Premium
On the nature of axial tensors
Author(s) -
Boyle L. L.,
Matthews P. S. C.
Publication year - 2009
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560050866
Subject(s) - tensor (intrinsic definition) , tensor contraction , symmetry (geometry) , permutation (music) , symmetric tensor , rank (graph theory) , physics , tensor product , cartesian tensor , tensor density , parity (physics) , tensor operator , point (geometry) , mathematics , mathematical physics , pure mathematics , tensor field , exact solutions in general relativity , quantum mechanics , combinatorics , geometry , spherical harmonics , acoustics
The case for representing axial tensor properties by polar tensors of a higher rank is examined with due reference to historical background. It is found that those properties which are determined by permutation symmetry, e.g. the determination of the number of independent components, require the latter description while the former description is equivalent from the point of view of all properties depending solely on spatial symmetry. The Levi‐Cività tensor, which was previously thought to be the analogue of the vector product operator, is replaced by three operators, which yield the correct parity for the vector product, and their construction is described. It is also shown that in certain cases an irreducible tensor component need not possess its full permutation symmetry to be nonzero, contrary to previous belief. Magnetic moments, angular momenta, the gyration tensor and magnetic susceptibility are discussed in detail.