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1. • CommentRowNumber1.
   • CommentAuthorMadeleine Birchfield
   • CommentTimeMar 20th 2025
   • PermaLink
   Author: Madeleine Birchfield
   Format: MarkdownItexWhat applications, if any, are there of graded differential cohesion and its modalities (rheonomy, bosonic, and fermionic) to non-supersymmetric quantum field theories like quantum electrodynamics or quantum chromodynamics?

   What applications, if any, are there of graded differential cohesion and its modalities (rheonomy, bosonic, and fermionic) to non-supersymmetric quantum field theories like quantum electrodynamics or quantum chromodynamics?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 21st 2025
   • (edited Mar 21st 2025)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOh, many: any QFT with fermions! Such as, yes, QED (with its electrons) and QCD (with its quarks). Namely, what (mostly in mathematics) is called "supergeometry" is much more general than what (mostly in physics) is called "supersymmetry": The phase space of a field theory is a "superspace" the moment that there are any fermions (reflecting their [[Pauli exclusion principle]]), independent of whether there is any supersymmetry! In particular the phase space of the standard model of particle physics, with its fermionic particles, hence of the physics that is experimentally observed ever since [[Stern-Gerlach experiment|Stern and Gerlach 1920s]], is supergeometric: The standard kinetic Lagrangian density of a fermionic field $\psi$, namely the Dirac term, schematically $\sim \psi D \psi$, would disappear if $\psi$ were not an odd-graded function on phase space (because for an ordinary function we'd have that $\psi D \psi = \tfrac{1}{2} D(\psi^2)$ is a total derivative and hence $\sim$zero as a Lagrangian density). This is "well-known", as one says, even if the different (but completely standard) use of "super"-terminology is bound to be misleading. The physicist's "super" is really shorthand for something more specific, namely for "super-Poincaré symmetry" or "super conformal symmetry". But a phase space may be supergeometric without admitting super-Poincaré-symmetry. In more recent exposition, this point is highlighted for instance in introduction and outlook of [Sati & Giotopoulos 2025](https://ncatlab.org/nlab/show/smooth+set#GiotopoulosSati23) and in the respective section of *[[schreiber:Higher Topos Theory in Physics|Higher Topos Theory in Physics]]*. (One may turn this around and argue that, given this state of affairs, the world being (locally) super-*symmetric* (hence: super-gravitational) would actually be "less surprising" than if not, since it's "more surprising" for a super phase space not to also support a super-symmetry. But that is a different discussion.)

   Oh, many: any QFT with fermions! Such as, yes, QED (with its electrons) and QCD (with its quarks).

   Namely, what (mostly in mathematics) is called “supergeometry” is much more general than what (mostly in physics) is called “supersymmetry”:

   The phase space of a field theory is a “superspace” the moment that there are any fermions (reflecting their Pauli exclusion principle), independent of whether there is any supersymmetry!

   In particular the phase space of the standard model of particle physics, with its fermionic particles, hence of the physics that is experimentally observed ever since Stern and Gerlach 1920s, is supergeometric:

   The standard kinetic Lagrangian density of a fermionic field $\psi$, namely the Dirac term, schematically $\sim \psi D \psi$, would disappear if $\psi$ were not an odd-graded function on phase space (because for an ordinary function we’d have that $\psi D \psi = \tfrac{1}{2} D(\psi^2)$ is a total derivative and hence $\sim$zero as a Lagrangian density).

   This is “well-known”, as one says, even if the different (but completely standard) use of “super”-terminology is bound to be misleading. The physicist’s “super” is really shorthand for something more specific, namely for “super-Poincaré symmetry” or “super conformal symmetry”. But a phase space may be supergeometric without admitting super-Poincaré-symmetry.

   In more recent exposition, this point is highlighted for instance in introduction and outlook of Sati & Giotopoulos 2025 and in the respective section of Higher Topos Theory in Physics.

   (One may turn this around and argue that, given this state of affairs, the world being (locally) super-symmetric (hence: super-gravitational) would actually be “less surprising” than if not, since it’s “more surprising” for a super phase space not to also support a super-symmetry. But that is a different discussion.)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 21st 2025
   • (edited Mar 21st 2025)
   • PermaLink
   Author: Urs
   Format: MarkdownItex[second message deleted, wrong thread]

   [second message deleted, wrong thread]