[last update 2000-2-4]




feynman diagrams and categorification



the mathematical formalism of feynman diagrams can be understood as
arising from the possibility of expressing physically important linear
operators as algebraic combinations of so-called "annihilation and
creation operators".  (i don't know enough about the history of
feynman diagrams to say to exactly what extent this is how the
formalism actually did arise, though.)

it's tempting to try to relate these creation and annihilation
operators (which in some representations can be seen as multiplication
and differentiation operators, respectively) to the categorified
multiplication and differentiation operators which act on so-called
combinatorial species [1], particularly since the whole idea of
feynman diagrams already suggests categorification (=, very roughly,
the progression/regression from the study of quantities to the study
of combinatorial objects counted by those quantities).

(the categorification idea implicit in the use of feynman diagrams
traces back in a way to einstein's re-interpretation of the ideas of
planck, who introduced discretization (that is the natural numbers in
contrast to the reals) but not categorification (that is the finite
sets in contrast to the natural numbers) into physics.  planck was (i
think just barely) willing to think in terms of a "ladder" with
regularly spaced rungs, but rejected einstein's idea that the rungs
represented the cardinalities of sets of "photons".  creation and
annihilation operators are operators which "step up" and "step down"
the ladder, respectively; their matrixes with respect to the "ladder"
basis have non-zero entries only along a diagonal next to the main
diagonal.)

however an attempt to carry out the above program runs into certain
difficulties, most of which seem to be related to the fact that the
de-categorification process from combinatorial species to power series
assigning to a monomial species x^m/g (where x is a formal variable, m
a finite set, and g a subgroup of m!) the de-categorified monomial
x^m/g (where now m is the cardinality of the finite set m and g the
cardinality of the finite group g) lacks sufficiently strong formal
properties.  in particular, preservation of substitution by the
de-categorification process, though holding somewhat generally, fails
in the crucial case where the substituted species is of zeroth degree.

with an eye towards repairing this failure, let's examine more closely
why it occurs.  substituting x^m/g into x^n/h means forming the
"wreath product" of the concrete groups (m,g>->m!) and (n,h>->n!).
that means taking the semi-direct product (call it "p") of h acting on
g^n, and letting it act in the hopefully obvious way on the cartesian
product mXn.  but in the special case where m=0, (mXn,p->[mXn]!) will
generally _not_ be a concrete group yet, because the group
homomorphism p->[mXn]! won't be faithful; thus to complete the
construction the group p must be replaced by it's image p' under the
homomorphism p->[mXn]!.  this last step is what spoils the
compatibility between substitution and de-categorification.

the apparent resolution of this difficulty is to change the formalism
so that the final step in the substitution process, the replacement of
p by p' that causes all the problems, is omitted.  thus we want to
enlarge the category of objects under study from just the groupoids
fibered faithfully over the groupoid of finite sets (that is,
essentially just the combinatorial species in the usual sense) to the
groupoids fibered over the groupoid of finite sets, faithfully or not.
(in the further development of the theory it may be desirable to
enlarge the category even further to include higher-dimensional
groupoids fibered over the groupoid of finite sets, or to go even
further; but it's probably unnecessary to consider such developments
for my present purposes.)

this new formalism is, according to my again imperfect understanding
of history, not actually that new.  as i understand it (based partly
on a conversation a long time ago with todd trimble, who's not
responsible if i misrembered what they said), kelly [2] considered
something like categories fibered over the category of categories
(possibly with the word "small" thrown in there somewhere) as
representing operations on categories built out of basic 2-limit and
2-colimit operations (with the 2-limits distributing over the
2-colimits in a certain way that generally wouldn't happen if
considered in some 2-category more general than the 2-category of
categories).  the groupoids fibered over the groupoid of finite sets
are included among kelly's objects, corresponding to operations on
groupoids built out of basic tensor product (actually just the usual
cartesian product of groupoids, but treated as just a tensor product
not assumed to be cartesian) and 2-colimit operations (with
distributivity of tensor product over 2-colimits).

thus the basic modification converting the formalism of combinatorial
species into the new formalism is the systematic replacement of
ordinary colimits by 2-colimits (perhaps it's better to say by
homotopy colimits, with an eye toward further developments).

now we define the "homotopy cardinality" of a connected groupoid to be
the reciprocal of the size of it's fundamental group, with additivity
under discrete sums to extend the definition to arbitrary groupoids.
the point of this concept of "homotopy cardinality" is to treat
groupoids (or at least those of finite homotopy cardinality) as
categorified positive real numbers, somewhat analogously to the way in
which the ordinary concept of cardinality amounts to a way of treating
finite sets as categorified natural numbers.  (though of course
groupoids of the same homotopy cardinality need not be equivalent, in
contrast to the way in which sets of the same cardinality must be
isomorphic.  perhaps that means that i'm stretching the meaning of the
concept "categorification" here, but i tend to consider it a concept
that needs some stretching anyway.)

consider now some well-known useful combinatorial species such as for
example the underlying combinatorial species of the commutative monoid
operad.  this is supposed to be the categorified analog of the
exponential function.  what a disappointment it is then to take the
free commutative monoid on a set of cardinality 1 and discover that
the cardinality of it's underlying set is not 2.718281828... .  no
such disappointment occurs when we examine the homotopy cardinality of
the underlying groupoid of the free symmetric monoidal groupoid on a
groupoid of homotopy cardinality 1 (for example the contractible
groupoid).

("underlying groupoid of the free symmetric monoidal groupoid on x" is
what the operation "underlying set of the free commutative monoid on
x" becomes when colimits are systematicly replaced by homotopy
colimits as discussed above.)

as in the above example, the process of de-categorification generally
behaves much more nicely in the new formalism than in the old
formalism of combinatorial species.  this means that we can, much more
easily than before, create de-categorified things with interesting
properties by first creating categorified things with analogous
properties and then systematicly de-categorifying.

for example, the simplest uses of feynman diagrams, to construct and
describe interesting linear operators on the fock space of a
1-dimensional hilbert space, can be analyzed in this way.  a groupoid
fibered over the groupoid of pairs of finite sets (the categorified
analog of a power series in 2 variables) can be treated as the
categorified "matrix" of a categorified linear operator on the
categorified "vector space" of groupoids fibered over the groupoid of
finite sets (the categorified analog of power series in 1 variable),
and the simplest kinds of feynman diagrams are basicly just pictures
of the objects in such a categorified-2-variable-power-series, which
when de-categorified becomes the "kernel" (if the 2-variable power
series is viewed as the taylor series of a 2-variable function) or
"matrix" (if treated formally) of a linear operator on the fock space.

i'd like to develop this program further, extending the formalism to
cover the fancier uses of feynman diagrams in physics, making contact
with other developments in mathematics and physics, and putting some
emphasis on the question of whether the occurrence in physics of
things with interesting categorifications might indicate that the
categorifications themselves have some sort of physical reality.



(sorry i didn't get around to describing a specific concrete example
of a "species of feynman diagrams" here; i may try to fix that
omission later.)






[1] joyal, ????

[2] kelly, ????