Mukadas Missarov Dynamical properties of renormalization group flow in projective space representationAbstract
We consider fourcomponent fermionic (Grassmannvalued) field on the hierarchical
lattice. The Gaussian part of the Hamiltonian in the model is invariant under the
blockspin renormalization group transformation with given degree of normalization
factor ( renormalization group parameter). The nonGaussian part of the Hamiltonian
is given by the selfinteraction forms of the 2nd and 4th order. Fermionic
hierarchical model has natural continuous version and can be described in terms of
the $p$adic quantum field theory approach and Wilson's renormalization group.
We investigate dynamics of the renormalization group in the space of the
coefficients determining the Grassmannvalued free measure density. This space is
treated as twodimensional projective space. The action of the renormalization group
(RG) transformation in this projective space is reduced to the quadratic
homogeneous mapping. If the RG parameter is more then 1, then the unique attractive
fixed point of RG transformation is given by the density of the Grassmann
deltafunction. We describe four different RGinvariant subsets in the small
neighborhood of deltafunction fixed point . Almost all points of the projective
plane tend to the $\delta$function under the iterations of RG mapping through only
one of these subsets. Using the results of this classification we give explicit
description of the global RGflow in the whole projective plane.
The exact solution of the fermionic hierarchical model generates nontrivial
conjectures for the different fermionic and bosonic hierarchical and Euclidean field
models.
