QCD problem
Peter Ta-Chen Chang
Hello sci.* netters,
A friend of mine wants to know the solution to the
problem, if there is one:
Consider a two dimensional lattice consisting of sites & links.
sites are labelled by a coordinate pair (x,y).
links connect the sites and are denoted L(i,x,y) where:
i = 1 if the link points in x direction and 2 if y direction
and (x,y) is the site from which L(i,x,y) originates.
for example, the link L(1,x,y) connects the sites (x,y) and (x+1,y)
and is "oriented" in the +x direction.
let links be (i,x,y)-dependent complex numbers of modulus 1
so that L(1,x,y) = exp(i*phi) for some real phase angle phi.
furthermore, suppose our lattice is finite and has N sites
in each directions and that the links are periodic:
L(1,N,y) connects (N,y) with (1,y);
L(1,x,N) connects (x,N) with (x,1).
for all sites (x,y), let the links touching that site satisfy:
L(1,x,y) * L^(1,x-1,y) * L(2,x,y) * L^(2,x,y-1) = 1
where L^ denotes complex conjugate.
define a "connector" P(x1,y1;x0,y0) between two sites to be
the product of all the oriented links joining the sites
(x1,y1) and (x0,y0).
for example:
L(1,x,y)*L(2,x+1,y)*L(2,x+1,y+1)*L(2,x+1,y+2) = P(x+1,y+3;x,y).
(of course, there are a large number of distinct connectors
between (x,y) and (x+1,y+3).)
Problem: consider the countable set of connectors {P}
(indexed by i)
between any two given points (x0,y0) and (x1,y1).
Determine a linear combination of these P's so that
sum over all i of c_i* P_i = 1.
(eg. determine the complex coefficients c_i).
Perhaps a good reference or pointers is also ok.
Matt Austern
Perhaps I'm misunderstanding the problem, since it seems somewhat
trivial to me. Just choose c_1 = (P_1)^(-1), and all of the other c's
0.
My confusion might be because I'm not sure whether or not you're
willing to allow the c's to depend on the lattice configuration. If
you're asking for a set of constants such that your relation holds for
every lattice configuration, I doubt that this can be done. (After
all, the different connectors are uncorrelated.)
I'm curious how this problem arises in lattice QCD. (Your gauge group
here is U(1), and in QCD it is SU(3).) Is there a simple explanation?
I'm also curious what results you're looking at on a 2-dimensional
lattice; I was under the impression that 2-dimensional pure gauge
theories had been solved exactly.
(My reference on lattice gauge theories is Creutz, by the way. Does
anyone else have references they like?)
--
Matthew Austern aus...@lbl.bitnet Proverbs for paranoids, 3: If
(415) 644-2618 aus...@ux5.lbl.gov they can get you asking the wrong
aus...@lbl.gov questions, they don't have to worry
about answers.
Iain Jameson
Some papers which may be of interest are
E.Fradkin & L.Susskind, Physical Review D17, p2637 (1978).
(4-dim v's 2-dim)
J.B.Kogut, Reviews of Modern Physics, Vol.51, p659 (1979).
(This is a review article.)
J.B.Kogut, Les Houches 1982 p319. Ed. J.B.Zuber & R.Stora.
(another review paper)
A.Hasenfratz & P.Hasenfratz, Annual Review of Nuclear particle Science,
Vol.35, p559 (1985).
(another review paper)
H.Satz, Ann.Rev.Nucl.Part.Sci., Vol.35, p245 (1985).
(hadron to quark-gluon plasma)
D.J.E.Callaway, Contemporary Physics Vol.26, p23 and P95 (1985)
(yet another review)
A number of papers apply LGT to the confinement problem.
A must read is
K.G.Wilson, Phys. Rev. D10, p2445 (1974).
Also try
A.M.Polyakov's "Gauge Fields and Strings"
An amazing book.
I'm not sure who published it.
Hope this helps.
Iain Jameson
Peter Ta-Chen Chang
>Perhaps I'm misunderstanding the problem, since it seems somewhat
>trivial to me. Just choose c_1 = (P_1)^(-1), and all of the other c's
>0.
>My confusion might be because I'm not sure whether or not you're
>willing to allow the c's to depend on the lattice configuration. If
>you're asking for a set of constants such that your relation holds for
>every lattice configuration, I doubt that this can be done. (After
>all, the different connectors are uncorrelated.)
>I'm curious how this problem arises in lattice QCD. (Your gauge group
>here is U(1), and in QCD it is SU(3).) Is there a simple explanation?
>I'm also curious what results you're looking at on a 2-dimensional
>lattice; I was under the impression that 2-dimensional pure gauge
>theories had been solved exactly.
>(My reference on lattice gauge theories is Creutz, by the way. Does
>anyone else have references they like?)
my friend (who doesn't have access to the net) replies:
c_i must be configuration dependent;
2D pure gauge theory in continuum is completely solved, but not
on lattice. (in creutz, he solves it by a perturbative expansion,
but of course, the expansion cannot be calculated to all orders;
one cannot even show that expansion converges.)
my problem arises as follows; it is essentially a generalization
of the "gauge fixing" chapter in creutz's book in which creutz
talks about "axial gauge" and "maximal trees."
i want to fix to landau gauge (not axial gauge which is trivial, but
has problems physically). my landau gauge condition is that
L(1,x,y)*L^(1,x-1,y)*... * = 1.
it is a known fact that <L(1,x,y)*L(1,x',y')> .ne. 0 in landau
gauge (it vanishes when we do not gauge fix, of course).
it is also obvious(?) that the only nonvanishing expectation values
are of the form <closed loop> or "wilson loops." therefore,
we would expect that
<L(1,x,y)*L(1,x',y')>_{Landau Gauge} = sum over some configuration of
<closed loops>_{no gauge fixing}.
so, i want to figure out the configuration of closed loops corresponding
to <L(1,x,y)*L(1,x',y')>; the strategy is to use the landau gauge
identity to fill in the space between L(1,x,y) and L(1,x',y') with
links to make a closed loop config.
Matt Austern
>on lattice. (in creutz, he solves it by a perturbative expansion,
>but of course, the expansion cannot be calculated to all orders;
>one cannot even show that expansion converges.)
In fact, it is possible to solve 2D pure gauge theory
nonperturbatively. If you choose an axial gauge, then it reduces to a
set of noninteracting one-dimensional chains, each of which is
basically an Ising model. (Or to be picky, it's an Ising model if
your gauge group is Z_2. Otherwise it's a generalization.) You can
do the path integral exactly if you change variables so that your
group elements are associated with plaquettes instead of links.
I admit that this isn't tremendously useful to you, since this trick
won't work in Landau gauge. (And I agree with you that a covariant
gauge is nice.) Still, we aren't stuck with perturbation theory!
I'm trying to learn about applications of lattice methods to the
SU(2)xU(1) Standard Model, to study the question of strongly
interacting (i.e., superheavy) Higgs bosons. Do you have any words of
wisdom about that subject?