Gauge Theory Explained

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Chapters:
Spacetime 01:23
Bosons 13:31
Fermions 14:52
Equations 15:28
Lagrangians 24:31
Gauge theories describe local interactions that occur in single spacetime points. Fields on spacetime often cannot be described simply by a map to a fixed vector space, but rather as sections of a non-trivial vector bundle. The fundamental geometric opbject in a gauge theory is a principal bundle over spacetime with structure group given by the gauge group.
Under gravity, a sphere of test particles deforms into an egg-shaped closed surface. From the Newtonian inverse-square law, we know that gravity preserves the volume of the surface delta R_{mu nu} = 0. Whenever matter is present R not equal to 0, then a volume-reducing force field deforms spacetime too.
Einstein's Spacetime has four degrees of freedom, where objects in free fall move along geodesic curves. If the loop integral of the gravitational work 1-from does not vanish, a trajectory is stretched. But, the stretch changes the spacing of the stack representing the 1-form. The deviation of geodesics means that a Hamiltonian flow lifts a circular path to a spiral trajectory. He, discards the Weyl curvature, as R_{mu nu} measures the stretch and its trace measures density. Finally, sets the total matter-energy equal to a combination of of the Ricci R_{mu nu} and scalar R curvatures. So, a stick figure walks along Penrose steps or the Escher staircase, seemingly going upward forever.
What are “horizontal subspaces” and what do they have to do with vector potentials and gauge fields?
A connection (denoted by A) on a principal G-bundle P assigns a horizontal subspace of the tangent space to each point q in P, in a smooth way. First, the tangent space of a principal bundle P at point q is equal to the direct sum of the vertical subspace and the horizontal subspace at q, for all q in P. Second, the push forward of the horizontal subspace of the Principal G-bundle P at point q by an action g of the structure group G is equal to the horizontal subspace of the principal G-bundle P at the transformation of the point q by the action g, for all g in G and q in P.
If the exterior derivative of the curvature 2-form vanishes, then there exists a 1-form potential that acts as a reference level. The vector representation of the 1-form is called a vector potential. But, the 1-form on spacetime is called a gauge field. So gemoterically, the vector potential is normal to the stack of horizontal planes of the 1-form.
References
Sir Roger Penrose on the The Portal with Eric Weinstein
Mark J.D. Hamilton, Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics, Springer Cham, DOI, published: 10 January 2018.
Sir Roger Penrose, The Road to Reality, (2004).
Roger Penrose, Wolfgang Rindler, Spinors and Space-Time, Volume 1: Two-spinor calculus and relativistic fields, (1984).
M.J.D. Hamilton, The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking, arXiv preprint arXiv:1512.02632, (2015).
Edward Witten, Physics and Geometry, (1987).
The iconic Wall of Stony Brook University.
Tristan Needham, Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts, Princeton University Press, 2021.

[@KiraandtheSurvivalPod]

So cool thanks for making this. I really enjoy the visuals

[@iamazadi]

I'm glad you enjoyed them.

[@Diryma-b9u]

thanks dude!

[@iamazadi]

Glad you found it helpful.

[@TheMemesofDestruction]

Fantastic!

[@iamazadi]

Thanks!

[@TheMemesofDestruction]

@@iamazadi You make it easier to follow. ^.^