Groups, continuous irreducible representations

Since we will continually be requiring the characters of the irreducible representations of the point groups, it is convenient to put them together in tables known as character tables- In the character table of a point group each row refers to a particular irreducible representation and, since the characters of operations of the same class are identical, only a single entry (C,) is made for all the operations of a given class. The columns are headed by a representative element from each class preceded by the number of elements or operations in that class gf. 

It can therefore be inferred that 0(3) electrodynamics is a theory of Rieman-nian curved spacetime, as is the homomorphic SU(2) theory of Barrett [50], Both 0(3) and SU(2) electrodynamics are substructures of general relativity as represented by the irreducible representations of the Einstein group, a continuous Lie group [117]. The Ba> field in vector notation is defined in curved spacetime by... 

Character of Classes of Cubic Symmetry in the (2L + 1)-Dimensional Representation DL of the Continuous Rotation Group and Their Resolution into Irreducible Representations of Cubic Symmetry... 

Since Vls has the form of equation 8, it is necessary to consider the spin as well as the orbital part of the wave function. For the free atom, the spin is represented by the continuous rotation group Ds appropriate to a given spin angular momentum S. In a crystal this is reduced to the irreducible representation appropriate for the... 

The symbols in parentheses for the irreducible representations of the Dsh symmetry group are those used in (7), the lower case indicating that they relate to the single-electron system. The spin-allowed one-electron transitions can take place between the a and e", a and e, and the e" and e orbitals. From the two-electron transitions, the only spin-allowed one is that in which 1 electron jumps from the e" into the e orbitals and 1 electron from the a/ into the e orbitals. This transition requires a considerable amount of energy and probably is not observed in our spectrum, except perhaps in the region of increasing continuous absorption between 30,000 and 40,000 cm"1. 

A free atom belongs to the continuous rotation group R3. The irreducible representations of group R3 are labelled by the quantum number /. The spherical harmonic functions Yl m form the basis of the irreducible representation of R3 with the dimension 2/ + 1. 

The reader will realize that the nature of the operator Tr has hardly been specified. The foregoing theory applies equally well to point groups and to permutation groups, and without much alteration to continuous point groups. It is clear that we require the irreducible representations of these groups before tackling actual applications of the theory to chemistry. These are the subject of Chapter 6. 

Frequencies of crystal vibrations diange continuously with the phase-difference vector (5) although the symmetry species of crystal vibrations change for special S vectors (fc-groups). Compatibility relations of the symmetry species of ft-groups may be derived from irreducible representations [Slater (1965)]. 

The covariance groups underlying the tensor forms of the respective Einstein and the Maxwell held equations are reducible. This is because they entail reflection symmetry, not required by relativity theory, as well as the required continuous symmetry of the Einstein group E. When the Einstein held equations are factorized, they yield the irreducible form, which are then in terms of the quaternion and spinor variables, rather than the tensor variables. Such a generalization must then extend the physical predictions of the usual tensor forms of general relativity of gravitation and the standard vector representation of the Maxwell theory (both in terms of second-rank tensor helds, one symmetric and the other antisymmetric) because the new factorized variables have more degrees of freedom than did the earlier version variables. 

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