Quantization of the Hamilton Equations

We quantize the Hamilton equations ignoring the Hamilton condition. The resulting equation has the simple form \(-{\varDelta u}=0\) in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler-DeWitt metric provided \(n \ne 4\) . Using then separation of variables the solutions can be expressed as products of temporal and spatial eigenfunctions, where the spatial eigenfunctions are eigenfunctions of the Laplacian in the symmetric space \(SL(n,\mathbb {R})/SO(n)\) . Since one can define a Schwartz space and tempered distributions in \(SL(n,\mathbb {R})/SO(n)\) as well as a Fourier transform, Fourier quantization can be applied such that the spatial eigenfunctions are transformed to Dirac measures and the spatial Laplacian to a multiplication operator.