%% slac-pub-7056: page file slac-pub-7056-0-0-2-3-2.tcx. %% subsubsection The Vacuum [slac-pub-7056-0-0-2-3-2 in slac-pub-7056-0-0-2: slac-pub-7056-0-0-2-3-3] %%%% latex2techexplorer block: %% latex2techexplorer page setup: \newmenu{slac-pub-7056::context::slac-pub-7056-0-0-2-3-2}{ \docLink{slac-pub-7056.tcx}[::Top]{Top}% \subsectionLink{slac-pub-7056-0-0-2}{slac-pub-7056-0-0-2-3}{Above: 2.3. Problems and Open Issues }% \subsubsectionLink{slac-pub-7056-0-0-2}{slac-pub-7056-0-0-2-3-1}{Previous: Renormalization }% \subsubsectionLink{slac-pub-7056-0-0-2}{slac-pub-7056-0-0-2-3-3}{Next: Tools for Practical Calculations }% } %%%% end of latex2techexplorer block. %%%% code added by add_nav perl script \docLink{slac-pub-7056.tcx}[::Top]{Top of Paper}% | \docLink{pseudo:previousTopic}{Previous Section}% \bigskip% %%%% end of code added by add_nav %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% author definitions added by nc_fix \renewcommand{\thefootnote}{\fnsymbol{footnote}} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \def\a{\alpha} \def\b{\beta} \def\D{\Delta} \def\G{\Gamma} \def\e{\epsilon} \def\g{\gamma} \def\d{\delta} \def\p{\phi} \def\vp{\varphi} \def\s{\sigma} \def\l{\lambda} \def\L{\Lambda} \def\th{\theta} \def\om{\omega} \def\del{\partial} \def\ha{\frac{1}{2}} \def\psibar{\overline{\psi}} \def\sla#1{#1\!\!\!/} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% end of definitions added by nc_fix \subsubsection{\usemenu{slac-pub-7056::context::slac-pub-7056-0-0-2-3-2}{ The Vacuum }}\label{subsubsection::slac-pub-7056-0-0-2-3-2} 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 initial-value surface contains points that are light-like separated. Mathematically, the subtleties arise because the LC initial-value 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 initial-value 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 mean-field 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}. %%%% code added by add_nav perl script \bigskip% \docLink{pseudo:nextTopic}{Next Section}% | \docLink{slac-pub-7056-0-0-10u1.tcx}[::Bottom]{Bottom of Paper}% %%%% end of code added by add_nav