论文标题
Chern-Simons状态和Hartle-Hawking和Vilenkin Wave功能的等效性
Equivalence of the Chern-Simons state and the Hartle-Hawking and Vilenkin wave-functions
论文作者
论文摘要
我们表明，降低到迷你苏普斯空间时的Chern-Simons（CS）是宇宙的Hartle-Hawking（HH）和Vilenkin（V）波函数的傅立叶双重双重。鉴于前者和后者求解了相同的约束方程，这是可以预期的，该方程是根据共轭变量（松散的扩展因子和哈勃参数）编写的。映射中的许多微妙之处，与连接整合的轮廓有关，阐明了量子宇宙学中边界条件问题。如果我们坚持使用一个{\ it Real} Hubble参数，则只能由CS状态表示HH波功能，而Hubble参数覆盖整个真实线。对于V（或隧道）波功能，哈勃参数仅限于正真实线（这是有道理的，因为状态仅接受传出波），但是轮廓也涵盖了整个负阴性轴。因此，如果将现实条件强加于联系，则国家是不可接受的。对V状态进行修改，要求对哈密顿的约束添加源术语，并发现更可口。在双图中，HH状态预测整个真实线上哈勃参数的均匀分布。修改后的V状态在正真实线上均匀分布。
We show that the Chern-Simons (CS) state when reduced to mini-superspace is the Fourier dual of the Hartle-Hawking (HH) and Vilenkin (V) wave-functions of the Universe. This is to be expected, given that the former and latter solve the same constraint equation, written in terms of conjugate variables (loosely the expansion factor and the Hubble parameter). A number of subtleties in the mapping, related to the contour of integration of the connection, shed light on the issue of boundary conditions in quantum cosmology. If we insist on a {\it real} Hubble parameter, then only the HH wave-function can be represented by the CS state, with the Hubble parameter covering the whole real line. For the V (or tunnelling) wave-function the Hubble parameter is restricted to the positive real line (which makes sense, since the state only admits outgoing waves), but the contour also covers the whole negative imaginary axis. Hence the state is not admissible if reality conditions are imposed upon the connection. Modifications of the V state, requiring the addition of source terms to the Hamiltonian constraint, are examined and found to be more palatable. In the dual picture the HH state predicts a uniform distribution for the Hubble parameter over the whole real line; the modified V state a uniform distribution over the positive real line.
