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Maurer-Cartan methods in perturbative quantum mechanics

Andrey Losev, Tim Sulimov

Abstract

We reformulate the time-independent Schrödinger equation as a Maurer-Cartan equation on the superspace of eigensystems of the former equation. We then twist the differential so that its cohomology becomes the space of solutions with a set energy. A perturbation of the Hamiltonian corresponds to a deformation of the twisted differential, leading to a simple recursive relation for the eigenvalue and eigenfunction corrections.

Maurer-Cartan methods in perturbative quantum mechanics

Abstract

We reformulate the time-independent Schrödinger equation as a Maurer-Cartan equation on the superspace of eigensystems of the former equation. We then twist the differential so that its cohomology becomes the space of solutions with a set energy. A perturbation of the Hamiltonian corresponds to a deformation of the twisted differential, leading to a simple recursive relation for the eigenvalue and eigenfunction corrections.
Paper Structure (12 sections, 5 theorems, 75 equations)

This paper contains 12 sections, 5 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Schrödinger equation in Maurer-Cartan form
  3. The superspace of eigensystems
  4. Maurer-Cartan equation
  5. Kernel and cohomology
  6. Homotopy
  7. Perturbation theory
  8. Non-perturbed problem
  9. Perturbation
  10. Recurrence relation
  11. Corrections as sums over tree diagrams
  12. Conclusion

Key Result

Theorem 1

The formula where $\mathop{\mathrm{ad}}\nolimits_\Phi = \{ \Phi , - \}$, defines a left action of the gauge group $\Gamma$ on $\mathrm{MC}(\hat{\mathscr{H}})$.

Theorems & Definitions (18)

  • Example 1
  • Example 2
  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • proof
  • ...and 8 more