$\mathrm{AdS}_2$中的标量场和$\widetilde{\mathrm{SL}}(2,\mathbb{R})$的表示
Scalar field in $\mathrm{AdS}_2$ and representations of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$
摘要：
研究了二维anti-de Sitter空间的泛覆盖空间中有质量标量场的Klein-Gordon方程的解。对于某些值的质量参数，我们施加一组合适的边界条件，使空间分量的Klein-Gordon算子自伴。这使得经典场的时间演化得到很好的定义。然后，我们使用的变换性质的标量场的等距群的理论，即$\mathrm{SL}(2,\mathbb{R})$的通用覆盖群，以确定哪些自伴边界条件下是不变的，并导致正频率的解决方案形成一个酉表示的这个群体，因此，在这个群体的真空状态不变。然后，我们研究了边界条件导致真空态不变量的不变量理论的情况，并确定了真空态所属的幺正表示。
英文摘要：
We study the solutions to the Klein-Gordon equation for the massive scalar field in the universal covering space of two-dimensional anti-de Sitter space. For certain values of the mass parameter, we impose a suitable set of boundary conditions which make the spatial component of the Klein-Gordon operator self-adjoint. This makes the time-evolution of the classical field well defined. Then, we use the transformation properties of the scalar field under the isometry group of the theory, namely, the universal covering group of $\mathrm{SL}(2,\mathbb{R})$, in order to determine which self-adjoint boundary conditions are invariant under this group, and which lead to the positive-frequency solutions forming a unitary representation of this group and, hence, to a vacuum state invariant under this group. Then we examine the cases where the boundary condition leads to an invariant theory with non-invariant vacuum state and determine the unitary representation to which the vacuum state belongs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
