Abstract
We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form −Δu=0 in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler–DeWitt metric provided n≠4. Using then separation of variables, the solutions u 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,R)/SO(n). Since one can define a Schwartz space and tempered distributions in SL(n,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.
Highlights
General relativity is a Lagrangian theory, i.e., the Einstein equations are derived as the Euler–Lagrange equation of the Einstein–Hilbert functional (R − 2Λ), (1)N where N = Nn+1, n ≥ 3, is a globally hyperbolic Lorentzian manifold, Ris the scalar curvature, and Λ is a cosmological constant
Using separation of variables, the solutions u 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, R)/SO(n)
Since one can define a Schwartz space and tempered distributions in SL(n, 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
Summary
General relativity is a Lagrangian theory, i.e., the Einstein equations are derived as the Euler–Lagrange equation of the Einstein–Hilbert functional (R − 2Λ),. We evaluated the resulting equation for a particular metric that we considered important to the problem and obtained a hyperbolic equation in the base space, which happened to be identical to the Wheeler–DeWitt equation obtained as a result of a canonical quantization of a Friedman universe if we only looked at functions that did not depend on x. This result can not be regarded as the solution to the problem of quantizing gravity in a general setting.
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