The Hartle–Hawking wave function is known to be the Fourier dual of the Chern–Simons or Kodama state reduced to mini-superspace, using an integration contour covering the whole real line. But since the Chern–Simons state is a solution of the Hamiltonian constraint (with a given ordering), its Fourier dual should provide a solution (i.e. beyond mini-superspace) of the Wheeler DeWitt equation representing the Hamiltonian constraint in the metric representation. We write down a formal expression for such a wave function, to be seen as the generalization beyond mini-superspace of the Hartle–Hawking wave function. Its explicit evaluation (or simplification) depends only on the symmetries of the problem, and we illustrate the procedure with anisotropic Bianchi models and with the Kantowski–Sachs model. A significant difference of this approach is that we may leave the torsion inside the wave functions when we set up the ansatz for the connection, rather than setting it to zero before quantization. This allows for quantum fluctuations in the torsion, with far reaching consequences.

众所周知，哈特尔-霍金波函数是陈-西蒙斯态或儿玉态的傅里叶对偶，使用覆盖整个实线的积分轮廓简化为迷你超空间。但是由于陈-西蒙斯状态是哈密顿约束的解（具有给定的排序），它的傅立叶对偶应该提供在度量表示中表示哈密顿约束的惠勒德威特方程的解（即超出微型超空间）。我们写下这样一个波函数的形式表达式，可以看作是 Hartle-Hawking 波函数在微型超空间之外的推广。它的显式评估（或简化）仅取决于问题的对称性，我们用各向异性 Bianchi 模型和 Kantowski-Sachs 模型说明了该过程。这种方法的一个显着区别是，当我们为连接设置 ansatz 时，我们可能会将扭转留在波函数内，而不是在量化之前将其设置为零。这允许扭转中的量子波动，产生深远的影响。