How can we fix the lightcone gauge in QCD?

[henry_m]

Hi, first post here so be nice!

I'd be interested to hear any input on a fairly minor point that I've come across with gauge fixing. Specifically, for a $$U(N)$$ gauge group, how to show it is possible to fix $$A_\mu=0$$ for some coordinate (the specific example I'm using is lightcone gauge in (1+1)-D, $$A_{-}=0$$).

So we have a gauge transformation $$A_\mu\to U A_\mu U^\dagger + i U\partial_\mu U^\dagger$$ for unitary $$U$$. Most of the coordinate are just spectators, so we can supress them, and it eventually reduces to solving a matrix ODE:

$$\dot{U}(t)=iU(t)A(t)$$

Here $$t$$ represents the coordinate whose component of the gauge field we're fixing to zero, and $$A(t)$$ is that component. $$A$$ is Hermitian. At first I thought $$U(t)=\exp\left[i\int^t A\right]$$ would do the trick, but that's no good since $$A$$ won't necessarily commute with its derivative. It's plausible that a solution exisits, but I haven't been able to find it. So:

1. Does someone know how to solve it explicitly?
2. The next best thing, can we prove a solution for unitary $$U$$ exists? Is there a uniqueness theorem to invoke? I suspect the main thing here is showing that we can take $$U$$ to be unitary.

Thanks,
Henry

 

[tom.stoer]

I guess what you are looking for is the path-order exponential

$$U(x^-, x^-_0) = \mathcal{P}\exp\left[-ig\int^{x^-}_{x^-_0} dz^-A(z^-)}\right] = \lim_{N\to\infty}\prod_{n=0}^{N-1}U(z^-_{n+1}, z^-_n)$$

with

$$U(z^-_{n+1}, z^-_n) = 1 - igA(z^-_{n+1})\,(z^-_{n+1} - z^-_n)$$

and where the product is ordered along the z_- path. This solves

$$\partial_-U^\dagger = igAU^\dagger$$

which is required to fix the light cone gauge.

btw.: I was thinking for a long time about using light-cone coordinates and light-cone gauge in 1+1 dim. QCD; I came to the conclusion that this is somehow ill-defined, that you lose certain fermionic d.o.f., that axial or light-cone gauge fixing (classical gauge fixing before quantization) is the wrong way to go at all, that you need a huge apparatus of constraint quantization a la Dirac to show that not everything is well-defined, ... So I think light-cone quantization of 1+1 dim QCD is a waste of time.

 

[cello]

Hi, first post here so be nice!

I'd be interested to hear any input on a fairly minor point that I've come across with gauge fixing. Specifically, for a $$U(N)$$ gauge group, how to show it is possible to fix $$A_\mu=0$$ for some coordinate (the specific example I'm using is lightcone gauge in (1+1)-D, $$A_{-}=0$$).

So we have a gauge transformation $$A_\mu\to U A_\mu U^\dagger + i U\partial_\mu U^\dagger$$ for unitary $$U$$. Most of the coordinate are just spectators, so we can supress them, and it eventually reduces to solving a matrix ODE:

$$\dot{U}(t)=iU(t)A(t)$$

Here $$t$$ represents the coordinate whose component of the gauge field we're fixing to zero, and $$A(t)$$ is that component. $$A$$ is Hermitian. At first I thought $$U(t)=\exp\left[i\int^t A\right]$$ would do the trick, but that's no good since $$A$$ won't necessarily commute with its derivative. It's plausible that a solution exisits, but I haven't been able to find it. So:

1. Does someone know how to solve it explicitly?
2. The next best thing, can we prove a solution for unitary $$U$$ exists? Is there a uniqueness theorem to invoke? I suspect the main thing here is showing that we can take $$U$$ to be unitary.

Thanks,
Henry

Yes, you can always solve it, at least in a finite domain. Essentially this is because we are dealing with a set of ordinary first-order equations, for which the existence and uniqueness of the solution was established long ago. Weinberg mentioned this fact in his second volume of QFT textbook, on page 5.

 

[henry_m]

Thanks Tom, that's what I was after.

btw.: I was thinking for a long time about using light-cone coordinates and light-cone gauge in 1+1 dim. QCD; I came to the conclusion that this is somehow ill-defined, that you lose certain fermionic d.o.f., that axial or light-cone gauge fixing (classical gauge fixing before quantization) is the wrong way to go at all, that you need a huge apparatus of constraint quantization a la Dirac to show that not everything is well-defined, ... So I think light-cone quantization of 1+1 dim QCD is a waste of time.

This is an interesting aside. Surely from the path-integral point of view we are just splitting the integral over the gauge connection into an integral over a representative of each gauge orbit (one which fixes the gauge) and an integral over the gauge group itself, and using gauge invariance to factor the latter out? I'd be interested to know where the inconsistencies arise. Not something I have much time to follow up right now unfortunately.

Incidentally, the context I'm using this in is 't Hoofts 'solution' of 2-D chromodynamics in the large-N limit. Without this kind of gauge fixing, is there another way to get the required simplifications (no gluon interactions) required for this to work? Otherwise it seems a little harsh to label the process a waste of time.

 

[tom.stoer]

I studied QCD in 1+1 dim. on the light-cone in canonical quantization. The large-N limit hides some of these problems (there is a way to recover the quark-condensate even on the light-cone, but it's a dirty trick were you have toplay around with products of delta- and step-functions, but in general it depends what you want to calculate.

I remember an artice of Bars & Green

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVC-472R1D0-BR&_user=10&_coverDate=09%2F11%2F1978&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1734223682&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=2562c519f5bd88fd95c2d5d849bc014a&searchtype=a

dealing with 1+1 dim. QCD, no light cone coordinates but in the continuum where you can see al the trouble you get when gauge fixing is implemented classically. This applies to the light cone as well. In addition you will find that additional problems due to the non-dynamical fermions on the light-cone. That's fundamentally wrong but seems to be uncritical in the large-N limit.

My recommendation for QCD (regardless if 3+1 or 1+1) in canonical quantization is:
- use periodic boundary conditions i.e. QCD on a 1-circle or 3-torus
- take care about the zero modes of the gauge field
- fix A°=0 as A° is unphyical and acts as a Lagrange multiplier
- impose Gauss-law on the physical states and
- use a unitary transformation to fix the Gauss-law = eliminate longitudinal gluons

http://www.adsabs.harvard.edu/abs/1994AnPhy.233..317L
QCD in the Axial Gauge Representation
Authors: Lenz, F.; Naus, H. W. L.; Thies, M.
Publication: Annals of Physics, Volume 233, Issue 2, p. 317-373.
Publication Date: 08/1994
Abstract: Within the canonical Weyl gauge formulation, the axial gauge representation of QCD on a torus is derived. The resolution of the Gauss law constraint is achieved by applying unitary gauge fixing transformations. The result of this formal development is a Hamiltonian explicitly formulated in terms of unconstrained degrees of freedom. Novel features of this Hamiltonian are the non-perturbative dynamics of two-dimensional degrees of freedom appearing in the gauge-fixing procedure, such as Jacobian and centrifugal barrier. These two-dimensional fields appear to be essential for the infrared properties of the theory. The global residual gauge symmetries of QCD are established in this representation. It is shown that SU(N) gauge theories may exhibit at most N-1 massless vector (gauge) bosons. The implications for the phase structure of non-abelian gauge theories (QCD, Georgi-Glashow model) are discussed.

This seems to be terribly complicated but of course it is much easier in 1+1 dim. The main idea is how to fix the gauge.