"Hyperspace"*: Physics as GeometryFritz Reitz, Ph.D. (its not in physics, as will become obvious shortly)5/18/09
*a la Michio Kaku's sensionalization of the invocation of unseen dimensions (e.g. isospin space) in physics
(hey Fritz, dont forget to run vis ahead of time)
Talk outline: examples of recasting phenomena as rotations in new spacesapplication to gaugesvisualizing the internal spaces of the gauge theories
familiar example: special relativity recasts/simplifies motionadd "time" direction to spacewith new concept of 4-velocity,dr/d =(c, vx, vy, vz), |dr/d| = c for everythingif someone's clock seems slow, it's because they've steered away (via boost) from your "time" direction, just as one goes north slower in an airplane pointed NNEadding a dimension has cast motion in a different light, and simplified things
unfamiliar example: Kaluza-Klein Theoryin ~1919, Kaluza (and others, long story) looked at "Christoffel symbols" used in general relativity, thought "wow, F and look similar!"hmm, F would need another index to match up properly, or one lesswhy, 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, 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, HyperspaceThen, when you calculate Christoffels involving extra dimension, 5 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 stress tensor u5 x velocity, so 0 u5!I don't get it either, but wow! 0 u5!long version is 0u J, J F/x, F/x R5, R5 T5, T5 u5u 0u, 0 u5 :
Kleins paperOskar Klein is late to the party again (story of his life -- long story), but then curls up the extra dimension tightly u5 and thus momentum, and thus 1 / (de Broglie wavelength h/p)he imagined the extra dimension wrapped in a circle, with an integer number of standing wavescharge thus quantized, and quantum of charge specifies radius of extra dimension < 10-30 in.bunch of other stuff including repeated use of the word simplyafter Greene, Fabric of the Cosmos, Fig. 12-7
Their immediate legacysurely, the genius of these giants of unification would be lauded by their peers for decades!
actually their theory was totally eclipsed by quantum mechanics for 60 years or soDOH!
But THEN their theory was much celebrated figure after Kaku, Hyperspace theories such as Supergravity & String theory invoke yet more compactified, ~ Planck-scale dimensions with 10 dimensions, you can fit everything!
(long story)figure after Greene, Fabric of the Cosmos
on to gauges, and D + iq/hbarc Aif we insist on local gauge invariance of the Lagrangian ( is function of x), theres trouble as (e-i) e-i
can redefine D, but isnt that cheating? aren't we just sweeping terms under the rug?
what is "D"now? is it still a "partial derivative" even?
classical analogy for D + iq/hbarc Aconsider a spinning top and a vector x in the top's frame such that x = xi ei , where ei are themselves ei(t)
dt x = (dt xi) ei + xi ( dt ei ) [need product rule]for rotating frame, dt ei = ei = ijkjek let i ijkjekdt x = (dt xi) ei + xi i
let t dt, local + , or, if hbarc/iq
t dt + iq/hbarc
"Why don't we call [choosing a gauge] choosing coordinates in the extra space? It's an unfortunate historical accident." -- C. Bloom
D + iq/hbarc A &t t + iq/hbarc an ant living in the rotating frame might not realize that ei changed with time, they might think dt x = (dt xi) ei was the whole story, would think Coriolis and centrifugal forces were real
tis thus the derivative of the "real" x
the suggestion is that we are like the ant, immersed in and yet oblivious to some mode of motion, like rotation in Kaluza's 5th dimension, and EM is like the "fictitious" Coriolis force
visualizing the internal spaces of the gauge theoriesdifferent gauge theories span different internal spaces, with differing numbers of generatorsgenerators contain the essence of their transformation, e.g. (x+a) = exp( a d/dx ) (x) (per Taylor series; see footnote 1)(+z) = exp( [01-10]) ()
in a sense, d/dx is translation; its the rule for how to go from here to there. similarly, [01-10] is how to rotategenerators of internal spaces correspond to bosons that act on their corresponding fermions what is the essence of a weak bosons? of a gluon?
(1): (x-a) = (x) - a d/dx (x) / 1! + a2 d2/dx2 (x) / 2! - ... = e-a d/dx (x)
simulations of SU(2), SU(3)spin, weak isospin are SU(2), generated by Pauli matrices, color SU(3) & Gell-MannSU(2) like 3D rotations, and quark state is possible to represent, but how meaningful is it really to draw a continuous, classical pictures of unpicturable quantum processes?answer: kinda (to simulation!)
A favorite quote:If I have seen further than others, it is by standing upon the shoulders of giants. -- Isaac Newton
Personal footnoteIf I have seen less than others, it is because I as yet but cling to the buttocks of giants. -- Fritz Reitz
Further readingVideo Lectures from ASTI conference, intro to symmetry, group theory, strings, supersymmetry, QFT at http://www.asti.ac.za/lectures.phpGriffiths has a particle physics text!: Introduction to Elementary Particles, 2nd ed. (just as accessible as his EM & QM texts)Popularizations re: particles, electro-weak mixing, U(1), SU(2), SU(3): Schumms Deep Down ThingsPopularizations re: Kaluza-Klein, string theory Halperns The Great Beyond (much biographical history), Kakus Hyperspace, Greenes The Elegant UniversePopularizations re: quantum gravity, critique of string theory: Smolin's Three Roads to Quantum Gravity, Smolin's The Trouble with PhysicsOriginal K-K papers: reprinted in Appelquist et al. Modern Kaluza-Klein Theories