Invariant Casimir operators

For each Lie algebra, one can construct a set of operators, called invariant (or Casimir, 1931) operators after the name of the physicist who first introduced them in connection with the rotation group. These operators play a very important role since they are associated with constants of the motion. They are defined as those operators that commute with all the elements of the algebra... 

They are constructed from powers of the operators Xs and can be linear, quadratic, cubic,. Quite often a subscript is attached to C in order to indicate the order. For example, C2 denotes a quadratic invariant. The number of independent Casimir invariants of an algebra is called the rank of the algebra. It is easy to see, by using the commutation relation (2.3) that the operator... 

We have stated several times that whenever the Hamiltonian can be written in terms of invariant (Casimir) operators of a chain, its eigenvalue problem can be solved analytically. This method can be applied to the construction of both local and normal Hamiltonians. For local Hamiltonians, one writes H in terms of Casimir invariants of Eq. (4.43). 

Invariant operators were introduced by Casimir (1931) for SO(3). Racah (1950) generalized them to all orders. 

In this section, we suggest a resolution of this > 70-year-old paradox using 0(3) electrodynamics [44]. The new method is based on the use of covariant derivatives combined with the first Casimir invariant of the Poincare group. The latter is usually written in operator notation [42,46] as the invariant P P 1, where P1 is the generator of spacetime translation ... 

In this expression, and 62 Casimir operators of the Oi(4) and 02(4) algebras. Their matrix elements in the local basis (4.6) have been already obtained in Section II.C.2 [see Eq. (2.111)]. We also recall that 0(4) possesses two invariant operators, C and C, which can be written. 

Thus the Casimir operator of SO(3) is the familiar square of the angular momentum (a constant of the motion when the Hamiltonian is invariant under rotation). One can show that SO(3) has only one Casimir operator, and it is thus an algebra of rank one. Multiplication of C by a constant a, which obviously satisfies (2.7), does not count as an independent Casimir operator, nor do powers of C (i.e., C2,...) count. Casimir operators can be constructed directly from the algebra. This construction has been done for the large majority of algebras used in physics. 

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... 

Dynamic symmetry corresponds to an expansion of the Hamiltonian in terms of Casimir operators. The Casimir operator of U(2) plays no role, since it is a given number within a given representation of U(2) and thus can be reabsorbed in a constant term E(). The algebra U(l) has a linear invariant... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

For any Lie algebra, one can construct a set of operators, called invariant or Casimir operators, C, such that... 

So Cl commutes with p. It follows from Proposition 8.5 that each eigenspace of Cl is an invariant space for the representation p. Because p is irreducible, we conclude that Ci has only one eigenspace, namely, all of V. Hence Ci must be a scalar multiple of the identity on V. Similarly, C2 must be a scalar multiple of the identity on V. By Proposition 8.9 and Equation 8.13, we know that the Casimir operators can take on only certain values on finite-dimensional representations, so we can choose nonnegafive half-integers 1 and 2 such that Cl = —fi(fi 1) and C2 = — 2( 2 + ) ... 

To appreciate the actual meaning of this procedure, we need to clarify briefly the concept of invariant or Casimir operator. Returning to the rotation group SO(3), we know that the square of the angular momentum... 

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