Dilatation Operator and Space-time Geometry

AdS/CFT correspondence [1] conjectures an equivalence between the string/gravity theory on AdS5 × S and the conformal gauge theory of N = 4 super Yang–Mills (SYM) in the four-dimensional Minkowski space, which appears as the boundary to the five-dimensional anti de Sitter (AdS5) space. With the correspondence reversed, the AdS string/gravity can be regarded as built up over the fourdimensional gauge theory, where the scale dependence of gauge invariant composite operators is translated into the dynamics of the dual theory [2, 3]. From this point of view one should recover the AdS5 × S space as the geometry of the target space of a dynamical model whose Hamiltonian is represented by the dilatation operator. This geometrical description is similar to one in terms of so called Connes’ triples, which is used in non-commutative geometry, see e.g. [4]. According to this approach, one can describe, say, a Riemann space in terms of a triple consisting of algebra of observables, Hamiltonian and a Hilbert space. (For a brief review of this description and examples the reader is referred to [5,6].) The Hamiltonian in this approach is given by the dilatation operator, while the algebra of observables is given by the algebra of automorphisms of the algebra of composite operators. Thus the Hilbert space is an important element of the description of the dynamical model. According to prescriptions of AdS/CFT correspondence, quantum states of the dual theory are represented by local gauge invariant composite operators in the gauge theory. For the linear space of such operators to be a Hilbert space one should endow it with a corresponding Hermitian metric. The main condition to be satisfied by the metric is to render the dilatation operator self-adjoint. This is required in order to have unitarity of the quantum dynamics. This condition does not fix the metric uniquely, however, it is still a non-trivial requirement. Thus the planar metric proposed in [7, 8] does not fulfill this condition for a finite N , but most likely it can be corrected to do so by inserting a twist operator proposed in [9] (see also [10, 11]). As an approach in [2, 3] it was proposed to interpret the letter insertion and removal operators in the SU(2) sector of N = 4 SYM as oscillator ladder operators. This interpretation leads unambiguously to a matrix quantum mechanics with well-defined Hermitian product and a self-adjoint Hamiltonian. The