2-Hilbert spaces
marcus said:
Did JB already discuss twoHilberts in one of those longer papers that I shrink from printing out for fear of total bafflement?
Higher-dimensional algebra II: 2-Hilbert spaces
http://arxiv.org/abs/q-alg/9609018"
I explained the basic idea in http://math.ucr.edu/home/baez/week99.html" .
Here's what I wrote... but you like punchlines, perhaps I should start
with the punchline:
When physicists do Feynman path integration - just like a shepherd
counting sheep - they are engaged in a process of decategorification!
They get a mere number saying the amplitude for something to happen,
when they could get a Hilbert space of ways for things to happen.
This amplitude is the inner product in a Hilbert space; the Hilbert space
they could have gotten would be the inner product in a 2-Hilbert space.
Mindblowing, huh?
In other words: the complex numbers are just a shrunken,
miniaturized version of the category of Hilbert spaces, and similarly
the category of Hilbert spaces is just a shrunken, miniaturized version
of the 2-category of 2-Hilbert spaces... and so on, ad infinitum. This
"shrinking down" is called
decategorification: it happens whenever
you pretend isomorphic things are equal. I believe that to do math and
physics wisely, we need to undo this pretense.
Anyway, here's what I wrote:
I want to say a bit about what category theory has to do with quantum
mechanics!
First remember the big picture: n-category theory is a language to
talk about processes that turn processes into other processes.
Roughly speaking, an n-category is some sort of structure with
objects, morphisms between objects, 2-morphisms between morphisms,
and so on up to n-morphisms. A 0-category is just a set, with its objects
usually being called "elements". Things get trickier as n increases.
For a precise definition of n-categories for n = 1 and 2, see "week73"
and "week80", respectively.
Most familiar mathematical gadgets are sets equipped with extra bells
and whistles: groups, vector spaces, Hilbert spaces, and so on all
have underlying sets. This is why set theory plays an important role
in mathematics. However, we can also consider fancier gadgets that
are *categories* equipped with extra bells and whistles. Some of the
most interesting examples are just "categorifications" of well-known
gadgets.
For example, a "monoid" is a simple gadget, just a set equipped with
an associative product and multiplicative identity. An example we all
know and love is the complex numbers: the product is just ordinary
multiplication, and the multiplicative identity is the number 1.
We may categorify the notion of "monoid" and define a "monoidal
category" to be a *category* equipped with an associative product and
multiplicative identity. I gave the precise definition back in
"week89"; the point here is that while they may sound scary, monoidal
categories are actually very familiar. For example, the category of
Hilbert spaces is a monoidal category where the product of Hilbert
spaces is the tensor product and the multiplicative identity is C, the
complex numbers.
If one systematically studies categorification one discovers an
amazing fact: many deep-sounding results in mathematics are just
categorifications of stuff we all learned in high school. There is a
good reason for this, I believe. All along, mathematicians have been
unwittingly "decategorifying" mathematics by pretending that
categories are just sets. We "decategorify" a category by forgetting
about the morphisms and pretending that isomorphic objects are equal.
We are left with a mere set: the set of isomorphism classes of
objects.
I gave an example in "week73". There is a category FinSet whose
objects are finite sets and whose morphisms are functions. If we
decategorify this, we get the set of natural numbers! Why? Well, two
finite sets are isomorphic if they have the same number of elements.
"Counting" is thus the primordial example of decategorification.
I like to think of it in terms of the following fairy tale. Long ago, if
you were a shepherd and wanted to see if two finite sets of sheep were
isomorphic, the most obvious way would be to look for an isomorphism.
In other words, you would try to match each sheep in herd A with a
sheep in herd B. But one day, along came a shepherd who invented
decategorification. This person realized you could take each set and
"count" it, setting up an isomorphism between it and some set of
"numbers", which were nonsense words like "one, two, three, four,..."
specially designed for this purpose. By comparing the resulting
numbers, you could see if two herds were isomorphic without explicitly
establishing an isomorphism!
According to this fairy tale, decategorification started out as the
ultimate stroke of mathematical genius. Only later did it become a
matter of dumb habit, which we are now struggling to overcome through
the process of "categorification".
Okay, so what does this have to do with quantum mechanics?
Well, a Hilbert space is a set with extra bells and whistles, so maybe
there is some gadget called a "2-Hilbert space" which is a *category*
with analogous extra bells and whistles. And maybe if we figure out
this notion we will learn something about quantum mechanics.
Actually the notion of 2-Hilbert space didn't arise in this
simple-minded way. It arose in some work by Daniel Freed on
topological quantum field theory:
5) Higher algebraic structures and quantization, by Dan Freed,
Comm. Math. Phys. 159 (1994), 343-398, preprint available as
hep-th/9212115; see also week48.
Later, Louis Crane adopted this notion as part of his program to
reduce quantum gravity to n-category theory:
6) Louis Crane: Clock and category: is quantum gravity algebraic?,
Jour. Math. Phys. 36 (1995), 6180-6193, preprint available as
gr-qc/9504038.
These papers made is clear that 2-Hilbert spaces are interesting
things and that one should go further and think about "n-Hilbert
spaces" for higher n, too. However, neither of them gave a precise
definition of 2-Hilbert space, so at some point I decided to do this.
It took a while for me to learn enough category theory, but eventually
I wrote something about it:
7)
John Baez, Higher-dimensional algebra II: 2-Hilbert spaces,
to appear in Adv. Math., available at q-alg/9609018.
To understand this requires a little category theory, so let
me explain the basic ideas here.
I'll concentrate on finite-dimensional Hilbert spaces, since the
infinite-dimensional case introduces extra complications. To define
2-Hilbert spaces we need to start by categorifying the various
ingredients in the definition of Hilbert space. These are: 1) the
zero element, 2) addition, 3) subtraction, 4) scalar multiplication,
and 5) the inner product. The first four have well-known categorical
analogs. The fifth one, which is really the essence of a Hilbert
space, may seem a bit more mysterious at first, but as we shall see,
it's really the key to the whole business.
1) The analog of the zero vector is a `zero object'. A zero object in
a category is an object that is both initial and terminal. That is,
there is exactly one morphism from it to any object, and exactly one
morphism to it from any object. Consider for example the category
Hilb having finite-dimensional Hilbert spaces as objects, and linear
maps between them as morphisms. In Hilb, any zero-dimensional Hilbert
space is a zero object.
Note: there isn't really a unique zero object in the "strict" sense of
the term. Instead, any two zero objects are canonically isomorphic.
The reason is that if you have two zero objects, say 0 and 0', there
is a unique morphism f: 0 -> 0' and a unique morphism g: 0' -> 0.
These morphisms are inverses of each other so they are isomorphisms.
Why are they inverses? Well, fg: 0 -> 0' must be the identity
morphism 1_0: 0 -> 0, because there is only one morphism from 0 to 0!
Similarly, gf is the identity on 0'. (Note that I am using category
theorist's notation, where the composite of f: x -> y and g: y -> z is
denoted fg: x -> z.)
This is typical in category theory. We don't expect stuff to be
unique; it should only be unique up to a canonical isomorphism.
2) The analog of adding two vectors is forming the "coproduct" of two
objects. Coproducts are just a fancy way of talking about direct
sums. Any decent quantum mechanic knows about the direct sum of
Hilbert spaces. But in fact, we can define this notion very generally
in any category, where it goes under the name of a "coproduct". (I
give the definition below; if I gave it here it would scare people
away.) As with zero objects, coproducts are typically not unique, but
they are always unique up to canonical isomorphism, which is what
matters. It's a good little exercise to show this.
3) The analog of subtracting vectors is forming the "cokernel" of a
morphism f: x -> y. If x and y are Hilbert spaces, the cokernel of f
is just the orthogonal complement of the range of f. In other words,
for Hilbert spaces we have "direct differences" as well as direct
sums. However, the notion of cokernel makes sense in any category
with a zero object. I won't burden you with the precise definition
here.
An important difference between zero, addition and subtraction and
their categorical analogs is that these operations represent extra
*structure* on a set, while having a zero object, coproducts of two
objects, or cokernels of morphisms is merely a *property* of a
category. Thus these concepts are in some sense more intrinsic to
categories than to sets. On the other hand, we've seen one pays a
price for this: while the zero element, sums, and differences are
unique in a Hilbert space, the zero object, coproducts, and cokernels
are typically unique only up to canonical isomorphism.
4) The analog of multiplying a vector by a complex number is tensoring
an object by a Hilbert space. Besides its additive properties (zero
object, binary coproducts, and cokernels), Hilb is also a monoidal
category: we can multiply Hilbert space by tensoring them, and there
is a multiplicative identity, namely the complex numbers C. In
fact, Hilb is a "ring category", as defined by Laplaza and Kelly.
We expect Hilb it to play a role in 2-Hilbert space theory analogous
to the role played by the ring C of complex numbers in Hilbert space
theory. Thus we expect 2-Hilbert spaces to be "module categories"
over Hilb, as defined by Kapranov and Voevodsky.
An important part of our philosophy here is that C is the primordial
Hilbert space: the simplest one, upon which the rest are modeled. By
analogy, we expect Hilb to be the primordial 2-Hilbert space. This is
part of a general pattern pervading higher-dimensional algebra; for
example, there is a sense in which the (n+1)-category of all (small)
n-categories, nCat, is the primordial (n+1)-category. The real
significance of this pattern remains mysterious.
5) Finally, what is the categorification of the inner product in a
Hilbert space? It's the `hom functor'! The inner product in a
Hilbert space eats two vectors v and w and spits out a complex number
<v,w>
Similarly, given two objects v and w in a category, the hom functor
gives a *set*
hom(x,y)
namely the set of morphisms from x to y. Note that the inner product
<v,w> is linear in w and conjugate-linear in y, and similarly, the hom
functor hom(x,y) is covariant in y and contravariant in x. This hints
at the category theory secretly underlying quantum mechanics. In
quantum theory the inner product <v,w> represents the *amplitude* to
pass from v to w, while in category theory hom(x,y) is the *set* of
ways to get from x to y. In Feynman path integrals, we do an integral
over the set of ways to get from here to there, and get a number, the
amplitude to get from here to there. So when physicists do Feynman
path integration - just like a shepherd counting sheep - they are engaged
in a process of decategorification!
To understand this analogy better, note that any morphism f: x -> y in
Hilb can be turned around or "dualized" to obtain a morphism f*: y -> x.
This is usually called the "adjoint" of f, and it satisfies
<fv,w> = <v,f*w>
for all v in x, and w in y. This ability to dualize morphisms is
crucial to quantum theory. For example, observables are represented
by self-adjoint morphisms, while symmetries are represented by unitary
morphisms, whose adjoint equals their inverse.
However, it should now be clear - at least to the categorically minded -
that this sort of adjoint is just a decategorified version of the
"adjoint functors" so important in category theory. As I explained in
"week79", a functor F*: D -> C is a "right adjoint" of F: C -> D if
there is, not an equation, but a natural isomorphism
hom(Fc,d) ~ hom(c,F*d)
for all objects c in C, and d in D.
Anyway, in the paper I proceed to use these ideas to give a precise
definition of 2-Hilbert spaces, and then I prove all sorts of stuff
which I won't describe here.
Let me wrap up by explaining the definition of "coproduct". This is
one of those things they should teach all math grad students, but for
some reason they don't. It's a bit dry but it'll be good for you. A
coproduct of the objects x and y is an object x+y equipped with
morphisms
i: x -> x+y
and
j: y -> x+y,
that is universal with respect to this property. In other words,
if we have any *other* object, say z, and morphisms
i': x -> z
and
j': y -> z,
then there is a unique morphism f: x+y -> z such that
i' = if
and
j' = jf.
This kind of definition automatically implies that the coproduct is
unique up to canonical isomorphism. To understand this abstract
nonsense, it's good to check that the coproduct of sets or topological
spaces is just their disjoint union, while the coproduct of vector
spaces or Hilbert spaces is their direct sum.
- so people keep rediscovering it...
