Mathematical Methods for Physics and Engineering - Matematica.NET

26.11 DUAL TENSORSformations, for which the physical system of interest is left unaltered, and onlythe coordinate system used to describe it is changed. In an active transformation,however, the system itself is altered.As an example, let us consider a particle of mass m that is located at a positionx relative to the origin O and hence has velocity ẋ. The angular momentum ofthe particle about O is thus J = m(x × ẋ). If we merely invert the Cartesiancoordinates used to describe this system through O, neither the magnitude nordirection of any these vectors will be changed, since they may be consideredsimply as arrows in space that are independent of the coordinates used to describethem. If, however, we perform the analogous active transformation onthe system, by inverting the position vector of the particle through O, thenitis clear that the direction of particle’s velocity will also be reversed, since itis simply the time derivative of the position vector, but that the direction ofits angular momentum vector remains unaltered. This suggests that vectors canbe divided into two categories, as follows: polar vectors (such as position andvelocity), which reverse direction under an active inversion of the physical systemthrough the origin, and axial vectors (such as angular momentum), whichremain unchanged. It should be emphasised that at no point in this discussionhave we used the concept of a pseudovector to describe a real physicalquantity. §26.11 Dual tensorsAlthough pseudotensors are not themselves appropriate for the description ofphysical phenomena, they are sometimes needed; for example, we may use thepseudotensor ɛ ijk to associate with every antisymmetric second-order tensor A ij(in three dimensions) a pseudovector p i given byp i = 1 2 ɛ ijkA jk ; (26.40)p i is called the dual of A ij . Thus if we denote the antisymmetric tensor A by thematrix⎛0 A 12 −A 31⎞A =[A ij ]= ⎝ −A 12 0 A 23⎠A 31 −A 23 0then the components of its dual pseudovector are (p 1 ,p 2 ,p 3 )=(A 23 ,A 31 ,A 12 ).§ The scalar product of a polar vector and an axial vector is a pseudoscalar. It was the experimentaldetection of the dependence of the angular distribution of electrons of (polar vector) momentump e emitted by polarised nuclei of (axial vector) spin J N upon the pseudoscalar quantity J N · p e thatestablished the existence of the non-conservation of parity in β-decay.949