Adjoint relation

The commutator relation (3 8) for the excitation operators together with the adjoint relation (3 9) leads to the following symmetry properties for the coupling coefficients (for real-valued wave functions) ... 

The Adjoint Relation—Role of Orthogonality of One-Particle States... 

Note that Eq. (2.48b) follows from Eq. (2.48a) as a consequence of Eq. (2.51). Similarly, Eq. (2.50b) follows from Eq. (2.50a). Note also that all the above properties, with the exception of the adjoint relation of Eq. (2.51), hold for the true annihilation operators even in the nonorthogonal case. 

The symmetry properties of the second-order density matrix are easy to establish. First, its Hermitian symmetry follows from the definition of Eq. (7.17) and from the adjoint relation ... 

The essential feature of the orthogonality of the underlying basis orbitals with respect to second quantization is the adjoint relation ... 

Both of these equations have their own significance. The adjoint relation is essential when putting down the second quantized correspondances of bra-vectors, cf. Eq. (2.53). The anticommutation rule is important in dealing with matrix elements where one often has to transpose creation and annihilation operators. 

Accordingly, the second quantized formalism can be generalized to the nonorthogonal case in two alternative manners one may keep either the adjoint relation of Eq. (13.1) or the simple anticommutation rule of Eq. (13.2). In the former case the commutation rules become more complicated, while in the latter case the annihilation operators will not be the adjoints of the corresponding creation operators. 

Application of operators %k is advantageous since we can use any results derived for the orthogonal case which are based on the anticommutation rule of Eq. (13.13). On the other hand, this story represents a typical example for the law of the conservation of the difficulty . As a matter of fact, in the absence of the adjoint relation, the construction of bra wave functions may be troublesome. It is possible only by means of the inverse transformation ... 

In this case the flucPiation-dissipation relation, ( A3.2.21T reduces to D = IcTa. It is also clear that GE = (A + S)/Mlct which is not self-adjoint. 

The components of the operator P are hermitian.2 In general, any differential operator Q has a hermitian adjoint Qf, defined by the integral relation... 

An operator is called the adjoint of if they are related as follows ... 

It should be noted that by virtue of Eq. (9-693), A x) is self-adjoint within the indefinite metric. The vacuum state can now be characterized by the relation... 

With this, for any self-adjoint operator A the following relations occur ... 

Due to the second Green formula it is self-adjoint. In turn, the first Green formula assures us of the validity of the relation... 

We thus have shown that under conditions (26) inequality (14) is sufficient for the stability of scheme (la) in the space Ha, that is, relation (29) occurs. Let us stress that the requirement of self-adjointness of the operator B is necessary here, while the energy method demands only the positivity of B and no more. 

Because of the rectangular form of the initial domain, the operators j4i and A2 are self-adjoint, positive and commutative. It is straightforward to verify the relations A1A2 = A2A1 and A1A2J/ = A2A1J/ = at all... 

Since the operators A, A2 and A > 0 are commutative and self-adjoint, the relations A AiA2 > 0 and B > A +0.bT E occur. Applying Theorem 10 from Chapter 6, Section 2 yields estimate (27). 

Still using the framework of the general theory, it is straightforward to verify that the operator B is self-adjoint (B = B ) with the aid of the relations R = R and T = T,n- We shall need yet the constants 7j and 7, of the energetic equivalence of the operators B and A. Because of (46), we thus have... 

Neither nor J is hermitian. Application of equation (3.33) shows that they are adjoints of each other. Using the definitions (5.18) and (5.14) and the commutation relations (5.13) and (5.15), we can readily prove the following relationships... 

The matrix represented in this chapter by A is usually called the adjoint matrix. It is obtained by constructing the matrix which is composed of all of the cofactors of the elements a,j in A and then taking its transpose. With the basic definition of matrix multiplication (Eq. (29)J and some patience, die reader can verify the relation... 

It is important to note that all operators of interest in quantum mechanics are Hermitian (or self-adjoint). This property is defined by the relation... 

A new class of effective Lagrangians have been constructed to show how the information about the center group symmetry is efficiently transferred to the actual physical states of the theory [12-15] and will be reviewed in detail elsewhere. Via these Lagrangians we were also able to have a deeper understanding of the relation between chiral restoration and deconfinement [15] for quarks in the fundamental and in the adjoint representation of the gauge group. 

The Holstein-Primakoff transformation also preserves the commutation relations (70). Due to the square-root operators in Eqs. (78a)-(78d), however, the mutual adjointness of S+ and 5 as well as the self-adjointness of S3 is only guaranteed in the physical subspace 0),..., i- -m) of the transformation [219]. This flaw of the Holstein-Primakoff transformation outside the physical subspace does not present a problem on the quantum-mechanical level of description. This is because the physical subspace again is invariant under the action of any operator which results from the mapping (78) of an arbitrary spin operator A(5i, 2, 3). As has been discussed in Ref. 100, however, the square-root operators may cause serious problems in the semiclassical evaluation of the Holstein-Primakoff transformation. 

Relation to the variational method. As we remarked in Introduction, we base our theory on the variational method in its generalized form.57) It will be convenient to give here a sketch of this relation. If FI is a self-adjoint operator for which we are to solve the eigenvalue problem, the cited variational method consists in making the quantity... 

The density rt M is closely related to the Green s function, GN lll, which satisfies the self-adjoint equation... 

Exercise. Solve (6.2). [Hint Determine the normal modes obeying the boundary conditions, find an orthonormality relation between them and their adjoints, and assume that they constitute a complete set in the interval (L, R) so that the initial condition can be satisfied.]... 

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