标题： 量子自旋动力学VIII。 主约束 显示英文标题

标题： Quantum Spin Dynamics VIII. The Master Constraint

摘要： 最近，圈量子引力（LQG）的主约束计划（MCP）被提出，该计划将无限数量的哈密顿约束替换为一个主约束。 MCP旨在克服与哈密顿约束的狄拉克代数非李代数结构相关的复杂性，并在各种场论模型中成功测试。 对于3+1引力本身的情况，到目前为止，仅得到了主约束算子的正二次型。 在本文中，我们弥补了这一空白，并证明该二次型是可闭合的，因此来源于唯一的自伴主约束算子。 该证明基于LQG中哈密顿约束构造的一般模式的一个简单特性，因此适用于任意物质耦合，并适用于任何度规签名。 有了这个结果，通过标准谱分析建立了LQG的物理希尔伯特空间的存在性。 显示英文摘要

摘要： Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single Master constraint. The MCP is designed to overcome the complications associated with the non -- Lie -- algebra structure of the Dirac algebra of Hamiltonian constraints and was successfully tested in various field theory models. For the case of 3+1 gravity itself, so far only a positive quadratic form for the Master Constraint Operator was derived. In this paper we close this gap and prove that the quadratic form is closable and thus stems from a unique self -- adjoint Master Constraint Operator. The proof rests on a simple feature of the general pattern according to which Hamiltonian constraints in LQG are constructed and thus extends to arbitrary matter coupling and holds for any metric signature. With this result the existence of a physical Hilbert space for LQG is established by standard spectral analysis.