A semi-infinite matrix analysis of the BFKL equation

A new mapping between the BFKL equation and Beisert's representation of the XXX Heisenberg ferromagnet with spin s = - 1/2 is given. The action of the Hamiltonian operator of a spin chain with SL(2) invariance on a symmetric double copy of a harmonic oscillator excited state is shown to be identical to the action of the BFKL Hamiltonian on the gluon Green function for the azimuthal-angle averaged forward scattering case. A natural mapping between the gluon Green function, discretized in virtuality space, and the double harmonic oscillator excited state emerges.The forward BFKL equation is discretised in virtuality space and it is shown that the diffusion into infrared and ultraviolet momenta can be understood in terms of a semi-infinite matrix. The square truncation of this matrix can be exponentiated leading to asymptotic eigenstates sharing many features with the BFKL gluon Green’s function in the limit of large matrix size. This truncation is closely related to a representation of the XXX Heisenberg spin $= - \frac{1}{2}$ chain with SL(2) invariance where the Hamiltonian acts on a symmetric double copy of the harmonic oscillator. A simple modification of the BFKL matrix suppressing the infrared modes generates evolution more compatible with the Froissart bound.The forward BFKL equation is discretised in virtuality space and it is shown that the diffusion into infrared and ultraviolet momenta can be understood in terms of a semi-infinite matrix. The square truncation of this matrix can be exponentiated leading to asymptotic eigenstates sharing many features with the BFKL gluon Green's function in the limit of large matrix size. This truncation is closely related to a representation of the XXX Heisenberg spin $= - \frac{1}{2}$ chain with SL(2) invariance where the Hamiltonian acts on a symmetric double copy of the harmonic oscillator. A simple modification of the BFKL matrix suppressing the infrared modes generates evolution with energy compatible with unitarity