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arXiv:1607.07685 · Theoretical and Mathematical Physics

Quantization of the Kadomtsev–Petviashvili equation

Karol Kajetan Kozlowski, E. K. Sklyanin, Алессандро Торриелли

Published Aug 1, 2017• 8 citations
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Abstract

We propose a quantization of the Kadomtsev–Petviashvili equation on a cylinder equivalent to an infinite system of nonrelativistic one-dimensional bosons with the masses m = 1, 2,.... The Hamiltonian is Galilei-invariant and includes the split and merge terms $$\Psi _{{m_1}}^\dag \Psi _{{m_2}}^\dag {\Psi _{{m_1} + {m_2}}}$$ and $$\Psi _{{m_1} + {m_2}}^\dag {\Psi _{{m_1}}}{\Psi _{{m_2}}}$$ for all combinations of particles with masses m 1, m 2, and m 1 + m 2 for a special choice of coupling constants. We construct the Bethe eigenfunctions for the model and verify the consistency of the coordinate Bethe ansatz and hence the quantum integrability of the model up to the mass M=8 sector.

DOI
10.1134/s0040577917080074

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We propose a quantization of the Kadomtsev–Petviashvili equation on a cylinder equivalent to an infinite system of nonrelativistic one-dimensional bosons with the masses m = 1, 2,.... The Hamiltonian is Galilei-invariant and includes the split and merge terms Ψm1†Ψm2†Ψm1+m2\Psi _{{m_1}}^\dag \Psi _{{m_2}}^\dag {\Psi _{{m_1} + {m_2}}}Ψm1​†​Ψm2​†​Ψm1​+m2​​ and Ψm1+m2†Ψm1Ψm2\Psi _{{m_1} + {m_2}}^\dag {\Psi _{{m_1}}}{\Psi _{{m_2}}}Ψm1​+m2​†​Ψm1​​Ψm2​​ for all combinations of particles with masses m 1, m 2, and m 1 + m 2 for a special choice of coupling constants. We construct the Bethe eigenfunctions for the model and verify the consistency of the coordinate Bethe ansatz and hence the quantum integrability of the model up to the mass M=8 sector.
We study the quantised version of the Kadomtsev-Petviashvili equation in the formulation proposed in [4]. We analyse two aspects. One is the conservation of the first higher charge H3H_3H3​ (besides the total mass H0H_0H0​, the momentum H1H_1H1​ and the Hamiltonian H2H_2H2​). We find the expression of H3H_3H3​ and prove commutativity, provided we regularise a specific ultraviolet divergence. The result then does not depend on the choice of a regulator, under fairly general assumptions. We then study the Bethe ansatz in finite volume and obtain a general solution to the two-body problem. We have used AI supervised by us to aid the proofs.
The method of q -oscillator lattices, proposed recently in Bazhanov and Sergeev 2005 ( Preprint hep-th/0509181), provides the tool for a construction of various integrable models of quantum mechanics in (2 + 1)-dimensional spacetime. In contrast to any one-dimensional quantum chain, its two-dimensional generalizations—quantum lattices—admit different geometrical structures. In this paper, we consider the q -oscillator model on a special lattice. The model may be interpreted as a two-dimensional lattice Bose gas. The most remarkable feature of the model is that it allows the coordinate Bethe ansatz: the p -particles' wavefunction is the sum of plane waves. Consistency conditions is the set of 2 p equations for p one-particle wave vectors. These 'Bethe ansatz' equations are the main result of this paper.
We develop a generalized coordinate Bethe-ansatz framework for the massive Thirring model with an explicitly time-dependent fermion mass. Unlike the generalized Bethe ansatz for chiral quantum field theories, where particle transport acts in internal spin spaces and leads to quantum Knizhnik-Zamolodchikov equations, the present construction acts directly on particle coordinates and physical rapidities. Its central new feature is a rapidity lattice. Because the time-dependent mass couples right- and left-moving components already in the one-particle sector, a fixed-rapidity block is not closed under spatial winding. Periodicity therefore requires all sectors generated by the finite shift β→β−pβ\toβ-pβ→β−p. Particle exchange relates amplitudes on this lattice through scalar analytic difference equations of Ruijsenaars type, whose solutions determine the complete time-dependent many-body wavefunction. Consistency of the coordinate construction also selects an exponential evolution of the bare mass, providing a new realization of the time dependent integrability-renormalization group correspondence. In the static limit, the rapidity-lattice construction reduces to the conventional massive-Thirring Bethe equations. Through the massive-Thirring/sine Gordon equivalence, the result also connects this exact quantum construction to the generalized space-time dependent Lax formalism.