Chers amis,

> The evaluation (hopefully positive) of the Institute by the international
> commission 

good news !


> is luckily over ...
>
> 1A. In order to speak about Hopf algebras I need to explain what the
> coproduct
> is. People who try to write something on Hopf algebras for physicists
> introduce coproduct in many ways - for example Brouder writes:
> 
> " it is possible to see the coproduct as giving all ways to split a
> differential operators D  into two operators D1 and D2 such that D1.D2 = D"
> 
> For me much clearer is the definition based on the duality concepts, i.e.
> introducing the coproduct as a dual to multiplication - such an approach
> (through representations) Gerard proposed in the Myczkowce paper.

this is exactly my feeling. Even following Brouder's choice, the best formulation 
would NOT be to split the product but to explain the scheme of how such an operator 
acts on the product of two functions. For example

coproduct(D)=D\otimes 1 + 1\otimes D

for a (first order) derivative

coproduct(D)=D\otimes 1 + \sum D_i\otimes D_j +1\otimes D

for a second order etc ...

> 1B. Let's think about the coproduct as a dual to multiplication: is it
> correct
> to formulate the following statement?
> 
> Assumptions:
> In quantum field theory, or more precisely in the quantum scattering theory,
> we have two notions:
> 
> --- formulas defining probability amplitudes of some physical processes,
> given
> by multiple integrals, each built from standard blocks, which we know how to
> write down using a priori given Feynman rules,
> 
> --- (Feynman) graphs which are drawings being in one to one correspondence
> with the integrals just mentioned. These drawings have a nice feature that
> they may be interpreted as pictures of real processes (up to
> some nonstandard notions, like "virtual" particles, particles moving
> backwards
> in time, etc.). The latter (sometimes) makes physical considerations easier.
> 
> Thesis:
> Two just described concepts may be treated as duals, in the sense that
> Feynman
> integrals are dual to the Feynman graphs and we can ask how manipulations on
> graphs are mirrored on integrals and vice-versa. In such a language  the
> Feynman integrals are "representation" of the Feynman graphs.
> 
> Comment: Renormalization provides us
> with an example of such a procedure - the scheme is consistent if one is able
> to show how some manipulations done on simple building blocks lead to well
> defined expressions in more complicated cases. Manipulations may be done and
> explained on graphs but these are integrals which must exist!
> 
> MY QUESTION IS - can we think in such a way?

for me it seems convincing. I am not a field theoretist (but would like to know more ...)
 
> 
> 1B. If the philosophy of "integrals-graphs" duality is reasonable then in the
> first step (and in maximally simplified model) one should look for a dual to
> concatenation of graphs. This is because concatenation of graphs expresses
> multiplication (of the second exponents) in the product formula, or rather
> the
> product formula itself because the second exponent with a series of V's is in
> fact a product of exponents with single V's in the exponential. If we expand
> all these exponetials then obviously resulting expression will contain
> concatenated graphs coming from all exponents. It is also clear that the
> product formula enforces us to treat the space spanned by graphs as a vector
> space with concatenation as multiplication.
> 
> 
> 2. Explanation of our graph philosophy was perfectly given by Allan and this
> I would like to repeat with an addition how graphs with arbitrary number of
> white and black spots come from the product formula depending on more L's
> then L1.
> 
> 3. In the "analytic language" all this may be rewritten using multivariate
> Bell polynomials and in fact is nothing more then the Faa-di Bruno formula.

"nothing more" as the eveluation is concerned (i.e. the result of the formula itself). 
But our diagrmmatic expansion is a sort of pull backof the product formula at the 
level of all incidences of two set-partitions.  

> I will try to write down these remarks in less chaotic form, with some
> general field theoretical background.
> 
> Best greetings
> Andrzej
>