Second Order Calculations of the O(N) σ-Model Laplacian

For slowly varying fields on the scale of the lightest mass the logarithm of the vacuum functional of a massive quantum field theory can be expanded in terms of local functionals satisfying a form of the Schrödinger equation, the principal ingredient of which is a regulated functional Laplacian. We extend a previous work to construct the next to leading order terms of the Laplacian for the Schrödinger equation that acts on such local functionals. Like the leading order the next order is completely determined by imposing rotational invariance in the internal space together with closure of the Poincaré algebra. 1 The O(N) σ-model Laplacian We are going to study the O(N) σ-model as a test model to construct the Schrödinger representation for non-linear systems. This model has a very interesting structure when it is quantised. Because of its non-linearity it gives characteristics similar to the ones we have in Yang-Mills or the Einstein cases (see [1]). The O(N) σ-model is defined as the infinite dimensional theory of a particle on an N dimensional sphere parametrised by the function z(σ, τ) as the variable τ varies. We ask the (σ, τ) plane to be a Minkowski space. The action of the theory has to be a scalar with respect to the Lorentz transformations and the reparametrisations on the sphere. So we can choose it to be S = 1 2α ∫ dσdτgμν(ż ż − źź) (1) where ′ and · denote differentiations with respect σ and τ respectively, and α is a coupling constant. We can quantise this model using the Schrödinger Representation (see [3]-[10]) by taking the field z(σ, τ) to be diagonalised at τ = 0 satisfying the relation z(σ, 0)Ψ[z] = z(σ)Ψ[z] (2) for Ψ the Schrödinger wave functional and its conjugate momentum π(σ, τ) to be at τ = 0 π(σ, 0)Ψ[z] = iαDν(σ)Ψ[z] (3) so that the equal time commutation relation [z(σ, 0), πν(σ , 0)] = iαδ ν δ(σ, σ ) (4) is satisfied. In (3) the differential operator is defined with respect to a covariant differentiation whose meaning and structure will be given later on. Though, as z is a ‘scalar’, Dν(σ) takes the usual functional derivative form δ/δz (σ). From (1) we can read the Hamiltonian H = α 2 ∆ + 1 2α ∫ dσgμν ź ź (5) The Laplacian given in (5) as ∆ = ∫ dσ g12 Dμ1(σ)Dμ2(σ) = ∫ dσ1dσ2 g (σ1, σ2) Dμ1(σ1) Dμ2(σ2) for g (σ1, σ2) = g (z(σ1))δ(σ1 − σ2), is not well defined because the two functional derivatives act at the same point σ. Also the determinant of the infinite dimensional metric, g, is ill-defined as the integral on its diagonal σ1 = σ2 gives infinity. We can get around this problem by defining the Laplacian to have the regulated expression ∆s = ∫ dσ1dσ2 G μ1μ2 s (σ1, σ2)Dμ1(σ1)Dμ2(σ2). (6) The Kernel G, which takes the place of the infinite dimensional metric g, can be determined by a number of physical requirements. It has been shown in [2] that this is possible at the leading order, when the Laplacian acts on local functionals. We will see here that this is the case also for the next order. Following similar steps with [2] we require that G is a regularisation of the inverse metric, so we will assume that it depends