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\subsubsection{\usemenu{slacpub7056::context::slacpub705600211}{Connection to the Constituent Quark Model}}\label{subsubsection::slacpub705600211}
The simplicity of the vacuum means that a powerful physical
intuition can be applied in the study of lightcone QCD: that of the
constituent quark model (CQM). Indeed, LCQ offers probably the only
realistic hope of deriving a constituent {\em approximation} to QCD,
as stressed particularly by Wilson \cite{24,12}. In
contrast, in equaltime quantization the physical vacuum involves
Fock states with arbitrary numbers of quanta, and a sensible
description of constituent quarks and gluons requires quasiparticle
states, i.e., collective excitations above a complicated ground
state. Thus an ET approach to hadronic structure based on a few
constituents, analogous to the CQM, is bound to fail.
On the LC, a simple cutoff on small longitudinal momenta suffices to
make the vacuum completely trivial. Thus we immediately obtain a
constituent picture in which all partons in a hadronic state are
connected directly to the hadron, instead of being disconnected
excitations in a complicated medium. Whether or not the resulting
theory allows reasonable approximations to hadrons to be constructed
using only a {\it few} constituents is an open question. However,
one might choose to regard the relative success of the CQM as a
reason for optimism.
The price we pay to achieve this constituent framework is that the
renormalization problem becomes considerably more complicated on the
LC. We shall discuss this in more detail in section 1.3; for the
moment let us merely note that this is where the familiar ``Law of
Conservation of Difficulty'' manifests itself in the LC approach.
Wilson and collaborators have recently advocated an approach to
solving the lightcone Hamiltonian for QCD which draws heavily on
the physical intuition provided by the CQM \cite{12,25}.
One begins by constructing a suitable effective Hamiltonian for QCD,
including the counterterms that remove cutoff dependence. At
present this can only be done perturbatively, so that the cutoff
Hamiltonian is given as a power series in the coupling constant
$g_\Lambda$:
\begin{equation}
P^_\Lambda = P^_{(0)} + g_\Lambda P^_{(1)}+g_\Lambda^2 P^
_{(2)}+ \dots\; .
\end{equation}
In the next step a similarity transformation is applied to this
Hamiltonian, which is designed to make it look as much like a CQM
Hamiltonian as possible. For example, we would seek to eliminate
offdiagonal elements that involve emission and absorption of gluons
or of $q\overline{q}$ pairs. It is the emission and absorption
processes that are absent from the CQM, so we should remove them by
the unitary transformation. This procedure cannot be carried out
for all such matrix elements, however. This is because the
similarity transformation is sufficiently complex that we only know
how to compute it in perturbation theory. Thus we can reliably
remove in this way only matrix elements that connect states with a
large energy difference; perturbation theory breaks down if we try
to remove, for example, the coupling of a lowenergy quark to a
lowenergy quarkgluon pair. We design the transformation
to remove offdiagonal matrix elements between sectors where the
lightcone energy {\it difference} between the initial and final
states is greater than some new cutoff $\lambda$. This procedure is
known as the ``similarity renormalization group'' method. For a
more detailed discussion and for connections to RG concepts see
Ref. \cite{26}.
The result of the similarity transformation is to generate an
effective Hamiltonian $P^_{\rm eff}$ which has fewer matrix
elements connecting states with different parton number, and
complicated potentials in the diagonal Fock sectors. The idea is
that the collective states generated in the similarity
transformation will correspond roughly to constituent quarks and
gluons, and the potentials in the different Fock space sectors will
dominate the physics. If this is correct, then the potentials
should give a reasonable description of hadronic structure, and the
offdiagonal interactions should represent small corrections. This
can be checked explicitly using boundstate perturbation theory.
The collective states and potentials would then furnish a
constituent approximation to QCD \cite{25}.
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