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Re: Queries about Species
Posted:
Jan 8, 2003 3:53 AM
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In article <av8rch$bki$1@news.ox.ac.uk>,
David Corfield <david.corfield@philosophy.oxford.ac.uk> wrote:
>I have a few questions about combinatorics and species, and would be
>grateful for any comments.
I'll take another crack at these now. By the way, you might have gotten
more replies if you'd phrased your question in terms of *generating
functions*, rather than *species*. Category theory is great, but most
people don't know this, so when trying to extract information from
them it's often best to play down this aspect of things, emphasizing
other buzzwords.
>X/(1 - e^X) looks like a simple composition of species - pick out a one
>element set and arrange a set of sets whose union is the remainder - yet
>it can't be that simple to get at the Bernoulli numbers. I guess lots of
>unwanted empty sets appear in the union. This would no doubt require
>some of Cartier's mathemagics to make sense of it.
Okay, I understand this remark now that you've expanded it...
and resummed to get a finite answer.
I'm getting more and more confused and tantalized by the relation
between Bernoulli numbers, species, statistical mechanics and quantum
theory. The reason I mentioned Connes' remarks is that he was
emphasizing the relation between Bernoulli numbers and *partition
functions*. Partition functions in statistical mechanics and
quantum theory are often just another way of looking at generating
functions of species. If you haven't thought about this yet, I can
perhaps shed a little light on what's going on - though not enough
to fully penetrate all the murk, yet.
If you have a system with allowed energy levels E_n, the probability
that it's in the nth one is proportional to
exp(-b E_n)
where b, usually written "beta", is the inverse temperature -
or really 1/kT where k is Boltzmann's constant and T is the
temperature. To get probabilities we need to normalize these
numbers by dividing by the "partition function"
Z(b) = sum_n exp(-b E_n)
You can do lots of great stuff in statistical mechanics if you
know the partition function of a system, mainly by differentiating
it with respect to b and playing various games.
Now, if we don't count the "zero-point energy", the harmonic
oscillator in quantum mechanics has allowed energy levels
0,1,2,3,4,... , so its partition function is
Z(b) = sum_n exp(-nb)
1
= -------------
1 - exp(-b)
This is where Connes' remark about Bernoulli numbers
comes in. The above partition function blows up as b -> 0
(the infinite temperature limit, where all energy levels
become equally probable). In fact it has a first-order pole
at b = 0. To deal with this, we can work with
b
b Z(b) = -------------
1 - exp(-b)
This is analytic at b = 0, and its Taylor series defines the
Bernoulli numbers, up to some signs:
b Z(b) = sum B_n (-x)^n / n!
What does all this have to do with species? I'm not sure.
But it's tantalizing, for two reasons! In my work with
Jim Dolan, the harmonic oscillator itself is categorified:
its Hilbert space becomes the category of species, and its
Hamiltonian (the "number operator") becomes a functor from
this category to itself. In your post, you try to interpret
B_n as the cardinality of a set of structures on the n-element
set. You get get a divergent series, basically the Riemann
zeta function evaluated at a negative integer, and you resum
this to get the right answer. This trick is common in quantum
field theory under the name of "zeta function regularization".
So, there seem to be two possible connections between
Bernoulli numbers and species via physics, but they don't
fit together neatly, and neither one is fully worked out -
though yours comes a lot closer.
To add to the excitement and murk read this:
http://www.maths.ex.ac.uk/~mwatkins/zeta/physics2.htm
On a different note:
>Now, if a species is a categorification of an element of N[[x]], why are,
>Fiore and Leinster interested in categorifying N[x]/ (some polynomial in x),
>where the polynomial is a relation for the species?
Yikes... another twisted loop of ideas?
Here's all I can say:
We can indeed define "multivariable species" that serve as a
categorification of rigs like N[[x_1,...,x_n]]. To do this, just define an
"n-variable species" to be a type of structure that we can put
on an ordered n-tuples of finite sets. In other words, it's a functor
F: FinSet_0 x ... FinSet_0 -> Set
where FinSet_0 is the groupoid of finite sets and bijections,
and we take an n-fold Cartesian product of this category.
So, yes, we could define "complex species" whose generating
functions have complex coefficients by starting with two-variable
species and then imposing Fiore and Leinster's equation on one
variable to make that variable act like "i".
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