mass gap in Yang-Mills theories

Haelfix

There are no gravity terms (either mass terms or interactions) in the theory, why do you think it would need quantum gravity?

marlon

Why is that ??? Never heard of this, though...
marlon

Kea

Heisenberg said that particles were not fundamental, because every
particle in some sense contained all others. It appears that the
same should be true of Bekenstein's atoms for spacetime. There is
no physical difficulty in thinking of spacetime degrees of freedom
in a quantum manner. The difficulty arises in coupling matter and
spacetime degrees of freedom in a mathematically sensible way. It
appears that this is not at all possible unless one addresses some
basic issues in quantum logic.

Categorical internalisation is an essential element here. There is
mounting support for this point of view from studies of, for
instance, the Hopf algebra structure of renormalisation (see
Connes and Marcolli) and its connection with non-commutative
geometry.

Twistor theory is one investigation that attempted to respect
background independence, and which played an important role in the
development of state sum models for quantum gravity.

The first interesting step towards a modern category theoretic
understanding of mass is perhaps the study of the Klein-Gordon
equation in the Hughston and Hurd paper, in which they combine two
solutions to the massless equations for spin s particles thought
of as elements of a sheaf cohomology group on a twistor space. The
Klein-Gordon equation solutions then belong to a second cohomology
group.

Naively at least, therefore, a quantisation of this origin of mass
involves a non-Abelian sheaf theoretic second cohomology group. And
an understanding of such an object leads one inexorably in the
direction of topos cohomology.

The first cocycle condition may be thought of as a triangle. Such
triangles make sense in any category, so the coefficients for H1
may be generalised, in particular to non-Abelian groups. The
difficulty arises in understanding categories deeply enough to
develop a sufficiently subtle higher dimensional analogue.

The interplay of categories and logic (ie. topos theory) in
physics has already been carefully considered by Markopoulou in
the context of causal sets, and Isham and others in the context of
quantum theory.

A topological space is a category of objects the open sets, with
inclusions for arrows. For example, the celestial sphere of the
twistor correspondence is considered as such a category.

Already in two dimensions, Yang-Mills theory involves some
beautiful combinatorics (see Witten's work). This uses a
generalisation of the Abelian localisation principle from
equivariant cohomology. Localisation reaches a pinnacle of
abstraction in an adjunction between the inclusion of a topos of
sheaves into the presheaf category and the so-called
sheafification functor (see Mac Lane and Moerdijk). Sheaves are
defined with respect to a topology on the base category.

As the String theorists like to tell us, path integrals are
heinously complex and unsmooth things. They are now telling us
that maybe 4D Yang-Mills is pretty amazing all on its own. And
they seem to be saying that twistors are cool too.

In other words, we want a higher categorical analogue of the
evaluation of path integrals like 2D Yang-Mills.

The intended interpretation of pieces of categories is that they
are geometric entities. Objects are zero dimensional and arrows
are one dimensional etc. I won't go into this now.

Objects in a category such as Rep(SU(2)) are representation spaces
rather than 'particle states', so to capture the notion of a state
properly in category theoretic terms it is necessary to
internalise this picture further than is normally considered and
to replace the Mac Lane pentagon by at least its tricategorical
analogue. The truly fascinating thing is that tensor products in
higher dimensional categories are no longer stable dimensionally.
For the pentagon this leads to a sort of symmetry breaking. This
has already been used to explain confinement RIGOROUSLY (see
Joyce).

selfAdjoint

Kea, the trouble with all your references is that nobody who doesn't have a graduate university library can see them. Isam's stuff is online at the arxive; is any of the rest of it. Anything at the level of Joyce?

Kea

Unfortunately, no.

On-line books on topos theory include Barr and Wells
and Goldblatt. John Baez's website is a good source
of references.

Sorry everybody.

Kea

Here

http://www.cwru.edu/artsci/math/wells/pub/ttt.html

marcus

Hello everybody, selfAdjoint asked for arxiv references for Joyce work because they are accessible to those of us who dont have a university library close by. Here are some:

http://arxiv.org/find/hep-th/1/au:+J.../0/1/0/all/0/1

hep-th/0307047
Quark confinement without a confining force
P.S. Isaac, W.P. Joyce, J. Links
5 pages

hep-th/0307046
An algebraic origin for quark confinement
P.S. Isaac, W.P. Joyce, J. Links
23 pages

hep-th/0306256
Quark State Confinement as a Consequence of the Extension of the Bose--Fermi Recoupling to SU(3) Colour
W. P. Joyce
15 pages, 4 figures

this list gives not only the one which Kea cited but also two others which are more recent---not yet in hardcopy
I have not examined these papers personally, but only helping as
an assistant librarian (I think I would not understand the papers in any case, or grasp their applicability)

regards to all

*

Kea

http://www.arxiv.org/find/math/1/au:.../0/1/0/all/0/1

selfAdjoint

Thank you both Kea and Marcus. I will start looking at these tomorrow. I also just found out about Google Scholar (www.scholar.google.com). I typed in topos quantization and got some very interesting results. There's hope for the old presheaf guy yet.

Chronos

Is there actually any missing mass? I don't think so. Yang-Mills suggests a slight CPT violation, but no missing energy [mass] that I can see. No matter how much you twist and turn space time around, mass does not go away. The observational evidence for its existence is fairly solid.

ohwilleke

The mass isn't missing from the observational universe. The mass is missing from the theory. The gap is between the presence of mass in the observational world, and the absence of mass in the theory, not the other way around. The issue is how to get mass into the theory (with a Higgs like mechanism a possible solution) so that it can fit what we observe, because expect for the mass problem Y-M does a good job of giving us to QCD that we observe.

Lester

You can find a simple definition of mass gap on Wikipedia. A Yang-Mills theory, in the limit of the coupling gauge going to infinity, displays at the classical level a mass gap. This because there is a theorem proved in

http://arxiv.org/abs/0709.2042 (appeared in Physics Letters B)

http://arxiv.org/abs/0903.2357 (appeared in Modern Physics Letters A)

that maps classical solutions of a massless quartic scalar field on the Yang-Mills field. These solutions appear to describe free massive fields notwithstanding we started from massless theories. One can use these solutions to build a quantum field theory and obtain an identical situation once is proved that quantum corrections do not modify it.