Let me ask several related questions on discretization of classical field theory:

  1. In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary operator is a "discrete analogue" of exterior derivative. Who invented this correspondence (and thus created now-popular "discrete exterior calculus")? de Rham in 1930s? What is a canonical reference (Geometric integration theory by Whitney?)

  2. This immediately gives discrete analogues of the Maxwell equation on any simplicial complex. Who invented that? I've seen physists' works on that from 1970s, but this must be an older story. Likewise, who first stated the Kirchhoff network laws in terms of (co)boundary operators? In 1920s Herman Weil published papers on foundations of both electrical networks and of combinatorial topology, but I cannot find fulltexts.

  3. There are two discrete analogues of the exterior product: the cochain cup-product and a version (actually, the original construction) by A.~Kolmogorov and J.~Alexander from 1930s. The former is associative, while the latter is anticommutative. Have a discretization of exterior calculus based on the former (the cup-product) ever been studied?

  4. A discrete analogue of a connection is parallel transport operators along edges, i.e., a matrix-valued 1-cochain. This construction is actively used by physicists, starting from Kenneth Wilson (1970s). Have a discretization of covariant differentiation based on this construction and the cup-product ever been studied? (Dennis Sullivan?) Was the resulting Yang-Mills equation have ever been studied?

  5. All the above concerns forms, i.e., antisymmetric tensor fields. Have discrete analogues of symmetric tensor fields ever been studied?

  6. Have discrete analogues of the Euler-Lagrange equations for functionals on simplicial cochains ever been studied? Has a general "discrete field theory" been ever built?

  • $\begingroup$ bobenko wrote a book which is called discrete differential geometry $\endgroup$ Dec 18, 2018 at 10:45
  • 1
    $\begingroup$ @IsetHawthorne: thanks, familiar with the book, it is more focused on different questions $\endgroup$ Dec 20, 2018 at 22:55
  • $\begingroup$ Written down all that in arxiv.org/abs/1709.04788 because no references are available (except for point 4) $\endgroup$ Jul 25, 2021 at 6:29


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy