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Quantum dispersionless KdV hierarchy revisited

Zhe Wang

Abstract

We quantize Hamiltonian structures with hydrodynamic leading terms using the Heisenberg vertex algebra. As an application, we construct the quantum dispersionless KdV hierarchy via a non-associative Weyl quantization procedure and compute the corresponding eigenvalue problem.

Quantum dispersionless KdV hierarchy revisited

Abstract

We quantize Hamiltonian structures with hydrodynamic leading terms using the Heisenberg vertex algebra. As an application, we construct the quantum dispersionless KdV hierarchy via a non-associative Weyl quantization procedure and compute the corresponding eigenvalue problem.

Paper Structure

This paper contains 14 sections, 23 theorems, 195 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Heisenberg vertex algebra and quantum Hamiltonian structure
  3. Basics of vertex algebra
  4. Quantum Hamiltonian structure
  5. Weyl quantization of dispersionless KdV hierarchy
  6. Construction of quantum Hamiltonians
  7. Another expression for quantum Hamiltonians
  8. Properties of quantum Hamiltonians
  9. Proposition \ref{['AZ']} for $\tilde{h}_n$
  10. Proposition \ref{['AW']} for $\tilde{h}_n$
  11. Theorem \ref{['AY']} for $\tilde{h}_n$
  12. Theorem \ref{['AX']} for $\tilde{h}_n$
  13. Proof of Theorem \ref{['BE']}
  14. Conclusion

Key Result

Theorem 1.1

The Lie bracket defines a deformation quantization of the Hamiltonian structure AC, here $\varphi(G)_{(0)}$ is the zeroth mode of the state $\varphi(G)$. In terms of differential polynomials, this bracket is of the form here $f, g\in \mathcal{A}$ are arbitrary densities of $F$ and $G$, respectively.

Figures (5)

  • Figure 1: The unique tree in $BT(2)$.
  • Figure 2: Trees in $BT(3)$. If we denote by $T$ the unique tree in $BT(2)$, then the first tree is $T_1$ and the second one is $T_2$.
  • Figure 3: Evaluation of each node for trees in $BT(3)$.
  • Figure 4: Trees $R_1$, $R_2$ and $R_3$.
  • Figure 5: Trees $S_1$, $S_2$ and $S_3$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Remark 2.1
  • Lemma 2.1
  • proof
  • ...and 32 more