Renormalization transformation

Because (7) has the same form as the original map, we can do the same analysis all over again, now regarding (7) as the fundamental map. In other words, we can renormalize ad infinitum. This allows us to bootstrap our way to the onset of chaos, using only the renormalization transformation (8). 

The notion of an effective Hamiltonian can be defined almost rigorously from the perspective of the renormalization group. One intriguing idea that has emerged from work in the critical phenomena arena is that in addition to the importance of renormalization transformations near the critical point, there may be merit to imitating the systematic way in which degrees of freedom are eliminated even when there is no critical point in question. 

Of course, each transformation from N to N + 1, is implemented by proper scale changes and by a re-numbering of the degrees of freedom. In this way, for an infinite system, it is possible to establish a one-to-one correspondence between the degrees of freedom of Nassociated with the same set [Pg.472]

The finite-size scaling approach to localization assumes that there is a unique function describing the length scaling of the renormalization transformation which is universal for systems of the same symmetry and dimension and that depends only on the Thouless number (22). This function is (3(g), where (for L a small multiple of L)... 

Figure I Scaling function for the renormalization transformation. The critical value, which is an unstable fixed point for the transformation, is indicated by gc. d is the number of dimensions. 3, the scaling function, and g, the Thouless number, are defined fully in the text.

Define R, = IC,S, as a renormalization transformation. Equation 16 then takes the form... 

It has to be specially noted that the set of S, for s real numbers is an Abelian group while that of >C, for s non-negative numbers is a commutative semigroup. S, and K, commute therefore, the sot of R, for s non-negative numbers is also a commutative semigroup called a renormalization transformation group (renormalization group). ... 

Discuss (/ ) from which I) = 2 aa Q (Oono and Freed, 1981a). Renormalization transformation 17 gives... 

The renormalization transformation in the problem of polymer chain conformations in the Kadanoff-Wilson fashion forms, in essence, a semigroup. A version of such transformations based on the true group (also called the renormalization group) was applied by Alkhimov (1991, 1994). This method provides an asymptotical solution of the exact equation for the eiid-to-eiid distance probability distribution of a self-avoiding trajectories. The following formula has been obtained for the critical index i/ in d-dimension space ... 

The first point of the renormalization transformation (Amit, 1978) is determined from the condition... 

A more fundamental but much more complex numerical approach is provided by the hierarchical reference theory (HRT). °° In the HRT the numeral implication of the renormalization transformation is applied on a microscopic model of the fluid. One starts from a reference system with short-range repulsive interactions and then formulates a hierarchy of integral equations accounting for successively longer-range fluctuations. The theory has also been extended to fluid mixtures. " The HRT provides estimates for both... 

Figure 2.68 Grid model of a porous medium (left) and renormalization group transformation replacing a cluster of grid cells by a unit cell of larger scale (right).

The coupling of electronic and vibrational motions is studied by two canonical transformations, namely, normal coordinate transformation and momentum transformation on molecular Hamiltonian. It is shown that by these transformations we can pass from crude approximation to adiabatic approximation and then to non-adiahatic (diabatic) Hamiltonian. This leads to renormalized fermions and renotmahzed diabatic phonons. Simple calculations on H2, HD, and D2 systems are performed and compared with previous approaches. Finally, the problem of reducing diabatic Hamiltonian to adiabatic and crude adiabatic is discussed in the broader context of electronic quasi-degeneracy. 

Here in eq. (38) "EpqfpQN a.pag is new Hartree-Fock operator for a new fermions (25), (26), operator Y,pQRsy>pQR a Oq 0s%] is a new fermion correlation operator and Escf is a new fermion Hartree-Fock energy. Our new basis set is obtained by diagonalizing the operator / from eq. (36). The new Fermi vacuum is renormalized Fermi vacuum and new fermions are renormalized electrons. The diagonalization of/ operator (36) leads Jo coupled perturbed Hartree-Fock (CPHF) equations [ 18-20]. Similarly operators br bt) corresponds to renormalized phonons. Using the quasiparticle canonical transformations (25-28) and the Wick theorem the V-E Hamiltonian takes the form... 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

The most direct influence on the current work is the recent canonical diago-nalisation theory of White [22]. This, in turn, is an independent redevelopment of the flow-renormalization group (flow-RG) of Wegner [23] and Glazek and Wilson [24]. As pointed out by Freed [25], canonical transformations are themselves a kind of renormalization, and our current theory may be viewed also from a renormalization group perspective. 

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