Zero-mode problem on the light front

A series of lectures are given to discuss the zero-mode problem on the light-front (LF) quantization with special emphasis on the peculiar realization of the trivial vacuum, the spontaneous symmetry breaking (SSB) and the Lorentz invariance. We first identify the zero-mode problem on the LF. We then discuss Discrete Light-Cone Quantization (DLCQ) which was first introduced by Maskawa and Yamawaki (MY) to solve the zero-mode problem and later advocated by Pauli and Brodsky in a different context. Following MY, we present canonical formalism of DLCQ and the zero-mode constraint through which the zero mode can actually be solved away in terms of other modes, thus establishing the trivial vacuum. Due to this trivial vacuum, existence of the massless Nambu-Goldstone (NG) boson coupled to the current is guaranteed by the non-conserved charge such that Q\0] = 0 and (Q) over dot not equal 0 but not by the NG theorem which in the equal-time quantization ensures existence of the massless NG boson coupled to the current with the charge Q\0] not equal 0 and (Q) over dot = 0. The SSB (NG phase) in DLCQ can be realized on the trivial vacuum only when an explicit symmetry-breaking mass of the NG boson m(pi) is introduced so that the NG-boson zero mode integrated over the LF exhibits singular behavior similar to 1/m(pi)(2) in such a way that (Q) over dot not equal 0 in the symmetric limit m(pi) --> 0. We also demonstrate this realization more explicitly in a concrete model, the linear sigma model, where the role of zero-mode constraint is clarified. We further point out, in disagreement with Wilson et al., that for SSB in the continuum LF theory, the trivial vacuum collapses due to the special nature of the zero mode which is no longer the problem of a single mode P+ = 0 but of the accumulating point P+ --> 0, in sharp contrast to DLCQ. Finally, we discuss the no-go theorem of Nakanishi and Yamawaki, which forbids exact LF restriction of the field theory that satisfies the Wightman axioms. The well-defined LF theory exists only at the sacrifice of the Lorentz invariance. Thus DLCQ as well as any other regularization on the exact LF has no Lorentz-invariant limit as the theory itself, although we can argue, based on an explicit solution of the dynamics (i.e., perturbation), that the Lorentz-invariant limit can be realized on the c-number quantity like S matrix which has no reference to the fixed LF.