"Hyperspace"*:
Physics as Geometry
Fritz Reitz, Ph.D.
(it’s 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, 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 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 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:
α
Christoffel symbols, “Γ βγ”
• 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 • Then,
when you
calculate
Christoffel
s involving
extra
dimension,

Γ5  F
•
figure after Kaku, “Hyperspace”
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 :
Klein’s paper
• Oskar 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 waves
• charge thus quantized, and quantum
of charge specifies radius of extra
dimension < 10-30 in.
• bunch of other stuff including
repeated use of the word “simply”
after Greene, Fabric of the Cosmos, Fig. 12-7
Their immediate legacy
• surely, the genius of these giants of unification
would be lauded by their peers for decades!
D’OH!
• actually their
theory was
totally eclipsed
by quantum
mechanics for
60 years or so
But THEN their
theory was much
celebrated
• theories such as
Supergravity & String
theory invoke yet more
compactified, ~ Planckscale dimensions
•
figure after Greene, Fabric of the Cosmos
• with 10
dimensions, you
can fit everything!
• sorta.
•
• (long story)
figure after Kaku, Hyperspace
on to gauges, and
D  + iq/hbarc A
• if we insist on local gauge invariance of the
Lagrangian ( is function of x), there’s
trouble as  (e-i)  e-i  
• can redefine D, but isn’t 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 A
• consider 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 = ijkjek
let i  ijkjek
dt 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
• t is 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 theories
• different gauge theories span different internal spaces,
with differing numbers of generators
• generators 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; it’s the rule for how to go
from here to there. similarly, [01-10] is how to rotate
• generators 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-Mann
• SU(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
footnote
• “If I have seen
less than
others, it is
because I as
yet but cling to
the buttocks of
giants.” -Fritz Reitz
Further reading
• Video Lectures from ASTI conference, intro to symmetry, group
theory, strings, supersymmetry, QFT at
http://www.asti.ac.za/lectures.php
• Griffiths 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): Schumm’s Deep Down Things
• Popularizations re: Kaluza-Klein, string theory Halpern’s The Great
Beyond (much biographical history), Kaku’s Hyperspace, Greene’s
The Elegant Universe
• Popularizations re: quantum gravity, critique of string theory:
Smolin's Three Roads to Quantum Gravity, Smolin's The Trouble with
Physics
• Original K-K papers: reprinted in Appelquist et al. Modern KaluzaKlein Theories
fin