Workshop on Causal Structure in Physics and Computer Science Introductory Remarks

Prakash Panangaden

McGill University

+ < > Causality Workshop 26th June 2003 – p.1/8 This workshop is a first step towards exploring these connections and making contacts between researchers in the two fields.

What is all this about?

Serendipity!

Partially ordered sets have emerged as a key mathematical ingredient in the causal set approach to quantum gravity (Sorkin) but also as a fundamental notion in the semantics of programming languages: domain theory. It is possible that ideas (especially topological and sheaf theoretic ideas) in the two fields are common. Causal structure in computer science was also captured in the order-theoretic framework (Winskel).

+ < > Causality Workshop 26th June 2003 – p.2/8 What is all this about?

Serendipity!

Partially ordered sets have emerged as a key mathematical ingredient in the causal set approach to quantum gravity (Sorkin) but also as a fundamental notion in the semantics of programming languages: domain theory. It is possible that ideas (especially topological and sheaf theoretic ideas) in the two fields are common. Causal structure in computer science was also captured in the order-theoretic framework (Winskel). This workshop is a first step towards exploring these connections and making contacts between researchers in the two fields.

+ < > Causality Workshop 26th June 2003 – p.2/8 Minkowskian: Absolute light cones; light speed bounds propagation of causality. Topology is R4 = R × R3. Flat geometry. General Relativistic: Absolute light cones as above; curved spacetime geometry. Topology can be nontrivial; in particular global topology is important. Quantum Gravity: Discrete spacetime structure; causal structure given by a partially ordered set. Can we recover the topology from the causal structure?

Spacetime Structure

Newtonian: Absolute space, absolute time. No bound on propagation of effects. Topology is R4 = R × R3.

+ < > Causality Workshop 26th June 2003 – p.3/8 General Relativistic: Absolute light cones as above; curved spacetime geometry. Topology can be nontrivial; in particular global topology is important. Quantum Gravity: Discrete spacetime structure; causal structure given by a partially ordered set. Can we recover the topology from the causal structure?

Spacetime Structure

Newtonian: Absolute space, absolute time. No bound on propagation of effects. Topology is R4 = R × R3. Minkowskian: Absolute light cones; light speed bounds propagation of causality. Topology is R4 = R × R3. Flat geometry.

+ < > Causality Workshop 26th June 2003 – p.3/8 Quantum Gravity: Discrete spacetime structure; causal structure given by a partially ordered set. Can we recover the topology from the causal structure?

Spacetime Structure

Newtonian: Absolute space, absolute time. No bound on propagation of effects. Topology is R4 = R × R3. Minkowskian: Absolute light cones; light speed bounds propagation of causality. Topology is R4 = R × R3. Flat geometry. General Relativistic: Absolute light cones as above; curved spacetime geometry. Topology can be nontrivial; in particular global topology is important.

+ < > Causality Workshop 26th June 2003 – p.3/8 Spacetime Structure

Newtonian: Absolute space, absolute time. No bound on propagation of effects. Topology is R4 = R × R3. Minkowskian: Absolute light cones; light speed bounds propagation of causality. Topology is R4 = R × R3. Flat geometry. General Relativistic: Absolute light cones as above; curved spacetime geometry. Topology can be nontrivial; in particular global topology is important. Quantum Gravity: Discrete spacetime structure; causal structure given by a partially ordered set. Can we recover the topology from the causal structure?

+ < > Causality Workshop 26th June 2003 – p.3/8 Topological space: nearness Smooth manifold: local vector space structure Conformal structure: light cones Affine structure: parallel transport Metric structure: distance

Structures on Spacetimes

Set of events

+ < > Causality Workshop 26th June 2003 – p.4/8 Smooth manifold: local vector space structure Conformal structure: light cones Affine structure: parallel transport Metric structure: distance

Structures on Spacetimes

Set of events Topological space: nearness

+ < > Causality Workshop 26th June 2003 – p.4/8 Conformal structure: light cones Affine structure: parallel transport Metric structure: distance

Structures on Spacetimes

Set of events Topological space: nearness Smooth manifold: local vector space structure

+ < > Causality Workshop 26th June 2003 – p.4/8 Affine structure: parallel transport Metric structure: distance

Structures on Spacetimes

Set of events Topological space: nearness Smooth manifold: local vector space structure Conformal structure: light cones

+ < > Causality Workshop 26th June 2003 – p.4/8 Metric structure: distance

Structures on Spacetimes

Set of events Topological space: nearness Smooth manifold: local vector space structure Conformal structure: light cones Affine structure: parallel transport

+ < > Causality Workshop 26th June 2003 – p.4/8 Structures on Spacetimes

Set of events Topological space: nearness Smooth manifold: local vector space structure Conformal structure: light cones Affine structure: parallel transport Metric structure: distance

+ < > Causality Workshop 26th June 2003 – p.4/8 The presence of matter curves the ambient spacetime. Other material objects (even light) follow the straight lines (geodesics) of the curved geometry. Geometry becomes a dynamical entity: expanding universes and gravity waves.

Gravity as Spacetime Geometry

Einstein’s Idea: there is no such thing as gravitational “force”.

+ < > Causality Workshop 26th June 2003 – p.5/8 Other material objects (even light) follow the straight lines (geodesics) of the curved geometry. Geometry becomes a dynamical entity: expanding universes and gravity waves.

Gravity as Spacetime Geometry

Einstein’s Idea: there is no such thing as gravitational “force”. The presence of matter curves the ambient spacetime.

+ < > Causality Workshop 26th June 2003 – p.5/8 Geometry becomes a dynamical entity: expanding universes and gravity waves.

Gravity as Spacetime Geometry

Einstein’s Idea: there is no such thing as gravitational “force”. The presence of matter curves the ambient spacetime. Other material objects (even light) follow the straight lines (geodesics) of the curved geometry.

+ < > Causality Workshop 26th June 2003 – p.5/8 Gravity as Spacetime Geometry

Einstein’s Idea: there is no such thing as gravitational “force”. The presence of matter curves the ambient spacetime. Other material objects (even light) follow the straight lines (geodesics) of the curved geometry. Geometry becomes a dynamical entity: expanding universes and gravity waves.

+ < > Causality Workshop 26th June 2003 – p.5/8 Typically (but not always) one has a completely uninformative element called bottom ⊥. Posets representing data types are called domains.

Computable functions between domains are order preserving (monotone).

Posets in Computer Science

Posets model partial information.

+ < > Causality Workshop 26th June 2003 – p.6/8 Posets representing data types are called domains.

Computable functions between domains are order preserving (monotone).

Posets in Computer Science

Posets model partial information. Typically (but not always) one has a completely uninformative element called bottom ⊥.

+ < > Causality Workshop 26th June 2003 – p.6/8 Computable functions between domains are order preserving (monotone).

Posets in Computer Science

Posets model partial information. Typically (but not always) one has a completely uninformative element called bottom ⊥. Posets representing data types are called domains.

+ < > Causality Workshop 26th June 2003 – p.6/8 Posets in Computer Science

Posets model partial information. Typically (but not always) one has a completely uninformative element called bottom ⊥. Posets representing data types are called domains. Computable functions between domains are order preserving (monotone).

+ < > Causality Workshop 26th June 2003 – p.6/8 Everything is the limit (supremum) of a sequence of finite elements. So certain suprema are required to exist. Computable functions are required to preserve these suprema. Intuition: finite information about the output requires finite information about the input. These are called continuous functions. There is a topology - the Scott topology - which captures the above notion.

Topology of Domains

Domains have a notion of finite element.

+ < > Causality Workshop 26th June 2003 – p.7/8 Computable functions are required to preserve these suprema. Intuition: finite information about the output requires finite information about the input. These are called continuous functions. There is a topology - the Scott topology - which captures the above notion.

Topology of Domains

Domains have a notion of finite element. Everything is the limit (supremum) of a sequence of finite elements. So certain suprema are required to exist.

+ < > Causality Workshop 26th June 2003 – p.7/8 There is a topology - the Scott topology - which captures the above notion.

Topology of Domains

Domains have a notion of finite element. Everything is the limit (supremum) of a sequence of finite elements. So certain suprema are required to exist. Computable functions are required to preserve these suprema. Intuition: finite information about the output requires finite information about the input. These are called continuous functions.

+ < > Causality Workshop 26th June 2003 – p.7/8 Topology of Domains

Domains have a notion of finite element. Everything is the limit (supremum) of a sequence of finite elements. So certain suprema are required to exist. Computable functions are required to preserve these suprema. Intuition: finite information about the output requires finite information about the input. These are called continuous functions. There is a topology - the Scott topology - which captures the above notion.

+ < > Causality Workshop 26th June 2003 – p.7/8 Rafael Sorkin: causal sets Glynn Winskel: event structures Bob Coecke: ??? Samson Abramsky: topology of domains Fotini Markopoulou Kalamara: quantum causal evolution Prakash Panangaden: possible problems

The Plan for Today

Sumati Surya: causal structures in spacetimes

+ < > Causality Workshop 26th June 2003 – p.8/8 Glynn Winskel: event structures Bob Coecke: ??? Samson Abramsky: topology of domains Fotini Markopoulou Kalamara: quantum causal evolution Prakash Panangaden: possible problems

The Plan for Today

Sumati Surya: causal structures in spacetimes Rafael Sorkin: causal sets

+ < > Causality Workshop 26th June 2003 – p.8/8 Bob Coecke: ??? Samson Abramsky: topology of domains Fotini Markopoulou Kalamara: quantum causal evolution Prakash Panangaden: possible problems

The Plan for Today

Sumati Surya: causal structures in spacetimes Rafael Sorkin: causal sets Glynn Winskel: event structures

+ < > Causality Workshop 26th June 2003 – p.8/8 Samson Abramsky: topology of domains Fotini Markopoulou Kalamara: quantum causal evolution Prakash Panangaden: possible problems

The Plan for Today

Sumati Surya: causal structures in spacetimes Rafael Sorkin: causal sets Glynn Winskel: event structures Bob Coecke: ???

+ < > Causality Workshop 26th June 2003 – p.8/8 Fotini Markopoulou Kalamara: quantum causal evolution Prakash Panangaden: possible problems

The Plan for Today

Sumati Surya: causal structures in spacetimes Rafael Sorkin: causal sets Glynn Winskel: event structures Bob Coecke: ??? Samson Abramsky: topology of domains

+ < > Causality Workshop 26th June 2003 – p.8/8 Prakash Panangaden: possible problems

The Plan for Today

Sumati Surya: causal structures in spacetimes Rafael Sorkin: causal sets Glynn Winskel: event structures Bob Coecke: ??? Samson Abramsky: topology of domains Fotini Markopoulou Kalamara: quantum causal evolution

+ < > Causality Workshop 26th June 2003 – p.8/8