Subalgebras

The fact that px is a semidirect product of these two subalgebras is a necessary condition to support such an interpretation. Indeed, since we have [p,p] = p, we see that the role played by the generators of symmetries p is to impress dynamical modification on the observables p giving rise to other observables. As a consequence, the non-commutativity between the observables is a matter of measurement. In the case we are studying in this section we have [p,p] = p resulting in a quantum theory. For the sake of consistency, we expect to derive a classical TFD theory with an algebra similar to pr but in which [p,p] = 0. (This result has been explored in Ref.(L.M. Silva et.al., 1997))... 

A Lie subalgebra is a subset G of operators of G, which, by itself, is closed with respect to commutation. In other words, the commutator of two elements is a linear combination of the same elements. In mathematical terms,... 

We shall use systematically the symbol z>, meaning containing , to denote this situation. In some cases, the subset G is trivial. For example, it is clear that the single operator Jz, i.e. the component of the angular momentum on a fixed z axis, forms a subalgebra of the angular momentum algebra SO(3) since... 

A problem that arises in connection with the construction of the basis is that of finding what are the allowed values of the quantum numbers of the subalgebra G contained in a given representation of G. For example, what are the allowed values of Mj for a given J in Eq. (2.12). In this particular case, the answer is well known from the solution of the differential (Schrodinger) equation satisfied by the spherical harmonics (see Section 1.4), that is,... 

The Hamiltonian (2.23) represents the general expansion in terms of the elements Gap, and it corresponds to a Schrodinger equation with a generic potential. In some special cases, one does not have in Eq. (2.23) generic coefficients e ap, apY8 but only those combinations that can be written as invariant Casimir operators of G and its subalgebras, GdG dG"D ", This situation... 

Dynamical symmetries for one-dimensional problems can be studied by considering all the possible subalgebras of U(2). There are two cases... 

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... 

An algebra (or subalgebra) is said to be Abelian if all its elements commute... 

The subalgebra SO(2) also has a (trivial) Casimir invariant, that is, Jz itself,... 

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... 

It follows from relations (15) that the basis elements of the Lie algebra c(l, 3) have the form (6), where the functions c a depend on x e X = Rp only and the functions r j are linear in u. We will prove that owing to these properties of the basis elements of c(l, 3), the ansatzes invariant under subalgebras of the algebra (15) admit linear representation. 

With the explicit forms of the matrices in hand, we can determine the structure of the matrices H = -1 for ansatz (22) invariant under a subalgebra g of the conformal algebra c(l, 3). 

Next, if g is a subalgebra of the conformal algebra c(l,3) with a nonzero projection on the vector space spanned by the operators D. Kq. K. Kt. K, then the corresponding matrices are linear combinations of the matrices E and. Sj(v. That is why the matrix should be sought in the more general form... 

As a second step of the algorithm of symmetry reduction formulated above, we have to describe the optimal system of subalgebras of the algebra c(l, 3) of the rank 5 = 3. Indeed, the initial system has p = 4 independent variables. It has to be reduced to a system of differential equations in 4 — s = 1 independent variables, so that 5 = 3. 

Classification of inequivalent subalgebras of the algebras p(l,3), p(1.3), c(l,3) within actions of different automorphism groups [including the groups P(l, 3), P(l, 3) and 0(1,3)] is already available [30]. Since we will concentrate on conformally invariant systems, it is natural to restrict our disscussion to the classification of subalgebras of c(l, 3) that are inequivalent within the action of the conformal group 0(1, 3). 

In order to get the full lists of the subalgebras in question we have to check that relation (7) with 5 = 3 holds for each element of the lists of inequivalent subalgebras of the algebras p, 3),p(l, 3), c(l, 3) given elsewhere [30]. Evidently, we can restrict our considerations to subalgebras having the dimension not less than 3. 

It follows from Lemma 2 that the validity of the relation (7) with s = 3 should be ascertained only for the three-dimensional subalgebras of the algebras p(l,3),p(l,3),c(l,3) given elsewhere [30]. Moreover, we need to check the first condition from (7) only. 

Assertion 1. The list of subalgebras of the algebra p, 3) of the rank 3, defined within the action of the inner automorphism group of the algebra c(l,3), is exhausted by the following subalgebras ... 

Now we turn to constructing C(l, 3)-invariant ansatzes that reduce conformally invariant systems of partial differential equations to systems of ordinary differential equations. To this end, we use the lists of subalgebras of the algebra c(l, 3) given in Assertions 1-3. Note that all the subsequent computations are... 

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