敷料形式中引力孤立子的转移矩阵、Poisson括号
Transition Matrix, Poisson Bracket for gravitational solitons in the dressing formalism
摘要：
本文将孤子理论中的哈密顿方法应用于引力孤子的dressing形式。本文定义并计算了李代数值单形式$A(ς, η, γ)=Ψ_{,γ}Ψ^{-1}$的Poisson括号，并与已有的结果一致.定义了用于$A$的转移矩阵${\cal T}=A(ς, η, γ) A^{-1}(η, ξ, -γ)$，该转移矩阵与入射光锥和出射光锥处的$A$相关。证明了在零坐标$η$所起的作用下，它满足与可积偏微分方程类似的方程。这在数学上是一个新结果，因为文献中还没有$A$的转移矩阵，而在物理上它提供了获得运动积分的可能性（对于适当的边界条件），从${\cal T}$相对于零坐标$η$的导数的迹线，根据经典的相对论联系，因为$A(γ= \pm 1)$可以用经典的Christoffel符号以简单的方式表示。这可能证明在量子化时有用，因为连接是量子引力的基本变量。$η$和$ς$的作用可以颠倒，以获得$ς$的运动积分，因此$ς$扮演时间的角色。这与以前已经确立的两次解释和方法密切相关。
英文摘要：
The Hamiltonian methods of the theory of solitons are applied to gravisolitons in the dressing formalism. The Poisson bracket for the Lie-algebra valued one-form $A(ς, η, γ)=Ψ_{,γ}Ψ^{-1}$, for gravisolitons in the dressing formalism, for a specific background solution, is defined and computed, agreeing with results previously obtained. A transition matrix ${\cal T}=A(ς, η, γ) A^{-1}(η, ξ, -γ)$ for $A$ is defined relating $A$ at ingoing and outgoing light cones. It is proved that it satisfies equations familiar from integrable pde's with the role of time played by the null coordinate $η$. This is a new result mathematically, since there has not been a transition matrix for $A$ in the litterature, while physically it presents the possibility of obtaining integrals of motion (for appropriate boundary conditions), from the trace of the derivative with respect to the null coordinate $η$, of ${\cal T}$, in terms of classical relativity connections, since $A(γ= \pm 1)$ can be expressed in a simple way in terms of the classical Christoffel symbols. This may prove of use upon quantization since connections are fundamental variables of quantum gravity. The roles of $η$ and $ς$ may be reversed to obtain integrals of motion for $ς$, thus $ς$ playing the role of time. This ties well with the two-time interpretation and approach already established before.
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