Quantum Fields on Causal Sets

Abstract

Causal set theory provides a model of discrete spacetime in which spacetime events are represented by elements of a causal set---a locally finite, partially ordered set in which the partial order represents the causal relationships between events. The work presented here describes a model for matter on a causal set, specifically a theory of quantum scalar fields on a causal set spacetime background. The work starts with a discrete path integral model for particles on a causal set. Here quantum mechanical amplitudes are assigned to trajectories within the causal set. By summing these over all trajectories between two spacetime events we obtain a causal set particle propagator. With a suitable choice of amplitudes this is shown to agree (in an appropriate sense) with the retarded propagator for the Klein-Gordon equation in Minkowski spacetime. This causal set propagator is then used to define a causal set analogue of the Pauli-Jordan function that appears in continuum quantum field theories. A quantum scalar field is then modelled by an algebra of operators which satisfy three simple conditions (including a bosonic commutation rule). Defining time-ordering through a linear extension of the causal set these field operators are used to define a causal set Feynman propagator. Evidence is presented which shows agreement (in a suitable sense) between the causal set Feynman propagator and the continuum Feynman propagator for the Klein-Gordon equation in Minkowski spacetime. The Feynman propagator is obtained using the eigendecomposition of the Pauli-Jordan function, a method which can also be applied in continuum-based theories. The free field theory is extended to include interacting scalar fields. This leads to a suggestion for a non-perturbative S-matrix on a causal set. Models for continuum-based phenomenology and spin-half particles on a causal set are also presented.

PhD Thesis, Imperial College London, September 2010

References in corpus (14)

• A Classical Sequential Growth Dynamics for Causal Sets
• The Scalar Curvature of a Causal Set
• The causal set approach to quantum gravity
• The causal hierarchy of spacetimes
• The mathematical basis for deterministic quantum mechanics
• Feynman Propagator for a Free Scalar Field on a Causal Set
• Particle propagators on discrete spacetime
• Evidence for a continuum limit in causal set dynamics
• Energy-momentum diffusion from spacetime discreteness
• Indications of de Sitter Spacetime from Classical Sequential Growth Dynamics of Causal Sets
• Light, Links and Causal Sets
• Dynamics for causal sets with matter fields: A Lagrangian-based approach
• Particle simulations in causal set theory
• Photon dispersion in causal sets

Cited by in corpus (6)

• Directions in Causal Set Quantum Gravity
• A preferred ground state for the scalar field in de Sitter space
• Electromagnetic Lagrangian on a causal set that resides on edges rather than points
• Entanglement Entropy of Disjoint Spacetime Intervals in Causal Set Theory
• The use of test functions to help define quadratic Lagrangian on a causal set
• Restoring locality of scalar fields on a causal set by avoiding the use of d'Alembertians