Paul Martin: Reference Section
: Projects...
Quantum field theory is the best currently
available/considered
framework
for modelling of fundamental physics
on the small scale.
In this framework gauge theories are the best currently
available models.
(Here `best' means providing the best fit to experimental data.)
A gauge model is prescribed by a gauge group for the gauge field.
And
observation indicates to use
certain specific abelian and non-abelian gauge
fields.
and so one eventually arrives at Yang-Mills theory,
and a very difficult calculation for the mass gap.
In the direction of performing the mass gap calculation, it is indicated to
explore various relatively simple but similar models first.
For example one can simplify the field to pure gauge.
One can simplify from 3+1-dimensions to 4-dimensions and then to
3-dimensions.
One can simplify the gauge group to a small cyclic group.
One can simplify space-time dynamics localised on a regular lattice
(allowing for a mechanism to recover the continuum limit).
Making all these simplifications, one arrives at the
cubic lattice Potts (or Ising) gauge model.
This is still impossible to do arbitrary computations on.
But some specific important computations start to become possible
in principle.
To turn principle into practice, a number of algebraic, geometric and
computational tools must be employed.
Geometrically, we have Kramers-Wannier duality which (in 3d) turns a
lattice gauge model into a Potts model
or a clock model.
-- Which are also good models more directly for certain systems
in condensed matter physics.
Then the first step is to compute the partition function (normalisation)
for the Potts model.
For arbitrary lattices this is still impossible, but for some finite
lattices it is possible.
The challenge then is to compute on lattices large enough to be in the
thermodynamic limit.
For this we need algebra, and in particular
algebras contained within the partition algebra.
More specifically we need control of the representation theory of such algebras.
For a comprehensive exposition of the technical details of the theory
of correlation functions and transfer matrix algebras see e.g.
[Martin08]
or
[Martin91].
For references on results obtained in this (and other) ways, see for example [MartinZakaria19].
The cubic-lattice 3-state Potts (or the 2-state= Ising) model exhibits an order-disorder phase transition at a critical temperature in the ferromagnetic region. (The proof of this is not rigorous but it is compelling --- coming precisely from the kind of calculations that we are discussing here.) In order to compute the mass gap in 4d Yang-Mills one would also require a (specific type of) phase transition there. Thus it is essential to study these phase transitions. They are also directly relevant for many applications.
Some nice looking post- [M91] papers on the Potts model, and in particular on related spin chains, indicating a lot of interesting open problems (!):