GR from generalized inner product?

friend

The kinetic energy of a free particle is sometimes viewed geometrically as the inner product of velocity with momentum, where velocity is seen as a vector in the tangent space to the configuration space of a particle, and momentum is viewed as a vector in the tangent space of the phase space of a particle. The kinetic energy is seen as the inner product between these spaces. And the mass is sometimes called the "mass metric". Then some generalize this inner product using a metric tensor and vectors. See for example eq (3) on page

http://users.physik.fu-berlin.de/~kl...00000000000000


This is Prof. Kleinert's work, where he uses this generalized inner product to derive the path integral for a free particle in curved spacetime. He starts his derivation on page:


http://users.physik.fu-berlin.de/~kl...252/node9.html


He comes up with an action in curved spacetime in eq (101), on page:


http://users.physik.fu-berlin.de/~kl...00000000000000


Here we see metric tensors and connections, etc.


This makes me wonder if this all could be manipulated into the Hilbert-Einstein action from which GR can be derived. Has such an effort ever been tried? Do we need to include more terms in eq (101)? Can we derive something from which the Hilbert-Einstein action is an approximation? Or do you see any inherent reason why such an effort can not be found? This would make QM an local interpretation of GR, right?


Any comments or guidance you might have would be very much appreciated. Thanks