Vector, axial

Polar vectors such as r = epic + ew + e3z change sign on inversion and on reflection in a plane normal to the vector, but do not change sign on reflection in a plane that contains the vector. Axial vectors or pseudovectors do not change sign under inversion. They occur as vector products, and in symmetry operations they transform like rotations (hence the name axial vectors). The vector product of two polar vectors 

Under a symmetry operator T, Rx transforms into Rx = / (e2 x e3 ) and similarly, so that 

The transformation properties of Rx Ry Rz are then readily worked out from eq. (6) using the primed equations (7) with e/ e2 e3 obtained from eq. (8) with the use, when necessary, of eq. (2), which simply states that reversing the order of the terms in a vector product reverses its sign. 

This is a matrix representation of the group D4 = E 2C4 C2 2C2 2C2 and it is clearly reducible. The character systems of the two representations in the direct sum 

Exercise 4.6-2 Show that F3 is an IR of D4. How many IRs are there in the character table of D4 Give the names of F2 and r5 in Mulliken notation. 

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From the invariance of the theory under space inversion, it follows that the axial vector and tensor amplitudes transform as follows ... 

Being the edge of a sheared area, a dislocation is a line, but does not, in general, lie on one plane, so its motion is usually three-dimensional. Since shear has two signs (plus and minus) so do dislocations and dislocations of like signs repel, while those of opposite signs attract. In some structures, the Burgers vector is an axial vector, so plus shear differs from minus shear (like a ratchet). 

The next diagram (FIG. 3) also does not contribute to the partial width for the 0 — W7r° decay. In this case the Lagrangian of the strong coupling of axial-vector mesons to vector and pseudoscalar mesons is derived in a similar way and has the form (Nasriddinov, 1994)... 

The nonvanishing components of the tensors y a >--eem and ya >-mee can be determined by applying the symmetry elements of the medium to the respective tensors. However, in order to do so, one must take into account that there is a fundamental difference between the electric field vector and the magnetic field vector. The first is a polar vector whereas the latter is an axial vector. A polar vector transforms as the position vector for all spatial transformations. On the other hand, an axial vector transforms as the position vector for rotations, but transforms opposite to the position vector for reflections and inversions.9 Hence, electric quantities and magnetic quantities transform similarly under rotations, but differently under reflections and inversions. As a consequence, the nonvanishing tensor components of x(2),eem and can be different... 

If we understand FM or magnetic properties of quark matter more deeply, we must proceeds to a self-consistent approach, like Hartree-Fock theory, beyond the previous perturbative argument. In ref. [11] we have described how the axial-vector mean field (AV) and the tensor one appear as a consequence of the Fierz transformation within the relativistic mean-field theory for nuclear matter, which is one of the nonperturbative frameworks in many-body theories and corresponds to the Hatree-Fock approximation. We also demonstrated... 

Figure 5. Axial-vector mean-field (VA) as a function of baryon number density Pb(po = 0.16fm 3). Solid (dashed) lines denote VA in the presence (absence) of CSC.

In this talk we have discussed a magnetic aspect of quark matter based on QCD. First, we have introduced ferromagnetism (FM) in QCD, where the Fock exchange interaction plays an important role. Presence of the axial-vector mean-field (AV) after the Fierz transformation is essential to give rise to FM, in the context of self-consistent framework. As one of the features of the relativistic FM, we have seen that the Fermi sea is deformed in the presence of... 

The reciprocal lattice of a mosaic crystal is a three-dimensional periodic system of points, each of which characterized by a vector Hhu = ha -l- kb -l-Ic, where a, b, c are axial vectors and h,k,l, are point indices. 

The dual axial vector in 4-space is constructed geometrically from the integral over a hypersurface, or manifold, a rank 3-tensor in 4-space antisymmetric in all three indices [101]. In three-dimensional space, the volume of the parallelepiped spanned by three vectors is equal to the determinant of the third rank formed from the components of the vectors. In four dimensions, the projections can be defined analogously of the volume of the parallelepiped (i.e., areas of the hypersurface) spanned by three vector elements < dl, dx and dx". They are given by the determinant... 

The handedness, or chirality, inherent in foundational electrodynamics at the U(l) level manifests itself clearly in the Beltrami form (903). The chiral nature of the field is inherent in left- and right-handed circular polarization, and the distinction between axial and polar vector is lost. This result is seen in Eq. (901), where , is a tensor form that contains axial and polar components of the potential. This is precisely analogous with the fact that the field tensor F, contains polar (electric) and axial (magnetic) components intermixed. Therefore, in propagating electromagnetic radiation, there is no distinction between polar and axial. In the received view, however, it is almost always asserted that E and A are polar vectors and that is an axial vector. 

Suppose one first considers electric-dipole and magnetic-dipole transitions. As is now well recognized, these are the major contributors to rare-earth absorption and emission spectra. We know that the electric-dipole operator transforms as a polar vector, that is, just as the coordinates (23, 24). This means that it has odd parity under an inversion operation. On the other hand, the magnetic-dipole operator transforms as an axial vector or pseudovector and of course must have even parity (23, 24). 

For the rTu and rT, combinations there are no central atom counterparts and consequently, as they therefore do not mix with the central atom orbitals, they do not take part in the bonding. Nonetheless, appropriate rTl combinations can be found. The axial vectors R9, wid Ra (see 9-6) form a basis for rTl and using this piece of information, we find the following T3 1 combinations which mirror Rm, R and Rt ... 

Fig. 9.8 The effect of reflection on (a) polar vectors representing translations (b) axial vectors representing rotations. In (b), the directions of the rotational-mode displacement vectors of the nuclei (symbolized by the curved lines) determine the directions of the axial vectors.

The angular momentum or an electron moving in an orbit of the type described by Bohr is ail axial vector L = r x p, formed from the radial distance r between electron and nucleus and the linear momentum p of the electron relative lo a fixed nucleus. Figure 2 shows the customary method used to illustrate the axial vector L in terms of the orbital morion of any object, of which the electron of the Bohr atom is only one example. Although Bohr s planetary model needed only circular orbits lo explain the spectral lines observed in the spectrum of a hydrogen atom, subsequent... 

In his interesting paper Professor Nicolis raises the question whether models can be envisioned which lead to a spontaneous spatial symmetry breaking in a chemical system, leading, for example, to the production of a polymer of definite chirality. It would be even more interesting if such a model would arise as a result of a measure preserving process that could mimic a Hamiltonian flow. Although we do not have such an example of a chiral process, which imbeds an axial vector into the polymer chain, several years ago we came across a stochastic process that appears to imbed a polar vector into a growing infinite chain. 

The metric coefficient in the theory of gravitation [110] is locally diagonal, but in order to develop a metric for vacuum electromagnetism, the antisymmetry of the field must be considered. The electromagnetic field tensor on the U(l) level is an angular momentum tensor in four dimensions, made up of rotation and boost generators of the Poincare group. An ordinary axial vector in three-dimensional space can always be expressed as the sum of cross-products of unit vectors... 

The unit vector components of the classical magnetic fields Ba>, Ba and B<3> in vacuo are all axial vectors by definition, and it follows that their unit vector components must also be axial in nature. In matrix form, they are, in the Cartesian basis... 

There is also a relation between polar unit vectors, boost generators, and electric fields. An electric field is a polar vector, and unlike the magnetic field, cannot be put into matrix form as in Eq. (724). The cross-product of two polar unit vectors is however an axial vector k, which, in the circular basis, is e<3>. In spacetime, the axial vector k becomes a 4 x 4 matrix related directly to the infinitesimal rotation generator /3) of the Poincare group. A rotation generator is therefore the result of a classical commutation of two matrices that play the role of polar vectors. These matrices are boost generators. In spacetime, it is therefore... 

One occurrence is a violation of the conservation of the axial vector current. We have that the 1 and 2 currents are conserved and invariant. On the high-energy vacuum we expect that currents should obey... 

The mass of the chiral 1,2 -bosons will then vanish, while the mass of the chiral 3-boson will be m. So rather than strictly setting A3 = 0, it is a separate chiral gauge field that obeys axial vector nonconservation and only occurs at short ranges. 

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