Generalized uncertainty principles and quantum field theory

TL;DR

This paper explores how generalized uncertainty principles modify quantum field theory, revealing a transition from Lorentz to Galilean invariance at small scales and long-wavelength recovery of Lorentz invariance.

Contribution

It applies deformed quantization to scalar fields with specific functions, uncovering new scale-dependent behaviors and invariance properties.

Findings

Green's function for f_+ shows Lorentz to Galilean transition at small scales

Modes for f_- do not propagate at small wavelengths

Lorentz invariance is restored at large wavelengths

Abstract

Quantum mechanics with a generalized uncertainty principle arises through a representation of the commutator $[\hat{x}, \hat{p}] = i f(\hat{p})$. We apply this deformed quantization to free scalar field theory for $f_\pm =1\pm \beta p^2$. The resulting quantum field theories have a rich fine scale structure. For small wavelength modes, the Green's function for $f_+$ exhibits a remarkable transition from Lorentz to Galilean invariance, whereas for $f_-$ such modes effectively do not propagate. For both cases Lorentz invariance is recovered at long wavelengths.