hyperbolic spacetimes. [CMP Nov’06] A (globally hyperbolic) spacetime can be given domain structure: approximate points. [CMP Nov’06] The space of causal curves in the Vietoris topology is compact (cf. Sorkin-Woolgar) [GRG ’06] Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 5 / 37 Introduction Summary of Results The causal order alone determines the topology of globally hyperbolic spacetimes. [CMP Nov’06] A (globally hyperbolic) spacetime can be given domain structure: approximate points. [CMP Nov’06] The space of causal curves in the Vietoris topology is compact (cf. Sorkin-Woolgar) [GRG ’06] The geometry can be captured by a Martin “measurement.” [AMS Symposia in Pure and Appliedd Math 2012] Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 5 / 37 Topology: 4 dimensional real manifold, Hausdorff, paracompact,... Differentiable structure: tangent spaces Causal structure: light cones, defines metric up to conformal 9 transformations. This is 10 of the metric. Parallel transport: affine structure. Lorentzian metric: gives a length scale. Causal Structure The layers of spacetime structure Set of events: no structure Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 6 / 37 Differentiable structure: tangent spaces Causal structure: light cones, defines metric up to conformal 9 transformations. This is 10 of the metric. Parallel transport: affine structure. Lorentzian metric: gives a length scale. Causal Structure The layers of spacetime structure Set of events: no structure Topology: 4 dimensional real manifold, Hausdorff, paracompact,... Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 6 / 37 Causal structure: light cones, defines metric up to conformal 9 transformations. This is 10 of the metric. Parallel transport: affine structure. Lorentzian metric: gives a length scale. Causal Structure The layers of spacetime structure Set of events: no structure Topology: 4 dimensional real manifold, Hausdorff, paracompact,... Differentiable structure: tangent spaces Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 6 / 37 Parallel transport: affine structure. Lorentzian metric: gives a length scale. Causal Structure The layers of spacetime structure Set of events: no structure Topology: 4 dimensional real manifold, Hausdorff, paracompact,... Differentiable structure: tangent spaces Causal structure: light cones, defines metric up to conformal 9 transformations. This is 10 of the metric. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 6 / 37 Lorentzian metric: gives a length scale. Causal Structure The layers of spacetime structure Set of events: no structure Topology: 4 dimensional real manifold, Hausdorff, paracompact,... Differentiable structure: tangent spaces Causal structure: light cones, defines metric up to conformal 9 transformations. This is 10 of the metric. Parallel transport: affine structure. Panangaden (McGill) Causality, Order, Information and Topology Benasque June 2013 6 / 37 Causal Structure The layers of spacetime structure Set of events: no structure Topology: 4 dimensional real manifold, Hausdorff, paracompact,... Differentiable structure: tangent spaces Causal structure: light cones, defines metric up to conformal