On the calculation rule of probability of relativistic free particle in quantum mechanics

TL;DR

This paper proposes a new calculation rule for the probability of measurement outcomes of relativistic free particles in quantum mechanics, based on logical reasoning, Dirac sea theory, and the non-observability of the vacuum.

Contribution

It introduces a novel probability calculation rule for relativistic particles, extending the standard quantum measurement framework to Dirac and Klein-Gordon equations.

Findings

New probability calculation rule derived for Dirac particles

Proof based on Dirac sea, hole theory, and vacuum non-observability

Discussion of implications for Klein-Gordon particles

Abstract

As is well known, in quantum mechanics, the calculation rule of the probability that an eigen-value a_n is observed when the physical quantity A is measured for a state described by the state vector |> is P(a_n)=<|A_n><A_n|> . However, in Ref.[1], based on strict logical reasoning and mathematical calculation, it has been pointed out, replacing <|A_n><A_n|>, one should use a new rule to calculate P(a_n) for particle satisfying the Dirac equation. In this paper, we first state some results given by Ref.[1]. And then, we present a proof for the new calculation rule of probability according to Dirac sea of negative energy particles, hole theory and the principle "the vacuum is not observable". Finally, we discuss simply the case of particle satisfying the Klein-Gordon equation.