Boson commutation relation

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... 

The coefficients Cpq iCpg) in eqs. (25, 26) are determined so that 5p (a ) satisfy fermion anticommutadon relation. The coefficients d pg d pg) in eqs. (27,28) are determined so that b bt) satisfy boson commutation relation. Finally we ask fermions dp dp) to commute with bosons br br)- This means that we can write similarly as in (5) the total wave function F(r, R) as a product of fermion wave function / (r, R) and boson wave function as 0( r, R)... 

The value of a a) is always the same, but the averaging procedure differs in each case. The relations (65)-(67) are a simple consequence of the boson commutation relation [ , = 1 and the definition... 

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

The operation must be understood as hermitian conjugation on B and P.) In addition, the new operators P, P must satisfy the boson commutation relation satisfied by B and B The relations (1.13) may also be expressed in a compact form through the use of a matrix q>, defined as... 

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 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... 

Here rjji = exp(—27T l j/N). Since the f s and tit s obey bosonic commutation relations up to corrections 0(1/N), one sees from (28c) that Z = 0 + 0(1 /N), i.e., within the bosonic quasi-particle approximation, the action of a phase flip Zi cannot be calculated. However one can draw the conclusion that a single-atom phase error only contributes in first order of 1/N. From the other equations one recognizes an important property if we assume that the initial state Wo is an ideal storage state, i. e., without bright polariton excitations, we find that after tracing out the bright polariton states only decoherence contributions of order 0(1/IV) survive, e.g.,... 

Therefore, whenever the normal form of the quadrature variance is negative, this component of the field is squeezed or, in other words, the quantum noise in this component is reduced below the vacuum level. For classical fields, there is no unity coming from the boson commutation relation, and the normal form of the quadrature component represents true variance of the classical stochastic variable, which must be positive. 

The fact that every state may be occupied by several particles shows that the second quantization particles are bosons. However, in terms of different commutation relations an equivalent scheme may be obtained for fermions. To achieve this objective the wave functions are written in decomposed form as before ... 

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... 

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 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 . 

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... 

Finally, we should note that all that has been said so far is valid for fermionic annihilation and creation operators only. In the case of bosons these operators need to fulfill commutation relations instead of the anticommutation relations. The fulfillment of anticommutation and commutation relations corresponds to Fermi-Dirac and Bose-Einstein statistics, respectively, valid for the corresponding particles. Accordingly, there exists a well-established cormection between statistics and spin properties of particles. It can be shown [65], for instance, that Dirac spinor fields fulfill anticommutation relations after having been quantized (actually, this result is the basis for the antisymmetrization simply postulated in section 8.5). Hence, in occupation number representation each state can only be occupied by one fermion because attempting to create a second fermion in state i, which has already been occupied, gives zero if anticommutation symmetry holds. 

The second term in Eq. (8.1) represents the energy of the photons. The operators and are the creation and annihilation operators for Bosons. They satisfy the commutation relation... 

The effect of inter-site hopping is then introduced into the system. The manifold of basis states are limited to those in which the local correlations have been diagonalized. The wave functions for the composite particles then obey Bloch s theorem, which results in the formation of a dispersion relation consisting of two bands for the quasi-bosons the first band describes spinless quasi-boson excitations, the second band describes the magnetic quasi-bosons. Although these composite particles are bosons in that they commute on different sites, they nevertheless have local occupation numbers which are Fermi-Dirac like. 

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