Newsgroups: sci.physics
Subject: Loop variables questions
From: baez@guitar.ucr.edu (john baez)
Date: 17 Nov 92 06:58:33 GMT

My friend Allen Knutson emailed the following to me, because in the
math department at Princeton (that place beloved to me) the news poster
is DEAD.   My comments are interwoven, marked with an initial JB:

Return-Path: <aknaton@math.Princeton.EDU>
Subject: Quantum GR questions
To: baez@ucrmath.ucr.edu
Date: Mon, 16 Nov 92 20:57:54 EST
Cc: aknaton@math.Princeton.EDU (Allen Knutson)

AK: I must have some deep misunderstanding about the A,R & S quantum gravity
stuff. I would ask you this on the net, but the nnposter is still broken,
dammit. Certainly feel free to answer on the net, if you don't want to
repeat yourself to other people asking about loopy QGR.

First question is about the loop representation of connections. If I
remember right, you said that to a connection and a representation of
the structure group one can associate a number, the trace of the
holonomy around the loop, and with luck one can reconstruct the connection
uniquely up to gauge equivalence from knowing all these numbers. When
is this the case? Obviously the representation must be faithful.

JB:  Yes, clearly the representation must be faithful or it won't work.

Here's how the loop transform goes again folks.  This will be fairly heavy
going for those not versed in differential geometry, but it's never too late
to learn!

First of all, recall the notion of holonomy.  Suppose we are
given a vector bundle E over the manifold M with a connection A.  Let
E_x denote the vector space sitting over some point x in M -- E_x is
called the "fiber" of the vector bundle E over the point x.
Using the connection to parallel translate a vector in E_x around
a loop in M, each loop based at the point x in M gives rise to a holonomy,
that is, a linear transformation of the vector space E_x.    If we take the
trace of this linear transformation we get a number.  This number doesn't
change if we do a gauge transformation on A.

Now the transform of a given connection A is the function on the space of loops
given by the trace of the holonomy.  So we may think of the loop transform
as a function from the space of connections modulo gauge transformations to
the space of functions on loops.

If for all loops this holonomy lies in a certain subgroup
G of End(E_x) (the space of linear transformations of E_x) we may say that
A is a G-connection.  Given any old Lie group G, we can define the
space of G-connections on E modulo gauge transformations (where we restrict
ourselves to G-valued gauge transformations).  The loop transform can be
regarded as a map from this space to the space of functions on the space
of loops!  (Whew.)

When is this one-to-one, AK askes.  I know it is for the defining rep of
SU(2) but not for SL(2,C).  Right now I am confused about the general criterion
for when it is.    I may be screwed up here, but part of what we need
is for the functions

tr(g^n)

to generate an algebra on G/[G,G] that separates points.  It is easy to see
that they do NOT for SL(2,C).  Take the matrices

1 1+a
0 1

These are conjugate for all a > 0 but not for a = 0.  (This is a good exercise -
they are conjugate by an element of SL(2,C), I mean.)   Thus no
continuous Ad-invariant function on SL(2,C) can separate the points

1 2
0 1

and

1 0
0 1

even though they are not conjugate in SL(2,C).  This is irritating but
it also implies that *no* continuous gauge-invariant function on the space
of connections (in any reasonable topology) can separate gauge equivalence
classes of connections for this gauge group.

Hmm, I should reread R. Giles' Reconstruction of gauge potentials from
Wilson loops, Phys Rev D24 (1981) 2160-2168.

AK: Second, if we have the numbers for knots, why do people want to know/have
the right to ask for numbers on links? And why is the right answer to demand
the product of the other numbers?

MP: a note from me - what I am looking for is a way for the discrete
entities like knots and links to be performing computations, thru loopyQG's
dynamical aspect [QM metaphysical probs aside]



JB: If one had a measure on the space of connections mod gauge transformations,
one could assign a number to any knot, by forming the trace of the holonomy
and then integrating over the space of connections mod gauge transformations.

One could also assign a number to any link, by forming the products of the
traces of the holonomies of each of the components of the link (which are
knots).  People do both.

AK: Third, it is claimed (if I am reading right) that one of the QG constraints
amounts to saying "The loop functionals must give the same answer on two
isotopic loops". Say we cross a loop through itself, and look at how the
holonomy changes, i.e. smoothly I would have thought.

JB:  In this context (quantum gravity) the space of states can be viewed
as a certain space of "measures" on the space of connections modulo
gauge transformations.  Measures on this space must satisfy two constraints to
define states of quantum gravity in the canonical quantization approach:
diffeomorphism-invariance, and the Hamiltonian constraint.

Now the loop transform can be extended to define a map from the space of
measures on the space of connections mod gauge transformations to the
space of functions on loops!  (First take the trace,
then integrate over the space of connections with respect to your measure.)
In quantum gravity one would like to say that the loop transform of
a diffeomorphism-invariant measure on the space of connections mod gauge
transformations is a link invariant, that is, only depends on the
ambient isotopy class of the link.  This is true!  But the "measures"
that people are interested in, like the Chern-Simons path integral, are
not really measures in the honest sense.  One must generalize the notion
of a measure, much as one does in the case of linear field theories by
introducing the notion of a "distribution" -- in the sense of the book
with Segal and Zhou.   I'm working on this now.

MP: JB's personal research - this notion of generalizing measures to
whatever sort of space the space of connections is - I think I
recall him saying once that his effort was to make CSlike path integrals
rigorous in this context??

AK: Fourth, say we have a state s, i.e. a loop functional, and an area operator.
When we apply the area operator A to the state, we get another loop functional,
such that when we then evaluate As on a loop we count the number of
intersections. Doesn't that mean that As isn't constant on loop classes,
and thus isn't a state? Oh dear, perhaps I shouldn't be asking this question,
since it's probably founded on so many misconceptions.

JB:  The area operators are not defined in the physical state space
of quantum gravity, in which the diffeomorphism-invariance constraint
has been taken into account!  They are defined in the space of all
"measures on the space of connections mod gauge transformations".

AK: Fifth, I notice that I see Ed Witten's name in the ends of many of these
papers; is it not true then that everybody at Princeton hates this approach
to GR? That'd certainly be encouraging.			 Allen K.

JB: No, it just shows that everyone drops Ed Witten's name.  You're at
Princeton - *you* see if everyone there hates loop variables!

Newsgroups: sci.physics
Subject: Re: Hidden variable theories, was: Uncertainty Princi
From: jbaez@riesz.mit.edu (John C. Baez)
Date: Sat, 12 Sep 92 02:06:32 GMT

From galois!snorkelwacker.mit.edu!spool.mu.edu!uunet!mtnmath!paul Fri Sep 11 20:17:18 EDT 1992
Article: 20187 of sci.physics
Path: galois!snorkelwacker.mit.edu!spool.mu.edu!uunet!mtnmath!paul
From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.physics
Subject: Re: Hidden variable theories, was: Uncertainty Princi
Message-ID: <271@mtnmath.UUCP>
Date: 11 Sep 92 14:16:36 GMT
References: <1992Sep5.071519.16554@asl.dl.nec.com> <1992Sep11.015614.28674@galois.mit.edu>
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070
Lines: 49

Paul Budnik writes:
>In article <1992Sep11.015614.28674@galois.mit.edu>, jbaez@riesz.mit.edu (John C. Baez) writes:
>> In a down-to-earth vein, I would only say information was being
>> transmitted by such a device if I could use it tell my grandma what I
>> was having for dinner, or at least do so with some statistically
>> significant chance (i.e., I'll accept a noisy channel as long as it's
>> not 100% noise).  It would only be if someone did THIS with quantum
>> trickery that I would get nervous.  But of course one can't.

>You are free to redefine information in any way you choose (at least
>in a posting) but what you are defining here is normally regarded as
>sending a signal. Information is often transferred in ways that no
>human being can control.

Fine.  I have no attachment to terminology in this issue as long as we
agree on what one can and can't do.  Typically, 99% of argument about
measurement in QM concerns the right way to talk about what happens,
rather than what actually happend, which is why I find it so boring.
Your latest post confirms that we have no argument over facts, just
massive disagreements about vocabulary, which reassures me that I can
let the matter rest.

>> ... you
>> are using the word "causality" in a certain way, which I think is the
>>"wrong" way, but in any case, that there's another sensible usage of it
>> such that causality is not violated.

>I would appreciate some references. In everything I have read in respected
>journals there is no disagreement that either causality or Lorentz invariance
>is violated. There is a lot of garbage written in popular accounts
>of this subject.

See any book which explains the various axiomatic approaches to quantum
field theory.  The best for this purpose is "An Introduction to Axiomatic
Quantum Field Theory" by Bogoliubov, Todorov, and Shirkov (there's also
a new book by roughly the same authors).  There are various axiom schemes,
notably the Garding-Wightman axioms for quantum fields as operator-valued
distributions on Minkowski space, the Haag-Kastler axioms for quantum fields in
terms of local algebras of observables, and the Osterwalder-Schroeder
axioms for Euclidean quantum field theory.  All these axioms schemes
contain axioms for causality and Lorentz invariance.  The point is,
what quantum field theorist mean by causality is quite different from
what you seem to mean!  There's not really any contradiction.  There
are, in fact, two distinct notions of causality in quantum field
theory, "microscopic causality," which is called "Local commutativity" on
p. 597 of the book cited above, and another sort of causality
sometimes called the "diamond property," also discussed on the same
page.  These notions are easiest to get ahold of in the Haag-Kastler
axiom scheme.   The first says that the operator algebras living on
spacelike separated open sets commute.  The second says that if one
open set is in the causal shadow of another (see my recent post for a
definition of that term), its operator algebra is contained in the
operator algebra of the other.   Roughly speaking, the latter says
that information, or signals, or whatever you want to call 'em, can't
propagate faster than the speed of light.  More precisely, the state in a
given open set is completely determined by the state in open set of which it's
in the causal shadow.  Here I'm using the word "state" in the usual
sense of (mathematical) quantum theory: a positive linear functional on a
C*-algebra.

If you can't get the above book, which is really very nice, I
recommend the original article

Haag and Kastler
An Algebraic Approach to Quantum Field Theory
Journal of Math. Phys.
5 (1964) p. 848.

Realize, however, that there is a vast literature on the subject since.
There is a book called "An Introduction to Algebraic Quantum Field
Theory" which treats some of the newer stuff - I forget the author -
not to be confused with "An Introduction to Algebraic and Constructive
Quantum Field Theory" by Segal, Zhou and myself; we have a discussion of
causality but do not treat the Haag-Kastler axioms (or any others) as such.

>Regarding the rest of your message, there is no absolute contradiction
>in quantum mechanic only some very unlikely predictions.

Unlikely, eh?  Care to state odds at which you'd bet some real money?
I'm already looking forward to 50 bucks from Dave Ring when his
"solid evidence for supersymmetry" fails to materialize.  (Say - I
forget when the deadline on this bet was!  I don't want to let him
weasel out via indefinite postponement!)  I look forward to the day
when an improved Aspect-type experiment rules out a Lorentz-invariant
wavefunction collapse theory by giving a real violation of Bell's
inequalities - but I'd look forward to it even more if I knew I would
make some money off it!


Newsgroups: sci.physics
Subject: Re: Question of Theory of Everything (or Grand Unified theory)
From: jbaez@riesz.mit.edu (John C. Baez)
Date: Mon, 7 Sep 92 04:04:45 GMT
References: <1992Sep7.022904.15484@nntpd.lkg.dec.com>

Don't ever believe anyone who claims that there WILL BE a shocking
scientific advance 10-20 years from now.  Scientific breakthroughs are
by definition unexpected until they happen.  The fans of superstrings
were very optimistic about 5 years ago and some of the more bold, or
if I may say so, arrogant, claimed that shortly we would have a fully
functioning "Theory of Everything".  The relative lack of hubbub about
string theory in the popular press lately is an indication of what has
happened in the meantime - lots of good mathematics but sort of a
quagmire when it comes to working out the physics.  In particular,
people had been excited for a while when it looked like there was an
almost unique sensible string theory, but now this no longer seems
true.

I have my own prejudices about what are the best directions to look
for a theory of everything, and if you had been reading sci.physics
for the last year you would have seen me go on and on about them.
Briefly, I favor the Ashtekar/Rovelli/Smolin loop variables approach
to quantum gravity, and its extensions to treat other forces.  Check
out the "Science and the Citizen" column of the latest Scientific
American (September 1992) for a nice introduction.  I am spending all
my time working on this stuff myself, however, so I am hardly an
unbiased party!  On good days I think, "Yeah, maybe in 10 or 20 years
we'll work out a theory of all the forces!"  But that's just my
natural optimism and good spirits.  We may never have a theory of
everything - indeed even if we get one we will never know for sure
that it is true - but all we can say for now is that physics is full, very
full, of mysterious puzzles!



Newsgroups: sci.physics
Subject: Re: What do we know or believe about generations?
From: nextc.Princeton.EDU!mdd (Mark D. Doyle)
Date: Fri, 11 Sep 1992 03:24:01 GMT
References: <1992Sep11.020409.28542@nuscc.nus.sg>

In article <1992Sep11.020409.28542@nuscc.nus.sg> matmcinn@nuscc.nus.sg
(Mcinnes B T (Dr)) writes:
> One of the really pressing problems in physics is the generation
> problem: why do we have three?
> What solutions have been proposed for this one?
> I know one: in string theory, the CAlabi-Yau people proposed that we
> have more than one generation essentially because of the way that the
> topology of the internal manifold affects the solutions of the Dirac
> equation. The number of generations turns out to be related to the Euler
> characteristic of the internal space. This extremely natural solution of
> the problem seems to be dying out along with string theory itself.

I wouldn't say that string theory is dying out. (Of course, I may be a
bit biased since my thesis defense is tomorrow morning in string theory.)
It has just become clear that there is a lot of things in string theory
that need to be understood better. One big problem is to understand the
nonperturbative definition of the theory so that one can address questions
such as supersymmetry breaking, number of generations, the low energy
spectrum (what we currently call high energy physics), etc. This has
always been known to be a tough problem. Some progress has been made in
the last few years with the introduction of matrix models and topological
theories. The current trend is to set aside the harder problems of working
with something like the heterotic string in ten dimensions and work with
low dimensional string theories (D < or = 1) that are exactly solvable.
The hope is to glean insight that can be applied to the more physically
relevant case. The last few years have seen an explosion of interest in
this approach and it has been quite fruitful. The most remarkable thing is
the breadth of different areas of mathematics that come up in these
pursuits. There is still a large contigent of people who are actively
seeking ways to get down from string theory to more accessible (low)
energies. What is clear is that it seems that it is no longer true that
there are only a handful of consistent string theories. And there has
always been a plethora of vacuum solutions (i.e. all those Calabi-Yau
manifolds). The hope is now that a nonperturbative formulation of the
theory will narrow down the choices. String theory is still the current
favorite (certainly in these parts) for unifying gravity with the other
forces. Anyway, I need to finish some transparancies, but I just want to
reiterate that string theory is not dead; the emphasis has merely shifted
in our approach to it and phenomenological questions have fallen into the
background somewhat pending new insights and data. Hopefully a nice boost
will come from the discovery of supersymmetry at the SSC. (Let's not start
that thread again :^).) The other problem is more practical. Jobs in
string theory are becoming very limited and the amount of activity will
probably decrease as people switch over to other areas of physics. This is
not to say that strings are not worth pursuing.

Mark Doyle, 11 hours to go...
mdd@puhep1.princeton.edu

Newsgroups: sci.physics
Subject: Re: What do we know about choice of groups?
From: matt@physics.berkeley.edu (Matt Austern)
Date: 10 Sep 92 23:13:27

In article <1992Sep11.021551.1744@nuscc.nus.sg> matmcinn@nuscc.nus.sg (Mcinnes B T (Dr)) writes:

> Suppose that the standard theory is right, and that the gauge group of
> the world is "SU3xSU2xU1"  [actually S[ U2xU3 ] of course]. Then we will
> have to understand why this particular group was chosen from infinitely
> many others. The same problem arises, albeit less urgently, if GUTs are
> correct: why SO[10]  [actually Spin[10] of course]  rather than SO[110]
> ?
> What ideas have been proposed to solve this problem?

Well, one constraint that most theorists believe in is that a gauge
theory has to be anomaly-free.  I really don't feel like explaining
what that means just now; for the moment, let's just say that it is a
technical property which is necessary for the theory to be
renormalizable.  This excludes most possible gauge groups---still
leaving an infinite number, but a much smaller infinity than without
that constraint.

We can hope that by imposing other physical principles, we might be
left with fewer possibilites---ideally, only one.  String theorists
have some optimism along those lines.


--
Matthew Austern		   Just keep yelling until you attract a
(510) 644-2618		    crowd, then a constituency, a movement, a
austern@lbl.bitnet		faction, an army!  If you don't have any
matt@physics.berkeley.edu	 solutions, become a part of the problem!

Newsgroups: sci.math
Subject: Categories (was: Re: Almost a group, or what?)
From: jbaez@banach.mit.edu (John C. Baez)
Date: 7 Aug 92 15:55:51 GMT
References: <18240@nntp_server.ems.cdc.com> <1992Aug6.224328.14971@pasteur.Ber
Keywords: Algebra, groups

In article <1992Aug6.224328.14971@pasteur.Berkeley.EDU> luzeaux@bellini.berkel

>As for R non associative and R commutative, it is a commutative magma
>(cf Bourbaki),
>which does not seem so interesting.

It is interesting however to note why associativity without
commutativity is studied so much more than commutativity without
associativity.  Basically, because most of our examples of binary
operations can be interpreted as composition of functions.  For example,
if write simply x for the operation of adding x to a real number (where
x is a real number), then x + y is just x composed with y.  Composition
is always associative so the + operation is associative!

Let me point out that one of the most interesting generalizations of a
group is a category.  In what follows I'll give a skimpy introduction to
category theory and hint at its applications to physics, though I want
to say more about that later.

-- All you ever needed to know about category theory in 2 pages. -----

Categories are some of the most basic structures in mathematics.  They
were created by Saunders MacLane, I believe.  (A lot of other people
were involved, but I know MacLane said: "I did not invent category
theory to talk about functors.  I invented it to talk about natural
transformations."  Huh?  Wait and see.)

What is a category?  Well, a category consists of a set of OBJECTS and
a set of MORPHISMS.  Every morphism has a SOURCE object and a TARGET
object.  (The example to think of is the category in which the objects
are sets and the morphisms are functions.  If f:X -> Y, we call X the
source and Y the target.)  Given objects X and Y, we write Hom(X,Y)
for the set of morphisms from X to Y (i.e., having X as source and Y
as target).

The axioms for a category are that it consist of a set of objects and
for any 2 objects X and Y a set Hom(X,Y) of morphisms from X to Y, and

1)  Given a morphism g in Hom(X,Y) and a morphism f in Hom(Y,Z), there
is morphism which we call fog in Hom(X,Z).  (This binary operation o is
called COMPOSITION.)

2)  Composition is associative:  (fog)oh = fo(goh).

3)  For each object X there is a morphism id|X from X to X, called the
IDENTITY ON X.

4)  Given any f in Hom(X,Y), foid|X = f and id|Yof = f.

The classic example is Set, the category with sets as objects and
functions as morphisms, and the usual composition as composition!
Or else

Vect --- vector spaces as objects, linear maps as morphisms
Group ---- groups as objects, homomorphisms as morphisms
Top ---  topological spaces as objects, continuous functions as morphisms
Diff --- smooth manifolds as objects, smooth maps as morphisms
Ring --- rings as objects, ring homomorphisms as morphisms

Note that in all these cases the morphisms are actually a special sort
of function.  That need not be the case in general!  For example, an
ordered set is a category with its elements as objects and one
morphism in each Hom(X,Y) if X is less than or equal to Y, but none
otherwise.  Weird, huh?

The golden rule of modern mathematics is that life takes place within
-- and between -- categories.  Many nice things in mathematics are
functors.  A functor is a kind of map between categories.  A FUNCTOR F
from a category C to a category D is a map from the set of objects of
C to the set of objects of D together with a map from the set Hom(X,Y)
for any objects X,Y of C to Hom(F(X),F(Y)).  That is, objects go to
objects and morphisms go to morphisms.

Category theory is popular among algebraic topologists.  Typically an
algebraic topologist will try to assign algebraic invariants to
topological structures.  The golden rule of such invariants is that
they should be FUNCTORIAL.  That is, they should be functors!  For
example, the fundamental group is functorial.  Topologists know how to
cook up a group called the fundamental group from any space.  (The
group keeps track of how many holes the space has.)  But ALSO, any map
between spaces determines a homomorphism of the fundamental groups.
So the fundamental group is really a functor from the category Top to
the category Group.

This allows us to completely transpose any situation involving spaces
and continuous maps between them to a parallel situation involving
groups and homomorphisms, and thus reduce some topology problems to
algebra problems!

There is a famous saying about quantization among mathematical
physicists: "First quantization is a mystery, but second quantization
is a functor!"  No one is a true mathematical physicist unless they
can explain that remark.  In second quantization we attach to each
Hilbert space H its Fock space K (another Hilbert space), and to each
unitary map between Hilbert spaces a unitary map between their Fock
spaces.  (Fock spaces come in two flavors: bosonic and fermionic.)

Now, there are NATURAL TRANSFORMATIONS between functors.
Suppose we have two functors F and G from the category C to the category
D.  A natural transformation n from F to G consists of: 1) for each
object X in C, a morphism n(X) from F(X) to G(X), such that 2) the
following diagram commutes:

			 F(X) -F(f)-> F(Y)
			  |	   |
		      n(X)|	   |n(Y)
			  v	   v
			 G(X) -G(f)-> G(Y)

An example would be "abelianization", which maps a group H to the
abelian group H/[H,H].  If F were the fundamental group and G were the
first homology group, we could say that abelianization is a natural
transformation from F to G.

An interesting object in physics is Minkowski space.  We can imagine a
category Mink which has only one object - Minkowski space!  And whose
morphisms are the Poincare transformations (i.e., rotations,
translations, Lorentz transformations, and composites thereof)!  This
shows that categories are a generalization of group representations,
by the way.  Then one can imagine a natural transformation from
Minkowski space to the category Spin with one object, the space of
spinors (fancy for 4-tuples of complex numbers), and morphisms given
by the representation of the Poincare group on this space.  Then what
expresses the principle of relativity most precisely is that the value
of any observable, e.g. a spinor, must define a FUNCTOR from Mink to
the relevant category, in this case Spin.   (We can also express the
principal of general covariance and the principal of gauge-invariance
most precisely by saying that observables are functorial.)

So physicists should regard functoriality as mathematical for "able to
be defined without reference to a particular choice of coordinate system."

Now what is the category of all categories?  As I said, it's a 2-category.
What's a 2-category?  And what do they have to do with quantum gravity?
Stay tuned....

Newsgroups: sci.math
Subject: Categories and Quantization
From: jbaez@nevanlinna.mit.edu (John C. Baez)
Date: 7 Aug 92 19:41:58 GMT
References: <1992Aug7.155551.20912@galois.mit.edu> <BsMHq1.JpH@cs.psu.edu>

Two more micro-essays on physics and category theory.  First, an
advertisement for the notion that group representations are really only
a special case of category representations.  This idea was sold to me by
Minhyong Kim.  He said: "Eventually people will see that group
representation theory is not such a big deal; what really matters is
representations of categories."  At first I thought he was trying to
sound slick (he always goes for the most abstract and elegant
viewpoint).  But then I wound up needing category representations in my
own work on quantum gravity.

Second, an explanation of the claim that "first quantization is a
mystery, but second quantization is a functor."

-----
In article <BsMHq1.JpH@cs.psu.edu> sibley@math.psu.edu writes:
>
>Speaking of groups and categories, I have always liked the category
>version of the definition of a group:
>
>  A group is a category with one object in which all the morphisms
>  are isomorphisms.

This is very important because it leads one to see that more general
than a representation of a group is a representation of a CATEGORY.  A
representation of a group, if we think of a group as a category as
Sibley suggests, is just a functor from that category to the category
Vect of vector spaces.  So we can define a representation of a category
to be a functor from the category to the category of vector spaces.

An example of an interesting category with interesting representations
is the category of TANGLES.  Tangles are like braids but the strands can
double back on themselves and there can also be closed loops.
The objects in the category Tang are {0,1,2,...} and the morphisms in
Hom(m,n) are (isotopy classes of) tangles with m strands going in and n
strands coming out.  A picture is worth a thousand words here.  Here is
an element of Hom(2,4):

	|   |
	\   /
	 \ /
	  \     /\
	 / \   /  \
	|   \ /    \
	|    \      |
	|   / \     |

2 in, 4 out!   Here is an element of Hom(4,0):

	|  |   |    |
	\  /   \ /\ /
	 \/     \  \
	       / \/ \
	       \____/

4 in, none out!  We can compose these morphisms to get a morphism in
Hom(2,0):

	|   |
	\   /
	 \ /
	  \     /\
	 / \   /  \
	|   \ /    \
	|    \      |
	|   / \     |
	|  |   |    |
	\  /   \ /\ /
	 \/     \  \
	       / \/ \
	       \____/


To be precise, a tangle is a 1-manifold X with boundary embedded in
[0,1] x R^2, such that boundary of X is mapped to the boundary of [0,1]
x R^2 and such that X intersects the boundary of [0,1] x R^2
transversally.  We also assume that the points in the boundary of X get
mapped to certain "standard" points (0,x_i) and (1,x_i) in the boundary
of [0,1] x R^2, so we can compose tangles by gluing them together as in
the picture above.  There is thus a category whose objects are
{0,1,2,....} and whose morphisms Hom(m,n) are isotopy classes of tangles
with m boundary points in {0}xR^2 and n boundary points in {1}xR^2.

Now it turns out that quantum gravity involves finding representations
of the category of tangles!  And it turns out that there is a way to get
a representation of the category of tangles from any finite-dimensional
representation of a semisimple Lie group.  This construction is due to
Reshetikhin and Turaev and involves quantum groups.  Try:

Turaev V G 1988 The Yang-Baxter equation and
invariants of links {\sl Invent.\ Math.\ }{\bf 92} 527

Reshetikhin N,  Turaev V 1990 Ribbon
graphs and their invariants derived from quantum groups
{\sl Comm.\ Math.\ Phys.\ }{\bf 127}  1

Turaev V G 1990 Operator invariants of tangles, and
R-matrices {\sl Math.\ USSR Izvestia} {\bf 35} 411

Reshetikhin N,  Turaev V 1991 Invariants of
3-manifolds via link-polynomials and quantum groups {\sl Invent.\
Math.\} {\bf 103}  547

For the quantum gravity application see my paper, and also

Br\"ugmann B, Gambini R, Pullin J 1992 Jones
polynomials for intersecting knots as physical states for quantum
gravity {\sl University of Utah preprint}

Crane L 1991 2-d physics and 3-d topology {\sl Comm.\Math.\ Phys.\ }{\bf
135}  615

-----
Someone asked me to explain first and second quantization.  In ten words
or less.  :-)  First quantization is a mystery.  It is the attempt to
get from a classical description of a physical system to a quantum
description of the "same" system.  Now it doesn't seem to be true that
God created a classical universe on the first day and then quantized it
on the second day.  So it's unnatural to try to get from classical to
quantum mechanics.  Nonetheless we are inclined to do so since we
understand classical mechanics better.   So we'd like to find a way to
start with a classical mechanics problem -- that is, a phase space and a
Hamiltonian function on it -- and cook up a quantum mechanics problem --
that is, a Hilbert space with a Hamiltonian operator on it.  It has
become clear that there is no utterly general systematic procedure for
doing so.

Mathematically, if quantization were "natural" it would be a FUNCTOR
from the category whose objects are symplectic manifolds (= phase spaces) and
whose morphisms are symplectic maps (= canonical transformations) to the
category whose objects are Hilbert spaces and whose morphisms are
unitary operators.  Alas, there is no such nice functor.  So
quantization is always an ad hoc and problematic thing to attempt.  A
lot is known about it, but more isn't.  That's why first quantization is
a mystery.

(By the way, I have seen many "no-go" theorems concerning quantization
but I have never seen one phrased quite like the above.  "There is no
functor from the symplectic category to the Hilbert category such that
...... holds."  Is anyone up to the challenge??  If this hasn't been done
yet it would clarify the situation.

Note that there IS a functor from the symplectic category to the
Hilbert category, namely one assigns to each symplectic manifold X the
Hilbert space L^2(X), where one takes L^2 w.r.t. the Liouville measure.
Every symplectic map yields a unitary operator in an obvious way.
This is called PREQUANTIZATION.  The problem with it physically is that
a one-parameter group of symplectic transformations generated by a
positive Hamiltonian is not mapped to a one-parameter group of unitaries
with a POSITIVE generator.  So my conjecture is that there is no
"positivity-preserving" functor from the symplectic category to the
Hilbert category.)

Second quantization is the attempt to get from a quantum description of
a single-particle system to a quantum description of a many-particle
system.  (There are other ways to think of it, but let's do it this
way.)  Starting from a Hilbert space H for the single particle system,
one forms the symmetric (or antisymmetric) tensor algebra over H and
completes it to form a Hilbert space K, called the bosonic (or
fermionic) FOCK SPACE over H.  Any unitary operator on H gives a unitary
operator on K in an obvious way.  More generally, one has a functor
called "second quantization" from the Hilbert category to itself, which
sends each Hilbert space to its Fock space, and each unitary map to an
obvious unitary map.  This functor *is* positivity-preserving.  (All the
weird problems with negative-energy states of the electron, Dirac's
"holes in the electron sea," and such, are due to thinking about things
the wrong way.)

Newsgroups: sci.physics,sci.math
Subject: Tangles
From: jbaez@riesz.mit.edu (John C. Baez)
Date: Thu, 8 Oct 92 03:46:04 GMT

Before I finally get around to revealing what a 2-category is and what
they might have to do with quantum gravity, I thought I should talk a
bit about the category of tangles, because it is so utterly beautiful.

I'm not going to be very formal, so anyone who wants the rigorous
details should take a look at

Yetter D N 1988 Markov algebras, in Braids, Contemp. Math. 78, 705.

Turaev V G 1990 Operator invariants of tangles, and
R-matrices, Math.\ USSR Izvestia 35 411.

I will just say that a tangle is a bunch of strands connecting
n points on the ceiling to m points on the floor, possibly with a bunch
of knots thrown in the middle:



	 |   |
	\   /
	 \ /
	  \     /\
	 / \   /  \
	/   \ /    \
       /     \      \
      /     / \      |
      \    /   \     |
       |  /     \ /\ /
      /   \      \  \
     /     |    / \/ \
    /      |    \____/
   |       |
   |       |

For us two tangles will be the same (technically, "isotopic") if one
can be deformed into the other; i.e., we think of the strands as being
infinitely flexible and are allowed to wiggle them around but not move
them over the ceiling or under the floor; we aren't allowed to move
the places where the strands touch the ceiling and floor, though.

Note that knots, links and braids are all special cases of tangles.
Tangles are great because the provide a nice algebraic structure to
study all of these things.

We say the above tangle is in Hom(2,2) because there are 2 points on the
ceiling and 2 on the floor.  Here is an element of Hom(2,4):

	 |   |
	\   /
	 \ /
	  \     /\
	 / \   /  \
	|   \ /    \
	|    \      |
	|   / \     |

and here is an element of Hom(4,0):

	|  |   |    |
	\  /   \ /\ /
	 \/     \  \
	       / \/ \
	       \____/

Note that we can "compose" these tangles to get one in Hom(2,0):

	 |   |
	\   /
	 \ /
	  \     /\
	 / \   /  \
	|   \ /    \
	|    \      |
	|   / \     |
	|  |   |    |
	\  /   \ /\ /
	 \/     \  \
	       / \/ \
	       \____/

In Hom(n,n) there is an "identity" tangle which is just a bunch of
vertical strands:

      |   |   |
      |   |   |
      |   |   |
      |   |   |
      |   |   |
      |   |   |

and if you compose any tangle x with the identity on the right or left you
get x again.  This, together with the associativity of composition, is
all we mean by saying that tangles form a category.

But we can also take the tensor product of two tangles.  The tensor
product of

	 |   |
	\   /
	 \ /
	  \     /\
	 / \   /  \
	|   \ /    \
	|    \      |
	|   / \     |

and

	|  |   |    |
	\  /   \ /\ /
	 \/     \  \
	       / \/ \
	       \____/

is

	 |   |		   |  |   |    |
	\   /		   \  /   \ /\ /
	 \ /		     \/     \  \
	  \     /\		     / \/ \
	 / \   /  \		    \____/
	|   \ /    \
	|    \      |
	|   / \     |

MP: all this is reminiscent of Penrose's spin-networks, at least
the diagrammatic representations of algebraic compositions, to me

Notice that the tensor product is associative and that composition
and tensor product satisfy certain obvious identities (just like the
identities that hold for tensor product and composition of linear maps
between vector spaces).

By now the physicists must be wondering how mathematicians get paid to
play around with this sort of thing.  I will try to head off such rude
remarks by noting that another name for tangles would be "Feynman
diagrams".  Of course Feynman diagrams are *labeled* tangles - the strands
carry spin, momentum, and other quantum numbers.  Also Feynman diagrams
have vertices, which tangles don't.  Also Feynman diagrams don't really
live in 3-dimensional space.  Nonetheless, there is a real relationship.
For starters, if we were going to talk about labeled tangles with
vertices we would be working with a generalization that has been studied
by Reshetikhin and Turaev in their paper:

Reshetikhin N,  Turaev V 1990 Ribbon
graphs and their invariants derived from quantum groups,
Comm. Math. Phys. 127, 1.

Their strands are labeled by representations of quantum groups and they
get nice topological invariants this way.

Anyway, here's the question of the day: how do we describe the category
of tangles.  Well, Turaev and Yetter both showed that it can be
described by generators and relations almost like a group can.  The
generators are as follows.  First, the identity 1 in Hom(1,1):

   |
   |
   |
   |
   |

Second, the basic "right-handed crossing" r in Hom(2,2):

 \   /
  \ /
   /
  / \
 /   \

and the left-handed crossing r^{-1} in Hom(2,2):

 \   /
  \ /
   \
  / \
 /   \

(Note that rr^{-1} and r^{-1}r are both the identity in Hom(2,2), so
the names are appropriate.)

Third, the "cup" in Hom(2,0):

 \      /
  \    /
   \  /
    \/

and the "cap" in Hom(0,2):

    /\
   /  \
  /    \
 /      \

These are really primordial things!  In what sense do they "generate"
the category of tangles.  Well, any tangle can be formed from these guys
by taking tensor products and composites.  For example, this guy

	 |   |
	\   /
	 \ /
	  \     /\
	 / \   /  \
	|   \ /    \
	|    /      |
	|   / \     |

can be written as

     (1 x r x 1)(r^{-1} x cap)

where I'm using x for tensor product and juxtaposition for composition.

Okay, so those are the generators (if you don't believe me, prove it!).
What are the relations?  This is the cool part.  Of course, there are a
bunch of relations which just come from the properties of the tensor
product and composition.  Tensor product and composition satisfy these
relations in any "monoidal category", but what we want are the relations
special to the category of tangles.  They were figured out by Turaev and
Yetter, who actually came up with slightly different, but equivalent,
sets of relations.  As I like them, they are as follows... I'll draw
them rather than write them as formulae:


 \   /       \   /       |    |
  \ /	 \ /	|    |
   /	   \	 |    |
  / \	 / \	|    |
 /   \    =  /   \    =  |    |
 \   /       \   /       |    |
  \ /	 \ /	|    |
   \	   /	 |    |
  / \	 / \	|    |
 /   \       /   \       |    |

(which just says that r^{-1} really lives up to its name, and is also
known as the 2nd Reidemeister move)


\   /    |	|     \ /
 \ /     |	|      \
  \      |	|     / \
 / \     |	|    /   \
/   \    /	\   /    |
|     \ /	  \ /     |
|      \     =	  \      |
|     / \	  \     |
|    /	 \	 /  \   /
\   /    |	|    \ /
 \ /     |	|     \
  \      |	  |    / \
 / \     |	|   /   \
/   \    |	|  /     \

(which is the 3rd Reidemeister move, also known as the Yang-Baxter
equation)

|       /\	   /\       |       |
|      /  \	 /  \      |       |
|     /    \       /    \     |       |
\    /      |  =  |      \    /  =    |
 \  /       |     |       \  /	|
  \/	|     |	\/	 |


   |	   |     /\	  |     /\
   |	   |    /  \	 |    /  \
   |	   \   /    \	\   /    \
   |	    \ /     |	 \ /     |
   |    =	\      |    =     /      |
   |	    / \     |	 / \     |
   |	   /   \    /	/   \    /
   |	   |    \  /	 |    \  /
   |	   |     \/	  |     \/
   |	   |		 |

(which is called the 1st Reidemeister move)

and then 4 closely related identities:

|     \   /      \   /     |
|      \ /	\ /      |
|       \	  /       |
|      / \	/ \      |
|     /   \  =   /   \     |
\    /    |     |     \    /
 \  /     |     |      \  /
  \/      |     |       \/


|     \   /      \   /     |
|      \ /	\ /      |
|       /	  \       |
|      / \	/ \      |
|     /   \  =   /   \     |
\    /    |     |     \    /
 \  /     |     |      \  /
  \/      |     |       \/


  /\      |     |      /\
 /  \     |     |     /  \
/    \    |     |    /    \
|     \   /     \   /     |
|      \ /       \ /      |
|       /    =    \       |
|      / \       / \      |
|     /   \     /   \     |


  /\      |     |      /\
 /  \     |     |     /  \
/    \    |     |    /    \
|     \   /     \   /     |
|      \ /       \ /      |
|       \    =    /       |
|      / \       / \      |
|     /   \     /   \     |


If you want to see these identities written algebraically, and you have
LaTeX, give this a try:

\def\tensor{\otimes}
\begin{array}  rr^{-1} &=& r^{-1}r \quad =\quad | \tensor | \nonumber\cr
  (| \tensor \cup)(\cap \tensor |) &=& (\cup \tensor |)(| \tensor
\cap) \quad =\quad |  \nonumber\cr
  (r \tensor |)(| \tensor r)(r\tensor |) &=& (| \tensor r)(r \tensor
|)(1 \tensor r) \nonumber\cr
(| \tensor \cup)(r \tensor |) &=& (\cup \tensor |)(| \tensor r^{-1})
\nonumber\cr
(| \tensor \cup)(r^{-1} \tensor |) &=& (\cup \tensor |)(| \tensor r)
\nonumber\cr
(r \tensor |)(| \tensor \cap) &=& (| \tensor r^{-1})(\cap \tensor |)
\nonumber\cr
(r^{-1} \tensor |)(| \tensor \cap) &=& (| \tensor r)(\cap \tensor |)
\nonumber\cr
(| \tensor \cup)(r \tensor |)(| \tensor \cap) &=& (| \tensor \cup)
(r^{-1} \tensor |)(| \tensor \cap = | \tensor |.\nonumber\end{array}

To see the power of these identities (together with the rules satisfied
by tensor product and composition), use them to deduce:

	   |     /\	    /\     |
	   |    /  \	  /  \    |
	   \   /    \	/    \   /
	    \ /     |       |      \ /
	     \      |    =  |       \
	    / \     |       |      / \
	   /   \    /	\    /   \
	   |    \  /	  \  /    |
	   |     \/	    \/     |
	   |			  |

There are endless fun and games to be had with these rules, which
encode a lot of the topology of 3-dimensional space.  Try Kauffman's
book "Knots and Physics" for some of this fun.  It may be surprising,
but it shouldn't be, that the category of tangles and its
representations constitutes a big hunk of conformal field theory (hence
string theory).  It also is practically the SAME THING as Chern-Simons
field theory (a 3-dimensional topological quantum field theory) and, I
attempt to show in my paper "Quantum Gravity and the Algebra of
Tangles," they have a lot to do with 4-dimensional quantum gravity.