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Light-front (LF) quantization of massless fields in two spacetime dimensions is a long-standing and much debated problem. Even though the classical wave-equation is well-documented for almost two centuries, either as problems with initial values in spacetime variables or with initial data on characteristics in light-cone variables, the way to a consistent quantization in both types of frames is still a puzzle in many respects. This is in contrast to the most successful Conformal Field Theoretic (CFT) approach in terms of complex variables $z$, $\bar{z}$ pioneered by Belavin, Polyakov and Zamolodchikov in the ’80s. It is shown here that the 2D-massless canonical quantization in both reference frames is completely consistent provided that quantum fields are treated as Operator-Valued Distributions (OPVD) with Partition of Unity (PU) test functions. We recall first that classical fields have to be considered as distributions (e.g. generalized functions in the Russian literature). Then, a necessary condition on the PU test function follows from the required matching of the classical solutions of the massless differential equations in both types of reference frame. Next we use a mathematical formulation for OPVD, developed in the recent past. Specifically, smooth ${{𝒞}}^{\infty }$ fields are introduced through the convolution operation in the distributional context. Due to the specific behavior of the Fourier-transform of the initial test function, this convolution transform has a well-defined integral in the dual space, whatever the initial choice of the reference frame. The relation to the conformal fields method follows immediately from the transition to Euclidean time and leads directly to explicit calculations of a few correlation functions of the scalar field and its energy–momentum tensor. The LF derivation of the Virasoro algebra is then obtained from the $z$ and $\bar{z}$ expansions of the canonical fields as distributional Laplace-transform in these variables. Finally, the popular and problematic Discretized Light Cone Quantization (DLCQ) method is scrutinized with respect to its zero mode and ultraviolet content as encompassed in the continuum OPVD formulation.

We report a simplification in the large N matrix mechanics of light-cone matrix field theories. The absence of pure creation or pure annihilation terms in the Hamiltonian formulation of these theories allows us to find their reduced large N Hamiltonians as explicit functions of the generators of the Cuntz algebra. This opens up a free-algebraic playground of new reduced models — all of which exhibit new hidden conserved quantities at large N and all of whose eigenvalue problems are surprisingly simple. The basic tool we develop for the study of these models is the infinite dimensional algebra of all normal-ordered products of Cuntz operators, and this algebra also leads us to a special number-conserving subset of these models, each of which exhibits an infinite number of new hidden conserved quantities at large N.

The recently formulated Bagger-Lambert-Gustavsson (BLG) theory in three dimensions is described in terms of a constrained chiral superfield in light-cone superspace. We discuss the use of Superconformal symmetry to determine the form of its interactions, in complete analogy with N = 4 SuperYang-Mills in four dimensions.

We investigate the quark angular momentum in a light-cone spectator model. The canonical angular momentum sum rule is satisfied in both the scalar and the axial-vector cases. Then we perform a calculation of the gravitational form factors which are expected to be related to the angular momentum of each constituent in this no gauge field model. We find this relation is satisfied for the total angular momentum as a direct conclusion of the momentum fraction sum rule and the anomalous gravitomagnetic moment sum rule, but for each constituent it is violated.

In the light-cone representation, vacuum effects are correctly introduced into the dynamics by the inclusion of induced operators. Recently, some of the induced operators in QCD have been derived. The knowledge of these operators makes it possible to perform calculations of the propertied of QCD resonances using essentially the QCD sum rules approach; but with the inclusion of the induced operators providing the nonperturbative physics rather than the operator product expansion that is used in the standard sum rules approach.

The recently formulated Bagger-Lambert-Gustavsson (BLG) theory in three dimensions is described in terms of a constrained chiral superfield in light-cone superspace. We discuss the use of Superconformal symmetry to determine the form of its interactions, in complete analogy with N = 4 Super Yang-Mills in four dimensions.