Boson operators method

Three-dimensional problems can also be addressed within the framework of the intensive boson operators method, by simply introducing, in place of the scalar quantities a and jS of Eq. (5.6), two (complex) scalars and two vectors associated with the scalar and vector boson operators of U(4), respectively. In this way it is possible to obtain the classical limit of... 

We also note that had we not used the method of intensive boson operators but rather evaluated Eq. (7.18) exactly, we would have obtained (van Roosmalen, 1982)... 

Table IV 1 7. Boson operators in the method of interacting bosons...

In the language of intensive boson operators, we simply have to double the definitions (5.6) and (5.7) to account for two families of boson operators and (f and i ), k =, 2. This method, when applied to Eq. (5.24), leads to the following potential surface ... 

As a final example of application of the intensive boson operator technique to the one-dimensional algebraic model, we consider the case of the n m Fermi operator introduced in Section III.E [Eq. (3.128)]. A straightforward use of the aforementioned method leads to the classical potential surface... 

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. 

In conclusion, a few words should be said about the equivalence between the ket-bra formalism frequently used in this article and the particle-hole formalism based on the ideas of second quantization T commonly used in the special propagator theories and the EOM method. Both formalisms are used to construct a basis for the operator space, and the essential difference is that the latter treats particles having specific symmetry properties—i.e., fermions or bosons—whereas the former is not yet adapted to any particular symmetry. In order to get a connection between the two schemes, it may be convenient in the ket-bra formalism to introduce a so-called Fock space for different numbers of particles... 

Particle statistics come in rather differently in PIMC. A permutation operation is used to project Bose and Fermi symmetry. (Remember that in DMC the fixed-node method with an antisymmetric trial function was used.) The permutations lead to a beautiful and computationally efficient way of understanding superfluidity for bosons, but for fermions, since one has to attach a minus sign to all odd permutations, as the temperature approaches the fermion energy a disastrous loss of computational efficiency occurs. There have been many applications of PIMC in chemistry, but almost all of them have been to problems where quantum statistics (the Pauli principle) were not important, and we do not discuss those here. The review article by Berne and Thirumalai [10] gives an overview of these applications. 

This expression can be obtained by applying the intensive boson method to the invariant operator of the coupled algebra Ui2(2). 

The chapter is organized as follows in the Section 7.2, we first present some details of spectral collocation method to develop space-time evolution of polarization plots for overall view of classical breathers in Section 7.2.1, then we present the mathematical model for TPBS parameters after second quantization in Section 7.2.2.1 and finally second quantization on K-G lattice is done with Bosonic field operators in Section 12.2.2. In Section 7.3, the results and discussion are also presented in three parts for the above three cases, hr Section 7.4, the conclusions are given. 

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