Another derivation of Weinberg's formula

TL;DR

This paper derives Weinberg's formula by examining how quantum effects alter Lorentz transformations, showing that quantum reference frames can propagate over spacelike intervals due to uncertainty principles.

Contribution

It presents a derivation of Weinberg's formula through a quantum Lorentz transformation involving symmetrization and operator replacement.

Findings

Quantum Lorentz transformation can be expressed in terms of quantum reference frame momentum.

Heisenberg's uncertainty allows particles to propagate over spacelike intervals.

The derivation aligns with results from quantum field theory.

Abstract

To investigate how quantum effects might modify special relativity, we will study a Lorentz transformation between classical and quantum reference frames and express it in terms of the four-dimensional (4D) momentum of the quantum reference frame. The transition from the classical expression of the Lorentz transformation to a quantum-mechanical one requires us to symmetrize the expression and replace all its dynamical variables with the corresponding operators, from which we can obtain the same conclusion as that from quantum field theory (given by Weinberg's formula): owing to the Heisenberg's uncertainty relation, a particle (as a quantum reference frame) can propagate over a spacelike interval.