Improper rotations and pseudotensors

TENSORS
Rotate by π/2 about the Ox3 -axis: L12 = −1, L21 = 1, L33 = 1, the other Lij = 0.
(d)
(e)
(f)
(g)
T111
T112
T221
T123
=
=
=
=
(−1) × (−1) × (−1) × T222 = −T222 ,
(−1) × (−1) × 1 × T221 ,
1 × 1 × (−1) × T112 ,
(−1) × 1 × 1 × T213 .
Relations (a) and (d) show that elements with all subscripts the same are zero. Relations
(e), (f) and (b) show that all elements with repeated subscripts are zero. Relations (g) and
(c) show that T123 = T231 = T312 = −T213 = −T321 = −T132 .
In total, Tijk differs from ijk by at most a scalar factor, but since ijk (and hence λijk )
has already been shown to be an isotropic tensor, Tijk must be the most general third-order
isotropic Cartesian tensor. Using exactly the same procedures as those employed for δij and ijk , it may be
shown that the only isotropic first-order tensor is the trivial one with all elements
zero.
26.10 Improper rotations and pseudotensors
So far we have considered rigid rotations of the coordinate axes described by
an orthogonal matrix L with |L| = +1, (26.4). Strictly speaking such transformations are called proper rotations. We now broaden our discussion to include
transformations that are still described by an orthogonal matrix L but for which
|L| = −1; these are called improper rotations.
This kind of transformation can always be considered as an inversion of the
coordinate axes through the origin represented by the equation
xi = −xi ,
(26.38)
combined with a proper rotation. The transformation may be looked upon
alternatively as one that changes an initially right-handed coordinate system into
a left-handed one; any prior or subsequent proper rotation will not change this
state of affairs. The most obvious example of a transformation with |L| = −1 is
the matrix corresponding to (26.38) itself; in this case Lij = −δij .
As we have emphasised in earlier chapters, any real physical vector v may be
considered as a geometrical object (i.e. an arrow in space), which can be referred
to independently of any coordinate system and whose direction and magnitude
cannot be altered merely by describing it in terms of a different coordinate system.
Thus the components of v transform as vi = Lij vj under all rotations (proper and
improper).
We can define another type of object, however, whose components may also
be labelled by a single subscript but which transforms as vi = Lij vj under proper
rotations and as vi = −Lij vj (note the minus sign) under improper rotations. In
this case, the vi are not strictly the components of a true first-order Cartesian
tensor but instead are said to form the components of a first-order Cartesian
pseudotensor or pseudovector.
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26.10 IMPROPER ROTATIONS AND PSEUDOTENSORS
x3
v
v
p
x1
x2
O
O
x2
x1
p
x3
Figure 26.2 The behaviour of a vector v and a pseudovector p under a
reflection through the origin of the coordinate system x1 , x2 , x3 giving the new
system x1 , x2 , x3 .
It is important to realise that a pseudovector (as its name suggests) is not a
geometrical object in the usual sense. In particular, it should not be considered
as a real physical arrow in space, since its direction is reversed by an improper
transformation of the coordinate axes (such as an inversion through the origin).
This is illustrated in figure 26.2, in which the pseudovector p is shown as a broken
line to indicate that it is not a real physical vector.
Corresponding to vectors and pseudovectors, zeroth-order objects may be
divided into scalars and pseudoscalars – the latter being invariant under rotation
but changing sign on reflection.
We may also extend the notion of scalars and pseudoscalars, vectors and pseudovectors, to objects with two or more subscripts. For two subcripts, as defined
previously, any quantity with components that transform as Tij = Lik Ljl Tkl under all rotations (proper and improper) is called a second-order Cartesian tensor.
If, however, Tij = Lik Ljl Tkl under proper rotations but Tij = −Lik Ljl Tkl under
improper ones (which include reflections), then the Tij are the components of
a second-order Cartesian pseudotensor. In general the components of Cartesian
pseudotensors of arbitary order transform as
= |L|Lil Ljm · · · Lkn Tlm···n ,
Tij···k
where |L| is the determinant of the transformation matrix.
For example, from (26.29) we have that
|L|ijk = Lil Ljm Lkn lmn ,
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(26.39)
TENSORS
but since |L| = ±1 we may rewrite this as
ijk = |L|Lil Ljm Lkn lmn .
From this expression, we see that although ijk behaves as a tensor under proper
rotations, as discussed in section 26.8, it should properly be regarded as a thirdorder Cartesian pseudotensor.
If bj and ck are the components of vectors, show that the quantities ai = ijk bj ck form
the components of a pseudovector.
In a new coordinate system we have
ai = ijk bj ck
= |L|Lil Ljm Lkn lmn Ljp bp Lkq cq
= |L|Lil lmn δmp δnq bp cq
= |L|Lil lmn bm cn
= |L|Lil al ,
from which we see immediately that the quantities ai form the components of a pseudovector. The above example is worth some further comment. If we denote the vectors with components bj and ck by b and c respectively then, as mentioned in
section 26.8, the quantities ai = ijk bj ck are the components of the real vector
a = b × c, provided that we are using a right-handed Cartesian coordinate system.
However, in a coordinate system that is left-handed the quantitites ai = ijk bj ck
are not the components of the physical vector a = b × c, which has, instead, the
components −ai . It is therefore important to note the handedness of a coordinate
system before attempting to write in component form the vector relation a = b×c
(which is true without reference to any coordinate system).
It is worth noting that, although pseudotensors can be useful mathematical
objects, the description of the real physical world must usually be in terms of
tensors (i.e. scalars, vectors, etc.).§ For example, the temperature or density of a
gas must be a scalar quantity (rather than a pseudoscalar), since its value does
not change when the coordinate system used to describe it is inverted through
the origin. Similarly, velocity, magnetic field strength or angular momentum can
only be described by a vector, and not by a pseudovector.
At this point, it may be useful to make a brief comment on the distinction
between active and passive transformations of a physical system, as this difference
often causes confusion. In this chapter, we are concerned solely with passive trans§
In fact the quantum-mechanical description of elementary particles, such as electrons, protons and
neutrons, requires the introduction of a new kind of mathematical object called a spinor, which is
not a scalar, vector, or more general tensor. The study of spinors, however, falls beyond the scope
of this book.
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