Chapter 6 Special Conformal Invariance Conformal transformation on the d-dimensional flat space-time manifold M is an invertible mapping of the space-time coordinate x x x the metric tensor invariant up to a local scale factor = such that it leaves [ ] ( x ) Ω η η = η ν ν ν (6.1) The set of all the conformal transformations on the flat Lorentz manifold M is the connected part of the smallest group containing the Poincaré group and the inversion on ( )( ) an unit hyperboloid [18]. Full conformal symmetry is a d + 1 d + -parameter Lie group isomorphic to the non-compact classical group SO( d, ). Among the ( d + 1)( d + ) ( ) generators of conformal group there are dd+ 1 -Poincaré group generators, a 1-generator for the scale or dilatation transformation, and the rest d generators are for the special-conformal transformation which defines non-trivial 36
Ω( x ). In a d 3 flat space-time manifold M the special-conformal transformation is induced by the following coordinate transformation [18, 11] x x b x x = 1 b x + b x (6.) Here b is an arbitrary Poincaré vector. In d =, there exists an infinite number of coordinate transformations that, although not well-defined everywhere in M, but are locally conformal. The above transformation law is still true for the 6-parameter global conformal group made up of one-to-one mapping of the whole -dimensional complex plain M onto itself. But locally, there are an infinite variety of conformal transformations which maps various parts of the -dimensional complex plain into itself. Therefore, in general, it is impossible to define a special-conformal transformation uniquely in d =. In d 3, the commutators of the generators of special-conformal transformation and the generators of translation is a linear combination of the generators of the Lorentz transformation and scale transformation [18]. Thus a Poincaré invariant field theory, which is also invariant under special-conformal transformation, should necessarily be scale invariant. Therefore we have the following theorem. 37
6.1 A Sufficient Condition in d 3 Theorem: If a Poincaré invariant field theory possesses the special-conformal symmetry on a flat space-time manifold of d 3, then there exists an improved stress-energy tensor [4]. Proof: Looking at the infinitesimal special conformal diffeormorphism on flat M, we find ( ) x x = x + b x x b x, b << x (6.3a) x x ν ( b x ) δ ν ( x bν xνb ) = δ + + (6.3b) ν From these equations we understand that an infinitesimal special-conformal transformation is a local infinitesimal dilatation of σ = b x followed by a local infinitesimal Lorentz transformation with ω ν = ( ν ν ) xb xb at each space-time point of M. Therefore the fields, which are representations of Lorentz group and have specific transformation properties under scale transformation, should transform as 1 β ϕ ϕ ( x ) = I+ ωβ Σ ( I σd) ϕ (6.4) under (8.3a). Hence 38
( ) ( ) ( ) ( ) β δϕ x = ϕ x ϕ x = b x x + b x ϕ b x Σ ϕ b x D ϕ (6.5) β The action is special-conformal invariant as well as Poincaré invariant. Therefore it must be scale invariant. Applying the conservation of scale current, we find ( ) ( ) Scale Invariance: S = 0 Π ϕ d + Π D ϕ = 0 (6.6) Simplification of the last equation yields the following identity ϕ ϕ ϕ D + Π D ϕ + Π = d (6.7) This equation is the generalization of the global scale invariance to the special-conformal invariance. It can be thought as a local scale invariance at each point in M. Also β δ = δϕ + Π ( δϕ ) = bπβσ ϕ b Π D ϕ db x b x x + b x (6.8) ϕ Now applying eqn. (6.3a) in eqn. (6.8), we find ν ( ) b ( Π Πν Σ ) δ + ε = D ϕ + ϕ (6.9) In section.1 we showed that the weakest requirement for the existence of a conserved current due to an infinitesimal coordinate symmetry x x ε ε +, << 1 is 39
the existence of a vector field V such that ( ) δ + ε = V. Hence there must exists a tensor field σ ν ( x ) such that Π ν ν D ϕ + Π Σ ϕ = σ (6.10) ν ν Therefore for a Poincaré invariant special-conformal field theory on a flat space-time manifold of d 3 there always exists an improved traceless stress-energy tensor. 6. Special Conformal Current Eqn. (6.10) is a necessary condition for a Poincaré invariant field theory in d 3 to be special-conformal invariant. Applying eqns. (6.5) and (6.10) to eqn. (.9), we obtain the following conserved canonical special-conformal current ν ν ν ν ν ( x x η x ) + x Π Σ ϕ + x Π ϕ σ j (6.11) ν = Θ D The conservation law is not a trivial identity. Substituting T (6.11), we find β = Θ + B into eqn. β [ ] ν β ν ( η ) σ σ β ( δ ν ) j ν T x x ν ν x x ν = + B x x x (6.1) 40
Substituting T = E 1 C ρ ρ where C ρ is defined by (5.4a,b,c), and then dropping out all the anti-symmetric divergence, we find an improved special-conformal current C ν as C ν ν ν ( x x η x ) = E (6.13) Now the conservation law is an identity, as expected, due to ν ν ν E = 0, E = 0, and E = E. 6.3 General Conformal Transformation If the stress-energy tensor of an arbitrary dimensional (including d = ) flat space-time conformal field theory is obtained by functional differentiation of a conformally invariant action with respect to the metric tensor in the corresponding Riemannian manifold M R, ν then the stress-energy conservation law T = 0 where g δs M R δ ν g gβ = η β = T have ν to be conformally invariant. This is possible if and only if [35] ( ) ( ) = Ω( ) d 1 [ ] ( ) T x T x x T x ν ν ν (6.14a) ( ) ( ) T x = 0 T x = 0 (6.14b) 41
under a general conformal transformation ( ) ( ) = Ω( ) [ ] ( ) g x g x x g x ν ν ν in M R. Therefore if one can generalize the flat space-time action of a conformal field theory to the corresponding Riemannian manifold via the minimal coupling prescription d d ( ν g ν,, gβ = 0, d x d x g ) η such that the action S M R on the Riemannian manifold becomes general conformal invariant, then g δs M R δ g ν gβ = η β = E (6.15) ν Hence σ ν 0 for general-conformal field theories. We will find a few examples of this result in the next chapter. 4