Conformal algebra in 2 dimensions admits a central extension and while the procedure itself is explained in many textbooks quite well, I haven't found a source in which it is reasoned as to why we should carry out the central extension.
In this Wikipedia article, it is stated that the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra.
Can someone please elaborate on this and explain why this is the case?
