Conformal Field Theories Near a Boundary in General Dimensions

Abstract

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions $d$. Calculations of the universal function of a conformal invariant $ξ$ which appears in the two point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the $\vep=4-d$ expansion for the the operator $ϕ^2$ in $ϕ^4$ theory. The form for the associated functions of $ξ$ for the two point functions for the basic field $ϕ^α$ and the auxiliary field $λ$ in the the $N\to \infty$ limit of the $O(N)$ non linear sigma model for any $d$ in the range $2<d<4$ are also rederived. These results are obtained by integrating the two point functions over planes parallel to the boundary, defining a restricted two point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two point function. Consistency of the results is checked by considering the limit $d\to 4$ and also by analysis of the operator product expansions for $ϕ^αϕ^β$ and $λλ$. Using this method the form of the two point function for the energy momentum tensor in the conformal $O(N)$ model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two point functions are also derived making essential use of conformal invariance.