Weyl representation

In this section we advocate a far more advantageous route to studying conceptual features of the classical-quantum correspondence, and indeed for each mechanics independently, in which phase space distributions are used in both classical and quantum mechanics, that is, classical Liouville dynamics50 in the former and the Wigner-Weyl representation in the latter. This approach provides, as will be demonstrated, powerful conceptual insights into the relationship between classical and quantum mechanics. The essential point of this section is easily stated using similar mathematics in both quantum and classical mechanics results in a similar qualitative picture of the dynamics. 

As discussed subsequently, introduction of the standard p, q representation in classical mechanics, and of the Wigner-Weyl representation in quantum mechanics, defines densities p(p,q) = (p,q p) that both lie in the same Hilbert space. Thus, the essential difference between quantum and classical mechanics... 

The Wigner-Weyl representation of quantum mechanics is that obtained by the expansion of operators in an orthogonal set of operators A(p,q) the resultant p p.q) lie within a Hilbert space of L2 functions. Specifically,51... 

The algebra of the Wigner-Weyl representation has been extensively reviewed51 and we only call attention to the form that Eqs. (3.5) and (3.6) take in this representation. The Liouville equation becomes... 

This particular choice of the (4 x 4)-matrices is neither unique nor is the dimension 4 the only dimension, which allows us to fulfill the required properties. Higher even dimensions are also possible. How one may find other representations of these four parameters will be discussed in section S.2.4.3. The particular choice of Eqs. (5.37) is called the standard representation of the Dirac matrices a. and j6. We give as a further example the Weyl representation... 

Basically, DKH theory starts from the standard representation of the Dirac Hamiltonian introduced in section 5.2. The fact that the Dirac matrices a and j6 are not uniquely defined suggests that a different representation of the Dirac operator might be more advantageous. In order to shed light on this question, we consider the Weyl representation of Eq. (5.38), which produces a Dirac Hamiltonian of a very different structure, namely the following... 

Hence, we do not gain anything from the Weyl representation when subjected to an elimination procedure. 

The relations 04,0 1 + 0.104- = 2Ski, Clifford algebra, for which we choose the Dirac representation. In a phase space language, employing Weyl quantisation, this Hamiltonian can be written as (see, e.g., (Dimassi and Sjostrand, 1999))... 

For the application of Theorem 6.8 to the Springer representation of the Weyl group, we refer to [10]. Another interesting application to the representation theory was given in [12]. 

Apart of historical reasons, there are several features of the Dirac-Pauli representation which make its choice rather natural. In particular, it is the only representation in which, in a spherically-symmetric case, large and small components of the wavefunction are eigenfunctions of the orbital angular momentum operator. However, this advantage of the Dirac-Pauli representation is irrelevant if we study non-spherical systems. It appears that the representation of Weyl has several very interesting properties which make attractive its use in variational calculations. Also several other representations seem to be worth of attention. Usefulness of these ideas is illustrated by an example. 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

It is interesting that Weyl had a deep conviction that the harmony of nature could be expressed in mathematically beautiful laws and an outstanding characteristic of his work was his ability to unite previously unrelated subjects. He created a general theory of matrix representation of continuous groups and discovered that many of the regularities of quantum mechanics could be best understood by means of group theory. 

The conceptual simplicity of the CASSCF model lies in the fact that once the inactive and active orbitals are chosen, the wave function is completely specified. In addition such a model leads to certain simplifications in the computational procedures used to obtain optimized orbitals and Cl coefficients, as was illustrated in the preceding chapters. The major technical difficulty inherent to the CASSCF method is the size of the complete Cl expansion, NCAS. It is given by the so-called Weyl formula, which gives the dimension of the irreducible representation of the unitary group U(n) associated with n active orbitals, N active electrons, and a total spin S ... 

Figure 6. Spin frame representation of a spin-vector by flagpole normalized pair representation a,b over the Poincare sphere in Minkowski tetrad (l,x,y,z) form (n representation) and for three timeframes or sampling intervals providing overall (t]. r ) a Cartan-Weyl form representation. The sampling intervals reset the clock after every sampling of instantaneous polarization. Thus polarization modulation is represented by the collection of samplings over time. Minkowski form after Penrose and Rindler [28]. This is an SU(2) Gd hx) m C over it representation, not an SO(3) Q(to, 8) in C representation over 2it. This can be seen by noting that an b or bt-z a over it, not 2n, while the polarization modulation in SO(3) repeats at a period of 2it.

H. Weyl. Classical Groups Their Invariants and Representations, Princeton University Press, Princeton, 1939. 

This produces a non-standard Rumer function as shown by the representation on the right hand side. These non-standard functions, which correspond to a Rumer diagram in which the lines cross, are not used in calculations. A standard set of functions can be selected using the following procedure the open shell orbitals are represented by a 1 if their number occurs in the left hand column of the Weyl tableau or a 2 if it appears in the right hand column. Hence for the example above... 

Therefore the dimension of the model space is equal to the dimension of the pertinent representation and is given by the Weyl-Paldus dimension formula [20, 21]... 

Weyl s rule [Eq. (322)], with their matrix elements in the q-representation given by... 

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