Natural Discrete Differential Calculus in Physics

@article{Rovelli2019NaturalDD,
  title={Natural Discrete Differential Calculus in Physics},
  author={Carlo Rovelli and V{\'a}clav Zatloukal},
  journal={Foundations of Physics},
  year={2019},
  pages={1-7}
}
Abstract We sharpen a recent observation by Tim Maudlin: differential calculus is a natural language for physics only if additional structure, like the definition of a Hodge dual or a metric, is given; but the discrete version of this calculus provides this additional structure for free. 
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References

SHOWING 1-10 OF 16 REFERENCES
Random Lattice Gauge Theories and Differential Forms
We provide a brief overview on the application of the exterior calculus of differential forms to the ab initio formulation of field theories on random simplicial lattices. In this framework, discrete
Elements of algebraic topology
Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in
Laplacians on discrete and quantum geometries
We extend discrete calculus for arbitrary (p-form) fields on embedded lattices to abstract discrete geometries based on combinatorial complexes. We then provide a general definition of discrete
Dirac-Kähler fields and the lattice shape dependence of fermion flavour
We investigate the Dirac-Kähler operator on a triangular lattice in two dimensions and show that the number of degrees of freedom which survive in the continuum limit is the same as in the case of a
Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory
Preface Part I. Foundations: 1. Spacetime as a quantum object 2. Physics without time 3. Gravity 4. Classical discretization Part II. The 3D Theory: 5. 3D Euclidean theory 6. Bubbles and cosmological
The Spin-Foam Approach to Quantum Gravity
TLDR
The present status of the spin-foam approach to the quantization of gravity is reviewed and the pedagogical presentation of the recently-introduced new models for four-dimensional quantum gravity is paid to.
Discrete exterior calculus for variational problems in computer vision and graphics
TLDR
The paper demonstrates how discrete exterior calculus tools may be useful in computer vision and graphics and shows some example applications using variational problems from computer graphics and mechanics to demonstrate that formulating the problem discretely and using discrete methods for solution can lead to efficient algorithms.
On the Hypotheses Which Lie at the Bases of Geometry
III.—Application to Space. § 1.—By means of these inquiries into the determination of the measure relations of an n-fold extent the conditions may be declared which are necessary and sufficient to
Confinement of Quarks
A mechanism for total confinement of quarks, similar to that of Schwinger, is defined which requires the existence of Abelian or non-Abelian gauge fields. It is shown how to quantize a gauge field
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