一阶Einstein-Hilbert Action与玻色物质耦合的正则结构
On the Canonical Structure of First Order Einstein-Hilbert Action Coupled to Bosonic Matter
摘要：
一阶形式的d维爱因斯坦-希尔伯特作用量的狄拉克哈密顿公式表明，除了第二第一类约束外，还出现了第三第一类约束，第一类约束中有一个不寻常的非局部泊松括号（PB）代数[8 ]。这种方法是从ADM的不同之处在于，运动方程是独立的时间导数的字段，只有那些对应于第二类约束（在狄拉克约束形式主义的意义上）被用来消除字段的行动。本文考虑了在这种形式下宇宙学项、有质量标量场、Maxwell规范场和Yang-Mills场与一阶EH作用量的耦合，并指出尽管在一阶EH作用量的哈密顿形式及其约束结构上与ADM结果有明显的差别，在玻色子物质存在下的一阶EH作用量的ADM哈密顿公式的一般性质可从该新颖的哈密顿公式导出。在EH作用量上增加一个质量标量场，约束的PB代数保持不变，当杨-米尔斯场或麦克斯韦规范场耦合到EH作用量上时，与EH作用量有关的约束的PB代数接收来自物质场作用量的规范变换的生成元的线性贡献，这些生成元之间形成一个封闭的代数。此外，我们还发现，对于封闭空间，在由第一类约束（包括由物质场产生的约束）定义的约束面上，耦合到玻色物质的EH作用量的哈密顿量弱为零.
英文摘要：
A Dirac Hamiltonian formulation of d-dimensional Einstein-Hilbert action in first order form, has shown that as well as secondary first class constraints, tertiary first class constraints also arise, with an unusual nonlocal Poisson bracket (PB) algebra among first class constraints [8 ]. This approach is different from that of ADM in that of the equations of motion which are independent of the time derivative of fields only those which correspond to second class constraints (in the sense of the Dirac constraint formalism) are used to eliminate fields from the action. In this paper, we consider coupling of a cosmological term, massive scalar fields, Maxwell gauge fields and Yang-Mills fields to the first order EH action in this formalism, and show that in spite of the apparent differences with the ADM results in the Hamiltonian formulation of the first order EH action and its constraint structure, the generic properties of the ADM Hamiltonian formulation of the first order EH action in the presence of Bosonic matter are derivable from this novel Hamiltonian formulation. Addition of a massive scalar field to the EH action leaves the PB algebra of constraints unaltered, and when the Yang-Mills fields or Maxwell gauge fields are coupled to the EH action, the PB algebra of the constraints pertaining to the EH action receives linear contributions from the generator of the gauge transformations of the action for matter fields, and those generators form a closed algebra amongst themselves. Moreover, it is found that for closed spaces, the Hamiltonian of the EH action coupled to Bosonic matter is weakly zero on the constraint surface defined by the first class constraints, including the constraints arising from the matter fields.
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