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 Q̇ 6= 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〉 6 = 0 and Q̇ = 0. The SSB (NG phase) in DLCQ can be realized on the trivial vacuum only when an explicit symmetrybreaking mass of the NG bosonmπ is introduced so that the NG-boson zero mode integrated over the LF exhibits singular behavior ∼ 1/mπ Lectures given at 10 th Annual Summer School and Symposium on Nuclear Physics (NUSS 97), “QCD, Lightcone Physics and Hadron Phenomenology”, Seoul, Korea, June 23-28, 1997. Email address: [email protected]