Notes for Quantum Gravitational Field

Abstract

We discuss the problems of dynamics of the gravitational field and try to solve them according to quantum field theory by suggesting canonical states for the gravitational field and its conjugate field. To solve the problem of quantization of gravitational field, we assume that the quantum gravitational field $e ^I$ changes the geometry of curved spacetime $x^\mu$, and relate this changing to quantization of the gravitational field. We introduce a field $\pi_I$ and consider it as a canonical momentum conjugates to a canonical gravitational field $\tilde e^I$. We use them in deriving the path integral of the gravitational field according to quantum field theory, we get Lagrangian with dependence only on the covariant derivative of the gravitational field $e^I$, similarly to Lagrangian of scalar field in curved spacetime. Then, we discuss the case of free gravitational field. We find that this case takes place only in background spacetime approximation of low matter density; weak gravity. Similarly, we study the Plebanski two form complex field $\Sigma ^{i}$ and derive its Lagrangian with dependence only on the covariant derivative of $\Sigma ^{i}$, which is represented in selfdual representation $\left| {\Sigma ^i } \right\rangle$. Then, We try to combine the gravitational and Plebanski fields into one field: $K_{\mu}^i$. Finally, we derive the static potential of exchanging gravitons between particles of scalar and spinor fields; the Newtonian gravitational potential.