Generalising de Broglie-Bohm Mechanics with Effective Action for Quantum Gravity

The research generalises the de Broglie-Bohm formulation of mechanics to quantum gravity (QG) using an effective action framework. By replacing dBB equations with effective action dynamics, violations of Heisenberg uncertainty are avoided, and classical trajectories for bound states emerge even in non-gravitational scenarios. The approach extends naturally to field theories, enabling the calculation of field configurations and particle trajectories. The QG effective action is constructed using piecewise flat (PFQG) theory, approximated by general relativity coupled to matter in quantum field theory, with a cutoff determined by spacetime triangulation edge lengths. This framework provides a consistent foundation for Bohmian mechanics within QG.

The integration of Bohmian mechanics with quantum gravity has been advanced through the use of effective action formalism, offering a novel approach that circumvents issues related to Heisenberg uncertainty and classical trajectories. This method enhances Bohmian mechanics’ applicability in non-gravitational contexts and extends its utility to field theories by deriving field configurations. The framework employs piecewise flat quantum gravity (PFQG) theory, approximating it with a quantum field theory effective action for General Relativity coupled with matter, where spacetime triangulation edge lengths determine the cutoff.

Aleksandar Miković, affiliated with Lusófona University and COPELABS, as well as the Mathematical Physics Group at Instituto Superior Técnico, has contributed significantly to this field. His work, titled Generalization of Bohmian Mechanics and Quantum Gravity Effective Action, explores these advancements, providing a comprehensive analysis that bridges quantum mechanics with gravitational theory.

Integrating Bohmian mechanics into PL models for quantum gravity.

The paper introduces an innovative approach to quantum gravity by integrating Bohmian mechanics with piecewise linear (PL) spacetime models. Quantum gravity seeks to unify quantum mechanics and general relativity, two fundamental yet incompatible theories in physics. Bohmian mechanics, a deterministic interpretation of quantum mechanics, posits that particles have definite positions guided by a wave function, offering an alternative to probabilistic interpretations.

The paper employs PL models, commonly used in quantum gravity theories like spin foam models or loop quantum gravity, which represent spacetime as discrete building blocks. By combining Bohmian mechanics with these models, the authors aim to describe the dynamics of spacetime at a quantum level. Coherent states, known for maintaining their shape over time evolution, are utilized to approximate classical behavior, bridging the gap between discrete PL structures and observed smooth spacetime.

Physical states must satisfy constraints such as the Hamiltonian condition to ensure consistency with general relativity. Transition amplitudes describe how these states evolve, crucial for understanding dynamics within this framework. State-sum models, which sum over possible geometries weighted by amplitudes, are adapted to the Bohmian framework, contributing to spacetime configuration probabilities.

The model’s finiteness is a significant advantage, avoiding divergences that require renormalization in other quantum field theories. This aligns with concepts like asymptotic safety, where high-energy behavior remains well-defined without infinities. The paper discusses trajectories in spacetime dynamics, abstracting Bohmian mechanics to spacetime configurations evolving under quantum amplitudes, offering insights into spacetime evolution.

Combining Bohmian mechanics with discrete spacetime via state-sum methods.

The paper introduces a groundbreaking approach to quantum gravity by merging Bohmian mechanics with a piecewise linear spacetime model. Bohmian mechanics, known for its deterministic and causal framework, posits that particles have definite positions guided by a pilot wave, offering an objective reality without observers. This interpretation is particularly appealing in the context of quantum gravity, where reconciling quantum mechanics with general relativity remains a significant challenge.

The spacetime model employed in this research draws from Regge calculus, discretizing spacetime into flat regions or simplices. This piecewise linear approach facilitates the approximation of curved spacetime and may provide a regularization scheme to address issues such as renormalization in quantum gravity. By modeling spacetime in this manner, the paper aims to simplify complex calculations while maintaining fidelity to the principles of general relativity.

State-sum models from quantum topology are utilized to compute transition amplitudes by summing over different spacetime configurations. This method mirrors Feynman’s path integral approach but is adapted for discrete geometries, offering a novel way to calculate probabilities for various spacetime histories. The integration of these models with Bohmian mechanics allows for the exploration of quantum gravity in a deterministic framework, potentially leading to unique predictions and methods for comparing results with existing theories.

The paper also introduces hidden variables that may encompass aspects of spacetime geometry or field configurations. By extending Bohmian mechanics into this domain, the model seeks to preserve determinism and causality, ensuring that the evolution of these variables is guided by a pilot wave. This approach not only maintains an observer-independent reality but also aligns with the probabilistic origins of Bohmian mechanics, suggesting a method to derive probabilities from an initial measure over geometries or fields.

A Hamilton-Jacobi equation for geometries is derived.

The paper presents a novel approach to quantum gravity by merging Bohmian mechanics with state-sum models on a piecewise linear spacetime. This integration leverages the de Broglie-Bohm interpretation, which posits that particles have definite positions guided by a pilot wave, thereby avoiding issues such as non-unitarity and observer dependence. The use of state-sum models allows for the computation of partition functions or transition amplitudes by summing over different states or geometries, while the piecewise linear spacetime, inspired by Regge calculus, facilitates the approximation of curved spacetime.

A significant contribution is the derivation of a Hamilton-Jacobi equation for geometries, extending Bohmian trajectories from particle mechanics to spacetime geometry. This framework ensures consistency with standard Bohmian mechanics when gravity is disregarded and avoids foundational issues inherent in other quantum gravity theories. Additionally, the paper outlines a method to couple gravitational fields with matter without introducing inconsistencies, maintaining the integrity of Bohmian principles.

Despite these advancements, the model remains largely conceptual at this stage. Future work could explore its potential for testable predictions and compare it with established theories like loop quantum gravity or string theory. Investigating the dynamics of spacetime evolution within this framework and further developing the coupling of matter fields are also promising directions for research.

The study offers a promising approach to quantum gravity with realistic particle trajectories.

The study presents a novel approach to quantum gravity by integrating piecewise linear (PL) spacetime models with Bohmian mechanics and state-sum models. This framework simplifies geometric calculations while introducing realism through definite particle trajectories guided by wave functions. The use of a state-sum model allows for the summation of all possible PL manifold configurations, weighted by transition amplitudes that are crucial for understanding geometric evolution.

The derivation of an equation using the quantum effective action incorporates quantum corrections into dynamics, ensuring consistency with standard quantum mechanics. By employing Regge calculus and a path integral approach for matter fields, the model avoids divergences and supports the emergence of classical spacetime geometry in semi-classical limits. This suggests a promising direction for maintaining realism while addressing gravitational complexities.

Future work will focus on testing specific models, such as 3D quantum gravity, and exploring the coupling of various matter fields to refine the theory further. These developments aim to expand the framework’s applicability and validate its predictions within broader physical contexts.