Noncommutative Heisenberg-Robertson-Schrodinger Uncertainty Principles

2025/01/01 by Krishna, K. Mahesh
#46L08 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2502.05154

Abstract

Let E be a Hilbert C*-module over a unital C*-algebra A. Let A: D(A) ⊆ E → E and B: D(B)⊆ E→ E be possibly unbounded self-adjoint morphisms. Then for all x ∈ D(AB)∩ D(BA) with ⟨ x, x ⟩ =1, we show that (1) Δ_x(B)²d_x(A)²+Δ_x(A)²d_x(B)²≥ \frac(⟨ \A,B\x, x ⟩ -\⟨ Ax, x ⟩,⟨ Bx, x ⟩\)²-(⟨ [A,B]x, x ⟩ +[⟨ Ax, x ⟩,⟨ Bx, x ⟩])²2 and (2) Δ_x(A)Δ_x(B)≥ \frac√‖(⟨ \A,B\x, x ⟩ -\⟨ Ax, x ⟩,⟨ Bx, x ⟩\)²-(⟨ [A,B]x, x ⟩ +[⟨ Ax, x ⟩,⟨ Bx, x ⟩])²‖2, where Δ_x(A):= ‖Ax-⟨ Ax, x ⟩ x ‖, d_x(A):= √(⟨ Ax, Ax ⟩ -⟨ Ax, x ⟩²), [A,B] := AB-BA, \A,B\:= AB+BA, \⟨ Ax, x ⟩,⟨ Bx, x ⟩\:= ⟨ Ax, x ⟩⟨ Bx, x ⟩ +⟨ Bx, x ⟩⟨ Ax, x ⟩, [⟨ Ax, x ⟩,⟨ Bx, x ⟩]:= ⟨ Ax, x ⟩⟨ Bx, x ⟩ -⟨ Bx, x ⟩⟨ Ax, x ⟩. We call Inequalities (1) and (2) as noncommutative Heisenberg-Robertson-Schrodinger uncertainty principles. They reduce to the Heisenberg-Robertson-Schrodinger uncertainty principle (derived by Schrodinger in 1930) whenever A=ℂ.

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