On relativistic elements of reality

TL;DR

This paper examines the possibility of defining Lorentz-invariant elements of reality in quantum mechanics, arguing they can exist but do not satisfy the product rule, leading to specific algebraic constraints.

Contribution

It clarifies the conditions under which Lorentz-invariant elements of reality can be defined, challenging previous assumptions about their impossibility.

Findings

Lorentz-invariant elements of reality can be defined.

Such elements do not satisfy the product rule.

Algebraic constraints are derived for commuting Hermitian operators.

Abstract

Several arguments have been proposed some years ago, attempting to prove the impossibility of defining Lorentz-invariant elements of reality. I find that a sufficient condition for the existence of elements of reality, introduced in these proofs, seems to be used also as a necessary condition. I argue that Lorentz-invariant elements of reality can be defined but, as Vaidman pointed out, they won't satisfy the so-called product rule. In so doing I obtain algebraic constraints on elements of reality associated with a maximal set of commuting Hermitian operators.