Dual tensors

26.11 DUAL TENSORS
formations, for which the physical system of interest is left unaltered, and only
the 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 position
x relative to the origin O and hence has velocity ẋ. The angular momentum of
the particle about O is thus J = m(x × ẋ). If we merely invert the Cartesian
coordinates used to describe this system through O, neither the magnitude nor
direction of any these vectors will be changed, since they may be considered
simply as arrows in space that are independent of the coordinates used to describe them. If, however, we perform the analogous active transformation on
the system, by inverting the position vector of the particle through O, then it
is clear that the direction of particle’s velocity will also be reversed, since it
is simply the time derivative of the position vector, but that the direction of
its angular momentum vector remains unaltered. This suggests that vectors can
be divided into two categories, as follows: polar vectors (such as position and
velocity), which reverse direction under an active inversion of the physical system through the origin, and axial vectors (such as angular momentum), which
remain unchanged. It should be emphasised that at no point in this discussion have we used the concept of a pseudovector to describe a real physical
quantity.§
26.11 Dual tensors
Although pseudotensors are not themselves appropriate for the description of
physical phenomena, they are sometimes needed; for example, we may use the
pseudotensor ijk to associate with every antisymmetric second-order tensor Aij
(in three dimensions) a pseudovector pi given by
pi = 12 ijk Ajk ;
(26.40)
pi is called the dual of Aij . Thus if we denote the antisymmetric tensor A by the
matrix


0
A12 −A31
A = [Aij ] =  −A12
0
A23 
A31 −A23
0
then the components of its dual pseudovector are (p1 , p2 , p3 ) = (A23 , A31 , A12 ).
§
The scalar product of a polar vector and an axial vector is a pseudoscalar. It was the experimental
detection of the dependence of the angular distribution of electrons of (polar vector) momentum
pe emitted by polarised nuclei of (axial vector) spin JN upon the pseudoscalar quantity JN · pe that
established the existence of the non-conservation of parity in β-decay.
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