Kaluza-Klein Theory - fritopiainfo：卡鲁扎-克莱因理论fritopiainfo.ppt

* * * * * * * * Hyperspace*: Physics as Geometry Fritz Reitz, Ph.D. (it’s not in physics, as will become obvious shortly) 5/18/09 *a la Michio Kakus sensionalization of the invocation of unseen dimensions (e.g. isospin space) in physics (hey Fritz, don’t forget to run vi’s ahead of time) Talk outline: examples of recasting phenomena as rotations in new spaces application to gauges visualizing the “internal spaces” of the gauge theories familiar example: special relativity recasts/simplifies motion add time direction to space with new concept of 4-velocity, dr/d? =?(?c, ?vx, ?vy, ?vz), |dr/d?| = c for everything if someones clock seems slow, its because theyve steered away (via boost) from your time direction, just as one goes north slower in an airplane pointed NNE adding a dimension has cast motion in a different light, and simplified things unfamiliar example: Kaluza-Klein Theory in ~1919, Kaluza (and others, long story) looked at Christoffel symbols Γαβγ used in general relativity, thought wow, F?? and Γαβγ look similar! hmm, F?? would need another index to match up properly, or Γαβγ one less why, that would only happen if there was another spatial dimension (? goes from 0 to 4), that was connected less intimately (?g??/?x? = 0) Brief aside:Christoffel symbols, “Γαβγ” directional derivative in Euclidean coordinates: directional derivative in Polar coordinates: Christoffel symbols account for the extra part of the derivative due to changing coordinates (e.g. d? scales with r, dr changes direction with ?) handy when spacetime itself is curved, hence its use in GR What Kaluza did: Kaluza added the vector potential along the sides of the metric tensor essentially like so - figure after Kaku, “Hyperspace” Then, when you calculate Christoffels involving extra dimension, Γ5?? ? F?? ? and charge is velocity in this extra direction (BONUS!) short version is charge x velocity is a current density which is the derivative of F ? Ricci curvature ? flat space