D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes

Abstract: The Euklidean correlation functions and vacuum expectation values of products of field operators of some Lorentz spin and dimension are expressed through Mellin amplitudes which depend on complex dimensions subject to linear constraints. The constraints can be solved in terms of conserved momenta whose squares are given by the field dimensions, and related Mandelstam variables s. The Mellin amplitudes furnish a universal representation of conformal field theories without explicit reference to D. The costumary principles of quantum field theory plus conformal invariance and operator product expansions (OPE) say that the Mellin amplitudes are amplitudes of dual resonance models with exact duality and a form of factorization which follows from OPE. Fields in the OPE with spin l and dimension d produce simple poles in the scalar 4-point Mellin amplitude at s=d-l+2n, n=0,1,2,3... with polynomial residues. The leading pole determines the satellites n=1,2,3...

• The paper introduces a novel transformation that recasts CFTs in any dimension into auxiliary dual resonance models using Mellin amplitudes.
• It demonstrates how Mellin transforms convert Euclidean correlation functions into momentum-dependent variables, streamlining the analysis of OPE and crossing symmetry.
• The work establishes conditions for unitarity and positivity in CFTs, suggesting profound links between higher-dimensional theories and dual resonance models.

Summary of the Paper: D-independent Representation of Conformal Field Theories

The paper by G. Mack presents a novel approach to representing Conformal Field Theories (CFTs) in any arbitrary dimension $D$ by transforming these into auxiliary Dual Resonance Models. This transformation is executed through a framework of Mellin amplitudes, which bypass the need for explicit reference to the space-time dimension. The work revolves around expressing Euclidean correlation functions and vacuum expectation values via Mellin amplitudes and subsequently elucidating the implications of this representation on CFT features like Operator Product Expansions (OPE) and crossing symmetry.

Conformal field theories have been extensively studied due to their rich symmetry properties and their relevance in various physical theories, including statistical mechanics and quantum field theories. The AdS/CFT correspondence further amplifies this importance, linking theories of gravity in Anti-de Sitter (AdS) space with CFTs on the boundary. Here, Mack explores the structural aspects of CFTs by employing an algebraic approach dependent upon the transformation to auxiliary models characterized by dual resonance amplitudes.

Mellin Representation and Constrained Variables

The primary innovation involves using Mellin transforms to express correlation functions of fields with spin $l_i$ and dimension $d_i$. The Mellin amplitudes $M_{k_n,...,k_1}$ are determined over complex variables $\delta_{ij}$, derived from momenta $p_i$ constrained by field dimensions. The central condition $\sum_j \delta_{ij} = d_i$ signifies a transformation from fields' dimensional constraints to momentum-dependent Mandelstam variables, permitting a field-theoretic description that accommodates dual resonance structures.

The Mellin transform, in this framework, serves as the bridge between field operators in CFTs and the resonance model perspectives, offering a new lens to view symmetries and dualities inherent in such theories. This comprehensive approach highlights the universality of representations across dimensions and the implications on dual models which possess properties consistent with dual resonance models.

Implications on Operator Product Expansions

A significant aspect of the paper is the establishment of an explicit link between Mellin amplitudes and operator product expansions. The paper demonstrates that factors determining the convergence properties of OPE can be mirrored by the factorization properties of amplitudes in dual resonance models. Importantly, the analysis shows that exact duality in crossing symmetry and meromorphy of the dual amplitudes are achieved, with simple poles indicating interactions of fields. The link to dual resonance models implies a potential for identifying shared structures between CFTs and string theory frameworks.

Positivity, Unitarity, and Higher-Dimensional Connections

The analysis provides conditions under which positivity and unitarity of two-point functions in CFTs are guaranteed, correlating with the unitarizable representations dictated by the conformal group. By establishing an overarching dimensional framework, the paper suggests dimensional reduction and connections to theories on lower-dimensional manifolds, demonstrating that Mellin amplitudes encode information consistent with dimensional induction.

Conclusion and Future Directions

Mack’s approach proposes a coherent methodology for describing CFTs in a dimension-independent manner that aligns closely with the theoretical underpinnings of quantum field theory and dual resonance models. This opens intriguing possibilities for future exploration, particularly in elucidating the deep connections between higher-dimensional theories and their low-dimensional counterparts via holographically-inspired methods. The mathematical transformations presented may ultimately serve as a unifying framework, promoting deeper insights into the structure and symmetries of conformally invariant systems.