Poincare' invariance for continuous-time histories

TL;DR

This paper explores the invariance properties of continuous-time histories in relativistic quantum theory, identifying two distinct Poincaré groups that correspond to different types of time transformations.

Contribution

It introduces a relativistic analogue of time translation in history theory, revealing two separate Poincaré groups with different roles in the theory.

Findings

Identification of internal and external Poincaré groups in relativistic history theory

The external group performs explicit spacetime foliation changes

Clarification of the role of Poincaré invariance in history-based approaches

Abstract

We show that the relativistic analogue of the two types of time translation in a non-relativistic history theory is the existence of two distinct Poincar\'{e} groups. The `internal' Poincar\'{e} group is analogous to the one that arises in the standard canonical quantisation scheme; the `external' Poincar\'{e} group is similar to the group that arises in a Lagrangian description of the standard theory. In particular, it performs explicit changes of the spacetime foliation that is implicitly assumed in standard canonical field theory.