EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions (1601.05000v2)

Abstract: This is a writeup of lectures given at the EPFL Lausanne in the fall of 2012. The topics covered: physical foundations of conformal symmetry, conformal kinematics, radial quantization and the OPE, and a very basic introduction to conformal bootstrap.

• The paper outlines the foundational role of conformal symmetry in three-dimensional quantum field theories, emphasizing local scaling and extended invariance.
• It demonstrates the use of embedding space techniques and radial quantization to derive correlation functions and implement the conformal bootstrap program.
• The lectures highlight the impact of unitarity bounds and the operator product expansion in characterizing critical phenomena and phase transitions.

Overview of EPFL Lectures on Conformal Field Theory in Three Dimensions

The paper "EPFL Lectures on Conformal Field Theory in Three Dimensions" by Slava Rychkov offers a comprehensive exposition of conformal field theory (CFT), focused on lectures given to graduate students. The paper delineates the foundational aspects and advanced topics in CFT with an emphasis on three-dimensional theories. This essay aims to provide an expert-level overview of the content covered in these lectures, highlighting the critical aspects of conformal symmetry, the operator spectrum, and the implications for theoretical physics.

Fundamental Aspects of Conformal Symmetry

The paper begins by detailing the physical basis of conformal symmetry and its significance in the landscape of quantum field theories (QFTs). Conformal symmetry, characterized by invariance under local scaling transformations, forms the cornerstone of CFT. In three dimensions, the conformal group encompasses transformations that go beyond the Poincaré invariance, including dilatations and special conformal transformations. A notable feature of three-dimensional CFTs is the richness introduced by the additional symmetry transformations, which fundamentally alter the behavior of correlation functions compared to theories with only scale invariance.

Conformal Kinematics and the Bootstrap Program

A significant portion of the lectures is dedicated to exploring the kinematics of conformally invariant theories. The formalism utilized includes tools such as the projective null cone and the embedding space approach, facilitating a more intuitive grasp of conformal invariants and tensor structures. The paper particularly emphasizes the utility of these methods in simplifying the derivation of two-point and three-point functions, which are fixed by conformal symmetry up to a few constants.

The lectures also delve into the conformal bootstrap program—a non-perturbative framework for potentially solving CFTs by leveraging the constraints imposed by conformal invariance and unitarity. The bootstrap approach, particularly potent in dimensions greater than two, aims to determine the operator dimensions and OPE coefficients self-consistently. In three dimensions, the bootstrap has demonstrated its strength in providing rigorous results and bounds that guide the identification of possible CFTs.

Operator Product Expansion and Radial Quantization

Radial quantization emerges as a pivotal concept in the paper, employed to systematically understand state-operator correspondence in CFT. This approach allows for the decomposition of states on spheres centered at the origin, leading to a clear exposition of how local operators create states within the Hilbert space framework of quantum field theories.

The operator product expansion (OPE) holds a central place in the analysis presented in the lectures, elucidating how products of operators can be expanded in terms of primary and descendant fields. The convergence properties of the OPE in conformal theories lead to profound implications, such as the derivation of crossing symmetry as a consistency condition.

Unitarity, Anomalies, and Practical Implications

The constraints imposed by unitarity, particularly the unitarity bounds on operator dimensions, are addressed with detailed discussions on their derivation and significance. The lectures present these bounds as pivotal criteria that any sensible unitary CFT must satisfy, reflecting deep connections between conformal invariance and the positivity conditions required in physical theories.

An important theoretical implication of the discussed paradigms is their application in critical phenomena and statistical mechanics, where CFTs describe universality classes at second-order phase transitions. Furthermore, the discussions extend to exploring potential CFT applications in condensed matter systems and their potential utility in defining the fixed points relevant for understanding quantum phase transitions.

Conclusion and Future Directions

Slava Rychkov's lectures, as encapsulated in this paper, provide a foundational understanding of conformal field theory in three dimensions, demonstrating its theoretical robustness and applicability. The insights from conformal kinematics to the profound implications of the bootstrap program outline a vibrant domain of paper with profound theoretical and practical implications. As researchers push the boundaries of CFT, the foundational principles covered in these lectures will likely continue to underpin advancements in broader contexts of quantum field theory and beyond. Future directions in the field may involve more nuanced explorations of non-unitary theories or extended symmetries, enriching the tapestry of theoretical physics with rigorous underpinnings.