Non-Heisenbergian quantum mechanics

TL;DR

By formulating quantum mechanics without a position operator and replacing it with a position POVM, the paper derives a generalized uncertainty principle Δx_i Δp_i ≥ sqrt(hbar^2/4 + l0^2 (Δp_i)^2) and a nonzero minimal position uncertainty l0, while preserving Galilean symmetry. It builds a general probability density via a radial kernel f(|p−k|), extends the formalism to external potentials and two-particle interactions, and computes observable consequences such as harmonic-oscillator and hydrogen-atom energy shifts. Experimental bounds on l0 are obtained from the bar detector AURIGA and the hydrogen 1S−2S transition, yielding l0 on the order of a few 10^16 times the Planck length, compatible with standard QM in the l0 → 0 limit. The approach remains distinct from deformed-commutator GUP frameworks by maintaining symmetry and predicting different dynamical and spectroscopic signatures, with future work aimed at relativistic extensions and connections to alternative foundational theories.

Abstract

Relaxing the postulates of an axiomatic theory is a natural way to find more general theories, and historically, the discovery of non-Euclidean geometry is a famous example of this procedure. Here, we use this way to extend quantum mechanics by ignoring the heart of Heisenberg's quantum mechanics -- We do not assume the existence of a position operator that satisfies the Heisenberg commutation relation, $[\hat x,\hat p]=i\hbar$. The remaining axioms of quantum theory, besides Galilean symmetry, lead to a more general quantum theory with a free parameter $l_0$ of length dimension, such that as $l_0 \to 0$ the theory reduces to standard quantum theory. Perhaps surprisingly, this non-Heisenberg quantum theory, without a priori assumption of the non-commutation relation, leads to a modified Heisenberg uncertainty relation, $Δx Δp\geq \sqrt{\hbar^2/4+l_0^2(Δp)^2}$, which ensures the existence of a minimal position uncertainty, $l_0$, as expected from various quantum gravity studies. By comparing the results of this framework with some observed data, which includes the first longitudinal normal modes of the bar gravitational wave detector AURIGA and the $1S-2S$ transition in the hydrogen atom, we obtain upper bounds on the $l_0$.