Reply to Hagen & Sudarshan's Comment

TL;DR

This paper confirms the correctness of a 1993 argument regarding quantum commutators and refutes Hagen & Sudarshan's inconsistent solution, emphasizing the importance of careful algebraic analysis in theoretical physics.

Contribution

It demonstrates the validity of the original argument and identifies errors in Hagen & Sudarshan's solution, clarifying the correct behavior of quantum commutators.

Findings

The original 1993 argument is correct and consistent.

Hagen & Sudarshan's solution contains algebraic inconsistencies.

Correct algebra shows non-vanishing commutators in physical states.

Abstract

We show that the argument in Phys Rev Lett 70 (1993) 1360 is correct and consistent, and that Hagen & Sudarshan's solution has inconsistency leading to non-vanishing commutators of $[P^1, P^2]$ and $[P^j, H]$ even in physical states. This proves that many of HS's statements in their Comment are based merely on incorrect guess, but not on careful algebra.