Klein-Gordon-Wheeler-DeWitt-Schroedinger Equation

TL;DR

This paper derives a combined Klein-Gordon and Wheeler-DeWitt-Schrödinger equation for quantum gravity with point particles and p-branes, proposing a mechanism for the small cosmological constant related to brane momentum.

Contribution

It introduces a unified wave functional framework satisfying both Klein-Gordon and Wheeler-DeWitt equations, extending to p-branes and linking brane momentum to the cosmological constant.

Findings

Wave functional satisfies combined Klein-Gordon and Wheeler-DeWitt equations.

Quantized spacetime filling brane leads to an effective cosmological constant.

The cosmological constant has two discrete values, positive and negative.

Abstract

We start from the Einstein-Hilbert action for the gravitational field in the presence of a "point particle" source, and cast the action into the corresponding phase space form. The dynamical variables of such a system satisfy the point particle mass shell constraint, the Hamilton and the momentum constraints of the canonical gravity. In the quantized theory, those constraints become operators that annihilate a state. A state can be represented by a wave functional $\Psi$ that simultaneously satisfies the Klein-Gordon and the Wheeler-DeWitt-Schr\"odinger equation. The latter equation, besides the term due to gravity, also contains the Schr\"odinger like term, namely the derivative of $\Psi$ with respect to time, that occurs because of the presence of the point particle. The particle's time coordinate, $X^0$, serves the role of time. Next, we generalize the system to $p$-branes, and find out that for a quantized spacetime filling brane there occurs an effective cosmological constant, proportional to the expectation value of the brane's momentum, a degree of freedom that has two discrete values only, a positive and a negative one. This mechanism could be an explanation for the small cosmological constant that drives the accelerated expansion of the universe.