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 massmthat is located at a positionxrelative to the originOand hence has velocity ̇x. The angular momentum of
the particle aboutOis thusJ=m(x× ̇x). If we merely invert the Cartesian
coordinates used to describe this system throughO, 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 de-
scribe them. If, however, we perform the analogous active transformation on
the system, by inverting the position vector of the particle throughO,thenit
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:polarvectors (such as position and
velocity), which reverse direction under an active inversion of the physical sys-
tem through the origin, andaxialvectors (such as angular momentum), which
remain unchanged. It should be emphasised that at no point in this discus-
sion have we used the concept of a pseudovector to describe a real physical
quantity.§
26.11 Dual tensorsAlthough pseudotensors are not themselves appropriate for the description of
physical phenomena, they are sometimes needed; for example, we may use the
pseudotensorijkto associate with everyantisymmetricsecond-order tensorAij
(in three dimensions) a pseudovectorpigiven by
pi=^12 ijkAjk; (26.40)piis called thedualofAij. Thus if we denote the antisymmetric tensorAby the
matrix
A=[Aij]=0 A 12 −A 31
−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 experimental
detection of the dependence of the angular distribution of electrons of (polar vector) momentum
peemitted by polarised nuclei of (axial vector) spinJNupon the pseudoscalar quantityJN·pethat
established the existence of the non-conservation of parity inβ-decay.