When classical degrees of freedom and quantum degrees of freedom are

consistently coupled, the former diffuse, while the latter undergo decoherence.

Here, we construct a theory of quantum matter fields and Nordstrom gravity in

which the space-time metric is treated classically. The dynamics is constructed

via the classical-quantum path integral and is completely positive, trace

preserving (CPTP), and respects the classical-quantum split. The weak field

limit of the model matches the Newtonian limit of the full covariant path

integral but it is easier to show that the theory is both diffeomorphism

invariant, CPTP, and has the appropriate classical limit.

The conclusions of the text are as follows:

- When classical degrees of freedom and quantum degrees of freedom are consistently coupled, the classical degrees of freedom diffuse, while the quantum degrees of freedom undergo decoherence.
- A theory of quantum matter fields and Nordstrom gravity is constructed in which the space-time metric is treated classically.
- The dynamics of the theory is constructed via the classical-quantum path integral and is completely positive, trace preserving (CPTP), and respects the classical-quantum split.
- The weak field limit of the model matches the Newtonian limit of the full covariant path integral.
- The theory is both diffeomorphism invariant, CPTP, and has the appropriate classical limit.

## Future Roadmap

Based on the conclusions of the text, there are potential challenges and opportunities on the horizon for further research and development in the field. A future roadmap can be outlined as follows:

### 1. Studying Diffusion of Classical Degrees of Freedom

Further investigation is needed to understand and explore the diffusion of classical degrees of freedom when consistently coupled with quantum degrees of freedom. This can help in determining the extent to which classical information spreads and diffuses in such systems.

### 2. Investigating Quantum Decoherence

The phenomenon of quantum decoherence observed in the text requires deeper exploration to understand its implications and consequences. Research can focus on identifying methods to mitigate or control decoherence, allowing for the preservation of quantum coherence in interacting systems.

### 3. Refining the Theory of Quantum Matter Fields and Nordstrom Gravity

The theory proposed in the text, which treats the space-time metric classically, needs further refinement and development. Researchers can focus on enhancing the accuracy and applicability of the theory by incorporating additional factors and variables that affect the interaction between quantum matter fields and Nordstrom gravity.

### 4. Exploring Alternative Dynamics Construction Methods

The classical-quantum path integral used for constructing the dynamics in the proposed theory may not be the only approach available. Exploring alternative methods for constructing the dynamics can provide insights into different aspects of the system and potentially uncover new phenomena or behaviors.

### 5. Extending the Model to Non-Weak Field Limits

The current model’s weak field limit matches the Newtonian limit, which opens possibilities for investigating other limits of the model. Extending the analysis to non-weak field limits can lead to a deeper understanding of the behavior of the theory in different regimes and scenarios.

### 6. Validating Diffeomorphism Invariance and Classical Limits

Validating that the theory is both diffeomorphism invariant and has the appropriate classical limit is crucial to ensure its consistency and accuracy. Further studies can focus on rigorous mathematical proofs and experimental validations to confirm these properties of the proposed theory.

### 7. Practical Applications and Technological Impact

Exploring potential practical applications and technological implications of the developed theory is an essential aspect of future research. Investigating how the theory can be utilized in various fields, such as quantum computing, cosmology, or particle physics, can lead to innovative technologies and advancements.

Overall, the conclusions drawn from the text present exciting prospects for further research in understanding the coupling of classical and quantum degrees of freedom. The outlined future roadmap highlights potential challenges to address and opportunities for groundbreaking discoveries in the field.