Exploring $\boldsymbol{2+2}$ Answers to $\boldsymbol{3+1}$ Questions

TL;DR

This paper investigates the use of Kleinian ($2+2$) signature spacetimes to gain insights into Lorentzian ($3+1$) physics, exploring analytic continuations, supersymmetry structures, and potential implications for the cosmological constant problem.

Contribution

It introduces the application of $2+2$ signature formulations to understand Lorentzian physics, highlighting supersymmetry constraints and novel symmetry breakings relevant to cosmology.

Findings

Kleinian signature provides a framework for constructing low particle flux states.

Supersymmetry algebra exists naturally in $2+2$ signature and constrains correlation functions.

Spontaneous Lorentz symmetry breaking leads to $ frac{1}{2}$-supersymmetry algebras relevant to non-supersymmetric systems.

Abstract

We explore potential uses of physics formulated in Kleinian (i.e., $2+2$) signature spacetimes as a tool for understanding properties of physics in Lorentzian (i.e., $3+1$) signature. Much as Euclidean (i.e., $4+0$) signature quantities can be used to formally construct the ground state wavefunction of a Lorentzian signature quantum field theory, a similar analytic continuation to Kleinian signature constructs a state of low particle flux in the direction of analytic continuation. There is also a natural supersymmetry algebra available in $2+2$ signature, which serves to constrain the structure of correlation functions. Spontaneous breaking of Lorentz symmetry can produce various $\mathcal{N} = 1/2$ supersymmetry algebras that in $3 + 1$ signature correspond to non-supersymmetric systems. We speculate on the possible role of these structures in addressing the cosmological constant problem.