Boson operator formalism

We next consider another example of quantum-mechanical relaxation. In this example an isolated harmonic mode, which is regarded as our system, is weakly coupled to an infinite bath of other harmonic modes. This example is most easily analyzed using the boson operator formalism (Section 2.9.2), with the Hamiltonian... 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

As discussed in Section VI.A for the case of spin systems, the formalism described above is not the only way to construct a mapping of a A -level system. First of all, it is clear that one may again eliminate one boson DoF by exploiting the operator (which corresponds to the identity operator in the physical... 

As a matter of fact, the Boson normal-ordering procedure allows us to get the formal solution of the IP time-evolution operator involving the Dyson timeordering operator P. Also, observe that within the Bosons representation, the IP coordinate is... 

Linear terms are absent because of the Brillouin theorem. The coefficients Ap p. and Bap p, can be calculated by equating the nonzero matrix elements of the RPA Hamiltonian [Eq. (122)], in the basis of Eq. (121), to the corresponding matrix elements of the exact Hamiltonian [Eq. (23)] in the same basis. From the translational symmetry of the mean field states it follows that the A and B coefficients do not depend on the complete labels P = n, i, K and P = n, /, K1, but only on the sublattice labels /, AT and /, K. The second ingredient of the RPA formalism is that we assume boson commutation relations for the excitation and de-excitation operators (Raich and Etters, 1968 Dunmore, 1972). 

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... 

The abstract formalism introduced in this chapter builds the fundament of the theory of extended two-particle Green s functions. Our approach is very general in order to allow for a unified treatment of the different species of extended Green s functions discussed in the main part of this paper. Since the discussed propagators can be applied to a wide variety of physical situations, the emphasis of this chapter lies on the unifying mathematical structure. The formalism is developed simultaneously for (projectile) particles of fermionic and bosonic character. We will define the general extended states which serve to define the primary or model space of the extended Green s functions. We also define the /.j-product under which the previously defined extended states fulfil peculiar orthonormality conditions. Finally we introduce a canonical extension of common Fock-space operators and super-operators to the space of the extended states. 

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