The Lie Group Structure of the $\eta-\xi$ Space-time and its Physical Significance

TL;DR

This paper explores the global structure of the $\\eta-\\xi$ space-time, demonstrating it has a Lie group structure and revealing its connections to Euclidean and Minkowskian spaces, with implications for quantum field theory.

Contribution

It proves that the $\\eta-\\xi$ space-time is homeomorphic to a product of complex spaces, establishing its Lie group structure and linking it to important space-time models.

Findings

The $\\eta-\\xi$ space-time is homeomorphic to $\mathbb{C}^{*}\times \mathbb{C}^{*}\times \mathbb{C}^{2}$.

The space-time admits a Lie group structure with transformation groups including a 2D Lorentz group.

Connections between Euclidean, Minkowskian, and $\\eta-\\xi$ spaces are clarified.

Abstract

The $\eta-\xi$ space-time is suggested by Gui for the quantum field theory in 1988. This paper consists of two parts. The first part is devoted to the discussion of the global properties of the $\eta-\xi$ space-time. The result contains a proof which asserts that the $\eta-\xi$ space-time is homeomorphic to $\mathbb{C}^{*}\times \mathbb{C}^{*}\times \mathbb{C}^{2}$ by means of two explicit maps, which shows that the $\eta-\xi$ space-time allows a Lie group structure. Thus some transformation groups, one of which is isomorphic to the Lorentz group in two dimensions, can be found. The other part of the paper is the discussion about the embedding of some subspaces in the $\eta-\xi$ space-time. In particular, it is pointed out that the Euclidean space-time and the Minkowskian space-time are linked in a way in the $\eta-\xi$ space-time such that the tilde field appears naturally. In addition some formulae in the $\eta-\xi$ space-time reappear in a more natural way.