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Critical behaviour of the 2D scalar field theory in the LC framework is reviewed. The notion of dynamical zero modes is introduced and shown to lead to a non trivial covariant dispersion relation only for Continuous LC Quantization (CLCQ). The critical exponent $\eta$ is found to be governed by the behaviour of the infinite volume limit under conformal transformations properties preserving the local LC structure. The $\beta$-function is calculated exactly and found non-analytic, with a critical exponent $\omega=2$, in agreement with the conformal field theory analysis of Calabrese et al., Comment: 6 pages, 1 figure

Compact canonical quantization on the light cone (DLCQ) is examined in the limit of infinite periodicity lenth L. Pauli Jordan commutators are found to approach continuum expressions with marginal non causal terms of order $L^{-3/4}$ traced back to the handling of IR divergence through the elimination of zero modes. In contrast direct quantization in the continuum (CLCQ) in terms of field operators valued distributions is shown to provide the standard causal result while at the same time ensuring consistent IR and UV renormalization., Comment: 9 pages, 3 figures, add references