Boson annihilation operators

If and a are boson creation and boson annihilation operators, respec-... 

In Chapter 1 we have discussed the familiar realization of quantum mechanics in terms of differential operators acting on the space of functions (the Schrodinger wave function formulation, also called wave mechanics ). A different realization can be obtained by means of creation and annihilation operators, leading to an algebraic formulation of quantum mechanics, sometimes called matrix mechanics. For problems with no spin, the formulation is done in terms of boson creation, b (, and annihilation, ba, operators, satisfying the commutation relations... 

To provide a realization for the algebra U(2) we take two boson creation and annihilation operators, which we denote by ot, Tr and 0,x. The algebra U(2) has four operators which can be realized as (Schwinger, 1965),... 

The basis states N,nx> or IF,FZ > can be written explicitly in terms of boson creation and annihilation operators... 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

The expectation value of H in the coherent state (7.17) can be evaluated explicitly for any Hamiltonian. However, an even simpler construction of Hd (valid to leading order in N) can be done (Cooper and Levine, 1989) by introducing intensive boson operators (Gilmore, 1981). In view of its simplicity, we report here this construction. If one divides the individual creation and annihilation operators by the square root of the total number of bosons, the relevant commutation relations become... 

Consider, furthermore, a (2i- - 1)-dimensional subspace of the Hilbert space with fixed 5. Then, according to Schwinger s theory of angular momentum [98], this discrete spin DoF can be represented by two bosonic oscillators described by creation and annihilation operators with commutation relations... 

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]. 

Here a and are the usual oscillator creation and annihilation operators with bosonic commutation relations (73), and 0i,..., 1 ,..., 0Af) denotes a harmonic-oscillator eigenstate with a single quantum excitation in the mode n. According to Eq. (80a), the bosonic representation of the Hamiltonian (79) is given by... 

Or is the frequency of the harmonic oscilator and b) are boson (phonon) creation (annihilation) operators. In order to use the perturbation theory we have to split the Hamiltonian (16) onto the unperturbed part Hq and the perturbation H ... 

Since our sets of boson creation and annihilation operators and fermion creation and annihilation operators commute we can write our unperturbed wavefuntion (po) as the product of the fermion state vector /o) and the boson state vector %o), i.e. 

Three distinct sets of linear mappings for the partial 3-positivity matrices in Eqs. (31)-(36) are important (i) the contraction mappings, which relate the lifted metric matrices to the 2-positive matrices in Eqs. (27)-(29) (ii) the linear interconversion mappings from rearranging creation and annihilation operators to interrelate the lifted metric matrices and (iii) antisymmetry (or symmetry) conditions, which enforce the permutation of the creation operators for fermions (or bosons). Note that the correct permutation of the annihilation operators is automatically enforced from the permutation of the creation operators in (iii) by the Hermiticity of the matrices. 

In the method of interacting bosons, one introduces boson creation, T t, and annihilation, [ operators satisfying boson commutation relations... 

These boson operators are related to those introduced in [20] as shown in Table 7. They transform under D4/, as the representation indicated by the appropriate letter. [An extended s-wave i can also be introduced.] One then constructs the Hamiltonian by expanding it into bilinear products of creation and annihilation operators, with the constraint that H must transform as the representation of... 

For practical calculations it is often convenient to use the occupation number representation. In terms of boson annihilation(a ) and creationfaja ) operators satisfying... 

In the interacting boson model-2, low lying collective states of nuclei are described in terms of 12 dynamical bosons [ARI77,0TS78], six proton and six neutron bosons. The six proton and neutron bosons have angular momentum J=0 (s-boson) and J=2 (d-boson). It is convenient to introduce creation (d, s ) (u = 2, 1, 0) and annihilation (du,s) operators. When proton ( tt ) and neutron (v) degrees of freedom are added, the creation and annihilation operators assume an extra label ( tt, v), d s, ... 

There is a second, alternative approach. One could assume that the unpaired neutron and the unpaired proton form a quasibound state. The total number of components of the angular momenta of this quasi-bound state is given by n n v. Then we introduce a pair of new bosonic creation and annihilation operators associated with each level of this subsystem, cj, Cj, I,J =... 

Having established that creation and annihilation operators are rank 1 covariant and contravariant tensors, respectively, with respect to the operator ( )L,S, we can define an rath-rank boson operator as consisting of a like number of fermion creation and annihilation operators. Then the normal product of an rath-rank boson operator is a natural definition for the irreducible tensor. 

Thus far we have only considered one (boson) vector field, namely, the direct product field R Xn of creation and annihilation operators. The coefficients of the creation and annihilation operator pairs in fact also constitute vector fields this can be shown rigorously by construction, but the result can also be inferred. Consider that the Hamiltonian and the cluster operators are index free or scalar operators then the excitation operators, which form part of the said operators, must be contracted, in the sense of tensors, by the coefficients. But then we have the result that the coefficients themselves behave like tensors. This conclusion is not of immediate use, but will be important in the manipulation of the final equations (i.e., after the diagrams have contracted the excitation operators). Also, the sense of the words rank and irreducible rank as they have been used to describe components of the Hamiltonian is now clear they refer to the excitation operator (or, equivalently, the coefficient) part of the operator. 

In order to rigorously describe the nonlinear interaction between the weak pulsed fields, we now turn to the fully quantum treatment of the system. The traveling-wave electric fields can be expressed through single mode operators as j(z, t) = dj(t) Cqz (j = 1, 2), where uj is the annihilation operator for the field mode with the wavevector kp + q. The singlemode operators a and aq possess the standard bosonic commutation relations... 

The interest for possible applications of quantum algebras in physics has been awaken by the introduction of g-deformed boson creation and annihilation operators [23]. 

Let us define three independent, g-deformed boson creation and annihilation operators, 6] and b( (i = +,0, -), and the corresponding number operators Nit which satisfy the commutation relations ... 

The first two of these commutation relations are identical with the commutation relations for Boson creation and annihilation operators, while the last relation in the set differs slightly from the commutation relation, which would hold for bosons. Thus the operators By and By can be thought of as creating or annihilating quasi-bosons . 

Where B (Bn) is boson creation (annihilation) operator of quantum (exciton)... 

Here and in the rest of this section 0 shall explicitly represent the quantised fermionic field and also the normal order product AB is written explicitly, which implies that all annihilation operators are to the right of all creation opeartors. All boson operators within an normal ordered product are treated as if they would commute, while all fermion operators are treated as if they would anticommute (see appendix). After applying Wick s theorem a number of terms arise that describe different scattering... 

The corresponding Hamiltonian operator will still be given in terms of proper expansions over bilinear forms of (boson) creation and annihilation operators. (The more complex situations including half-spin particles can be addressed as well by using fermion operators [20].) The general rule is that one introduces a set of (n -I-1) boson operators b, and b (/, y = 1,. . . , n + 1) satisfying the commutation relations... 

A final comment about wavefunctions. We already have analytical expressions for wavefunctions of the harmonic, Morse, and Poschl-Teller potentials in the one-dimensional case. They can be obtained in terms of the single coordinate in a conventional differential approach. Instead, in the algebraic framework (or in a second-quantization scheme, more generally speaking), wavefunctions are expressed in terms of boson annihilation-creation operators acting on the Fock vacuum state, as shown in Eq. (2.29). In the specific case of U(2), for example, the basis state N,m) can be written (created) as... 

In (1) a ic.b Kjaic and b f are boson creation and annihilation operators for the a and b Hartree-Fock particle states with momentum hn and kinetic energy K-h k /2m. pa and pb denote the densities of the component holon gases while pa and pb denote their respective chemical potentials. In equilibrium, pa=Pb-Pt where u is determined by the condition that the statistical average of A Eic (a icaic b Kbic) be equal to p=pa+pb=N/A, the total number of holons per unit area (N , A ). V is taken to satisfy V<pairing interaction. Finally, V is restricted to operate between holons with k[Pg.45]

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