Perturbation theory of the space-time non-commutative real scalar field theories

TL;DR

This paper develops a perturbative framework for space-time non-commutative scalar field theories, ensuring unitarity and establishing Feynman rules, revealing divergence structures similar to space-space non-commutative theories without UV-IR mixing.

Contribution

It formulates a unitary perturbation theory for space-time non-commutative scalar fields, emphasizing minimal time-ordering and $ au$-time ordering, and clarifies divergence and UV-IR mixing properties.

Findings

Unitarity of the S-matrix is verified order by order.

Divergence structure matches that of space-space non-commutative theories.

No UV-IR mixing problem is present in space-time non-commutative theories.

Abstract

The perturbative framework of the space-time non-commutative real scalar field theory is formulated, based on the unitary S-matrix. Unitarity of the S-matrix is explicitly checked order by order using the Heisenberg picture of Lagrangian formalism of the second quantized operators, with the emphasis of the so-called minimal realization of the time-ordering step function and of the importance of the $\star$-time ordering. The Feynman rule is established and is presented using $\phi^4$ scalar field theory. It is shown that the divergence structure of space-time non-commutative theory is the same as the one of space-space non-commutative theory, while there is no UV-IR mixing problem in this space-time non-commutative theory.