John Baez
Jan27-05, 03:02 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Also available at http://math.ucr.edu/home/baez/week210.html\n\nJanuary 25, 2005\nThis Week\'s Finds in Mathematical Physics - Week 210\nJohn Baez\n\nAs you\'ve probably heard, the Huygens probe has successfully landed\non Saturn\'s moon Titan and is sending back pictures:\n\n1) Huygens Probe Descent\nhttp://saturn.jpl.nasa.gov/news/events/huygensDescent/index.cfm\n\nTitan averages a chilly -180 degrees Celsius, and its smoggy orange\natmosphere is thicker than the Earth\'s, mostly nitrogen but 6 percent\nmethane, together with substantial traces of all sorts of other\nhydrocarbons. The orange color may come from "tholins": polymers made\nby irradiating a mix of nitrogen and methane. Some other icy moons in\nthe outer solar system are covered with this goop, but Titan is the\nonly moon in the Solar System to have a substantial atmosphere. It\neven has clouds.\n\nAs the Huygens probe parachuted to the surface, it photographed\nwhat look like twisty riverbeds flowing into a large lake! People\nhave long suspected that Titan has lakes of made of methane and/or\nethane, but now we may be seeing them. And when Huygens landed, its\nsensors reported that it broke through a crusty surface and sunk\nabout 20 centimeters into something mushy: probably methane mud!\n\nThe first color photo of the surface looks disappointingly like Mars\nat first sight. But, the surface is pumpkin-colored due to tholins\nor something, not rust red. The sky is orange too! The "rocks" could\nbe water ice. And they\'ve detected hints of volcanos that spew molten\nwater and ammonia! So, it\'s a strange new world.\n\nBack here on Earth, there was a conference in December in honor of\nLarry Breen\'s 60th birthday:\n\n2) Arithmetic, Geometry and Topology: Conference on occasion of Larry\nBreen\'s sixtieth birthday, http://www-math.univ-paris13.fr/~lb2004/\n\nIt was in Paris. This was my first visit to that city, but luckily\nI got to stay there an extra week after the conference, so I could focus\non the math while it was going on.\n\nBut I can\'t resist a digression! Paris won my heart, despite my suspicions\nthat it had somehow been hyped all along. First of all, it\'s beautiful.\nSecond, it\'s nice to be someplace where people take simple foods like\nbread, cheese, fruits and vegetables really seriously, and don\'t settle\nfor the tasteless crud we so often eat in the US.\n\nNone of this came as a surprise, of course. What surprised is that I\'ve\nnever seen a city with so many bookstores - and good ones, too! They\'re\nclustered thick near the Sorbonne, but the Latin Quarter is dotted with\nthem, and there are even lots along the Boulevard St-Germain, which is the\nbiggest most fashionable shopping street. I don\'t think there\'s any place\nin the English speaking world with so many bookstores. Not London, not\nNew York... Cambridge Massachusetts used to have lots near Harvard Square,\nback when I was a grad student, but the high rents have long since squeezed\nthem out, replacing bohemian diversity with clothing shops for boring rich\npeople, like Abercrombie and Fitch. Somehow in Paris fancy clothing and\nbooks coexist.\n\nUmm, but what about the conference?\n\nWell, Breen\'s work is mainly on algebraic geometry a la Grothendieck, with\na strong emphasis on category theory. Beautiful stuff, and lately it\'s\nit\'s begun to find applications to string theory - especially his work on\ngerbes. People at his conference spoke on all sorts of topics, most of\nwhich I didn\'t understand very well - some heavy-duty number theory,\nfor example. I understood a few well enough to really enjoy them, like\nMike Hopkins\' talk on derived algebraic geometry, Clemens Berger\'s talk on\ngeometric Reedy categories, and Ieke Moerdijk\'s talk on the homotopy theory\nof operads. But I won\'t try to explain these - I want to explain what a\n"gerbe" is, so I have my work cut out for me.\n\nOne way to get going on the idea of gauge theory is to start with\nelectromagnetism, where the concept of "phase" turns out to play a\ncrucial role. If you move a charged particle through an electromagnetic\nfield, its wavefunction gets multiplied by a unit complex number, or\n"phase" - and it turns out, rather wonderfully, that all effects of\nelectricity and magnetism on charged particles is due to this!\n\nHowever, phases are funny. You can\'t actually measure the phase of\na charged particle - at least, there\'s no such thing as a "phasometer"\nwhere you stick in a particle and the dial on the meter points to\na unit complex number. Of course a unit complex number is just a fancy\nname for a point on the circle, and a dial is precisely the right shape\nfor that... but you just can\'t build this machine.\n\nInstead, you can only measure the *change* in phase of a particle as\nit goes around a loop. Or, equivalently, the *difference* in phases\nwhen a particle takes two different paths from here to there. See,\nin quantum mechanics you can play tricks like the "double slit experiment",\nwhere you coax a particle\'s wavefunction to smear out and take two routes\nfrom here to there... and then when it arrives, it interferes with itself,\nand if you\'re smart you can see by these interference effects what the\nrelative phase of the two paths is.\n\nPretty weird, eh? I\'m so used to this that it seems completely normal to\nme, but I should admit that this way of understanding the electromagnetic\nfield came fairly late. Weyl had a hint of it in 1918 when he invented\nthe term "gauge theory" in his quest to unify electromagnetism and gravity,\nbut he was mixed up in some crucial ways that only got sorted out quite\na bit later. For more details, try O\'Raiferteagh\'s book "The Dawning of\nGauge Theory", which I discussed in "week116".\n\nAnyway, the concept of relative phase, or difference in phase, is nicely\ncaptured by the concept of a "torsor". A unit complex number is a point\non the unit circle in the complex plane. This circle is a group since\nwe can multiply unit complex numbers and get unit complex numbers back.\nThis group is called U(1). Like a dial, U(1) has standard names for\nall the points on it - and it has one god-given special point, the\nidentity element, namely the number 1.\n\nA "U(1) torsor" is a lot like U(1), but subtly different. It\'s a circle\nwhere the points aren\'t given these standard names... but where you can\nstill tell measure angles, and tell the difference between clockwise and\ncounterclockwise.\n\nYou can\'t get an element of U(1) from *one* point on a U(1) torsor. But,\nif you have *two* points on a U(1) torsor, you can say how much rotation\nit takes to get from one to the other, and this give an element of U(1).\nIn other words, you can describe the "difference in phase" between these\ntwo points.\n\nFor more on torsors, try this:\n\n3) John Baez, Torsors made easy, http://math.ucr.edu/home/baez/torsors.html\n\nAnyway, the real idea behind electromagnetism is that sitting over\neach point in spacetime is a U(1) torsor. If a particle is sitting at\nsome point in spacetime, its phase is not really a numbers: it\'s an\nelement of the U(1) torsor sitting over that point! To get a *number*,\nwe have to carry the particle around a loop! Its phase will change when\nwe do this, so we get *two* points in a U(1) torsor, and their difference\nis an element of U(1).\n\nSo while it sounds far-out, the key mathematical structure in electromagnetism\nis a bunch of U(1) torsors, one over each point in spacetime. This is called\na "principal U(1) bundle" or sometimes just a "U(1) bundle" for short.\n\nIf we wanted to describe some force other than electromagnetism, we could\ntake this whole setup and replace U(1) with some other group. In fact,\nthis idea works great: it\'s the main idea behind gauge theories, which do\nan excellent job of describing all the forces in nature.\n\nTo set up a gauge theory, the first thing you need to do is pick a\ngroup G and pick a "principal G-bundle" over spacetime. Spacetime\nwill be some manifold X. A principal G-bundle over X is gadget that\nassigns a G-torsor to each point of X. A G-torsor is a space where if\nyou pick two points in it, you get an element of G which describes their\n"difference".\n\nI\'m being fairly sloppy here, so don\'t take these as precise definitions!\nI give a precise definition of a G-torsor in the above webpage, and any\ndecent book on differential geometry will give you a definition of\nprincipal G-bundles. However, only rather highbrow books define principal\nG-bundles with the help of G-torsors... which is sad, because it\'s not\nthat hard, and rather enlightening.\n\nAnyway, in gauge theory the forces of nature are described by "connections"\non principal G-bundles. Let\'s say we have a principal G-bundle P which\nassigns to each point x of our manifold a G-torsor P(x). Then a\n"connection" on P is a gadget that says how any path from x to y gives a\nmap from P(x) to P(y). If G is U(1), for example, this gadget says how the\nphase of a charged particle changes as we move it along any path from x to y.\n\nNow suppose we have a loop that starts and ends at x. Then our connection\ngives a map from P(x) to itself. If we start with a point in P(x), and\napply this map, we get another point in P(x). Since P(x) is a G-torsor,\nthese two points determine an element of G. This is how we get group\nelements from loops in gauge theory!\n\nNow let me sketch how gerbes enter the game. First I\'ll do the case where\nthe group G is abelian, for example U(1). It\'s the nonabelian gerbes that\nreally interest me... but the abelian case is a lot easier. The reason\nis that when G is abelian, the group element we get in the previous\nparagraph doesn\'t depend on the choice of a point of P(x).\n\nGerbes show up when we try to invent a kind of "higher gauge theory"\nthat describes how not just point particles but 1-dimensional objects\ntransform when you move them around. For example, the strings in string\ntheory, or the loops in loop quantum gravity.\n\nThis leads to a mind-boggling self-referential twist, which is just the\nkind of thing I love:\n\nAs we\'ve seen, a connection describes how a point particle transforms when\nyou carry it along a path:\n\nf\nx------------>-----y a path f from the point x to the point y:\nwe write this as f: x -> y\n\n\nNow we need a gadget that\'ll describe how a *path* transforms when you\ncarry it along a PATH OF PATHS:\n\nf\n----------->-----\n/ || \\\nx ||F y a path-of-paths F from the path f to the path g:\n\\ \\/ / we write this as F: f => g\n----------->-----\ng\n\nTo do this, we need to boost our level of thinking a notch, working not\nwith "G-torsors" and "principal G-bundles" but instead with "G-2-torsors"\nand "G-gerbes".\n\nHere\'s how it goes:\n\nWe start by picking an abelian group G and a manifold X.\n\nThen we pick a "G-gerbe" over M, say P.\n\nWhat\'s that? It\'s a thing that assigns to each point x of X a "G-2-torsor",\nsay P(x).\n\nWhat\'s that? Well, it\'s a thing where if you pick two points in it, you\nget a G-TORSOR describing their difference!\n\nGet it? This is the beginning of a story that goes on forever:\n\nTwo points in a G-torsor determine an element of G;\ntwo points in a G-2-torsor determine a G-torsor;\ntwo points in a G-3-torsor determine a G-2-torsor;\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Also available at http://math.ucr.edu/home/baez/week210.html
January 25, 2005
This Week's Finds in Mathematical Physics - Week 210
John Baez
As you've probably heard, the Huygens probe has successfully landed
on Saturn's moon Titan and is sending back pictures:
1) Huygens Probe Descent
http://saturn.jpl.nasa.gov/news/events/huygensDescent/index.cfm
Titan averages a chilly -180 degrees Celsius, and its smoggy orange
atmosphere is thicker than the Earth's, mostly nitrogen but 6 percent
methane, together with substantial traces of all sorts of other
hydrocarbons. The orange color may come from "tholins": polymers made
by irradiating a mix of nitrogen and methane. Some other icy moons in
the outer solar system are covered with this goop, but Titan is the
only moon in the Solar System to have a substantial atmosphere. It
even has clouds.
As the Huygens probe parachuted to the surface, it photographed
what look like twisty riverbeds flowing into a large lake! People
have long suspected that Titan has lakes of made of methane and/or
ethane, but now we may be seeing them. And when Huygens landed, its
sensors reported that it broke through a crusty surface and sunk
about 20 centimeters into something mushy: probably methane mud!
The first color photo of the surface looks disappointingly like Mars
at first sight. But, the surface is pumpkin-colored due to tholins
or something, not rust red. The sky is orange too! The "rocks" could
be water ice. And they've detected hints of volcanos that spew molten
water and ammonia! So, it's a strange new world.
Back here on Earth, there was a conference in December in honor of
Larry Breen's 60th birthday:
2) Arithmetic, Geometry and Topology: Conference on occasion of Larry
Breen's sixtieth birthday, http://www-math.univ-paris13.fr/~lb2004/
It was in Paris. This was my first visit to that city, but luckily
I got to stay there an extra week after the conference, so I could focus
on the math while it was going on.
But I can't resist a digression! Paris won my heart, despite my suspicions
that it had somehow been hyped all along. First of all, it's beautiful.
Second, it's nice to be someplace where people take simple foods like
bread, cheese, fruits and vegetables really seriously, and don't settle
for the tasteless crud we so often eat in the US.
None of this came as a surprise, of course. What surprised is that I've
never seen a city with so many bookstores - and good ones, too! They're
clustered thick near the Sorbonne, but the Latin Quarter is dotted with
them, and there are even lots along the Boulevard St-Germain, which is the
biggest most fashionable shopping street. I don't think there's any place
in the English speaking world with so many bookstores. Not London, not
New York... Cambridge Massachusetts used to have lots near Harvard Square,
back when I was a grad student, but the high rents have long since squeezed
them out, replacing bohemian diversity with clothing shops for boring rich
people, like Abercrombie and Fitch. Somehow in Paris fancy clothing and
books coexist.
Umm, but what about the conference?
Well, Breen's work is mainly on algebraic geometry a la Grothendieck, with
a strong emphasis on category theory. Beautiful stuff, and lately it's
it's begun to find applications to string theory - especially his work on
gerbes. People at his conference spoke on all sorts of topics, most of
which I didn't understand very well - some heavy-duty number theory,
for example. I understood a few well enough to really enjoy them, like
Mike Hopkins' talk on derived algebraic geometry, Clemens Berger's talk on
geometric Reedy categories, and Ieke Moerdijk's talk on the homotopy theory
of operads. But I won't try to explain these - I want to explain what a
"gerbe" is, so I have my work cut out for me.
One way to get going on the idea of gauge theory is to start with
electromagnetism, where the concept of "phase" turns out to play a
crucial role. If you move a charged particle through an electromagnetic
field, its wavefunction gets multiplied by a unit complex number, or
"phase" - and it turns out, rather wonderfully, that all effects of
electricity and magnetism on charged particles is due to this!
However, phases are funny. You can't actually measure the phase of
a charged particle - at least, there's no such thing as a "phasometer"
where you stick in a particle and the dial on the meter points to
a unit complex number. Of course a unit complex number is just a fancy
name for a point on the circle, and a dial is precisely the right shape
for that... but you just can't build this machine.
Instead, you can only measure the *change* in phase of a particle as
it goes around a loop. Or, equivalently, the *difference* in phases
when a particle takes two different paths from here to there. See,
in quantum mechanics you can play tricks like the "double slit experiment",
where you coax a particle's wavefunction to smear out and take two routes
from here to there... and then when it arrives, it interferes with itself,
and if you're smart you can see by these interference effects what the
relative phase of the two paths is.
Pretty weird, eh? I'm so used to this that it seems completely normal to
me, but I should admit that this way of understanding the electromagnetic
field came fairly late. Weyl had a hint of it in 1918 when he invented
the term "gauge theory" in his quest to unify electromagnetism and gravity,
but he was mixed up in some crucial ways that only got sorted out quite
a bit later. For more details, try O'Raiferteagh's book "The Dawning of
Gauge Theory", which I discussed in "week116".
Anyway, the concept of relative phase, or difference in phase, is nicely
captured by the concept of a "torsor". A unit complex number is a point
on the unit circle in the complex plane. This circle is a group since
we can multiply unit complex numbers and get unit complex numbers back.
This group is called U(1). Like a dial, U(1) has standard names for
all the points on it - and it has one god-given special point, the
identity element, namely the number 1.
A "U(1) torsor" is a lot like U(1), but subtly different. It's a circle
where the points aren't given these standard names... but where you can
still tell measure angles, and tell the difference between clockwise and
counterclockwise.
You can't get an element of U(1) from *one* point on a U(1) torsor. But,
if you have *two* points on a U(1) torsor, you can say how much rotation
it takes to get from one to the other, and this give an element of U(1).
In other words, you can describe the "difference in phase" between these
two points.
For more on torsors, try this:
3) John Baez, Torsors made easy, http://math.ucr.edu/home/baez/torsors.html
Anyway, the real idea behind electromagnetism is that sitting over
each point in spacetime is a U(1) torsor. If a particle is sitting at
some point in spacetime, its phase is not really a numbers: it's an
element of the U(1) torsor sitting over that point! To get a *number*,
we have to carry the particle around a loop! Its phase will change when
we do this, so we get *two* points in a U(1) torsor, and their difference
is an element of U(1).
So while it sounds far-out, the key mathematical structure in electromagnetism
is a bunch of U(1) torsors, one over each point in spacetime. This is called
a "principal U(1) bundle" or sometimes just a "U(1) bundle" for short.
If we wanted to describe some force other than electromagnetism, we could
take this whole setup and replace U(1) with some other group. In fact,
this idea works great: it's the main idea behind gauge theories, which do
an excellent job of describing all the forces in nature.
To set up a gauge theory, the first thing you need to do is pick a
group G and pick a "principal G-bundle" over spacetime. Spacetime
will be some manifold X. A principal G-bundle over X is gadget that
assigns a G-torsor to each point of X. A G-torsor is a space where if
you pick two points in it, you get an element of G which describes their
"difference".
I'm being fairly sloppy here, so don't take these as precise definitions!
I give a precise definition of a G-torsor in the above webpage, and any
decent book on differential geometry will give you a definition of
principal G-bundles. However, only rather highbrow books define principal
G-bundles with the help of G-torsors... which is sad, because it's not
that hard, and rather enlightening.
Anyway, in gauge theory the forces of nature are described by "connections"
on principal G-bundles. Let's say we have a principal G-bundle P which
assigns to each point x of our manifold a G-torsor P(x). Then a
"connection" on P is a gadget that says how any path from x to y gives a
map from P(x) to P(y). If G is U(1), for example, this gadget says how the
phase of a charged particle changes as we move it along any path from x to y.
Now suppose we have a loop that starts and ends at x. Then our connection
gives a map from P(x) to itself. If we start with a point in P(x), and
apply this map, we get another point in P(x). Since P(x) is a G-torsor,
these two points determine an element of G. This is how we get group
elements from loops in gauge theory!
Now let me sketch how gerbes enter the game. First I'll do the case where
the group G is abelian, for example U(1). It's the nonabelian gerbes that
really interest me... but the abelian case is a lot easier. The reason
is that when G is abelian, the group element we get in the previous
paragraph doesn't depend on the choice of a point of P(x).
Gerbes show up when we try to invent a kind of "higher gauge theory"
that describes how not just point particles but 1-dimensional objects
transform when you move them around. For example, the strings in string
theory, or the loops in loop quantum gravity.
This leads to a mind-boggling self-referential twist, which is just the
kind of thing I love:
As we've seen, a connection describes how a point particle transforms when
you carry it along a path:
f
x------------>-----y a path f from the point x to the point y:
we write this as f: x -> y
Now we need a gadget that'll describe how a *path* transforms when you
carry it along a PATH OF PATHS:
f
----------->-----
/ || \x ||F y a path-of-paths F from the path f to the path g:
\ \/ / we write this as F: f => g
----------->-----
g
To do this, we need to boost our level of thinking a notch, working not
with "G-torsors" and "principal G-bundles" but instead with "G-2-torsors"
and "G-gerbes".
Here's how it goes:
We start by picking an abelian group G and a manifold X.
Then we pick a "G-gerbe" over M, say P.
What's that? It's a thing that assigns to each point x of X a "G-2-torsor",
say P(x).
What's that? Well, it's a thing where if you pick two points in it, you
get a G-TORSOR describing their difference!
Get it? This is the beginning of a story that goes on forever:
Two points in a G-torsor determine an element of G;
two points in a G-2-torsor determine a G-torsor;
two points in a G-3-torsor determine a G-2-torsor;
January 25, 2005
This Week's Finds in Mathematical Physics - Week 210
John Baez
As you've probably heard, the Huygens probe has successfully landed
on Saturn's moon Titan and is sending back pictures:
1) Huygens Probe Descent
http://saturn.jpl.nasa.gov/news/events/huygensDescent/index.cfm
Titan averages a chilly -180 degrees Celsius, and its smoggy orange
atmosphere is thicker than the Earth's, mostly nitrogen but 6 percent
methane, together with substantial traces of all sorts of other
hydrocarbons. The orange color may come from "tholins": polymers made
by irradiating a mix of nitrogen and methane. Some other icy moons in
the outer solar system are covered with this goop, but Titan is the
only moon in the Solar System to have a substantial atmosphere. It
even has clouds.
As the Huygens probe parachuted to the surface, it photographed
what look like twisty riverbeds flowing into a large lake! People
have long suspected that Titan has lakes of made of methane and/or
ethane, but now we may be seeing them. And when Huygens landed, its
sensors reported that it broke through a crusty surface and sunk
about 20 centimeters into something mushy: probably methane mud!
The first color photo of the surface looks disappointingly like Mars
at first sight. But, the surface is pumpkin-colored due to tholins
or something, not rust red. The sky is orange too! The "rocks" could
be water ice. And they've detected hints of volcanos that spew molten
water and ammonia! So, it's a strange new world.
Back here on Earth, there was a conference in December in honor of
Larry Breen's 60th birthday:
2) Arithmetic, Geometry and Topology: Conference on occasion of Larry
Breen's sixtieth birthday, http://www-math.univ-paris13.fr/~lb2004/
It was in Paris. This was my first visit to that city, but luckily
I got to stay there an extra week after the conference, so I could focus
on the math while it was going on.
But I can't resist a digression! Paris won my heart, despite my suspicions
that it had somehow been hyped all along. First of all, it's beautiful.
Second, it's nice to be someplace where people take simple foods like
bread, cheese, fruits and vegetables really seriously, and don't settle
for the tasteless crud we so often eat in the US.
None of this came as a surprise, of course. What surprised is that I've
never seen a city with so many bookstores - and good ones, too! They're
clustered thick near the Sorbonne, but the Latin Quarter is dotted with
them, and there are even lots along the Boulevard St-Germain, which is the
biggest most fashionable shopping street. I don't think there's any place
in the English speaking world with so many bookstores. Not London, not
New York... Cambridge Massachusetts used to have lots near Harvard Square,
back when I was a grad student, but the high rents have long since squeezed
them out, replacing bohemian diversity with clothing shops for boring rich
people, like Abercrombie and Fitch. Somehow in Paris fancy clothing and
books coexist.
Umm, but what about the conference?
Well, Breen's work is mainly on algebraic geometry a la Grothendieck, with
a strong emphasis on category theory. Beautiful stuff, and lately it's
it's begun to find applications to string theory - especially his work on
gerbes. People at his conference spoke on all sorts of topics, most of
which I didn't understand very well - some heavy-duty number theory,
for example. I understood a few well enough to really enjoy them, like
Mike Hopkins' talk on derived algebraic geometry, Clemens Berger's talk on
geometric Reedy categories, and Ieke Moerdijk's talk on the homotopy theory
of operads. But I won't try to explain these - I want to explain what a
"gerbe" is, so I have my work cut out for me.
One way to get going on the idea of gauge theory is to start with
electromagnetism, where the concept of "phase" turns out to play a
crucial role. If you move a charged particle through an electromagnetic
field, its wavefunction gets multiplied by a unit complex number, or
"phase" - and it turns out, rather wonderfully, that all effects of
electricity and magnetism on charged particles is due to this!
However, phases are funny. You can't actually measure the phase of
a charged particle - at least, there's no such thing as a "phasometer"
where you stick in a particle and the dial on the meter points to
a unit complex number. Of course a unit complex number is just a fancy
name for a point on the circle, and a dial is precisely the right shape
for that... but you just can't build this machine.
Instead, you can only measure the *change* in phase of a particle as
it goes around a loop. Or, equivalently, the *difference* in phases
when a particle takes two different paths from here to there. See,
in quantum mechanics you can play tricks like the "double slit experiment",
where you coax a particle's wavefunction to smear out and take two routes
from here to there... and then when it arrives, it interferes with itself,
and if you're smart you can see by these interference effects what the
relative phase of the two paths is.
Pretty weird, eh? I'm so used to this that it seems completely normal to
me, but I should admit that this way of understanding the electromagnetic
field came fairly late. Weyl had a hint of it in 1918 when he invented
the term "gauge theory" in his quest to unify electromagnetism and gravity,
but he was mixed up in some crucial ways that only got sorted out quite
a bit later. For more details, try O'Raiferteagh's book "The Dawning of
Gauge Theory", which I discussed in "week116".
Anyway, the concept of relative phase, or difference in phase, is nicely
captured by the concept of a "torsor". A unit complex number is a point
on the unit circle in the complex plane. This circle is a group since
we can multiply unit complex numbers and get unit complex numbers back.
This group is called U(1). Like a dial, U(1) has standard names for
all the points on it - and it has one god-given special point, the
identity element, namely the number 1.
A "U(1) torsor" is a lot like U(1), but subtly different. It's a circle
where the points aren't given these standard names... but where you can
still tell measure angles, and tell the difference between clockwise and
counterclockwise.
You can't get an element of U(1) from *one* point on a U(1) torsor. But,
if you have *two* points on a U(1) torsor, you can say how much rotation
it takes to get from one to the other, and this give an element of U(1).
In other words, you can describe the "difference in phase" between these
two points.
For more on torsors, try this:
3) John Baez, Torsors made easy, http://math.ucr.edu/home/baez/torsors.html
Anyway, the real idea behind electromagnetism is that sitting over
each point in spacetime is a U(1) torsor. If a particle is sitting at
some point in spacetime, its phase is not really a numbers: it's an
element of the U(1) torsor sitting over that point! To get a *number*,
we have to carry the particle around a loop! Its phase will change when
we do this, so we get *two* points in a U(1) torsor, and their difference
is an element of U(1).
So while it sounds far-out, the key mathematical structure in electromagnetism
is a bunch of U(1) torsors, one over each point in spacetime. This is called
a "principal U(1) bundle" or sometimes just a "U(1) bundle" for short.
If we wanted to describe some force other than electromagnetism, we could
take this whole setup and replace U(1) with some other group. In fact,
this idea works great: it's the main idea behind gauge theories, which do
an excellent job of describing all the forces in nature.
To set up a gauge theory, the first thing you need to do is pick a
group G and pick a "principal G-bundle" over spacetime. Spacetime
will be some manifold X. A principal G-bundle over X is gadget that
assigns a G-torsor to each point of X. A G-torsor is a space where if
you pick two points in it, you get an element of G which describes their
"difference".
I'm being fairly sloppy here, so don't take these as precise definitions!
I give a precise definition of a G-torsor in the above webpage, and any
decent book on differential geometry will give you a definition of
principal G-bundles. However, only rather highbrow books define principal
G-bundles with the help of G-torsors... which is sad, because it's not
that hard, and rather enlightening.
Anyway, in gauge theory the forces of nature are described by "connections"
on principal G-bundles. Let's say we have a principal G-bundle P which
assigns to each point x of our manifold a G-torsor P(x). Then a
"connection" on P is a gadget that says how any path from x to y gives a
map from P(x) to P(y). If G is U(1), for example, this gadget says how the
phase of a charged particle changes as we move it along any path from x to y.
Now suppose we have a loop that starts and ends at x. Then our connection
gives a map from P(x) to itself. If we start with a point in P(x), and
apply this map, we get another point in P(x). Since P(x) is a G-torsor,
these two points determine an element of G. This is how we get group
elements from loops in gauge theory!
Now let me sketch how gerbes enter the game. First I'll do the case where
the group G is abelian, for example U(1). It's the nonabelian gerbes that
really interest me... but the abelian case is a lot easier. The reason
is that when G is abelian, the group element we get in the previous
paragraph doesn't depend on the choice of a point of P(x).
Gerbes show up when we try to invent a kind of "higher gauge theory"
that describes how not just point particles but 1-dimensional objects
transform when you move them around. For example, the strings in string
theory, or the loops in loop quantum gravity.
This leads to a mind-boggling self-referential twist, which is just the
kind of thing I love:
As we've seen, a connection describes how a point particle transforms when
you carry it along a path:
f
x------------>-----y a path f from the point x to the point y:
we write this as f: x -> y
Now we need a gadget that'll describe how a *path* transforms when you
carry it along a PATH OF PATHS:
f
----------->-----
/ || \x ||F y a path-of-paths F from the path f to the path g:
\ \/ / we write this as F: f => g
----------->-----
g
To do this, we need to boost our level of thinking a notch, working not
with "G-torsors" and "principal G-bundles" but instead with "G-2-torsors"
and "G-gerbes".
Here's how it goes:
We start by picking an abelian group G and a manifold X.
Then we pick a "G-gerbe" over M, say P.
What's that? It's a thing that assigns to each point x of X a "G-2-torsor",
say P(x).
What's that? Well, it's a thing where if you pick two points in it, you
get a G-TORSOR describing their difference!
Get it? This is the beginning of a story that goes on forever:
Two points in a G-torsor determine an element of G;
two points in a G-2-torsor determine a G-torsor;
two points in a G-3-torsor determine a G-2-torsor;