Equivalence of the Klein-Gordon random field and the complex Klein-Gordon quantum field

TL;DR

This paper demonstrates that the Klein-Gordon random field and the complex Klein-Gordon quantum field are equivalent because they can be constructed from the same algebra, despite differences in their treatment of negative frequency modes.

Contribution

It shows the equivalence of the Klein-Gordon random field and the complex quantum field through their shared algebraic construction, clarifying their conceptual relationship.

Findings

Both fields can be built from the same creation and annihilation operator algebra.

The difference in negative frequency modes does not affect their equivalence.

Theories are shown to be equivalent in their algebraic structure.

Abstract

The difference between a Klein-Gordon random field and the complex Klein-Gordon quantum field is characterized, explicitly comparing the roles played by negative frequency modes of test functions in creation and annihilation operator presentations of the two theories. The random field and the complex quantum field can both be constructed from the same creation and annihilation operator algebra, making them equivalent in that sense.