Working Notes and Derivations on Conformal Field Theory from First Principles (Part I)

These notes arose from my attempt to learn conformal field theory immediately after standard courses in quantum mechanics and quantum field theory. The aim is to develop CFT, and eventually holography, in a self-contained manner, assuming only basic familiarity with QM and QFT and avoiding reliance on unexplained results.

The notes are planned in \textbf{three parts}, of which the present document constitutes the first. This part develops the foundations of conformal field theory directly from quantum field theory. Later parts are intended to build on this material and progress toward string theory and holography in a logically continuous way.

The material is compiled primarily from Polchinski’s \emph{String Theory}, Nastase’s textbook on AdS/CFT, and various arXiv lecture notes (with appropriate citations), together with detailed derivations that are often omitted in standard references. These notes are intended to complement standard QFT texts, serving as a detailed continuation beyond treatments such as \emph{No-Nonsense Quantum Field Theory} by explicitly bridging in missing steps and calculations.

The notes are incomplete and evolving, and some sections will be added or revised in future versions. In places where discrepancies with standard references arise, the standard results are adopted to maintain continuity, with unresolved issues left for later revision.

The long-term goal is to provide a coherent pathway from quantum field theory to conformal field theory and holography, emphasizing derivations, logical continuity, and physical understanding. Errors and typos are likely, and feedback is welcome.

To reduce unnecessary tedium, limited use of AI tools(Claude and Gemini) was made for index labeling, equation formatting, and visual structuring of expressions.