A possibly idle question about lattice QCD.
Greg Kuperberg
Here is a possibly idle question about lattice QCD that I've always wondered
about:
Which lattice? The three possibilities that come to mind are
A_4, A_4^*, and D_4. D_4 is the set of all points with integer
coordinates with even sum and has many interesting properties,
including best lattice sphere packing. A_4 is the lattice you get
if you take the cubic lattice in five dimensions and intersect
with the hyperplane of points whose coordinates sum to zero. A_4^*
is what you get if you project onto the same hyperplane, and
it is the optimal lattice for sphere coverings.
The cubic lattice has no interesting properties that I can think
of other than convenience.
(By a lattice I mean a periodic arrangement of points with exactly one
point in each cell. If to you a lattice is any periodic arrangement
of points, optimal packings and coverings are only conjectured to be D_4
and A_4^*.)
Since I am asking my question, let me give my opinion as an uninformed
and narrow-minded pure mathematician on how much faith one should have
in the results of lattice QCD: Not much. I've heard that you have
to do lattice QCD in Euclidean space (where all my lattices above live)
rather than Lorentzian space. If you have to make several
approximations of this order, I don't see how the results can be much
better than numerology. Numerology has its place; for example the
numerology of the periodic table was very important to early quantum
mechanics. But it should not be confused with predictive calculation.
l...@landau.phys.washington.edu
>Since I am asking my question, let me give my opinion as an uninformed
>and narrow-minded pure mathematician on how much faith one should have
>in the results of lattice QCD: Not much. I've heard that you have
>to do lattice QCD in Euclidean space (where all my lattices above live)
>rather than Lorentzian space. If you have to make several
>approximations of this order, I don't see how the results can be much
Calculations in Euclidean space are in no way "approximations" of
Minkowski space! Functional integrals in Euclidean space represent
(correctly!) ground state expectation values. Functional integrals in
Minkowski space instead represent matrix elements of the time evolution
operator. Euclidean space calculations are relevant for questions
involving static properties - such as determining masses of particles.
>better than numerology. Numerology has its place; for example the
>numerology of the periodic table was very important to early quantum
>mechanics. But it should not be confused with predictive calculation.
True, but irrelevant to the discussion.
--
------------------------------------------------------------------------
Laurence G. Yaffe Internet: l...@newton.phys.washington.edu
University of Washington Bitnet: ya...@uwaphast.bitnet
Matt Austern
> Since I am asking my question, let me give my opinion as an uninformed
> and narrow-minded pure mathematician on how much faith one should have
> in the results of lattice QCD: Not much. I've heard that you have
> to do lattice QCD in Euclidean space (where all my lattices above live)
> rather than Lorentzian space.
Oh, that's not really a problem: relating results in Euclidean space
to results in Minkowski space is a technical issue that doesn't seem
to pose any real problems. In perturbation theory you do it by
Wick-rotating the propagators; you can't do it quite that way in
nonperturbative calculations, but there is a formalism that allows you
to relate the Euclidean results to physical Green's functions anyway.
I might add: there's nothing unique about lattices in this respect.
Path integrals often look more convenient in Euclidean space (i.e.,
they almost look like they might actuallly converge), and it's quite
common when you're doing formal work to use Euclidean space, or to go
back and forth between Euclidean and Minkowski space without giving
the matter much thought. (In fact, most people who use field theory
for particle physics are doing something equivalent to using Euclidean
space, whether or not they realize it.)
--
Matthew Austern Maybe we can eventually make language a
ma...@physics.berkeley.edu complete impediment to understanding.
Greg Kilcup
gk...@midway.uchicago.edu (Greg Kuperberg) posed a couple
of questions concerning lattice field theory.
At least two people have addressed Greg's concerns about
rotating to Euclidean space. I would just add the points that
(1) quantum field theory is really *defined* in Euclidean space,
and (2) the lattice is really the only regulator we have once we
step outside of perturbation theory. So lattice field theory is
about the only thing on solid ground, and it might be more
appropriate to worry about the status of Minkowski space theories.
Secondly, concerning the original "possibly idle question",
namely "Which lattice?", the answer is: any will do, thank you.
In the end we work in the limit where correlation lengths
diverge and all details of the original lattice structure
are lost. However, since the question was posed, the hypercubical
lattice is used most often just for convenience both in position
space and in momentum space. Some studies have been done using
the lattice Greg labels D_4, though in the physics literature
it is referred to as the lattice generated by the roots of F_4.
Now that you mention it, I don't see why people don't refer to it
as SO(8) as well. I am not aware of anyone using the A_4 or
SU(5) root lattice, and I am not even sure what algebra A_4^*
corresponds to, if any. But again, since the details of our
lattices don't really matter, we generally opt for grids which
fit conveniently in the computer, and whose Fourier transforms
are nice simply shaped objects such as torii.
-- Greg Kilcup