[Paper Review] Spectral Dimension of kappa-deformed space-time

V. Anjana, E. Harikumar|arXiv (Cornell University)|Jan 1, 2015

Noncommutative and Quantum Gravity Theories13 references3 citations

TL;DR

This paper investigates the spectral dimension of $κ$-deformed space-time using a $κ$-deformed diffusion equation with two distinct Laplacians in $n$-dimensional Euclidean space. It shows that the spectral dimension decreases with decreasing probe scale, flowing to $-\infty$ as $\sigma \to 0$, and becomes negative below critical scales ($\sigma < 0.78a^2$ and $\sigma < 0.031a^2$), indicating non-physical behavior unless the deformation parameter $a$ is bounded. The results suggest a potential multi-scale structure and highlight the need for reinterpreting spectral dimension in non-local, higher-derivative frameworks.

ABSTRACT

We investigate the spectral dimension of $κ$-space-time using the $κ$-deformed diffusion equation. The deformed equation is constructed for two different choices of Laplacians in $n$-dimensional, $κ$-deformed Euclidean space-time. We use an approach where the deformed Laplacians are expressed in the commutative space-time itself. Using the perturbative solutions to diffusion equations, we calculate the spectral dimension of $κ$-deformed space-time and show that it decreases as the probe length decreases. By introducing a bound on the deformation parameter, spectral dimension is guaranteed to be positive definite. We find that, for one of the choices of the Laplacian, the non-commutative correction to the spectral dimension depends on the topological dimension of the space-time whereas for the other, it is independent of the topological dimension. We have also analysed the dimensional flow for the case where the probe particle has a finite extension, unlike a point particle.

Motivation & Objective

• To investigate the spectral dimension of $\kappa$-deformed space-time using a diffusion equation approach.
• To analyze how non-commutativity, encoded via $\kappa$-deformed Laplacians, affects the effective dimensionality felt by a probe particle.
• To determine whether the spectral dimension remains positive and physically meaningful at trans-Planckian scales.
• To examine the impact of finite probe extension on dimensional flow in $\kappa$-deformed space-time.
• To explore the implications of negative spectral dimensions and propose constraints on the deformation parameter $a$ for physical viability.

Proposed method

• Constructing two distinct $\kappa$-deformed Laplacians in $n$-dimensional Euclidean space, both ensuring the correct commutative limit.
• Formulating a perturbative solution to the $\kappa$-deformed diffusion equation using these Laplacians in the commutative space-time framework.
• Defining the return probability $P(\sigma)$ from the trace of the diffusion kernel to compute the spectral dimension via $D_s = -2 \partial_\sigma \ln P(\sigma)$.
• Analyzing the spectral dimension's dependence on the diffusion time $\sigma$ and the deformation parameter $a$.
• Introducing a finite probe extension to assess robustness of dimensional flow under non-point-like probing.
• Applying bounds on the deformation parameter $a$ to ensure the spectral dimension remains positive definite at small scales.

Experimental results

Research questions

• RQ1 How does the spectral dimension of $\kappa$-deformed space-time evolve with decreasing probe scale $\sigma$?
• RQ2 What is the dependence of the non-commutative correction to the spectral dimension on the topological dimension of space-time?
• RQ3 Can the spectral dimension remain positive for all $\sigma$ if the deformation parameter $a$ is constrained?
• RQ4 How does the finite extension of the probe particle affect the dimensional flow in $\kappa$-deformed space-time?
• RQ5 Why does the spectral dimension diverge to $-\infty$ at small $\sigma$, and what does this imply for the physical interpretation of the model?

Key findings

• For both choices of $\kappa$-deformed Laplacian, the spectral dimension decreases with decreasing probe scale $\sigma$ and diverges to $-\infty$ as $\sigma \to 0$.
• The spectral dimension becomes negative when $\sigma < 0.78a^2$ for the first Laplacian choice and $\sigma < 0.031a^2$ for the second, indicating unphysical behavior at very small scales.
• For a 4-dimensional space-time, the spectral dimension reaches zero at $\sigma = 25a^2/32$ for the first Laplacian and $\sigma = a^2/32$ for the second.
• The non-commutative correction to the spectral dimension depends on the topological dimension for one Laplacian choice, but is independent of it for the other.
• The dimensional flow behavior is robust under finite probe extension, preserving the same qualitative behavior at both large and small $\sigma$.
• A bound on the deformation parameter $a^2 < 32\sigma/25$ and $a^2 < 32\sigma$ ensures the spectral dimension remains positive, suggesting a multi-scale structure in $\kappa$-space-time.