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Combining gravity with quantum theory is still work in progress. On the one
hand, classical gravity, is the geometry of space-time determined by the
energy-momentum tensor of matter and the resulting nonlinear equations; on the
other hand, the mathematical description of a quantum system, is Hilbert space
with linear equations describing evolution. In this paper, various measures in
Hilbert space will be presented. In general, distance measures in Hilbert space
can be divided into measures determined by energy and measures determined by
entropy. Entropy measures determine quasi-distance because they do not satisfy
all the axioms defining distance. Finding a general rule to determine such a
measure unambiguously seems to be fundamental.

Examine the conclusions of the following text and outline a future roadmap for readers, indicating potential challenges and opportunities on the horizon.

## Introduction

The article discusses the ongoing work of combining gravity with quantum theory. It highlights the differences between classical gravity, which is determined by the geometry of space-time, and the mathematical description of a quantum system, which is based on Hilbert space.

## Current State

Currently, various measures in Hilbert space are being presented to understand the relationship between gravity and quantum theory. These measures can be classified into two types: measures determined by energy and measures determined by entropy. However, it is important to note that entropy measures only provide quasi-distance and do not satisfy all the axioms defining distance.

Despite the challenges, it is essential to find a general rule that can unambiguously determine a measure in Hilbert space. This will be fundamental in solving the problem of combining gravity and quantum theory. Here is a potential roadmap for readers:

1. Understanding Measures Determined by Energy: Readers should familiarize themselves with the concept of measures in Hilbert space determined by energy. This will involve studying the mathematical equations and methods used to determine these measures.
2. Exploring Measures Determined by Entropy: A comprehensive understanding of measures determined by entropy is crucial. Readers should delve into the mathematical framework of entropy measures and its limitations in providing complete distance information.
3. Finding Unambiguous Measures: The challenge lies in finding a general rule that can determine a measure unambiguously in Hilbert space. Readers should stay updated with current research and breakthroughs in this area, as scientists work towards this goal.
4. Application of Measures: Once a general rule for determining measures is established, the focus will shift to applying these measures to the problem of combining gravity and quantum theory. This could involve investigating the implications of different measures on the geometry of space-time and the evolution of quantum systems.
5. Future Challenges and Opportunities: The roadmap should include a section that discusses potential challenges and opportunities on the horizon. These may include technological limitations, theoretical complexities, interdisciplinary collaborations, and the potential impact of successful integration of gravity and quantum theory on fields such as cosmology and quantum computing.

## Conclusion

The roadmap outlined above provides a structured approach for readers to navigate the complexities of combining gravity with quantum theory. By understanding the measures determined by energy and entropy in Hilbert space and keeping up with the advancements in determining unambiguous measures, readers can contribute to this groundbreaking field. Challenges and opportunities should be anticipated and explored along the way, opening doors for further research and discoveries.