I did follow a course on Hopf algebras, but that's already some time
ago. I will read the articles you mentioned, should be interesting!
B.t.w. Kreimer has also written some papers with David Broadhurst. He
has done some quite amazing work, see his homepage:
To answer your question about the nonexistence of ghosts, I would first
have to define what ghosts are. Simply put, a field theory is defined
by some path integral. You have to integrate over all field
configurations. By taking functional derivatives you can then compute
(in principle) all possible correlations. In some theories there are
complications because instead of the field, potentials are used. Fields
are certain derivatives of the potentials. Now, for a given field
configuration, there are many different potentials (they are related by
so-called gauge transformations). So, you can't just integrate over all
possible potential configurations. You must make sure that all field
configurations only appear once in the path integral. However, it is
very inconvenient to deal with such restrictions. Now, there exists a
certain trick to deal with this problem. One introduces certain (ghost)
fields in the path integral (that also have to be integrated over),
such that the integral over the potentials is unrestricted. The
integration over the ghosts takes care of the ''overcounting''.
If you now perform perturbative computations, you find the usual
Feynman rules plus some extra rules for the ghosts. They can appear
only in internal loops. The fields are anti-commutative, and yet have
spin zero. That seems impossible. Maybe you know the famous spin
statics theorem: half-integer spin particles are described by anti-
commuting fields, integer spin fields by commuting fields. This can be
derived by showing that if this were not the case you would violate
locality. So, the appearence of the ghosts seems to indicate that the
field theory is non-local, but that's false. The field theory can be
shown to be local. The ghosts are just a mathematical abstraction to
facilitate computations. This is reflected by the fact that they only
appear in internal loops. Thus, ghosts don't exist.
Maybe this analogy helps. First year students often get this problem:
Suppose you have a spherical planet of uniform density, with a cavity.
Suppose the cavity is also spherical. The radii of the planet and the
cavity are given, also the position of the center of the cavity w.r.t.
the center of the planet is given.
The problem is to compute the acceleration due to gravity (g) on the
surface of the planet. This will depend on the position on the surface,
Now, the following trick can be used: Thye planet with cavity is just a
superposition of the planet without cavity and a small planet with
negative density in the place of the cavity. Since the two objects are
spherical, you can immediately write down the result!
I think that everyone will agree that the appearance of negative
density, and thus of negative mass, in a formulae describing a real
physical situation as above, doesn't mean that negative mass objects
't Hooft's theory is one in which quantum mechanics also appears in an
analogous way as the ghosts in field theory. So, to answer Charles'
question, the Copenhagen interpretation means in this case that
although there is an underlying theory, this underlying theory says
nothing about elementary particles, because they don't exist. Just like
ghosts can do strange things in Feynman diagrams so can an electron go
through two slits at once. Ghosts don't exists. How do you know that an
electron does exist?
----- Origineel Bericht -----
Van: Marchal <[EMAIL PROTECTED]>
Datum: Woensdag, Februari 6, 2002 8:29 pm
Onderwerp: Re: Bell, Aspect & Copenhagen vs. MWI
> Hi Saibal,
> Your thesis is very interesting, as far as I understand it.
> I am sure you are right about the fact that renormalisation
> theory can put light ... on the question "what is a physical
> system and what is a physical simulation of a physical
> I see more and more my own Z1* as a sort of renormalisation
> on the computationalist indeterminacy (*Too* many worlds with
> comp). The quantum would be the result of renormalising
> classical comp indeterminacy!
> Do you know the Hopf algebra of renormalisation? (cf
> the Connes and Kreimer "Lessons from Quantum Field Theory"?
> My interest for Hopf algebra grew from the reading of the
> the Kitaev "fault-tolerant quantum computation by
> anyons" (quant-ph/9707021) (what crazy paper!), but your
> remark + your thesis + Connes and Kreimer adds still deeper
> BTW you said (to the FOR list):
> >In quantum field theory Feynman diagrams are used to perform
> >computations. Now, here it is clear from the start that virtual
> >and ghosts, as they appear in these diagrams, don't exist. Only
> >external lines represent real physical particles.
> What makes you so sure? I guess it's my incompetence in Quantum
> Field Theory, but this is not clear for me. I would appreciate
> a short comment on the MWI view of QFT, perturbation and
> >All this is very clear
> >because the Feynman rules are derived from a well defined theory
> that is
> >well understood.
> Ah ?