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Commutators of spectral projections of spin operators

Ood Shabtai

TL;DR

We establish the semiclassical limit of the commutator of spectral projections of noncommuting spin observables in irreducible SU(2) representations, proving lim_{n→∞} ||C_n||_{op} = 1/2 with the exact 1/2 value known for n ≡ 2 (mod 4). The method recasts matrix elements into an integral form via the one-parameter subgroup e^{-iθJ_y} and employs Szegő-type asymptotics for Wigner d-functions to connect to a Hankel operator H_E of norm 1/2, including convergence of truncated Hankel submatrices to [H_E]_N. The work further extends to analogues in R, T, finite Heisenberg groups, and SE(2), indicating a general phenomenon explained by Berezin-Toeplitz quantization and a conjectured principle for transversal-domain observables in 2D quantizations. These results illuminate a precise noncommutativity bound in semiclassical limits and provide a unified framework for spectral-projection commutators across disparate quantum models. The findings have potential significance for quantum geometry and quantization, offering a rigorous link between discontinuous classical observables and their quantum commutators.

Abstract

We present a proof that the operator norm of the commutator of certain spectral projections associated with spin operators converges to $\frac 1 2$ in the semiclassical limit. The ranges of the projections are spanned by all eigenvectors corresponding to positive eigenvalues. The proof involves the theory of Hankel operators on the Hardy space. A discussion of several analogous results is also included, with an emphasis on the case of finite Heisenberg groups.

Commutators of spectral projections of spin operators

TL;DR

We establish the semiclassical limit of the commutator of spectral projections of noncommuting spin observables in irreducible SU(2) representations, proving lim_{n→∞} ||C_n||_{op} = 1/2 with the exact 1/2 value known for n ≡ 2 (mod 4). The method recasts matrix elements into an integral form via the one-parameter subgroup e^{-iθJ_y} and employs Szegő-type asymptotics for Wigner d-functions to connect to a Hankel operator H_E of norm 1/2, including convergence of truncated Hankel submatrices to [H_E]_N. The work further extends to analogues in R, T, finite Heisenberg groups, and SE(2), indicating a general phenomenon explained by Berezin-Toeplitz quantization and a conjectured principle for transversal-domain observables in 2D quantizations. These results illuminate a precise noncommutativity bound in semiclassical limits and provide a unified framework for spectral-projection commutators across disparate quantum models. The findings have potential significance for quantum geometry and quantization, offering a rigorous link between discontinuous classical observables and their quantum commutators.

Abstract

We present a proof that the operator norm of the commutator of certain spectral projections associated with spin operators converges to in the semiclassical limit. The ranges of the projections are spanned by all eigenvectors corresponding to positive eigenvalues. The proof involves the theory of Hankel operators on the Hardy space. A discussion of several analogous results is also included, with an emphasis on the case of finite Heisenberg groups.

Paper Structure

This paper contains 23 sections, 15 theorems, 145 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Numerical simulations and preliminary results
  3. Outline of the proof of Theorem \ref{['maintheorem']}
  4. Analogues of Theorem \ref{['maintheorem']}
  5. Preliminaries on Hankel operators
  6. On the norm of Hankel operators
  7. The proof of Theorem \ref{['maintheorem']}
  8. An extension of Theorem \ref{['maintheorem']}
  9. Matrix elements of spectral projections
  10. The Wigner small d-matrix
  11. Integral formula for $P_{x,j,m',m}$
  12. Matrix elements of $\mathbbm{1}_{\left(a\left(j+\frac{1}{2} \right),\ \infty\right)}(J_x)$
  13. Limits of central matrix elements
  14. Asymptotic approximation of Wigner d-functions
  15. Integrals involving Wigner d-functions
  16. ...and 8 more sections

Key Result

Theorem 1.1

$\Vert C_{4n+2} \Vert_{\mathop{\mathrm{op}}\nolimits} = \frac{1}{2}$ for every $n \in \mathbb N$,

Figures (7)

  • Figure 1: $\Vert C_n \Vert_{\mathop{\mathrm{op}}\nolimits}$ as a function of $n$.
  • Figure 2: $\ln\left(\frac{1}{2} - \Vert C_{4k} \Vert_{\mathop{\mathrm{op}}\nolimits} \right)$ as a function of $\ln(4k)$.
  • Figure 3: The norm of $C^{(3)}_{n}$ as a function of $n$ for the Heisenberg groups $H(\mathbb Z_n)$. Note the similarity to the graph for $SU(2)$. In particular, the graph also appears to depend on $n\mod 4$.
  • Figure 4: (originally by Y. Le Floch) The modulus of (unit) eigenvectors of $C_{101}$ corresponding to extremal eigenvalues, realized as polynomials on $\mathbb C$.
  • Figure 5: The image above to the left, reproduced with the eigenvector realized as a function on $S^2$ using the stereographic projection.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Theorem 1.1: L. Polterovich
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7: Y. Le Floch
  • Theorem 1.8
  • Definition 2.1
  • Lemma 2.2
  • ...and 13 more