🚀 go-pugleaf

Here's a slightly reworked version of 2 recent posts of mine on
sci.math.

Quantum theory can be thought of as the generalization of classical
mechanics you get by dropping the assumption that observable quantities
like position and momentum commute.  In quantum theory one thus learns
to like noncommutative, but still associative, algebras.

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!  

If we try to generalize the heck out of the concept of a group, keeping
associativity as a sacred property, we get the notion of a category.
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.  So, let me explain that remark!

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.)

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)! 
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."

Physicists are entitled to regard this as a bunch of high-falutin'
abstract nonsense until I give them an example in which "functoriality"
really has more to it than just covariance under a group action.  The
point is, that a category is really a generalization of a group. 
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 just 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. 

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.

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

I should also cite Moore and Seiberg's influential paper on modular
tensor categories but I don't have the reference on me.

Okay, now back to MacLane's cryptic remark.  What's a natural
transformation?  Well, 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.  This example is a bunch of mumbo-jumbo
unless you know some algebraic topology, but I should be able to think
of a more physical one.

In weeks to come (maybe):

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