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\subsectionLink{slacpub7056002}{slacpub70560021}{2.1. Basic Formalism }%
\subsectionLink{slacpub7056002}{slacpub70560022}{2.2. Applications }%
\subsectionLink{slacpub7056002}{slacpub70560023}{2.3. Problems and Open Issues }%
\subsectionLink{slacpub7056002}{slacpub70560024}{2.4. The Road to QCD }%
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\section{\usemenu{slacpub7056::context::slacpub7056002}{LightCone Quantization}}\label{section::slacpub7056002}
In any practical calculation based on diagonalizing a
fieldtheoretic Hamiltonian, truncation of the space of states to a
finite subspace is inevitable. The simplest approach might be to
truncate to the most physically important states, and (numerically)
diagonalize the canonical Hamiltonian on this subspace. If the
subspace truly contains the states that are most important for
whatever structure is of interest, then the resulting eigenvalues
and wavefunctions should be a reasonably good approximation to the
full solution of the theory. Furthermore, the approximation can be
improved by allowing more and more states into the truncated theory
and verifying that the results converge.
In a more refined approach one would include the effects of the
discarded states in effective interactions. This step is essential
if one does not have a reliable way of identifying a physically
important subspace {\it a priori}, as in QCD. It is also very
likely to be the more practical approach. A useful analogy here
might be with the use of improved actions for lattice gauge theory.
The lattice spacing $a$ plays the role of an ultraviolet cutoff,
which removes states from the theory with momenta greater than
$\pi/a$. The problem is that one needs to make $a$ small enough
that lowenergy quantities become independent of $a$, but the cost
of a simulation increases rapidly with decreasing $a$, roughly as
$1/a^{\sim(47)}$ \cite{18}. Thus it makes sense to attempt to
remove the dependence on $a$ by modifying the Lagrangian, that is,
by including effective interactions or ``counterterms'' that
incorporate the physics of the states excluded by the cutoff. This
allows one to work at a larger value of $a$ for a fixed numerical
accuracy, drastically reducing the cost of the simulation. Of
course, one has to determine the effective interactions to be
included in the Lagrangian. For QCD this may be done using
perturbation theory if the cutoff is not too low. Asymptotic freedom
implies that the effects of highenergy states are governed by an
effective coupling constant that is small, so that if we eliminate
states of sufficiently high energies then perturbation theory should
suffice. The resulting perturbatively constructed action can then
be solved nonperturbatively using Monte Carlo techniques.
This kind of Hamiltonian approach is in fact the method of choice in
virtually every area of physics and quantum chemistry. It has the
desirable feature that the output of such a calculation is
immediately useful: the spectrum of states and wavefunctions.
Furthermore, it allows the use of intuition developed in the study
of simple quantum systems, and also the application of, e.g.,
powerful variational techniques. The one area of physics where it
is {\em not} widely employed is relativistic quantum field theory.
The basic reason for this is that in a relativistic field theory one
has particle creation/annihilation in the vacuum. Thus the true
ground state is in general extremely complicated, involving a
superposition of states with arbitrary numbers of bare quanta, and
one must understand the complicated structure of this state before
excitations can be considered. Furthermore, one must have a
nonperturbative way of separating out disconnected contributions to
physical quantities, which are physically irrelevant. Finally, the
truncations that are required inevitably violate Lorentz covariance
and, for gauge theories, gauge invariance. It is not clear how to
construct a viable renormalization scheme for this type of problem.
These difficulties (along with the development of covariant
Lagrangian techniques) eventually led to the almost complete
abandonment of fixedtime Hamiltonian methods in relativistic field
theories.
Lightcone quantization (LCQ) \cite{1} is an alternative to
the usual formulation of field theories in which some of these
problems appear to be more tractable. This raises the prospect of
developing a practical Hamiltonian approach to solving field
theories, based on diagonalizing LC Hamiltonians. In the next few
sections we shall give a brief overview of this approach. We begin
by describing the basic formalism and how it might allow a
connection to be established between QCD and the constituent quark
model. We then review some existing calculations in toy models, and
finally we discuss the remaining barriers that block progress in
QCD. Our presentation will necessarily be brief and thus somewhat
superficial. Our goal is primarily to give a flavor of the LC
approach and why it is of interest, and to set the stage for the
discussion of QCD phenomenology in the following sections. The
interested reader is advised to consult one of the more extensive
reviews on this subject for detailed discussions of the topics
mentioned here \cite{19}.
\subsectionInput{slacpub7056002}{slacpub70560021}{2.1. Basic Formalism }%
\subsectionInput{slacpub7056002}{slacpub70560022}{2.2. Applications }%
\subsectionInput{slacpub7056002}{slacpub70560023}{2.3. Problems and Open Issues }%
\subsectionInput{slacpub7056002}{slacpub70560024}{2.4. The Road to QCD }%
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