Digital Physics Regge Calculus

  Digital Physics Regge Calculus is an approach that involves discretizing spacetime into simplicial complexes, providing a framework that aligns with the discrete nature of digital physics. Here are ten essential equations defining various aspects of Digital Physics Regge Calculus:

1. Simplicial Complex Definition:

   • Define a simplicial complex $\mathcal{�}$ consisting of vertices ${�}_{�}$, edges ${�}_{��}$, triangles ${\mathrm{\Delta }}_{���}$, and higher-dimensional simplices, representing discretized spacetime regions.

2. Regge Action:

   • The Regge action $�$ for the simplicial complex, considering the deficit angles ${�}_{�}$ at each vertex:
   $$�=\sum _{�}{�}_{�}-�\sum _{\text{simplices}}\text{Volume}(\text{simplex})$$

3. Deficit Angle Calculation:

   • Calculate the deficit angle ${�}_{�}$ at each vertex due to the curvature mismatch, considering the surrounding triangles:
   $${�}_{�}=2�-\sum _{\text{adjacent triangles}}\text{angle}(\text{triangle})$$

4. Simplicial Volume Calculation:

   • Compute the volume of a higher-dimensional simplex in terms of its edge lengths and spacetime dimensionality.

5. Regge Equations of Motion:

   • The equations governing the dynamics of simplicial complexes based on the stationary action principle:
   $$\frac{\mathrm{\partial }�}{\mathrm{\partial }\text{vertex position}}=0$$

6. Simplicial Einstein Equations:

   • Equations that approximate Einstein's equations in discrete spacetime, incorporating curvature and topology information of simplicial complexes.

7. Discrete Curvature Calculation:

   • Calculate the discrete curvature at a vertex in terms of the deficit angles and simplicial volumes:
   $$\text{Curvature}({�}_{�})=\frac{\sum _{\text{adjacent simplices}}\text{Volume}(\text{simplex})}{\sum _{\text{adjacent triangles}}\text{Volume}(\text{triangle})}$$

8. Dynamical Triangulation Operator:

   • Operator representing the dynamic evolution of the simplicial complex, capturing changes in topology and geometry over discrete time steps.

9. Simplicial Metric Tensor:

   • Define the discrete metric tensor ${�}_{��}$ on simplices, approximating the spacetime metric in the Regge Calculus framework.

10. Quantum Regge Calculus:

    • Extend the Regge Calculus formalism into a quantum theory, incorporating discrete quantum states associated with simplices and defining transition amplitudes between these states.

These equations constitute a foundational framework for Digital Physics Regge Calculus, offering a discrete and computational perspective on spacetime geometry and dynamics. They are essential for exploring the emergent properties of spacetime within the digital physics paradigm.