Commutation relations commutator identities

The following general commutator identities are quite useful for simplifying the commutation relations, which occur in the determination of realizations of a Lie algebra. The well-known 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 . 

This operator can now be shown to be identical with the operator for an infinitesimal rotation of the vector field multiplied by i, i.e. J = — M. The components of the angular momentum operator satisfy the commutation relations... 

Some algebras have identical commutation relations. They are therefore called isomorphic algebras. A list of isomorphic algebras (of low order) is shown in Table A.3. In this table, the sign denotes direct sums of the algebra, that is, addition of the corresponding operators. There is also the trivial case U(l) SO(2). 

The mapping preserves the commutation relations (70) of the spin operators. As can be seen from (75d), the image of the 2s + 1)-dimensional spin Hilbert space is the subspace of the two-oscillator Hilbert space with 2s quanmm of excitation— the so-called physical subspace [218, 220], This subspace is invariant under the action of any operator which results by the mapping (75a)-(75d) from an arbitrary spin operator A(5 i, S2, S3). Thus, starting in this subspace the system will always remain in it. As a consequence, the mapping yields the following identity for the matrix elements of an operator A ... 

Since an arbitrary fc-matrix is orthogonal to S, the matrix representation for a Hermitian operator is far from unique. Suppose that H is the matrix representation of some Hamiltonian, and that S is an arbitrary matrix in S. Then H + S is an equally valid representation since the identity (p,h) = (P, H)j = (P, H - - S) clearly holds for all fe-matrices P. The explanation for such a large number of representations is straightforward the operator s corresponding to a matrix S E S is equal to the zero operator. The operator s is constructed by taking the matrix elements to be coefficients in an expansion but can then be reduced to the zero operator using the commutation relations. 

The Jacobi identity gives a relation between double commutators... 

In (23.80) and (23.81) the rank sum y+k is an odd number, otherwise these operators are identically equal to zero. We shall separate sets of operators that are scalars in the space of total angular momentum but tensors in isospin space. If we go through a similar procedure for one subshell of equivalent electrons we shall end up with the quasispin classification of its states. It turns out that ten operators l/(00), U 0 vffl, F 0) are generators of a group of five-dimensional quasispin, wnich can be easily verified by comparing their commutation relations with the standard commutation relations for generators of that group. 

As in section 2, we introduce the identity, position and momentum operators, labelling the two subsystems with a = 1,2, Ia,Xj,aJlaDjta (as before, j = 1,..., n is a vector index in Rn), whose commutation relations are... 

These two identities are called Kubo-Martin-Schwinger boundary conditions [47]. From the commutation relation, it follows that... 

In fact we need only consider one of these three commutation relations since using the Jacobi identity (cf. Section II) it can be shown that... 

The evolution of the remaining term a (0) under can be found by noting that this problem is formally identical to a well-known problem for which the solution is known (Muller and Ernst, 1979). The equivalence is based on the fact that there is a one-to-one correspondence between the commutator relations... 

The commutator of two square matrices is defined as [A, B] = AB — BA. If [A, B] = 0 the matrices A and B are said to commute. All diagonal matrices commute, every matrix commutes with itself, every matrix commutes with its inverse, and every matrix commutes with the identity matrix. If A and B are Hermitian, then [A, B] = 0 if and only if both matrices may be diagonalized by the same unitary matrix. This does not mean that every matrix that diagonalizes A will diagonalize B but that at least one such matrix exists that will diagonalize both. This relation may be used to determine the classes of matrices that can be diagonalized by a unitary transformation. Let A be an arbitrary matrix and define A + = (A -I- A )/2 and A = (A - AV2i. Then... 

To simplify this expression, consider a commutator of the form [a, ala ]. This appears to result in the difference of two terms each containing the product of three operators. However, the anticommutation relations of Eqs (57) and (58) may be used to produce the following sequence of identities ... 

It is easy to show that the mapping of the operators (54) preserves the commutation relations and leads to the exact identity of the electronic matrix elements of the propagator... 

Equation (D.4) is verified at once by differentiation of uf according to the product rule. The relations (D.3b) are satisfied trivially. Since F is an arbitrary (differentiable) function, we can consider the relations (D.3) as identities. We have thus shown that the representation (D.2) is equivalent to the commutator relations (D.3). 

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