The boundary S-matrix and the AdS to CFT dictionary

An S-matrix is defined for anti-de Sitter space by constructing “in” and “out” states that asymptote to the timelike boundary. An analog of the LSZ formula shows that this boundary S-matrix is given directly by correlation functions in the boundary conformal theory. This provides a key entry in the AdS to CFT dictionary. † Email address: [email protected] The conjectured AdS/CFT correspondence[1] has offered a promising new window into the dynamics of string/M theory. But in order to exploit this powerful framework, we must decipher the holographic relationship between the bulk and boundary theories. Gubser, Klebanov, and Polyakov[2] and Witten[3] made important progress in this regard by providing a CFT to AdS dictionary: they show how to derive CFT correlation functions from the bulk theory in AdS. This has allowed the successful calculation of various CFT correlators. However, in order to study bulk physics, and in particular to understand the undoubtedly profound implications of holography, a reverse dictionary is needed: we need to know which bulk quantities can be calculated, and how to calculate them, from the boundary CFT. Another important and closely related question is how to treat scattering in AdS. Due to the periodicity of particle orbits and lack of ordinary asymptotic states in AdS, a conventional S-matrix cannot be defined. However, [4] outlined the definition of an AdS analog of the S-matrix in terms of scattering of states from the timelike infinity. Refs. [5,6] gave a related definition in the infinite-N limit. The purpose of the present note will be to go further and provide an intrinsic and explicit definition of this “boundary” S-matrix for arbitrary N , and to give a precise relation between it and the CFT correlators. The discussion also clarifies the relation between the framework of [5,6] and that of [4]. Other recent treatments of related aspects of the AdS/CFT dictionary include[7,8,9]. To summarize in advance, the boundary S-matrix will be defined as an overlap of certain “in” and “out” states. These will be defined so that they correspond to particles asymptotic to the timelike boundary of AdS in the past/future. An AdS analog of the LSZ formula can then be derived and relates this S-matrix to the bulk correlation functions. Finally the results of [3] are used to rewrite the boundary S-matrix in terms of the CFT correlation functions. An extremely simple relationship results: the boundary S-matrix equals the corresponding CFT correlator. This serves as a key entry in the AdS to CFT dictionary. For simplicity we will consider scalar fields, with action