Chain of subalgebras

For any given quantum mechanical problem one needs to find the complete set of quantum numbers that characterize uniquely the states of the system. This corresponds to finding a complete chain of subalgebras... 

As an example, the chain of subalgebras (2.7) based on U(4) has, as a complete basis set associated to symmetric irreducible representations, the ket... 

This is called a chain. Each subalgebra has one (or more) Casimir operator(s) C(G,) which commute with all the operators of that subalgebra. The Casimir operator is usually bilinear in the generators and the number of linearly independent Casimir operators is the rank of the algebra. In (59) the Casimir operator of the last subalgebra necessarily commutes with all the Casimir operators of the earlier subalgebras. The Hamiltonian is given as a linear combination of the Casimir operators for the chain of Eq. (59). 

Before providing more detail on this procedure, we prefer to outline the strategy specifically adopted in the U(4) problem. By analogy with the U(2) case, we need to perform two distinct steps (1) to identify the subalgebra chains of U(4) closing in SO(3), and (2) to construct the Hamiltonian operator in a dynamical symmetry sense. The first step is a strictly Lie algebraic question, which can be solved by making explicit use... 

After the assignment of the three vibron numbers A, i = 1, 2, 3, the Hilbert model space of the physical problem is given by the basic states of the symmetric irreducible representations [NJ0[N2]0[N3] of the SGA (4.121). To specify these basic states unambiguously, we choose a complete subalgebra chain of (4.121). The first step is the usual local (bond) assignment of Morse rovibrating units, U,(4) D 0,(4) (i = 1, 2, 3), leading to... 

Dynamical symmetries for three-dimensional problems can be studied by the usual method of considering all the possible subalgebras of U(4). In the present case, since one wants states to have good angular momentum quantum numbers, one must always include the rotation algebra, 0(3), as a subalgebra. One can show then that there are only two possibilities, corresponding to the chains... 

We begin with a brief summary of exact results. For one-dimensional problems we have used the algebraic structure of U(2), with two subalgebra chains... 

Thus for our hydrogenic realization all Casimir operators have constant values so we are dealing with a single unirrep of so(4, 2) symmetry adapted according to the subalgebra chain... 

If the Hamiltonian now contains the Casimir operators of both G, and G[, which do not commute, then the labels of neither provide good quantum numbers. Of course, in general such a Hamiltonian has to be diagonalized numerically. In this way one can proceed to break the dynamical symmetries in a progressive fashion. In (61) all the quantum numbers of G, up to G remain good. If we add another subalgebra beside Gz only those quantum numbers provided by G, on will be conserved, etc. In applications, the different chains are found to correspond to different limiting cases such as the normal versus the local mode limits for coupled stretch vibrations (99). 

By making explicit use of the commutation relations defining the U(4) structure, one can show that two subalgebra chains can be identified, namely those generated by... 

The correspondence between rotations and unitary subalgebra chains can be made more precise by taking into account the explicit branching laws for the labels of the involved irreducible representations. Within the usual angular momentum framework, one has... 

The next step is the construction of Hamiltonian operators in the dynamical symmetry framework. The general procedure is to restrict the expansion (3.6) to invariant operators of the subalgebra chains, thus leading to two distinct models ... 

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