Conformal Invariance and the Stress-Energy Tensor

Abstract

We discuss in this chapter the geometrical consequences of the scale invariance property which characterizes a second order critical point. For systems with short-ranged interactions, it is reasonable to suppose that global scale invariance (1'" = br) generalizes to local scale invariance (1'" = b(r)f). However such transformations may change the global geometry of the system, therefore affecting the local covariant transformation properties of the scaling operators. This will be shown to lead to the conformal anomaly, parametrized by the central charge c, the existence of which describes the presence of long-ranged fluctuations. We define the transformation properties of physical scaling operators under smooth local transformations. This fixes the two-point and three-point correlation functions in d dimensions. The stress-energy tensor will be shown to act as a generating function for thes~ local transformations. 2.1 Conformal group In the infinite size limit, a critical system gains the invariance properties of a larger geometrical group. Dilatation invariance complements now the usual translational and rotational invariance. Such transformations are called conformal. By definition, conformal transformations preserve the angle of intersection of two arbitrary curves as illustrated in Figure 11. The assumption that the fundamental interaction is short-ranged suggests that Fig. 11. Conformal transformation in d dimensions. local smooth transformations do not affect the global behaviour of the critical system. This is verified for large classes of physical claSsical systems. This is not true any more if long distance interactions or strongly anisotropic effects occur. Suppose that a system is invariant under local combinations of rotations, translations, and dilatations. In d > 2 dimensions, the conformal group is finite-dimensional. It is generated by global rotations 1"-+ 1' " = Ar, global translations. 1"-+ 1' " = 1"+ ii, global dilatations 1"-+ 1' " = br, and the so-called special conformal transformation (2.1)