Abstract
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 ...
Abstract
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, (z) over bar 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-infinity 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 (z) over bar 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.
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