Lagrangian formalism of the field theory

There exist several transformations t for any initiaJ Hamiltonian Tio, which lead, generally speaking, to different fixed points H, Hg, or each of them characterizing a certain type of cooperative interaction. 

Universality, i.e, independence on the initial Tio, exists only for a given type of cooperative interaction. 

When calculations by means of renormalization group method are made, the space dimensionality d 2ind the order parameter dimensions (the number of components) n play an important role. 

for a ferromagnet with n = 1, spins can be placed only along one direction. If they are on a film surface, d = 2 if in bulk material, d = 3. n = 2 in the so-called XY model of a ferromeignet. If the magnetic momentum vector can be oriented along any direction in space, then n = 3 (Heisenberg s model) (Table 2.5). 

For a system with a scalar order parameter (density, concentration difference), n = 1. 

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I inally, section 2.6 represents the Lagrangian formalism of general field theory, following Amit (1978). This formalism was developed in the quantum field theory and has recently come into use in polymer theory. 

The recent, most significant achievements in studying the structure of polymer systems are based on applying the field theory formalism. This section contauns the definitions and brief characterization of the main quantities and terms of the field theory in the Lagrangian form. Hereinafter, we follow Amit (1978). 

Section 2.6 gives the Lagrangian formalism of field theory. The properties of the correlation function of the order parameter G (Green s function) are treated in detail, slncp many experimentally determined values are expressed in terms of this function. 

The Kohn-Sham theory made a dramatic impact in the field of ab initio molecular dynamics. In the 1985, Car and Parrinello38 introduced a new formalism to study dynamics of molecular systems in which the total energy functional defined as in the Kohn-Sham formalism proved to be instrumental for practical applications. In the Car-Parrinello method (CP), the equations of motion are based on a Lagrangian (Lcp) which includes fictitious degrees of freedom associated with the electronic state. It is defined as ... 

All aspects of Newtonian mechanics can equally well be formulated within the more general Lagrangian framework based on a single scalar function, the Lagrangian. These formal developments are essential prerequisites for the later discussion of relativistic mechanics and relativistic quantum field theories. As a matter of fact the importance of the Lagrangian formalism for contemporary physics cannot be overestimated as it has strongly contributed to the development of every branch of modem theoretical physics. We will thus briefly discuss its most central formal aspects within the framework of classical Newtonian mechanics. 

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