Fluctuating dimension in a discrete model for quantum gravity based on the spectral principle

Area of Science:

• Theoretical Physics
• Quantum Gravity
• Mathematical Physics

Background:

• The spectral action principle provides a foundation for unifying gravity and quantum mechanics.
• Exploring discrete models is crucial for understanding quantum gravity phenomena.
• Generalized geometries allow for fluctuations in fundamental spacetime properties.

Purpose of the Study:

• To define a discrete model for Euclidean quantum gravity based on the spectral principle.
• To investigate generalized geometries with fluctuating topology, metric, and dimension.
• To analyze the phase structure and critical behavior of the model.

Main Methods:

• Utilizing the spectral principle of Connes and Chamseddine.
• Summing over generalized geometries with fluctuating properties.
• Relating the model to the Gaussian unitary ensemble of Hermitian matrices.

Main Results:

• The discrete model exhibits two distinct phases.
• The average number of points () is finite in one phase and diverges in the other.
• Spacetime dimension (δ) is a dynamical observable, with an upper bound <δ> < 2.

Conclusions:

• The developed discrete model offers a novel approach to quantum gravity.
• The model's phase structure and critical exponents provide insights into spacetime properties.
• The dynamical nature of spacetime dimension suggests new avenues for theoretical physics research.