Discrete Causal Theory Emergent Spacetime and the Causal Metric Hypothesis 1st Edition by Benjamin Dribus 9783319500812 3319500813

Part I Foundations of Discrete Causal Theory

1 Introduction

1.1 A Picture and an Equation

1.2 The Causal Metric Hypothesis

1.3 Emergent Spacetime

1.4 Practice with Toy Histories

1.5 The Relative Viewpoint

1.6 Background Independence

1.7 Particles and Fields

1.8 Kinematics and Dynamics

1.9 Phenomenology

1.10 Looking Ahead: Twenty Questions

References

2 The Causal Metric Hypothesis

2.1 General Version of the Hypothesis

2.2 Classical Version of the Hypothesis

2.3 Structure on Relativistic Spacetime

2.4 Metric Structure

2.5 Conformal Structure

2.6 Causal Structure

2.7 Causality Conditions

2.8 Metric Recovery

2.9 Order Good, Continuum Bad

2.10 The Philosopher’s Peril

References

3 Causal Sets and Generalizations

3.1 Causal Set Theory: Conceptual Roots

3.2 Causal Sets and the Causal Metric Hypothesis

3.3 Quantum Causal Set Theory

3.4 Causal Set Dynamics and Phenomenology

3.5 Axioms of Causal Set Theory

3.6 Generalized Causal Sets: Directed Sets and Multidirected Sets

3.7 Building Blocks of Structure in Multidirected Sets

3.8 The Shadow of Grothendieck

3.9 Independent Modes of Influence

3.10 The Causal Relation

References

4 Local Properties

4.1 Local Behavior

4.2 Topology

4.3 Four Topologies for Multidirected Sets

4.4 Interval Finiteness and Star Finiteness

4.5 Sprinkled Causal Sets; Lorentz Invariance; Spatiotemporal Locality

4.6 Relative Multidirected Sets Over a Fixed Base

4.7 Seven Arguments Questioning Interval Finiteness

4.8 Nine Arguments in Favor of Star Finiteness

4.9 Hierarchy of Finiteness Conditions

4.10 New Axioms for Discrete Classical Causal Theory

References

Part II Discrete Quantum Causal Theory

5 Relation Space and Generalizations

5.1 Relation Space Over a Multidirected Set

5.2 Abstract Element Space

5.3 Preservation of Information

5.4 Permeability

5.5 Analogy Involving Morphism Categories

5.6 Power Spaces

5.7 Causal Atomic Resolution

5.8 Top–Down Causation; Classical Holism

5.9 Causal Path Spaces

5.10 Causal Path Algebras

References

6 Quantum Spacetime

6.1 Quantum Preliminaries

6.2 Path Summation

6.3 Iteration of Structure

6.4 Co-Relative Histories

6.5 Quantum Causal Generalities

6.6 Feynman Revisited

6.7 Quantum Theory in Relation Space

6.8 Schrödinger’s Equation via Path Summation

6.9 Causal Schrödinger-Type Equations

6.10 Generalized Quantum Amplitudes via Causal Path Functionals

References

7 Kinematics and Dynamics

7.1 Frames of Reference

7.2 Refinements

7.3 Kinematic Accounts

7.4 Kinematic Schemes

7.5 Positive Sequential Kinematic Scheme

7.6 Quantum Causal Metric Hypothesis

7.7 Path Summation Over a Kinematic Scheme

7.8 Generational Kinematics and Dynamics

7.9 Completions of Kinematic Schemes

7.10 Hyperquantum Theory

References

8 Toward Phenomenology

8.1 Axioms and Technical Methods Revisited

8.2 Phase Theory

8.3 Covariance

8.4 Emergence of the Poincaré Group

8.5 Elementary Graph Dynamical Considerations

8.6 Algebraic Structure and Hierarchy

8.7 Miscellany Regarding Alternative Formulations

8.8 Connections with Other Physical Theories

8.9 Connections with Physics-Related Mathematics

8.10 Final Remarks