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\subsubsection{\usemenu{slacpub7056::context::slacpub705600232}{ The Vacuum }}\label{subsubsection::slacpub705600232}
The problem of how to incorporate a nontrivial vacuum in LCQ is
closely related to the renormalization problem; all of the structure
of the vacuum is removed by a small$p^+$ cutoff, and putting this
physics back is one purpose of the infrared counterterms. We prefer
to consider it separately, however, because conceptually it is a
much different problem than that of removing dependence on a
transverse momentum cutoff. The vacuum problem is in fact one
aspect of a whole range of puzzles regarding LC field theory, which
can all be traced to the fact that the LC initialvalue surface
contains points that are lightlike separated.
Mathematically, the subtleties arise because the LC initialvalue
surface is a surface of characteristics \cite{41}.
Physically, they arise because points on the surface can be causally
connected. Thus one may not be completely free to impose initial
conditions on such a surface, for example. Furthermore, there is a
danger of missing degrees of freedom; in general, initial conditions
on one characteristic surface are not sufficient to determine a
general solution to the problem \cite{42,43}. These
difficulties are compounded by the fact that the vacuum lives at a
very singular point in the theory. Near $p^+=0$ states have
diverging free energies, but the density of states and couplings to
other states are also singular.
One way of addressing these issues is to carefully treat the LC
initialvalue problem with an infrared regulator that does not make
the vacuum trivial \cite{44,45}. The idea is to formulate
the theory with the vacuum degrees of freedom (sometimes called
``zero modes,'' though this phrase has several distinct meanings
among the experts) present, and then to integrate them out. This is
essentially the small$p^+$ part of the renormalization problem
discussed above. The goal is to obtain either an effective
Hamiltonian for use with a trivial vacuum or an explicit description
of the vacuum structure in terms of the LC degrees of freedom.
In the past few years there has been significant progress on
understanding the ways in which vacuum structure can be manifest on
the LC. A consistent meanfield description of spontaneous symmetry
breaking in the $\phi^4_{1+1}$ theory has been obtained \cite{46},
as well as a better understanding of certain topological properties
of gauge theories \cite{47}. McCartor's operator solution of the
Schwinger model on the LC is also instructive \cite{23}. In
particular the structure of the $\theta$vacua, while not trivial,
is considerably simpler in the LC representation than in ETQ
\cite{48}.
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