A complete set of Lorentz-invariant wave packets and modified uncertainty relation

TL;DR

This paper introduces a Lorentz-invariant set of wave packets that form a complete basis for quantum states and modifies the uncertainty relation to be consistent with relativity, reducing to Heisenberg's principle non-relativistically.

Contribution

It constructs a Lorentz-invariant wave packet basis and derives a corresponding modified uncertainty relation that aligns with relativistic principles.

Findings

Defines a Lorentz-invariant wave packet set spanning the Hilbert space

Establishes a Lorentz-invariant completeness relation

Derives a modified uncertainty relation reducing to Heisenberg's in the non-relativistic limit

Abstract

We define a set of fully Lorentz-invariant wave packets and show that it spans the corresponding one-particle Hilbert subspace, and hence the whole Fock space as well, with a manifestly Lorentz-invariant completeness relation (resolution of identity). The position-momentum uncertainty relation for this Lorentz-invariant wave packet deviates from the ordinary Heisenberg uncertainty principle, and reduces to it in the non-relativistic limit.