# A Novel Approach to Quantum Gravity in the Presence of Matter without the Problem of Time

## TL;DR

This paper introduces a new method for quantizing gravity with matter that results in a Schrödinger-like equation and a Klein-Gordon equation, providing a fresh perspective on the problem of time in quantum gravity.

## Contribution

It presents a novel approach starting from classical Einstein-Hilbert action with point particles, leading to a combined Schrödinger and Klein-Gordon framework in quantum gravity.

## Key findings

- Hamilton constraint becomes Schrödinger equation with matter
- Wave function satisfies Klein-Gordon equation
- Approach complements scalar field methods

## Abstract

An approach to the quantization of gravity in the presence matter is examined which starts from the classical Einstein-Hilbert action and matter approximated by "point" particles minimally coupled to the metric. Upon quantization, the Hamilton constraint assumes the form of the Schr\"odinger equation: it contains the usual Wheeler-DeWitt term and the term with the time derivative of the wave function. In addition, the wave function also satisfies the Klein-Gordon equation, which arises as the quantum counterpart of the constraint among particles' momenta. Comparison of the novel approach with the usual one in which matter is represented by scalar fields is performed, and shown that those approaches do not exclude, but complement each other. In final discussion it is pointed out that the classical matter could consist of superparticles or spinning particles, described by the commuting and anticommuting Grassmann coordinates, in which case spinor fields would occur after quantization.