格点上玻色子场论的可积性和散射
Integrability and scattering of the boson field theory on a lattice
摘要：
晶格上的自由玻色子是人们所能想到的最简单的场论。它的配分函数可以很容易地在动量空间中计算。然而，这个简单的解决方案隐藏了其可积性属性。在这里，我们使用精确可解模型的方法，目前适用于自旋系统，一个无质量和有质量的自由玻色子的二维晶格。模型的玻尔兹曼权满足杨-巴克斯特方程，在无质量的情况下，由三角函数给出的统一，和Jacobi椭圆函数在大规模的情况下。我们对角化的行到行的传输矩阵，导出守恒量，并实现量子逆散射方法。最后，我们构造了两个分解散射$S$矩阵模型连续自由度使用三角函数和椭圆函数。这些结果将2D中的自由玻色子模型置于与杨-巴克斯特完全可解的其他模型相同的位置，提供了量子计算中的可能应用。
英文摘要：
A free boson on a lattice is the simplest field theory one can think of. Its partition function can be easily computed in momentum space. However, this straightforward solution hides its integrability properties. Here, we use the methods of exactly solvable models, that are currently applied to spin systems, to a massless and massive free boson on a 2D lattice. The Boltzmann weights of the model are shown to satisfy the Yang-Baxter equation with a uniformization given by trigonometric functions in the massless case, and Jacobi elliptic functions in the massive case. We diagonalize the row-to-row transfer matrix, derive the conserved quantities, and implement the quantum inverse scattering method. Finally, we construct two factorized scattering $S$ matrix models for continuous degrees of freedom using trigonometric and elliptic functions. These results place the free boson model in 2D in the same position as the rest of the models that are exactly solvable à la Yang-Baxter, offering possible applications in quantum computation.
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