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Generating and Computing Quantum Periods in Exact WKB

Max Meynig

TL;DR

This work develops a finite, algebraic framework to compute all-orders semiclassical traces $\mathrm{tr}\,\delta(\hat{H}-E)$ for polynomial Hamiltonians by translating the problem into a matrix representation of a ring generated by the noncommuting operators $g_k$. Central to the method is combining the semiclassical symbol calculus with Griffiths–Dwork reduction to express all integrals as residues on a finite set of de Rham generators, and then mapping these into a finite-dimensional matrix action that controls pole-order growth. The paper proves two main theorems: (i) an explicit, constructive expression for the integrals in terms of noncommuting polynomials and a resolvent expansion, and (ii) a finite, ring-theoretic reduction that yields a constant bound on pole orders across all orders, enabling practical computation of WKB periods and exact quantization data. Through detailed examples, including the harmonic oscillator and a cubic potential in one dimension, the approach yields Picard–Fuchs equations and explicit residue relations that connect WKB periods to rational integrals on the associated Riemann surface. Overall, the framework provides a concrete, algebraic pipeline for obtaining WKB periods and exact spectral data for polynomial Hamiltonians across dimensions, with implications for exact quantization and related geometric structures.

Abstract

Periods of rational integrals appear in quantum mechanics through asymptotic expansions of traces computed with the semiclassical symbol calculus. We develop a novel formal series expansion for the trace of the Dirac delta of a differential operator. Restricting to operators which arise as the quantizations of polynomials, we are able to apply the Griffiths-Dwork reduction to the integrals. By developing this perspective, we find the reduction of all integrals in the asymptotic series to normal form through a finite calculation. In the case of one degree of freedom, the two dimensional residue formula relates the rational integrals to the quantum actions in the exact WKB formalism.

Generating and Computing Quantum Periods in Exact WKB

TL;DR

This work develops a finite, algebraic framework to compute all-orders semiclassical traces for polynomial Hamiltonians by translating the problem into a matrix representation of a ring generated by the noncommuting operators . Central to the method is combining the semiclassical symbol calculus with Griffiths–Dwork reduction to express all integrals as residues on a finite set of de Rham generators, and then mapping these into a finite-dimensional matrix action that controls pole-order growth. The paper proves two main theorems: (i) an explicit, constructive expression for the integrals in terms of noncommuting polynomials and a resolvent expansion, and (ii) a finite, ring-theoretic reduction that yields a constant bound on pole orders across all orders, enabling practical computation of WKB periods and exact quantization data. Through detailed examples, including the harmonic oscillator and a cubic potential in one dimension, the approach yields Picard–Fuchs equations and explicit residue relations that connect WKB periods to rational integrals on the associated Riemann surface. Overall, the framework provides a concrete, algebraic pipeline for obtaining WKB periods and exact spectral data for polynomial Hamiltonians across dimensions, with implications for exact quantization and related geometric structures.

Abstract

Periods of rational integrals appear in quantum mechanics through asymptotic expansions of traces computed with the semiclassical symbol calculus. We develop a novel formal series expansion for the trace of the Dirac delta of a differential operator. Restricting to operators which arise as the quantizations of polynomials, we are able to apply the Griffiths-Dwork reduction to the integrals. By developing this perspective, we find the reduction of all integrals in the asymptotic series to normal form through a finite calculation. In the case of one degree of freedom, the two dimensional residue formula relates the rational integrals to the quantum actions in the exact WKB formalism.
Paper Structure (24 sections, 11 theorems, 130 equations)

This paper contains 24 sections, 11 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Description of main results
  3. Restating the reduction problem
  4. Main result on the reduction of the integrals
  5. Notation
  6. Reduction of the integrals
  7. The Griffiths-Dwork reduction
  8. Complete reduction of the integrals
  9. Proof of theorem \ref{['thm::MainResultTwo']}
  10. Remarks on the ring of matrix valued functions
  11. The Generators of the algebra
  12. Connection to WKB
  13. The residue calculus
  14. WKB periods
  15. WKB periods as residues of rational integrals
  16. ...and 9 more sections

Key Result

Theorem 2.3

Let $E \in \mathbb{C}$. Let $H:\mathbb{C}^{2n} \to \mathbb{C}$ be a polynomial and let $\hat{H} =\mathop{\mathrm{Op}}\nolimits_1 (H)$ be the standard quantization of $H$. Then there is a cycle $\Gamma$ such that $\tr \delta(\hat{H} - E)$ has a formal series expansion in $\hbar$ given by where for each $r = 1,\dots,\deg H$ and for each $k$ in the sum.

Theorems & Definitions (46)

  • Definition 2.1
  • Example 1
  • Definition 2.2
  • Theorem 2.3
  • Example 2
  • Remark
  • Remark
  • Remark
  • Definition 2.4
  • Remark
  • ...and 36 more