Boson field

It is important to stress that use of the generalised Bogoliubov transformatin provides an elegant physical interpretation of the Casimir effect as a consequence of the condensation in the vacuum of the fermion or the boson field. The method can be extended to other geometries such as spherical or cylindrical. 

By expanding the effective Lagrangian with the respect to the Goldstone boson fields, one sees that g is also connected to the vector meson coupling to two pions, through the relation... 

Davies16) has given a rigorous mathematical analysis of the variational estimate of the ground state of the Hamiltonian K based on trial functions consisting of the tensor product x of a molecular wavefunction 0, and a coherent state 0 for the boson field, i.e. one searches for the minimum value of... 

In order to guarantee that K actually possesses a ground-state it is necessary to state the conditions that ensure that it is bounded below. This turns out to be the requirement that the energy of the boson field be finite, so we must have... 

As a result of the interaction between the molecule and the boson field, the bare molecule becomes dressed with a cloud of boson particles in the language of Sect. 2 the dressed molecule is an elementary excitation in the many-body system of molecules and boson particles. The number density of dressing boson particles is given by... 

The model Hamiltonian [Eq. (3)] defined on a continuum has some exact solutions [35]. These have culminated in what is now known as the bosonization technique, in which the interacting fermion fields can be expressed in terms of boson field operators. This method is reviewed in Refs. 15, 16, and 31. 

We consider a two-level system coupled to a bath of harmonic oscillators that will be referred to as a boson field. Two variations of this model, which differ from each other by the basis used to describe the two-level system, are frequently encountered. In one, the basis is made of the eigenstates of the two-state Hamiltonian that describes the isolated system. The full Hamiltonian is then written... 

The polaron transformation, executed on the Hamiltonian (12.8)-( 12.10) was seen to yield a new Hamiltonian, Eq. (12.15), in which the interstate coupling is renormalized or dressed by an operator that shifts the position coordinates associated with the boson field. This transformation is well known in the solid-state physics literature, however in much of the chemical literature a similar end is achieved via a different route based on the Bom-Oppenheimer (BO) theory of molecular vibronic stmcture (Section 2.5). In the BO approximation, molecular vibronic states are of the form (/) (r,R)x ,v(R) where r and R denote electronic and nuclear coordinates, respectively, R) are eigenfunctions of the electronic Hamiltonian (with corresponding eigenvalues E r ) ) obtained at fixed nuclear coordinates R and... 

Tliese quantum thermal averages over an equilibrium boson field can be evaluated by applying the raising and lowering operator algebra that was introduced in Section 2.9.2. 

Equations (12.55), sometime referred to as multiphonon transition rates for reasons that become clear below, are explicit expressions for the golden-rule transitions rates between two levels coupled to a boson field in the shifted parallel harmonic potential surfaces model. The rates are seen to depend on the level spacing 21, the normal mode spectrum mo,, the normal mode shift parameters Ao-, the temperature (through the boson populations ) and the nonadiabatic coupling... 

The golden-rule rate expressions obtained and discussed above are very useful for many processes that involve transitions between individual levels coupled to boson fields, however there are important problems whose proper description requires going beyond this simple but powerful treatment. For example, an important attribute of this formalism is that it focuses on the rate of a given process rather than on its full time evolution. Consequently, a prerequisite for the success of this approach is that the process will indeed be dominated by a single rate. In the model of Figure 12.3, after the molecule is excited to a higher vibrational level of the electronic state 2 the relaxation back into electronic state 1 is characterized by the single rate (12.34) only provided that thermal relaxation within the vibrational subspace in electronic state 2 is faster than the 2 1 electronic transition. This is... 

Details of the derivation of general expressions for energy shifts at a given order can be found in Mohr et al. (1998). Contractions between pairs of fermion or boson field operators AM lead to electron and photon propagator functions. The exact electron propagator in a static external field is homogeneous in time and appears as... 

Within the Higgs mechanism two complex scalar boson fields 4> x) and (t x) are introduced. These two fields form a SU(2)i, doublet, which is called the Higgs field (x). For this field a Lagrange density of the type... 

The computational technique used to treat the generalized Anderson impurity model in the slave boson representation will be described in some detail. For an extensive discussion see Coleman (1984). In Appendix A we represented the CEF states of stable 4f" shells (i.e., with integer occupation n) by pseudofermions. In the present case of unstable shells with possible 4f and 4f configurations we need an additional slave boson field for the 4f° state. The interesting physical quantities, static as well as dynamic, can be calculated in terms of the fully renormalized fermion and slave boson Matsubara Green s functions... 

In a first approximation the molecule length is assumed infinite and corrections due to the finite length are introduced as a second step. The intramolecular excitation brought by an external supply of energy (e.g., the output of a chemical reaction) is described by a complex Heisenberg boson field t), solution of a nonlinear Schrodinger equation... 

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