Science > Physics > This Week's Finds in Mathematical Physics (Week 223)
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Science > Physics |
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"John Baez" |
| Date: |
14 Nov 2005 08:47:36 PM |
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This Week's Finds in Mathematical Physics (Week 223) |
Also available as http://math.ucr.edu/home/baez/week223.html
November 14, 2005
This Week's Finds in Mathematical Physics - Week 223
John Baez
This week I'd like to talk about two aspects of higher gauge theory:
p-form electromagnetism and nonabelian cohomology. Lurking behind both
of these is the mathematics of n-categories, but I'll do my best to hide
that until the end, to build up the suspense.
But first, some cool pictures. Astronomy is booming these days, and it's
a great way to see beautiful complexity emerging from simple laws in this
wonderful universe of ours. So, I'd like the freedom to occasionally start
This Week's Finds with some pictures from the skies. Think of it as an
appetizer before the main course. Sometimes I'll explicitly relate these
pictures to math and physics; other times not.
Here's Saturn's moon Hyperion, photographed up close by the Cassini probe:
1) Cassini-Huyghens Mission, Hyperion: Odd World,
http://saturn.jpl.nasa.gov/multimedia/images/image-details.cfm?imageID=1762
It seems to be a huge pile of rubble loosely held together by gravity and
heavily cratered by meteor bombardments.
Hyperion is interesting because it's the only known moon that tumbles
chaotically on a short time scale, thanks to its eccentric shape and
gravitational interactions with Saturn and Titan.
This leads to some interesting math. We can think of Hyperion's angular
momentum vector as a point on a sphere. If we started out knowing
this point lay inside some small disk, time evolution would warp this disk
into an ever more complicated region as time passed. This region would
always have the same area, thanks to the wonders of symplectic geometry.
But it would sprout ever more complicated tendrils, with its perimeter
growing by a factor of e about every 100 days or so!
That's chaos for you.
Indeed, only quantum mechanics would stop the intricacy from growing forever,
by blurring it out. After about 37 years, the area of a typical tendril
would equal Planck's constant. At this point, classical mechanics would
no longer be accurate. You'd really need to describe Hyperion's spin
state using quantum theory: for example, a holomorphic section of some
line bundle on the sphere.
Well... at least you would if it weren't for decoherence caused by the
interaction of Hyperion with its environment, for example solar radiation!
For an explanation of how this changes the story, try:
2) Michael Berry, Chaos and the semiclassical limit of quantum mechanics
(is the moon there when somebody looks?), in Quantum Mechanics: Scientific
Perspectives on Divine Action, CTNS Publications, Vatican Observatory, 2001.
Also available at
http://www.phy.bris.ac.uk/people/berry_mv/the_papers/berry337.pdf
Here's another great picture:
3) The Hubble Heritage Project, Cat's Eye Nebula - NGC 6543,
http://heritage.stsci.edu/2004/27/index.html
This is a star about the size of the Sun, nearing the end of its life,
emitting pulses of gas and dust. Astronomers call such a thing a
"planetary nebula", though it has nothing to do with planets. It's in our
galaxy, about 3000 light years from us. When it's done shedding its outer
layers, all that's left of this star will be a white dwarf.
Our own Sun will become a planetary nebula in about 6.9 billion years,
after two separate stages of being a red giant - one as it runs out
hydrogen, and one as it runs out of helium. When the helium is all gone,
the Sun will start to pulsate every 100,000 years, ejecting more and more
mass in each pulse, eventually throwing off all but the hot inner core made
of heavier elements. The astronomer Bruce Balick has written eloquently
on what this will mean for the Earth:
Here on Earth, we'll feel the wind of the ejected gases sweeping
past, slowly at first (a mere 5 miles per second!), and then
picking up speed as the spasms continue - eventually to reach
1000 miles per second!! The remnant Sun will rise as a dot of
intense light, no larger than Venus, more brilliant than 100
present Suns, and an intensely hot blue-white color hotter than
any welder's torch. Light from the fiendish blue "pinprick"
will braise the Earth and tear apart its surface molecules and
atoms. A new but very thin "atmosphere" of free electrons
will form as the Earth's surface turns to dust.
So, don't keep procrastinating - enjoy life now!
For other pictures of planetary nebulae, try Balick's webpage:
4) Bruce Balick, Hubble Space Telescope images of planetary nebulae,
http://www.astro.washington.edu/balick/WFPC2/index.html
For a timeline of the universe, including the future life of our
Sun, try:
5) John Baez, A brief history of the universe,
http://math.ucr.edu/home/baez/timeline.html
Now... on to p-form electromagnetism!
In ordinary electromagnetism, the secret star of the show turns out
to be not the electromagnetic field but the "vector potential", A.
At least locally, we can think of this as a 1-form. A 1-form is just
a gadget that you can integrate along a path and get a number. In the
case of the vector potential, this number describes the change in
phase that a charged particle acquires as it moves along this path.
The 1-form A gives rise to a 2-form F called the "electromagnetic field".
A 2-form is a gadget you can integrate over a surface and get a
number. Here's how we get F from A. Suppose we move a charged particle
around a loop that's the boundary of some surface. Then the integral
of F over this surface is defined to be the integral of A around the loop!
We summarize this by saying that F is the "exterior derivative" of A,
and writing
F = dA.
F is called the electromagnetic field because... that's what it is!
It contains both the electric and magnetic fields in a single neat
package. In 4d spacetime, the magnetic field describes the change in
a phase of a charged particle that loops around a surface in the
xy, yz or zx planes. The electric field describes the change in phase
of a charged particle that loops around a surface in the xt, yt or zt
planes.
If you don't know this stuff, you're missing some of the best fun
life has to offer. For an easy introduction with lots of gorgeous
pictures, see:
6) Derek Wise, Electricity, magnetism and hypercubes, available at
http://math.ucr.edu/~derek/talks/050916bw.pdf
The idea of p-form electromagnetism is to replace point particles
by strings or higher-dimensional membranes. To see how this goes,
it's enough to look at 2-form electromagnetism.
In 2-form electromagnetism, the star of the show is a 2-form, A.
As already mentioned, a 2-form is a gadget you can integrate over a
surface and get a number. In 2-form electromagnetism, this number
describes the change in phase that a charged string acquires as it
moves along, tracing out a surface in spacetime.
The 2-form A gives rise to a 3-form, F. A 3-form is a gadget
you can integrate over a 3-dimensional region and get a number.
Suppose we move a charged string and let it trace out a surface
that's the boundary of some 3-dimensional region. Then the integral
of F over this region is defined to be the integral of A over the
surface! Again we write this as:
F = dA.
So, we're just adding one to the dimensions of things. This makes it
easy to keep on going. In fact, for any integer p, we can write down
a generalization of Maxwell's equations.
It goes like this. We start with a p-form A. We define a (p+1)-form
F = dA
This automatically implies some of Maxwell's equations:
dF = 0
but the nontrivial Maxwell equations say that
*d*F = J
where * is the Hodge star operator and J is a p-form called the "current",
which is produced by charged matter.
What does this mean, physically? The idea is that we have charged matter
consisting of (p-1)-dimensional membranes. These trace out p-dimensional
surfaces in spacetime as time passes. The current J is a p-form that's
concentrated on these surfaces. The current affects the A field in a
manner governed by Maxwell's equations. Conversely, the A field affects
the motion of the membranes. Classically, we just integrate the A field
over the surface traced out by a membrane and add the result to the
*action* for the membrane. In the path integral approach to quantum
mechanics, this number gives a change in phase, as already mentioned.
Maxwell's equations and their p-form generalization make sense when
spacetime is any Lorentzian manifold. However, to get a theory
where initial data determine a unique global solution, we want our
spacetime to be "globally hyperbolic", which means that it has a
"Cauchy surface": roughly, a spacelike surface that any sufficiently
long timelike curve hits precisely once. To get a good *quantum* theory
of p-form electromagnetism with a Hilbert space of states on which
time evolution acts as unitary operators, we need more: our spacetime
should be "stationary", meaning that it has time translation symmetry.
Otherwise there's no way to define energy and the vacuum state - which
is defined to be the least-energy state.
My student Miguel Carrion-Alvarez tackled an important special case
in his thesis, namely "static" globally hyperbolic spacetimes:
7) Miguel Carrion-Alvarez, Loop quantization versus Fock quantization
of p-form electromagnetism on static spacetimes, available as
math-ph/0412032.
There's a lot of interesting analysis involved, especially when space
(the Cauchy surface) is noncompact. When it's compact, we can use
"Hodge's theorem" to relate its deRham cohomology to its topology,
and this turns out to be crucial for understanding p-form electromagnetism -
especially issues like the p-form Bohm-Aharonov effect. When it's
noncompact we need something called "twisted L^2 cohomology" instead,
and Miguel proved a generalization of Hodge's theorem for this.
With the analysis under control, Miguel was able to set up a very
beautiful approach to "loop quantum electromagnetism" and its p-form
generalization. Here the idea is to write Maxwell's equations in terms
of the integrals of A around all possible loops in space - or more
generally, over all p-dimensional surfaces. People interested in loop
quantum gravity should like this.
As you can guess, either from seeing all the "d" operators or seeing all
the buzzwords I'm throwing around, p-form electromagnetism is really just
cohomology incarnated as physics! My student Derek Wise made this very
precise for a version of the theory where spacetime is *discrete* -
so-called "lattice p-form electromagnetism":
8) Derek Wise, Lattice p-form electromagnetism and chain field
theory, available as gr-qc/0510033. Version with better graphics
and related material at http://math.ucr.edu/~derek/pform/index.html
In this paper, he shows lattice p-form electromagnetism is a "chain
field theory": something like a topological quantum field theory, but
where what matters is not spacetime itself so much as the cochain
complex of differential forms *on* spacetime, equipped with just enough
extra geometrical structure to write down the p-form version of Maxwell's
equations.
Both Miguel's thesis and Derek's papers are great if you want to learn
lots of math and physics. I seem to attract students who enjoy explaining
things.
Speaking of which....
Next I want to explain some stuff Danny Stevenson told me at a mall in the
little town of Cabazon while we were recovering from a hike in the desert
followed by pancakes at the Wheel Inn - a roadside restaurant famous for
its enormous statues of dinosaurs.
Danny works on gerbes, stacks, and higher gauge theory. Last year we
wrote a paper with Alissa Crans and Urs Schreiber constructing 2-groups
(categorified groups) from the math of string theory - more precisely,
from central extensions of loop groups. Since then I've been spending a
lot of time writing a paper with Urs on higher gauge theory, where we set
up a theory of parallel transport along surfaces. 2-form electromagnetism
is the simplest case of this theory. Meanwhile, Danny has been thinking
about connections on 2-vector bundles and their relation to the cohomology
of Lie 2-algebras.
This has led him to generalize Schreier theory in some interesting ways.
So, let me tell you about Schreier theory!
Schreier theory is a way to classify short exact sequences of groups.
I'll say what I mean by that in a minute... but what makes Schreier theory
special is that avoids some simplifying assumptions you might have seen
if you've studied short exact sequences before.
Normally people water down their short exact sequences by assuming some
of the groups in question are *abelian*. This lets them use "cohomology
theory" to do the classification. See "week210" for a nice book that
takes this approach.
This standard approach is great - I'm not knocking it - but Schreier theory
is more general: it's really a branch of "nonabelian cohomology theory".
It's not all that hard to explain, either. So, I'll explain it and then
talk about various simplifying assumptions people make.
The goal of Schreier theory is to classify short exact sequences of groups:
1 -> F -> E -> B -> 1
for a given choice of F and B. "Exact" means that the arrows stand
for homomorphisms and the image of each arrow is the kernel of the next.
Here this just means that F is a normal subgroup of E and B is the quotient
group E/F. Such a short exact sequence is also called an "extension of B
by F", since E is bigger than B and contains F. The simplest choice is to
let E be the direct sum of F and B. Usually there are other more interesting
extensions as well.
To classify these, the trick is to use the analogy between group theory
and topology.
As I explained in "week213", you can think of a group as a watered-down
version of a connected space with a chosen point. The reason is that
given such a space, we get a group consisting of homotopy classes of
loops based at the chosen point. This is called the "fundamental group"
of our space. There's a lot more information in our space than this group.
But pretty much anything you can do for groups, you can do for such spaces.
It's usually harder, but it's completely analogous!
In particular, classifying short exact sequences is a lot like
classifying "fibrations":
1 -> F -> E -> B -> 1
where now the letters stand for connected spaces with a chosen point, and
the arrows stand for continuous maps. If you're a physicist or geometer
you may prefer fiber bundles to "fibrations" - but luckily, they're so
similar we can ignore the difference in a vague discussion like this.
The idea is basically just that E maps onto B, and sitting over each point
of B we have a copy of F. We call B the "base space", E the "total space"
and F the "fiber".
If we want to classify such fibrations we can consider carrying the fiber
F around a loop in B and see how it twists around. For example, if all our
spaces are smooth manifolds, we can pick a connection on the total space
E and see what parallel transport around a loop in the base space B does
to points in the fiber F. This gives a kind of homomorphism
Omega(B) -> Aut(F)
sending loops in B to invertible maps from F to itself. And, the cool
thing is: this homomorphism lets us classify the fibration!
Here I say "kind of homomorphism" since Omega(B), the space of loops in B
based at the chosen point, is only "kind of" a topological group: the
group laws only hold up to homotopy. But let's not worry about this
technicality - especially since I'm being vague about all sorts of other
equally important issues!
The reason I can get away with not worrying about these issues is that
I'm trying to explain a very robust powerful principle - one that can
easily survive a dose of vagueness that would kill a lesser idea. Namely,
if B is a connected space with a chosen basepoint,
FIBRATIONS OVER THE BASE SPACE B WITH FIBER F
ARE "THE SAME" AS
HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F.
This could be called "the basic principle of Galois theory", for reasons
explained in "week213". There I explained the special case where the
fiber is discrete. Then our fibration called a "covering space", and
the basic principle of Galois theory boils down to this:
COVERING SPACES OVER B WITH FIBER F
ARE "THE SAME" AS
HOMOMORPHISMS FROM THE FUNDAMENTAL GROUP OF B TO AUTOMORPHISMS OF F.
Okay. Now let's use the same principle to classify extensions of a group
B by a group F:
1 -> F -> E -> B -> 1
The group B here acts like "loops in the base". But what acts like
"automorphisms of the fiber"?
You might guess it's the group of automorphisms of F. But, it's
actually the *2-group* of automorphisms of F!
A 2-group is a categorified version of a group where all the usual group
laws hold up to natural isomorphism. They play a role in higher gauge
theory like that of groups in ordinary gauge theory. In higher gauge
theory, parallel transport along a path is described by an *object* in
a 2-group, while parallel transport along a path-of-paths is described
by a *morphism*. In 2-form electromagnetism we use a very simple "abelian"
2-group, which has one object and either the real line or the circle as
morphism. But there are other more interesting "nonabelian" examples.
If you want to learn more about 2-form electromagnetism from this
perspective, try "week210". For 2-groups in general, try this paper:
9) John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-groups,
Theory and Applications of Categories 12 (2004), 423-491. Available online at
http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html or as math.QA/0307200.
Anyway: it turns out that any group F gives a 2-group AUT(F) where the
objects are automorphisms of F and the morphisms are "conjugations" -
elements of F acting to conjugate one automorphism and yield another.
And, extensions
1 -> F -> E -> B -> 1
are classified by homomorphisms
B -> AUT(F)
where we think of B as a 2-group with only identity morphisms. More
precisely:
EXTENSIONS OF THE GROUP B BY THE GROUP F
ARE "THE SAME" AS
HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F)
It's fun to work out the details, but it's probably not a good use of
our time together grinding through them here. So, I'll just sketch
how it works.
Starting with our extension
i p
1 --> F --> E --> B --> 1
we pick a "section"
s
E <-- B
meaning a function with
p(s(b)) = b
for all b in B. We can find a section because p is onto. However,
the section usually *isn't* a homomorphism.
Given the section s, we get a function
alpha: B -> Aut(F)
where Aut(F) is the group of automorphisms of F. Here's how:
alpha(b) f = s(b) f s(b)^{-1}
However, usually alpha *isn't* a homomorphism.
So far this seems a bit sad: functions between groups want to be
homomorphisms. But, we can measure how much s fails to be a homomorphism
using the function
g: B^2 -> F
given by
g(b,b') = s(bb') s(b')^{-1} s(b)^{-1}
Note that g = 1 iff s is a homomorphism.
We can then use this function g to save alpha. The sad thing about
alpha is that it's not a homomorphism... but the good thing is, it's
a homomorphism up to conjugation by g! In other words:
alpha(bb') f = g(b,b') [alpha(b) alpha(b') f] g(b,b')^{-1}
Taken together, alpha and g satisfy some equations ("cocycle conditions")
which say precisely that they form a homomorphism from B to the 2-group
AUT(F). Conversely, any such homomorphism gives an extension of B by F.
In fact, isomorphism classes of extensions of B by F correspond
in a 1-1 way with isomorphism classes of homorphisms from B to AUT(F).
So, we've classified these extensions!
In fact, something even better is true! It's evil to "decategorify" by
taking isomorphism classes as we did in the previous paragraph. To avoid
this, we can form a groupoid whose objects are extensions of B by F, and a
groupoid whose objects are homomorphisms B -> AUT(F). I'm pretty sure
that if you form these groupoids in the obvious way, they're equivalent.
And that's what this slogan really means:
EXTENSIONS OF THE GROUP B BY THE GROUP F
ARE "THE SAME" AS
HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F)
Next, let me say how Schreier theory reduces to more familiar ideas in two
special cases.
People have thought a lot about the special case where F is abelian and
lies in the center of E. These are called "central extensions". This
is just the case where alpha = 1. The set of isomorphism classes of
central extensions is called H^2(B,F) - the "second cohomology" of B
with coefficients in F.
People have also thought about "abelian extensions". That's an even
more special case where all three groups are abelian. The set of
isomorphism classes of such extensions is called Ext(B,F).
Since we don't make any simplifying assumptions like this in Schreier
theory, it's part of a subject called "nonabelian cohomology". It was
actually worked out by Dedecker in the 1960's, based on much earlier
work by Schreier:
10) O. Schreier, Ueber die Erweiterung von Gruppen I, Monatschefte fur
Mathematik and Physick 34 (1926), 165-180. Ueber die Erweiterung von
Gruppen II, Abh. Math. Sem. Hamburg 4 (1926), 321-346.
11) P. Dedecker, Les foncteuers Ext_Pi, H^2_Pi and H^2_Pi non abeliens,
C. R. Acad. Sci. Paris 258 (1964), 4891-4895.
More recently, Schreier theory was pushed one step up the categorical
ladder by Larry Breen. As far as I can tell, he essentially classified
the extensions of a 2-group B by a 2-group F in terms of homomorphisms
B -> AUT(F), where AUT(F) is the *3-group* of automorphisms of F:
12) Lawrence Breen, Theorie de Schreier superieure, Ann. Sci. Ecole Norm.
Sup. 25 (1992), 465-514. Also available at
http://www.numdam.org/numdam-bin/feuilleter?id=ASENS_1992_4_25_5
We can keep pushing Schreier theory upwards like this, but we can also
expand it "sideways" by replacing groups with groupoids. You should have
been annoyed by how I kept assuming my topological spaces were connected
and equipped with a specified point. I did this to make them analogous
to groups. For example, it's only spaces like this for which the fundamental
group is sufficiently powerful to classify covering spaces. For more general
spaces, we should use the fundamental *groupoid* instead of the fundamental
group. And, we can set up a Schreier theory for extensions of groupoids:
13) V. Blanco, M. Bullejos and E. Faro, Categorical non abelian cohomology,
and the Schreier theory of groupoids, available as math.CT/0410202.
In fact, these authors note that Grothendieck did something similar
back in 1971: he classified *all* groupoids fibered over a groupoid
B in terms of weak 2-functors from B to Gpd, which is the 2-groupoid of
groupoids! The point here is that Gpd contains AUT(F) for any fixed
groupoid F:
14) Alexander Grothendieck, Categories fibrees et descente (SGA I),
Lecture Notes in Mathematics 224, Springer, Berlin, 1971.
Having extended the idea "sideways" like this, one can then continue
marching "upwards". I don't know how much work has been done on this,
but the slogan should be something like this:
n-GROUPOIDS FIBERED OVER AN n-GROUPOID B
ARE "THE SAME" AS
WEAK (n+1)-FUNCTORS FROM B TO THE (n+1)-GROUPOID nGpd
Grothendieck also studied this kind of thing with categories replacing
groupoids, so there should also be an n-category version, I think...
but it's more delicate to define "fibrations" for categories than
for groupoids, so I'm a bit scared to state a slogan suitable for
n-categories.
However, I'm not scared to go from n-groupoids to omega-groupoids, which
are basically the same as spaces. In terms of spaces, the slogan goes
like this:
SPACES FIBERED OVER THE SPACE B
ARE "THE SAME" AS
MAPS FROM B TO THE SPACE OF ALL SPACES
This is how James Dolan taught it to me. Most mortals are scared of "the
space of all spaces" - both for fear of Russell's paradox, and because we
really need a *space* of all spaces, not just a mere set of them. To avoid
these terrors, you can water down Jim's slogan by choosing a specific space
F to be the fiber:
FIBRATIONS WITH FIBER F OVER THE SPACE B
ARE "THE SAME" AS
MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F)
where AUT(F) is the topological group of homotopy self-equivalences of F.
The fearsome "space of all spaces" is then the disjoint union of the
classifying spaces of all these topological groups AUT(F). It's too large
to be a space unless you pass to a larger universe of sets, but otherwise
it's perfectly fine. Grothendieck invented the notion of a "Grothendieck
universe" for precisely this purpose:
14) Wikipedia, Grothendieck universe,
http://en.wikipedia.org/wiki/Grothendieck_universe
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
.
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| User: "Dr Tim" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
16 Nov 2005 03:23:09 PM |
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"The gauge invariance will look like
Wmn -> Wmn + Bn,m - Bm,n
so you now have the problem of forming a meaningful covariant
derivative without sacrificing isotropy"
Why is this a problem?
.
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| User: "" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
30 Nov 2005 02:44:12 AM |
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Gerard Westendorp wrote:
(snip)
The exterior derivative (d) can take on the role of
grad, curl and div, depending on what it operates on.
I made the following diagram (valid for 3D only):
scalar <----------> 3-form
| ^
(grad) |
| (div)
V |
1-vector <----------> 2-form
| ^
(curl) |
| (curl)
V |
2-vector <----------> 1-form
| ^
(div) |
| (grad)
V |
3-vector <----------> 0-form
(snip)
OK, that's a keeper, Gerard. Print and save. Nice!
Now, In GA, 2xy is a 2-vector, right?
5xyz is a 3-vector?
There can be 1 basis for a scalar, 3 bases for 1- or 2- vectors, and 1
basis for a 3-vector.
Let me know if I got that wrong. Or if I am not even wrong anywhere
here.
(a^2 + b^2 + c^2) is a nice, simple, scalar field. I thinkt is. It has
been 27 years since I did this last...I am using a,b, and c as
coordinates. x,y, and z as unit vectors. In vector calc, back in the
day, we'd write x-hat, etc for the unit vectors. x-hat = (0,0,1)
So you can take the grad of a scalar field and get a vector field, but
if you try to take the grad of a scalar number, I think that's 0 or
undefined. I can easily visualize the above field and its grad.
In GA, however, 4xy+2x+3 is a multivector, and is not a suitable
argument for the external derviative. Right? (I'm really trying here)
(4xy + 2x + 3 ) * (a^2 + b^2 + c*2) is a multivector field. I'm pretty
sure of that. And so, it is a suitable argument for the ED. Yes?
I'll have to post later after I compute the dual of that multivector.
The vertical arrows are all instance of the exterior
derivative. The horizontal arrows represent a dualty relation
between an n-vector and a (3-n) form.
I understand dual graphs. You explain duals below.
[Note: Applying 'd' 2 times in succesion always gives zero.
So for example curl(grad() )= 0. So a 2-vector that is
the curl of a 1-vector is only non-zero if the 1-vector
is not the gradient of a scalar]
I follow the logic, and would need to memorize all the cases to become
proficient.
A form is really just the gadget that complements a vector
to get a "number". Forms seem confusingly similar to vectors,
but it is good to understand that they are different. It is
a bit like the difference between voltage gradient and current:
They are related through Ohm's law, but combining currents
is fundamentally different from combining voltage gradients.
Voltage gradients are vectors, and they are the gradient
of a scalar. Current(densities) are 2-forms. The 'div' of
this 2-form is the net increase-rate of charge density. And
Charge density increase-rate is the dual of Voltage,
together they multiply to form power.
Ahh. I remember a simulation I did for my ultracapacitor bicycle...it
used a solver.
(See my web page for more on this,
http://www.xs4all.nl/~westy31/Electric.html
I also describe a lattice version of geometrical algebra
called "cell algebra". Hope I find time to work on it...)
One way to look at it, which seems to give a reasonably
consistent picture, is to say that to go from a n-vector
to a (3-n) form, multiply by the Clifford term e_xyz.
I'm sorry, I don't get that. Is that e as in e^x, the base of the
natural logarithm, and what's that underscore? This seems like an
important point, so I'd like to get it right.
I do understand multiplying by exyz or as one might write e*xyz;
this is an operation I can do with the calculator at elf.org, which
runs
on my Pocker PC Phone under a third party browser.
Or on paper. Ugh. Sitzfleisch required. I have some...
For vectors, the exterior derivative is the part of the
Clifford derivative that *increases* the grade of the
multivectors.
But for forms, the net effect is to *decrease* the grade
of the multivector.
I'd pick at that and say for vector fields, the ED increases the grade
of the vector field, while maintaining the vector nature of the field.
Am I close?
The problem is, Clifford algebra does not really seem to
care if something is a form or a vector.
No, it doesn't, but a Clifford calculator could easily display both a
Clifford number and its dual for every result and flag vectors and
forms to one's attention.
Let's leave 4-dimensional GA unexplored for now!
Looking forward to taking vector calculus in the Spring.
The Dougster
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| User: "Shmuel Seymour J. Metz" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
27 Nov 2005 04:50:36 PM |
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In <1132868781.501801.303280@g49g2000cwa.googlegroups.com>, on
11/26/2005
at 12:19 PM, said:
so it seems very natural to interrupt at this point and ask:
"Does dA = A or perhaps dF=F define anything using the rules
above?"
No, because dA is of higher degree than A, and hence cannot equal it.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to
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| User: "Ralph Hartley" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
28 Nov 2005 02:03:06 PM |
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Shmuel (Seymour J.) Metz wrote:
DGoncz@aol.com said:
so it seems very natural to interrupt at this point and ask:
"Does dA = A or perhaps dF=F define anything using the rules
above?"
No, because dA is of higher degree than A, and hence cannot equal it.
Unless they are both zero.
Ralph Hartley
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| User: "Robert Low" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
29 Nov 2005 03:24:54 PM |
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Ralph Hartley wrote:
Shmuel (Seymour J.) Metz wrote:
DGoncz@aol.com said:
"Does dA = A or perhaps dF=F define anything using the rules
above?"
No, because dA is of higher degree than A, and hence cannot equal it.
Unless they are both zero.
They're different zeroes, though. The zero scalar is not
equal to the zero one-form, even though it is common
to use the same symbol for both.
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| User: "Shmuel Seymour J. Metz" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
02 Dec 2005 12:47:24 AM |
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In <1133269325.923060.13170@g43g2000cwa.googlegroups.com>, on
11/30/2005
at 08:44 AM, said:
I'm sorry, I don't get that. Is that e as in e^x, the base of the
natural logarithm, and what's that underscore?
No; he's talking about an antisymmetric 3-tensor with e_123 = 1, and
the underscore indicates that x, y and z are subscripts rather than
superscripts.
BTW, his notation is confusing; 2-form and 2-vectors are the same[1].
In an oriented n-dimensional vector space with a metric, there's a
natural equivalence between m-forms and (n-m)-forms. For n=3, that
means that there's an equivalence between 2-forms and vectors.
[1] There are some technicalities connected with the difference
between a vector space and its dual, or, equivalently, between
covariant and contravariant.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to
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| User: "" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
15 Nov 2005 05:59:52 PM |
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John Baez skrev:
And, extensions
1 -> F -> E -> B -> 1
...
People have thought a lot about the special case where F is abelian and
lies in the center of E. These are called "central extensions". This
is just the case where alpha = 1. The set of isomorphism classes of
central extensions is called H^2(B,F) - the "second cohomology" of B
with coefficients in F.
People have also thought about "abelian extensions". That's an even
more special case where all three groups are abelian. The set of
isomorphism classes of such extensions is called Ext(B,F).
I am pretty sure that is not standard terminology. In an abelian
extension, only F is required to be abelian, not E and B. This makes
sense, because we don't require E and B to commute with everything
if the extension is central.
To lowest order, only abelian extensions matter, at least for the
associated Lie algebra extension
0 -> Lie F -> Lie E -> Lie B -> 0.
If we denote the generators of B by x,y,z,w
and the bracket in B by [x,y]_0, the brackets in E are
[x,y] = [x,y]_0 + h c(x,y) + O(h^2)
[x, h c(y,z)] = h d(x,y,z) + O(h^2)
[h c(x,y), h c(z,w)] = O(h^2)
Thus, to order O(h) we can set all non-abelian terms to zero. We
cannot in general assume that the extension is central to O(h),
though. E.g., the Moyal Lie algebra is a non-abelian deformation of
the algebra of Hamiltonian vector fields. To lowest order, it
defines an abelian extension.
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| User: "John Baez" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
19 Nov 2005 08:49:17 AM |
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In article <1132065631.635708.191820@g14g2000cwa.googlegroups.com>,
<thomas_larsson_01@hotmail.com> wrote:
John Baez wrote:
And, extensions
1 -> F -> E -> B -> 1
....
People have also thought about "abelian extensions". That's an even
more special case where all three groups are abelian. The set of
isomorphism classes of such extensions is called Ext(B,F).
I am pretty sure that is not standard terminology. In an abelian
extension, only F is required to be abelian, not E and B.
Hmm. Maybe you're right. But then I'm not sure what the standard
terminology *is* for the kind of extension where all three groups
are abelian.
There should be *some* name for them, since they're wildly popular
in homological algebra, which is a kind of algebra used by topologists.
When kids take an introductory course in homology and cohomology, they
typically learn a bit of this - mainly stuff about "Tor" and "Ext".
Ext(B,F) is the set - actually an abelian group - of isomorphism
classes of extensions
1 -> F -> E -> B -> 1
where F, E and B are all abelian. But, when I pulled down Rotman's book
"An Introduction to Homological Algebra" and tried to see what he
called these extensions, I found he just called them "extensions" -
because in that chapter, he's not considering any other kind!
Well, actually he's considering short exact sequences of R-modules
for any ring R. When R = Z such R-modules are abelian groups - so
nonabelian groups don't even get in the door, in this chapter. There's
lots of nice machinery that works only in this "completely abelian"
case.
My excitement with Schreier theory came from seeing more clearly
than before the nice machinery that works when you're trying to
classify extensions without assuming ANY abelianness. It's
less familiar, because to see how nice it is, you need to understand
2-groups.
Btw, I put a more detailed explanation of this stuff in the "addendum"
to week223 here:
http://math.ucr.edu/home/baez/week223.html
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| User: "" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
23 Nov 2005 04:07:18 AM |
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John Baez skrev:
I am pretty sure that is not standard terminology. In an abelian
extension, only F is required to be abelian, not E and B.
classes of extensions
1 -> F -> E -> B -> 1
where F, E and B are all abelian. But, when I pulled down Rotman's book
"An Introduction to Homological Algebra" and tried to see what he
called these extensions, I found he just called them "extensions" -
because in that chapter, he's not considering any other kind!
After I wrote that, I pulled down Pressley-Segal's book Loop Groups,
to see what they call the Mickelsson-Faddeev extension in chapter 4.
I was pretty sure that they did call it "abelian extension", but they
used the term "extension by an abelian ideal". I am positive that
Mickelsson uses the term "abelian extension" in my way, though,
since I first saw the term in one of his papers. I may ask him if I see
him.
I once invented the term "abelian charge" for the c-number parameter
multiplying the suitably normed extension. It felt silly to talk about
central charges in the case were the extension is not central.
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| User: "Ken S. Tucker" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
23 Nov 2005 02:41:18 PM |
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John Baez wrote:
....
Now... on to p-form electromagnetism!
In ordinary electromagnetism, the secret star of the show turns out
to be not the electromagnetic field but the "vector potential", A.
At least locally, we can think of this as a 1-form. A 1-form is just
a gadget that you can integrate along a path and get a number. In the
case of the vector potential, this number describes the change in
phase that a charged particle acquires as it moves along this path.
The 1-form A gives rise to a 2-form F called the "electromagnetic field".
A 2-form is a gadget you can integrate over a surface and get a
number. Here's how we get F from A. Suppose we move a charged particle
around a loop that's the boundary of some surface. Then the integral
of F over this surface is defined to be the integral of A around the loop!
We summarize this by saying that F is the "exterior derivative" of A,
and writing
F = dA.
F is called the electromagnetic field because... that's what it is!
It contains both the electric and magnetic fields in a single neat
package. In 4d spacetime, the magnetic field describes the change in
a phase of a charged particle that loops around a surface in the
xy, yz or zx planes. The electric field describes the change in phase
of a charged particle that loops around a surface in the xt, yt or zt
planes.
Hi John and all,
I think one can find an interesting asymmetry without
recourse to EM at a more basic GR level. Pardon the
ascii tensors, I'll try to be brief...
1 = g_uv U^u U^ = U^u U_u
i.e. U^u being contravariant 4-velocity...
From that I deduce, by covariant derivative wrt "dx^w",
U_u;w U^u = 0 (1)
Now I wish to compare that to the classical GR geodesic,
U_w;u U^u (2)
and please note the subtle difference.
A well known technique is to set a tensor to a sum of
Symmetric and Asymmetric tensors like,
U_u;w U^u = S_uw U^u + A_uw U^u (1a)
and
U_w;u U^u = S_wu U^u + A_wu U^u (2a).
Given S_wu = S_uw we can subtract
(1a) - (2a) = 0 + A_uw U^u - A_wu U^u (3)
where A_uw U^u = - A_wu A^u,
and (3) becomes, 0 + 2*A_uw U^u .
So we find the classical geodesic (2) is equal to,
U_w;u U^u = 2*A_uw U^u , (4)
and is "purely" relativistic, as is magnetism, but
magnetism uses charges to "physically" measure
that effect, every day.
We can re-write the LHS of (4) as the "absolute
derivative" of 4-velocity, (absolute acceleration),
DU_w /ds = U_w;u U^u = 2*A_uw U^u ,
and then respect the postulate of the General Theory
of Relativity where absolute acceleration vanishes to
provide two distinct geodesical forms, the familiar
U_w;u U^u = 0
and the less familiar (I term the quantum geodesic),
2*A_uw U^u =0 , (5)
each in accord with GR.
Putting (5) into a Lorentz force form like,
f_w = qF_uw U^u = 0 (6)
when examined, produces stable orbitals, when (5) is
measured using charged particles in the atom so that
(6) may be regarded as a "quantum geodesic" in GR.
The above is in accord with classical GR and the normal
application of tensor logic.
Regards
Ken S. Tucker
kxsxt
....
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| User: "Ken S. Tucker" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
25 Nov 2005 03:34:15 AM |
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Awaiting comments here...
Ken S. Tucker wrote:
John Baez wrote:
...
Now... on to p-form electromagnetism!
In ordinary electromagnetism, the secret star of the show turns out
to be not the electromagnetic field but the "vector potential", A.
At least locally, we can think of this as a 1-form. A 1-form is just
a gadget that you can integrate along a path and get a number. In the
case of the vector potential, this number describes the change in
phase that a charged particle acquires as it moves along this path.
The 1-form A gives rise to a 2-form F called the "electromagnetic field".
A 2-form is a gadget you can integrate over a surface and get a
number. Here's how we get F from A. Suppose we move a charged particle
around a loop that's the boundary of some surface. Then the integral
of F over this surface is defined to be the integral of A around the loop!
We summarize this by saying that F is the "exterior derivative" of A,
and writing
F = dA.
F is called the electromagnetic field because... that's what it is!
It contains both the electric and magnetic fields in a single neat
package. In 4d spacetime, the magnetic field describes the change in
a phase of a charged particle that loops around a surface in the
xy, yz or zx planes. The electric field describes the change in phase
of a charged particle that loops around a surface in the xt, yt or zt
planes.
Hi John and all,
I think one can find an interesting asymmetry without
recourse to EM at a more basic GR level. Pardon the
ascii tensors, I'll try to be brief...
1 = g_uv U^u U^ = U^u U_u
i.e. U^u being contravariant 4-velocity...
From that I deduce, by covariant derivative wrt "dx^w",
U_u;w U^u = 0 (1)
Now I wish to compare that to the classical GR geodesic,
U_w;u U^u (2)
and please note the subtle difference.
A well known technique is to set a tensor to a sum of
Symmetric and Asymmetric tensors like,
U_u;w U^u = S_uw U^u + A_uw U^u (1a)
and
U_w;u U^u = S_wu U^u + A_wu U^u (2a).
Given S_wu = S_uw we can subtract
(1a) - (2a) = 0 + A_uw U^u - A_wu U^u (3)
where A_uw U^u = - A_wu A^u,
and (3) becomes, 0 + 2*A_uw U^u .
So we find the classical geodesic (2) is equal to,
U_w;u U^u = 2*A_uw U^u , (4)
and is "purely" relativistic, as is magnetism, but
magnetism uses charges to "physically" measure
that effect, every day.
We can re-write the LHS of (4) as the "absolute
derivative" of 4-velocity, (absolute acceleration),
DU_w /ds = U_w;u U^u = 2*A_uw U^u ,
and then respect the postulate of the General Theory
of Relativity where absolute acceleration vanishes to
provide two distinct geodesical forms, the familiar
U_w;u U^u = 0
and the less familiar (I term the quantum geodesic),
2*A_uw U^u =0 , (5)
each in accord with GR.
Putting (5) into a Lorentz force form like,
f_w = qF_uw U^u = 0 (6)
when examined, produces stable orbitals, when (5) is
measured using charged particles in the atom so that
(6) may be regarded as a "quantum geodesic" in GR.
The above is in accord with classical GR and the normal
application of tensor logic.
Regards
Ken S. Tucker
kxsxt
...
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| User: "Androcles" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
25 Nov 2005 06:01:13 AM |
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"Ken S. Tucker" <dynamics@vianet.on.ca> wrote in message
news:1132911255.522755.287860@z14g2000cwz.googlegroups.com...
Awaiting comments here...
Ok, if you insist.
It's load of old crap.
Androcles.
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| User: "" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
26 Nov 2005 06:19:00 AM |
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John Baez wrote:
Also available as http://math.ucr.edu/home/baez/week223.html
November 14, 2005
This Week's Finds in Mathematical Physics - Week 223
John Baez
This week I'd like to talk about two aspects of higher gauge theory:
p-form electromagnetism and nonabelian cohomology. Lurking behind both
of these is the mathematics of n-categories, but I'll do my best to hide
that until the end, to build up the suspense.
(snip)
OK. Read on, MacDuff!
Now... on to p-form electromagnetism!
In ordinary electromagnetism, the secret star of the show turns out
to be not the electromagnetic field but the "vector potential", A.
At least locally, we can think of this as a 1-form. A 1-form is just
a gadget that you can integrate along a path and get a number. In the
case of the vector potential, this number describes the change in
phase that a charged particle acquires as it moves along this path.
Gadget, path, number. I follow.
The 1-form A gives rise to a 2-form F called the "electromagnetic field".
A 2-form is a gadget you can integrate over a surface and get a
number. Here's how we get F from A. Suppose we move a charged particle
around a loop that's the boundary of some surface. Then the integral
of F over this surface is defined to be the integral of A around the loop!
We summarize this by saying that F is the "exterior derivative" of A,
and writing
F = dA.
Gadget, surface, number. OK, I can see where this is going.
F is called the electromagnetic field because... that's what it is!
(snip)
If you don't know this stuff, you're missing some of the best fun
life has to offer. For an easy introduction with lots of gorgeous
pictures, see:
6) Derek Wise, Electricity, magnetism and hypercubes, available at
http://math.ucr.edu/~derek/talks/050916bw.pdf
You're right. This is fun. Whee!
The idea of p-form electromagnetism is to replace point particles
by strings or higher-dimensional membranes. To see how this goes,
it's enough to look at 2-form electromagnetism.
(snip)
So, we're just adding one to the dimensions of things. This makes it
easy to keep on going. In fact, for any integer p, we can write down
a generalization of Maxwell's equations.
It goes like this. We start with a p-form A. We define a (p+1)-form
F = dA
This automatically implies some of Maxwell's equations:
dF = 0
but the nontrivial Maxwell equations say that
*d*F = J
In a Clifford algebra, this looks equally concise, and carries equal
power.
where * is the Hodge star operator and J is a p-form called the "current",
which is produced by charged matter.
What does this mean, physically? The idea is that we have charged matter
consisting of (p-1)-dimensional membranes. These trace out p-dimensional
surfaces in spacetime as time passes. The current J is a p-form that's
concentrated on these surfaces. The current affects the A field in a
manner governed by Maxwell's equations. Conversely, the A field affects
the motion of the membranes.
Hmm. I'd like to read more on this coupling.
Classically, we just integrate the A field
over the surface traced out by a membrane and add the result to the
*action* for the membrane. In the path integral approach to quantum
mechanics, this number gives a change in phase, as already mentioned.
Oops. (p-1) dimensional hyperstrings tracing p dimensional
hypersurfaces
in (p+1) dimensional hyperspaces is what I get, but
I guess membrane has been defined elsewhere. So here I start to
wander....here we go.
Gadget, paths, surfaces, volumes, integrals, and phhht, out comes a
number. OK.
I learned in elementary calculus that a definition of e is
d(f(x)) = f(x) ==> f(x) = e^x
and that this is a fine definition of d as well.
so it seems very natural to interrupt at this point and ask:
"Does dA = A or perhaps dF=F define anything using the rules above?"
Maybe some kind of dimensional e, or maybe
the cuff of my sleeve? Maybe a hyperpotato? Or just d?
Heck, I don't know!
Let's say, for simplification's sake, that
dA=A ==> A=A(e,something)
Does that take us anywhere? What would "something" be?
Thanks in advance, readers of spr, sp, and sm.
Yours,
Doug Goncz
Replikon Reseach (Where some problems solve themselves)
Falls Church, VA 22044-0394
DGoncz at aol dot com email, if you get my meaning, wink, wink, nudge,
nudge....
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| User: "" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
27 Nov 2005 02:02:03 PM |
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wrote:
In a Clifford algebra, this looks equally concise, and carries equal power.
Talking about Clifford algebra, I'd like to gauge Baez's opinion, or
that of anyone else qualified to judge, if they're reading this, on the
merits or otherwise of geometric algebra, which claims to subsume vector
algebra, Clifford algebra, tensor algebra, and Hopf algebra, among other
things.
(It's promoters use phrases such as a "tower of Babel" and an "appalling
amount of wheel reinventing" of related linear algebra formalisms over
the 20th century.)
Geometric algebra was developed from Clifford algebra and Grassman
algebra, and seems to me a neat and compact formalism. The only
potential snags, which may have hampered its widespead adoption by
physicists, are that its proponents have used it to develop a variant of
GR based on gauges in a Euclidean spacetime. Also, the more advanced
aspects may not have even been finalized. (For example I gather a famous
mathematician called Gian Carlo Rota was working on what he felt was an
essential revamp of it in the 1990s shortly before he unfortunately
died.)
N.B. Geometric algebra shouldn't be confused with (modern) algebraic
geometry, which is the theory of schemes developed by Grothendieck.
A few links:
http://www.mrao.cam.ac.uk/%7Ecjld1/pages/book.htm
Geometric Algebra for Physicists
Chris Doran & Anthony Lasenby
CUP
http://www.mrao.cam.ac.uk/~clifford
Geometric algebra research group
http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/chris_thesis.html
C J. L. Doran.
Geometric Algebra and its Application to Mathematical Physics
Ph.D. thesis, University of Cambridge (1994).
Cheers
John R Ramsden (jhnrmsdn@yahoo.com.uk)
* Remove m from com to reply
* From address is defunct
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| User: "" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
28 Nov 2005 02:03:06 PM |
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Thanks to Greg Egan and Shmuel (Seymour J.) Metz for confirming in
their Nov 27 posts that
dA=A has no meaning.
Now to go back to:
John Baez provided:
6) Derek Wise, Electricity, magnetism and hypercubes, available at
http://math.ucr.edu/~derek/talks/050916bw.pdf
and really *study*!
For discussion, p can have *any* value?
Or are 0 and 1 excluded, or something like that?
The Dougster
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| User: "Gerard Westendorp" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
29 Nov 2005 02:43:53 AM |
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wrote:
Thanks to Greg Egan and Shmuel (Seymour J.) Metz for confirming in
their Nov 27 posts that
dA=A has no meaning.
Now to go back to:
John Baez provided:
6) Derek Wise, Electricity, magnetism and hypercubes, available at
http://math.ucr.edu/~derek/talks/050916bw.pdf
and really *study*!
I'll try to sketch a relationship between differential forms,
with the exterior derivative "d", and Clifford algebra, which
you seem to like.
I am not renowned for my mathematical rigor, but I'll
just explain how I understand this, and maybe someone will
correct me if necessary.
A good thing about differential forms is that they are
reveal coordinate independent relations, such as
F = dA.
A good thing about Clifford algebra is that is is easy to
learn, and it works.
So it would be nice if we could understand how they are related.
The exterior derivative (d) can take on the role of
grad, curl and div, depending on what it operates on.
I made the following diagram (valid for 3D only):
scalar <----------> 3-form
| ^
(grad) |
| (div)
V |
1-vector <----------> 2-form
| ^
(curl) |
| (curl)
V |
2-vector <----------> 1-form
| ^
(div) |
| (grad)
V |
3-vector <----------> 0-form
The vertical arrows are all instance of the exterior
derivative. The horizontal arrows represent a dualty relation
between an n-vector and a (3-n) form.
[Note: Applying 'd' 2 times in succesion always gives zero.
So for example curl(grad() )= 0. So a 2-vector that is
the curl of a 1-vector is only non-zero if the 1-vector
is not the gradient of a scalar]
A form is really just the gadget that complements a vector
to get a "number". Forms seem confusingly similar to vectors,
but it is good to understand that they are different. It is
a bit like the difference between voltage gradient and current:
They are related through Ohm's law, but combining currents
is fundamentally different from combining voltage gradients.
Voltage gradients are vectors, and they are the gradient
of a scalar. Current(densities) are 2-forms. The 'div' of
this 2-form is the net increase-rate of charge density. And
Charge density increase-rate is the dual of Voltage,
together they multiply to form power.
(See my web page for more on this,
http://www.xs4all.nl/~westy31/Electric.html
I also describe a lattice version of geometrical algebra
called "cell algebra". Hope I find time to work on it...)
One way to look at it, which seems to give a reasonably
consistent picture, is to say that to go from a n-vector
to a (3-n) form, multiply by the Clifford term e_xyz.
For vectors, the exterior derivative is the part of the
Clifford derivative that *increases* the grade of the
multivectors.
But for forms, the net effect is to *decrease* the grade
of the multivector.
The problem is, Clifford algebra does not really seem to
care if something is a form or a vector.
Gerard
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| User: "Bossavit" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
29 Nov 2005 03:24:56 PM |
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G. Westendorp:
One way to look at it [the relation between forms and vectors], which
seems to give a reasonably consistent picture, is to say that to go
from a n-vector to a (3-n) form, multiply by the Clifford term e_xyz.
This is correct, but why force Clifford algebras (and so, by
implication, a metric structure) into this picture when much weaker
structure will do it? All we need to establish this p-vector to (n -
p)-form correspondence is a *volume* in n-space, that is, a
distinguished n-form. (Alternatively, a reference n-simplex does the
trick just as well.) Since n-forms make a one-dimensional space, this
is much less information than specifying what it takes to build the
Clifford algebra.
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| User: "Han de Bruijn" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
29 Nov 2005 05:48:59 AM |
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Gerard Westendorp wrote:
(See my web page for more on this,
http://www.xs4all.nl/~westy31/Electric.html
The electrical analogue is certainly useful for heat conduction:
http://www.xs4all.nl/~westy31/Electric.html#Heat
Here is some of my (numerical) work, as it is related to the above:
http://hdebruijn.soo.dto.tudelft.nl/jaar2004/purified.pdf
Especially read the subsection "2-D Resistor Model". Also done in 3-D:
http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/belgisch.pdf
Han de Bruijn
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| User: "Han de Bruijn" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
30 Nov 2005 02:44:00 AM |
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Gerard Westendorp wrote:
(See my web page for more on this,
http://www.xs4all.nl/~westy31/Electric.html
The electrical analogue is certainly useful for heat conduction:
http://www.xs4all.nl/~westy31/Electric.html#Heat
Here is some of my (numerical) work, as it is related to the above:
http://hdebruijn.soo.dto.tudelft.nl/jaar2004/purified.pdf
Especially read the subsection "2-D Resistor Model". Also done in 3-D:
http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/belgisch.pdf
Han de Bruijn
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| User: "DRLunsford" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
15 Nov 2005 02:21:29 AM |
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John Baez wrote:
This week I'd like to talk about two aspects of higher gauge theory:
p-form electromagnetism and nonabelian cohomology. Lurking behind both
of these is the mathematics of n-categories, but I'll do my best to hide
that until the end, to build up the suspense.
The gauge invariance will look like
Wmn -> Wmn + Bn,m - Bm,n
so you now have the problem of forming a meaningful covariant
derivative without sacrificing isotropy* - in fact this seems like
evidence that the string idea is questionable as physics.
An interaction will be
Wmn Jmn
or is it
*Wmn Jmn
or .. i.e. you're immediately in problems just from algebra (which
complexification do I pick?)
-drl
*If one had a special vector field X then you could form Xm dn - Xn dm
+ k Wmn. Presumably in the string world that direction would be along
the string itself, but now there's a horrible constraint, ugh.
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| User: "Gerard Westendorp" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
16 Nov 2005 11:34:49 PM |
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John Baez wrote:
[..]
The idea of p-form electromagnetism is to replace point particles
by strings or higher-dimensional membranes. To see how this goes,
it's enough to look at 2-form electromagnetism.
In 2-form electromagnetism, the star of the show is a 2-form, A.
As already mentioned, a 2-form is a gadget you can integrate over a
surface and get a number. In 2-form electromagnetism, this number
describes the change in phase that a charged string acquires as it
moves along, tracing out a surface in spacetime.
Could string theory really be that easy?
My understanding till now has been:
scalar field
<-> spin 0 particles
<-> potential (phi) defined on vertices
<-> field defined on edges, d(phi)
vector field
<-> spin 1 particles
<-> potential (A) defined on edges
<-> field (F) defined on faces (loops of edges), F= dA
The next step up, in which the potential is defined on faces,
and the field is defined on "solids", ( "loops" of faces), did
seem natural to me, but I thought this would just be a field of
spin 2 particle wave functions.
[..]
but the nontrivial Maxwell equations say that
*d*F = J
where * is the Hodge star operator and J is a p-form called the "current",
which is produced by charged matter.
What does this mean, physically? The idea is that we have charged matter
consisting of (p-1)-dimensional membranes.
Does this mean that field of (p-1)-dimensional membranes are just spin
p wave functions? Or has my associating spin been based on superstition...
I admit I don't really understand the relationship between spin and
"grade", ie. scalar=spin0=vertex, vector=spin1=edge, etc. Except that
of course the information content is roughly right. A spin 0 field has
only one number per chunk of space-time, so it can only describe a
scalar field.
A spin 1 particle has 3 complex components, but apparently
it can not only describe a 3-vector field, but also a 4-vector.
(I need some time to think about this)
A spin 2 particle wave function would have 5 complex components,
and in 3+1 D space, there are 6 components of the 2-form
(xy, yz, zz, xt, yt, zt).
I don't see the logic yet, but on the other hand, there must
be 2-form fields, and there must be spin 2 waves, they seem to
have little choice but to be related.
Gerard
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| User: "John Baez" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
17 Nov 2005 06:15:03 PM |
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In article <437BBA7A.5020804@xs4all.nl>,
Gerard Westendorp <westy31@xs4all.nl> wrote:
John Baez wrote:
The idea of p-form electromagnetism is to replace point particles
by strings or higher-dimensional membranes. To see how this goes,
it's enough to look at 2-form electromagnetism.
In 2-form electromagnetism, the star of the show is a 2-form, A.
As already mentioned, a 2-form is a gadget you can integrate over a
surface and get a number. In 2-form electromagnetism, this number
describes the change in phase that a charged string acquires as it
moves along, tracing out a surface in spacetime.
Could string theory really be that easy?
No. But, the *bosonic* string is quite easy.
I've described the action that a string gets by interacting with
the 2-form analogue of the electromagnetic field - string theorists
call this the B field, or sometimes the "Kalb-Ramond" field. There's
also another term in the action, which is just proportional to the
*area* of the string worldsheet. So, for a string of tension m and
charge q,
string action = m (area of worldsheet)
+ q (integral of B field over worldsheet)
which is just like the action for a point particle in general relativity:
particle action = m (length of worldline)
+ q (integral of A field over worldline)
where now m is the particle's mass.
The fun starts when you try to quantize this theory, and see that
nasty stuff happens except in 26 dimensions - and that even then,
the string has tachyonic vibrational modes.
And, the fun *really* gets going when you consider the supersymmetric
version of the string.
But, the basic idea is pretty simple. Anyone who wants to learn
this stuff should read Zweibach's book "A First Course in String Theory".
It's aimed at smart undergraduates. I'm not sure undergrads should be
spending time on a theory for which there's no evidence, but it's a
very good book.
My understanding till now has been:
scalar field
<-> spin 0 particles
<-> potential (phi) defined on vertices
<-> field defined on edges, d(phi)
vector field
<-> spin 1 particles
<-> potential (A) defined on edges
<-> field (F) defined on faces (loops of edges), F= dA
The next step up, in which the potential is defined on faces,
and the field is defined on "solids", ( "loops" of faces), did
seem natural to me, but I thought this would just be a field of
spin 2 particle wave functions.
This is the right story for "discretized p-form electromagnetism",
which you can read about here:
http://math.ucr.edu/~derek/pform/
You are talking about
0-form electromagnetism (the scalar field phi),
then
1-form electromagnetism (the vector potential A),
then
2-form electromagnetism (the Kalb-Ramond field B),
and so on...
and you're getting fooled into thinking there's a certain
pattern that's not really there.
In 4d spacetime a 0-form acts like a spin-0 particle since it
transforms in the spin-0 representation of the rotation group
(i.e. a scalar doesn't change when you rotate it).
and
In 4d spacetime a 1-form acts like a spin-1 particle since it
transforms in the spin-1 representation of the rotation group
(i.e. a 1-form transforms like a vector when you rotate it).
but you don't get spin-2 particles this way, since:
In 4d spacetime a 2-form acts like a spin-1 particle since it
transforms in the spin-1 representation of the rotation group
(i.e. a 2-form transforms like a vector when you rotate it).
So, gravity is very different than the B field. String theory
involves both. Certain string theorists love to proclaim that
"string theory predicts gravity", which is true in a sense -
though the word "postdict" is more appropriate. But, string
theory also predicts the B field... which you don't tend to hear
string theorists boast about, for some mysterious reason.
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| User: "magic math tricks" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
19 Nov 2005 08:50:45 AM |
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so how is maxwell's equations related to F=ma? what's the symmetry
relating electricity to gravity?
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| User: "Kwok Man Hui" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
19 Nov 2005 08:49:22 AM |
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On Tue, 15 Nov 2005, John Baez wrote:
Also available as http://math.ucr.edu/home/baez/week223.html
Maxwell's equations and their p-form generalization make sense when
spacetime is any Lorentzian manifold. However, to get a theory
where initial data determine a unique global solution, we want our
spacetime to be "globally hyperbolic", which means that it has a
"Cauchy surface": roughly, a spacelike surface that any sufficiently
long timelike curve hits precisely once. To get a good *quantum* theory
of p-form electromagnetism with a Hilbert space of states on which
time evolution acts as unitary operators, we need more: our spacetime
should be "stationary", meaning that it has time translation symmetry.
Otherwise there's no way to define energy and the vacuum state - which
is defined to be the least-energy state.
Does the time translation symmetry get rid of the dynamics? Constant time
evolution?
My student Miguel Carrion-Alvarez tackled an important special case
in his thesis, namely "static" globally hyperbolic spacetimes:
Yes, I think so.
7) Miguel Carrion-Alvarez, Loop quantization versus Fock quantization
of p-form electromagnetism on static spacetimes, available as
math-ph/0412032.
As you can guess, either from seeing all the "d" operators or seeing all
the buzzwords I'm throwing around, p-form electromagnetism is really just
cohomology incarnated as physics! My student Derek Wise made this very
precise for a version of the theory where spacetime is *discrete* -
so-called "lattice p-form electromagnetism":
8) Derek Wise, Lattice p-form electromagnetism and chain field
theory, available as gr-qc/0510033. Version with better graphics
and related material at http://math.ucr.edu/~derek/pform/index.html
It seems to me the discretization is input by hands rather than by any
quantization scheme. I may buy more on the p-form electromagnetism if
there is a quantization scheme that leads to a p-form electromagnetism.
In this paper, he shows lattice p-form electromagnetism is a "chain
field theory": something like a topological quantum field theory, but
where what matters is not spacetime itself so much as the cochain
complex of differential forms *on* spacetime, equipped with just enough
extra geometrical structure to write down the p-form version of Maxwell's
equations.
I remember you mentioned that a topological quantum field theory has no
local degrees of freedom so no dynamics endowed in it. Is this claim still
holds here.
I believe sooner or later you guys still needs to face the question "what
does your parallel transport do/define to your curvature?" if you want to
describe some dynamics in your algebraic way or homotopic way.
So far, I do appreciate all the efforts and the statements made by you
guys, and hope I will be able to join the club in the arena of pursuing
the true understanding of quantum gravity and the fundamental four forces
of nature.
Charles Hui
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| User: "" |
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| Title: Re: This Week's Finds in Mathematical Physics (Week 223) |
04 Dec 2005 05:27:51 AM |
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Shmuel (Seymour J.) Metz wrote:
In <1133269325.923060.13170@g43g2000cwa.googlegroups.com>, on
11/30/2005
at 08:44 AM, said:
I'm sorry, I don't get that. Is that e as in e^x, the base of the
natural logarithm, and what's that underscore?
No; he's talking about an antisymmetric 3-tensor with e_123 = 1, and
the underscore indicates that x, y and z are subscripts rather than
superscripts.
(snip)
Oh! That tensor, is available in Mathcad. I can try this out. Many
thanks.
The Dougster
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