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Graded colour Lie superalgebras for solving Lévy-Leblond equations

Mitchell Ryan

TL;DR

The paper demonstrates that the Lévy-Leblond equation admits colour Lie superalgebras with gradings beyond $\mathbb{Z}_2\times\mathbb{Z}_2$ and that colour Lie structures can arise for potentials beyond the free case. It constructs a $\mathbb{Z}_2^{3}$-graded colour Lie superalgebra of eigenspace-preserving operators for the time-independent free LL equation, showing that LL solutions can be expressed as simultaneous eigenstates of two square-root operators. In the harmonic-oscillator setting, ladder operators generate a $\mathbb{Z}_2\times\mathbb{Z}_2$-graded colour Lie superalgebra, which is used to derive the discrete spectrum, with non-symmetry operators playing a crucial role. Overall, the work broadens the algebraic framework for LL equations and underscores the utility of higher-graded colour Lie structures in nonrelativistic quantum systems with nontrivial potentials.

Abstract

The Lévy-Leblond equation with free potential admits a symmetry algebra that is a $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded colour Lie superalgebra (see arXiv:1609.08224). We extend this result in two directions by considering a time-independent version of the Lévy-Leblond equation. First, we construct a $ \mathbb{Z}_2^3 $-graded colour Lie superalgebra containing operators that leave the eigenspaces invariant and demonstrate the utility of this algebra in constructing general solutions for the free equation. Second, we find that the ladder operators for the harmonic oscillator generate a $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded colour Lie superalgebra and we use the operators from this algebra to compute the spectrum. These results illustrate two points: the Lévy-Leblond equation admits colour Lie superalgebras with gradings higher than $ \mathbb{Z}_2\times\mathbb{Z}_2 $ and colour Lie superalgebras appear for potentials besides the free potential.

Graded colour Lie superalgebras for solving Lévy-Leblond equations

TL;DR

The paper demonstrates that the Lévy-Leblond equation admits colour Lie superalgebras with gradings beyond and that colour Lie structures can arise for potentials beyond the free case. It constructs a -graded colour Lie superalgebra of eigenspace-preserving operators for the time-independent free LL equation, showing that LL solutions can be expressed as simultaneous eigenstates of two square-root operators. In the harmonic-oscillator setting, ladder operators generate a -graded colour Lie superalgebra, which is used to derive the discrete spectrum, with non-symmetry operators playing a crucial role. Overall, the work broadens the algebraic framework for LL equations and underscores the utility of higher-graded colour Lie structures in nonrelativistic quantum systems with nontrivial potentials.

Abstract

The Lévy-Leblond equation with free potential admits a symmetry algebra that is a -graded colour Lie superalgebra (see arXiv:1609.08224). We extend this result in two directions by considering a time-independent version of the Lévy-Leblond equation. First, we construct a -graded colour Lie superalgebra containing operators that leave the eigenspaces invariant and demonstrate the utility of this algebra in constructing general solutions for the free equation. Second, we find that the ladder operators for the harmonic oscillator generate a -graded colour Lie superalgebra and we use the operators from this algebra to compute the spectrum. These results illustrate two points: the Lévy-Leblond equation admits colour Lie superalgebras with gradings higher than and colour Lie superalgebras appear for potentials besides the free potential.
Paper Structure (16 sections, 4 theorems, 89 equations)

This paper contains 16 sections, 4 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. The Lévy-Leblond equation and colour Lie superalgebras
  3. Free equation
  4. Lévy-Leblond operator with potential
  5. Time-independent Lévy-Leblond equation
  6. Spectrum formalism
  7. Colour Lie superalgebras
  8. Eigenspace-preserving operators of the free equation
  9. Eigenvalue conditions
  10. Finding operators
  11. Grading the operators
  12. Relationship between the algebra and solutions
  13. Solving the free equation
  14. Ladder operators of the harmonic oscillator
  15. Conclusions
  16. ...and 1 more sections

Key Result

Proposition 2.4

Assume that $H = -(1/\beta)\Delta + V(x)$ is self-adjoint and that $H_{\textup{SL}} = H - \gamma^j\gamma_+ (i/\beta) \pdv{V}{x_j}(\mathbf{x})$ is a closed operator with the same domain as $H$. Then, the spectrum of $H_{\textup{SL}}$ is contained in the spectrum of $H$.

Theorems & Definitions (12)

  • Remark 2.1
  • Remark 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 4.1
  • ...and 2 more