Formal relation to field theory

Much of our understanding of critical phenomena is based on the Landau-Ginzburg model of a ferromagnet. This model concentrates on the local magnetization, represented by an m-component vector field Sa (r), a = 1. m. often called a classical spin field . The interaction of the spin field is described by the Landau-Ginzburg Hamiltonian 

Evaluating the model (A 7.1) one finds that it describes a second order pha.se transition occurring for h = 0 and a critical value r = re. The transition is signaled by long range correlations of the spin field or power-type 

There exists a formal relation among polymer theory and the Landau-Ginzburg model. Specifically for the Greensfunction of a single continuous chain one finds 

The relation (A 7.5) first was derived by de Gennes, initiating the renormalization group or scaling approach to polymer solutions. Again these expressions need some explanation. Equation (A 7.6) defines G/ (r, r 7 q) as path integral , summing over all continuous paths r(s), 0 s Rh r(0) = r r(Rg) = r. It is properly defined as the continuous chain limit of the discrete 

Introducing the segment density generalized to a continuous chain 

Hiiigularities occurring for h O r r - Furthermore macroscopic observ-ablcjH like the free energj or the spin correlation functions show scaling in T = r - T(. and /i, most similar to the scaling laws in polymer solutions. 

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In order to formulate a theory for the evaluation of vibrational intensities within the framework of continuum solvation models, it is necessary to consider that formally the radiation electric field (static, Eloc and optical E[jc) acting on the molecule in the cavity differ from the corresponding Maxwell fields in the medium, E and Em. However, the response of the molecule to the external perturbation depends on the field locally acting on it. This problem, usually referred to as the local field effect, is normally solved by resorting to the Onsager-Lorentz theory of dielectric polarization [21,44], In such an approach the macroscopic quantities are related to the microscopic electric response of... 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

After having explained the relation between AIT and the LFT formalism, we now turn to a brief outline of the various parameterizations of V (p). The effective ligand field Hamiltonian (77) consists of one-electron terms, the one-electron ligand field Hamiltonian (vi>(/j). and two-electron terms (G(i,j)), which take account of the Coulomb interactions between the d-electrons summations is carried out over the d-electrons i <7 = 1, Nd. In difference to crystal field theory, these operators are left unspecified. Various LF models differ in the way they approxi-... 

The discussion of fhe previous 11 sections included introductory-historical remarks that are related to the work described here, and focused, on the one hand on certain aspects of formal properties of field-free and field-induced unstable sfafes (mainly Sections 3-5, 11), and on the other hand, on the theory and methodology for fhe sysfematic computation and... 

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