Volume 190, number 1,2 PHYSICS LETTERS B 21 May 1987 THE CASIMIR EFFECT IN CONFORMAL FIELD THEORIES Bernardo BARBIELLINI-AMIDEI D~partement de Physique, Universit~ de Gen~ve, CH-1211 Geneva4, Switzerland Received 13 October 1986 The aim of this letter is to give, by studying the zero-point energy of two-dimensional conformal tensor fields, a simple deriva- tion of two well-known formulae occurring in two-dimensional conformal field theory (CFT): the spectrum of the conformal central charge c and the critical dimension in string theory. In order to find the formula for c < 1, we inspire ourself by the Regge- pole model. 1. Con formal tensors. In CFT thinks a space as the complex plane, then the conformal transformations consist of the analytic mapping z~f(z), and the powerful machinery of complex analysis can be brought into play, Let us consider a conformal tensor field on the unit circle transforming under the rotations according to x'(o) =e-i:x(0), X'(2n) =X(2n) ~---e -i21rj , where j is the (conformal) spin (see fig. 1 ). One can expand Xin normal modes or oscillators: X(O) = ~ Xn_j e 2nitn-j)O . -- oo To avoid branch points, we require that j be a posi- tive integer. The quantum zero-point energy (i.e. the Casimir effect) is given, after the (-function regula- tion (see appendix), by 1 ~ (n-j)=½((-1;-j) 2 n=O =-,~+U(j+ l)=-hc, o o= 0 Fig. 1. where c= 1 + 6j0'+ 1 ) is called the conformal anom- aly parameter or central charge. With the central charge, one can parametrise the representations of the conformal group. For a tensor repr8sentation of spin j, c is an integer t> 1. We argue that the conformal tensors given all the unitary irreducible representations (UIR's) of the conformal group having c i> 1. 2. UIR's with c< 1 [I]. Regge in 1959 proposed to treat the angular momentum as a continuous com- plex variable - although, deafly, physically observa- ble states must have integral or half-integral angular momentum. Therefore inspired by Regge, let us con- sider the analytic continuation of c c(z)=l+6z, z=j(j+ 1)~C, which can be considered as a conformal field! A definition of hermitean conjugation for an observable A(z) in CFT's is given by [A(z)] + =A(- 1/z). In fact if the locations of the asymptotic states are z=0 and z=~, the conformal transformation f(z) = - 1/z maps 0 on ~, hence we get the exchange of the initial and final states. Note that in these theories z= 0 is a singularity, because of radial causality [ 1 ] ! Therefore the hermitian conjugate for c(z) is given by c+(z)=l-6/z. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 137