From: jbaez@zermelo.mit.edu (John C. Baez)
Subject: Re: Symmetries, groups, and categories
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Fri, 14 Aug 92 19:56:51 GMT

In article <4537@news.duke.edu> dmrrsn@math.duke.edu (David R. Morrison) writes:
>In article <1992Aug13.182411.11593@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes:
>>(In practice various sneaky things - infrared slavery and Higgses -
>>conspire to cloak the conformal invariance of the fields in these
>>cases, so the practiclal relevance of conformal symmetry is limited.)

>OK, John, I'll bite:  what's "infrared slavery"?

I will give a brief explanation since probably someone else can give a
better one.  The strong force so important in nuclei is explained in the
standard model as being due to the exchange of gluons.  Gluons are
vector bosons for a gauge field theory with gauge group SU(3), so-called
"chromodynamics".  As a gauge field they would classically satisfy the
Yang-Mills equations, which are conformally invariant.  But we never see
free gluons zipping through empty space like photons, so it's hard to
notice the conformal invariance of the Yang-Mills equations.  Why do we
never see free gluons?  They call it "infrared slavery".  Quantum
effects SEEM to ensure that in SU(3) gauge theory one will never find a
free "colored" particle, that is, one which carries the SU(3) charge.
That's the "explanation" for why we never see quarks.  But since
chromodynamics is a nonabelian gauge theory (the group SU(3) is
nonabelian) the gauge fields are themselves charged.  (Unlike photons
which are uncharged and arise from an abelian theory, with gauge group
U(1).)  So we never see free gluons either.  We just see mesons and
baryons, which are colorless (=zero net charge) composites of quarks and
gluons.  

Why is infrared slavery - or "confinement" - a consequence of SU(3)
quantum gauge field theory?  This is one of those funny things due to
renormalization that nobody quite understands yet.  (Many people
understand it better than me, though.)  There is no "proof" of
confinement, indeed for mathematical physicists this would be the holy
grail of constructive quantum field theory.  There are numerical
calculations that seem to show that confinement occurs when you
simulate chromodynamics.  In any event, the crude story they always tell
beginners is that colored particles in chromodynamics attract each other
with a force that is *independent of distance*, as opposed to the
good-old inverse square law.  This is because the gluons carrying the
force tend to form a "string" joining the two particles, rather than
radiating out in all directions like photons do.  (Why?  Well, that's
confinement.  :-))  So to pull apart the quark-antiquark pair in a meson
requires an energy proportional to how far you've pulled them apart,
like sticky taffy.  :-)  Past a certain point the energy is enough so
that it takes less energy just to create a new quark-antiquark pair:

                  quark----------antiquark

becomes

            quark---antiquark  quark---antiquark

So we never see lone quarks (or gluons).

A better explanation involves an effective action for the gluon fields
that one gets by doing renormalization theory, but this is technical and
I forget enough details so that I don't dare trying to run through it.

I should warn the reader of various cool possibilities, such as
"glueballs" colorless composites made solely of gluons -- which have not
been observed, but could (for all of our lousy understanding of
chromodynamics) possibly exist.   Also, at very high relative momenta,
i.e. at very short distance scales, quarks and gluons ARE approximated
well by free particles.  This is so-called "asymptotic freedom" - the
opposite of "infrared slavery".  Some experimentalist can remind us when
they first found evidence for this, which was a big boost for Feynman's
"parton" model.

Another cool thing is that superdense superheated plasmas of baryons and
mesons can get so hot that the quarks become effectively free... I think
Scott is working on torturing particles in this manner.

>>We can think of this as a spacetime where two 1-dimensional circular
>>universes collide and form one!  Or we can think of it as a tubular
>>Feynman diagram.  Weird, huh?  
>
>Gee, John, I hate to point this out but the tubular Feynman diagram
>interpretation leads to string theory!  I'm happy with it, being a 
>string theory fan myself, but from you...?

I should make clear that there are only certain things about string
theory that bug me, aside from the fact that people make such a big deal
about it.  One is that traditionally it has treated strings wiggling
around in a background spacetime with a given metric.  (So that the
tubular Feynman diagrams are really surfaces IN, say, Minkowski space or
some 26-dimensional analog thereof.)  This is rather unfortunate if one
is advocating string theory as an explanation of gravity, since in
gravity it is the dynamics of the metric one is trying to explain!
Writing string field actions that depend on a background metric sounds
awfully like trying to do quantum gravity perturbing around a flat
background metric.  Maybe a string fan can say what the current state of
the art is.

In any event, in 2d TQFTs the tubular Feynman diagrams just represent
the 2d spacetime itself, not the worldsheet of a string floating around
"in" some spacetime.  But there are many mathematical relationships
between string theory and TQFTs and that's why these days I think there
is something to some of the *math* of string theory.  Also in the loop
representation of quantum gravity one quickly runs into Chern-Simons
theory, Wess-Zumino-Witten models, fusion rules and other pieces of
jargon invented by string theorists.