# Minimum length scenarios that maintain continuous symmetries

### APA

Pye, J. (2019). Minimum length scenarios that maintain continuous symmetries. Perimeter Institute. https://pirsa.org/19110054

### MLA

Pye, Jason. Minimum length scenarios that maintain continuous symmetries. Perimeter Institute, Nov. 07, 2019, https://pirsa.org/19110054

### BibTex

@misc{ pirsa_19110054, doi = {10.48660/19110054}, url = {https://pirsa.org/19110054}, author = {Pye, Jason}, keywords = {Quantum Gravity}, language = {en}, title = {Minimum length scenarios that maintain continuous symmetries}, publisher = {Perimeter Institute}, year = {2019}, month = {nov}, note = {PIRSA:19110054 see, \url{https://pirsa.org}} }

Jason Pye University of Waterloo

## Abstract

It has long been argued that combining the uncertainty principle with gravity will lead to an effective minimum length at the Planck scale. A particular challenge is to model the presence of a smallest length scale in a manner which respects continuous spacetime symmetries. One path for deriving low-energy descriptions of an invariant minimum length in quantum field theory is based on generalized uncertainty principles. Here I will consider the question how this approach enables one to retain Euclidean or even Lorentzian symmetries. The Euclidean case yields a ultraviolet cutoff in the form of a bandlimit, and this then allows one to apply the powerful Shannon sampling theorem of classical information theory which establishes the equivalence between continuous and discrete representations of information. As a consequence, one obtains discrete representations of fields which are more subtle than a simple discretization of space, and are in fact equivalent to a continuum representation. Quantum fields in this model exhibit a finite density of information and a corresponding regularization of the entanglement of the vacuum, as I will demonstrate in detail. We then examine the Lorentzian symmetry generalization. This case leads to a Lorentz-invariant analogue of bandlimitation, and we discuss the nature of the corresponding sampling theory.