Embedding Space Construction of Conformal Field Theory From First Principles

This article presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to $\mathrm{SO}(d+1,1)$, and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra $[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$ from infinitesimal generators $\ell_n = -z^{n+1}\partial_z$, computing the central charge explicitly for free theories (boson $c=1$, Majorana fermion $c=1/2$, $bc$-ghosts $c = 1-3(2\lambda-1)^2$) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states $|h,\bar{h}\rangle$, deriving the transformation law $T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]$ that reveals the geometric origin of the conformal anomaly.