VECTORS and PSEUDO(?)VECTORS
INTRODUCTION
A vector is defined to be a quantity that has a length (magnitude) and a direction. Vectors are defined
independently of coordinate systems, but when we actually work problems, it is usually convenient to
choose a coordinate system. After choosing a coordinate system, we can represent a vector by its
components along the coordinate axes. These components determine the length and direction of the
vector and the definition of a vector is often stated in terms of the properties of the vector components
under rotations of the coordinate axes. We will always use Cartesian coordinate systems. We distinguish
between two different types of rotations. In a “passive” rotation the physical system is unchanged, but we
rotate the coordinate axes. In an “active” rotation we rotate the physical system and leave the coordinate
axes unchanged. We will first treat passive coordinate rotations and discuss active rotations later.
Under passive rotations of coordinate axes, vectors themselves are unchanged. After choosing a
coordinate system, we define unit vectors, ei (i=1,2,3), along the coordinate axes. We can represent a
vector A by A=A1e1+A2e2+A3e3 where we indicate vectors by bold faced print. We will use a short hand
notation by writing A=Aiei where we introduce the convention that we sum over repeated indices. The
unit vectors are chosen to be perpendicular to each other. They satisfy
where , the
Kronecker delta, is 1 if i=j and zero otherwise. We still have to specify the orientation of the coordinate
axes. We usually do this by using a right handed coordinate system, requiring that
where
we use a right hand rule to determine the direction of the cross product. In making a general passive
“rotation” to another Cartesian coordinate system we assume that the unit vectors in the new system
satisfy
and
, but we do not require that
. The orientation is not
necessarily preserved. If
, the new coordinate system is left handed and we say that the
rotation is “improper.” An improper rotation from one coordinate system to another is one that cannot be
made continuously starting from the identity rotation. An example of an improper rotation is a
coordinate inversion
. Since a vector is independent of coordinate system, it must be true that
For an inversion,
and
.
VECTOR CROSS PRODUCTS
Vector cross products are very useful in formulating the laws of classical physics. Many textbooks1,2,3,4
state that the transformation properties of cross products under passive improper rotations are different
from “true” vectors. Feynman5, for example, says that “The original quantities, like the angular momenta
and the angular velocities, and things of this kind were not really vectors at all!” He goes on to say that
“there are ‘honest’ vectors, for example a r in space…. But the other kind of vector, which has to do
with rotations, is of a different nature.” The “honest” vector is called a “polar” vector while the other
kind of vector is called an “axial” vector or a “pseudovector.” We will show that cross products have
precisely the same transformation properties as polar vectors under any passive orthogonal
transformation. This property of the cross product has been pointed out previously6,7, but has been largely
ignored. The distinction between polar and axial vectors arises only when an improper, active rotation of
the physical system occurs and the coordinate system remains fixed.
Before proving that cross products transform in the same way as polar vectors under any passive rotation,
we will first argue that the claim that they transform differently cannot be true. Let A and B be two polar
vectors and let
Under a change of coordinates, A and B are invariant but the above texts1-5
claim that if the coordinate change is an improper rotation then
Commonsense tells us that this
is unreasonable since A and B are invariant. Commonsense is not always a reliable guide, but in this case
it turns out to be right.
can only be true if we must switch from a right hand rule to a left hand
rule if we change from a right hand coordinate system to a left hand coordinate system. Many authors5,8
say it is necessary to link the handedness of the cross product to the handedness of the coordinate system.
These authors use the same symbol for the cross product in any coordinate system. This creates the
illusion that the equations are independent of coordinates, when in fact their rule for the cross product
depends on the coordinate system. The principle of relativity states that the laws of physics are the same
in all inertial frames of reference. That means that the same definition of cross product must be used in
all inertial coordinate systems. Switching from one definition to another depending on the orientation of
the coordinate system violates this principle. It also makes it impossible to find the direction of angular
momentum or a magnetic dipole moment without first choosing a coordinate system. We will always
interpret “ ” to mean that a right hand rule is used.
If the cross product of two polar vectors is a “pseudovector,” this will be true for the cross product of any
two polar vectors. However, it is obvious that if C is a polar vector, it always possible to find two other
polar vectors A and B perpendicular to C so that
. This is just a geometric accident. Does
imply that e3 is a pseudovector? If it did, then it would be impossible to express a vector in
component form because the unit vectors would behave differently under improper rotations.
TRANSFORMATION PROPERTIES OF CROSS PRODUCTS
Arfken and Weber5 define cross products first by using a right hand rule for determining the direction of
and defining
where θ is the smallest angle between the two vectors.
They call this the “geometric” definition of a vector cross product. Then they express the cross product
using the familiar determinant,
This leads to the equation C3= A1B2-A2B1 with similar expressions for the other components. Arfken and
Weber call this the “algebraic” definition of a cross product. They conclude that under an inversion
If C were a polar vector, under an inversion
we would have
. This leads them to state that C does not transform like a polar vector and
hence
under an inversion. If we make the replacements
the expression above, then
in
This gives the correct result that
Arfken and Weber’s algebraic definition is consistent with
the geometric definition in a right hand coordinate system, but it is inconsistent with the geometric
definition in a left hand coordinate system. Other texts1,2,3 do the same thing, and this seems to be the
source of the confusion. Using the “algebraic” definition gives
for any orientation of the
coordinate axes. This is inconsistent with using a right hand rule in a left hand coordinate system.
We will show below that the cross product of two polar vectors transforms exactly the same way as polar
vectors under any passive rotation, proper or improper. To do this it is useful to introduce the Levi-Civita
symbol
. It is defined so that
and
changes sign if any two indices are interchanged. This
means that
if any two indices are equal. There are only six non-zero elements:
This is useful because the determinant of a 3X3
matrix R is defined by
and satisfies (this can be generalized to an NXN matrix)
.
(1)
This relation can be used to prove that the determinant of the product of two matrices is the product of the
determinants of the matrices and that the determinant of the transpose of a matrix is equal to the
determinant of the matrix. It can also be used to find the inverse of a matrix. Since we can represent the
cross product of two vectors as a determinant, we can use the Levi-Civita symbol as a short hand way to
represent a cross product in component form. The determinant expansion for the cross product can be
conveniently written as
, remembering to sum over repeated indices. You should
remember that this is true only in a right hand coordinate system. We will derive the appropriate form for
a left hand coordinate system.
We will consider transformations of vector components between two Cartesian coordinate systems with
a common origin. We will assume that the initial coordinate system is a right-handed system with unit
basis vectors that satisfy (using a right hand rule for the cross product)
,
and
(2)
(3)
We can transform to another set of orthonormal basis vectors by using the “rotation” matrix
(4)
where
(5)
It is also true that
(6)
We can expand a vector using either set of basis vectors.
(7)
The components of the vector in the different coordinate systems are related by
(8)
The rotation matrix itself is an orthogonal matrix
and
If
If
is the transpose of R, then
the rotation is improper and the orientation
of the coordinate system changes.
It is easy to determine cross products of the new basis vectors. We have
We use Eq.(1) to obtain
.
For an improper rotation
(9)
and in a left hand coordinate system
We assume that A and B are two polar vectors and
. We can find the components of C in the
two different coordinate systems, starting from the right-hand system.
(10)
Thus
(11)
and
.
(12)
We see that
(13)
Thus the components of C transform exactly like the components of a polar vector and e3 has the same
transformation properties as e1 and e2 under any passive coordinate transformation. Many authors8 claim
that under an inversion
. It is clear from Eq. (12) that this is not correct and
under an inversion.
PSEUDOSCALAR PRODUCT
A related claim is often made that the triple scalar product,
, where A, B, and C are all polar
vectors, is a “pseudoscalar” under inversions of a coordinate system and it is called the pseudoscalar
product. If this were true, it would mean that
changes signs if the coordinate system is
inverted. Again, this defies commonsense and is wrong. This is proved incorrect by using the equations
derived above. Start from a right hand coordinate system and make a transition to another orthogonal
coordinate system. We note that
and
.
The same result is obtained for the triple scalar product regardless of the orientation of the coordinate
system. Inverting the coordinate system gives
.
The triple scalar product does not change sign if the rotation is improper and it is not a pseudoscalar.
ACTIVE ROTATIONS
We will now consider active rotations which means that the physical system is rotated, keeping the
coordinate system fixed. Vectors are invariant under coordinate rotations, but they change under rotations
of the system. Here we have
after the rotation. In order to determine , it is convenient to
also rotate the coordinate system along with the active rotation as we did for passive rotations, but in the
end we calculate everything in the original system. Then the components of in the rotated coordinate
system are the same as the components of A in the original system.
We want to compute the
. What we get is, using the same rotation matrix as before (Eq.(5)),
.
(14)
Thus the rotation matrix that gives us the components of the rotated vector in the original coordinate
system, is the transpose (or inverse since the matrix is orthogonal) of the matrix that previously gave us a
passive rotation of the coordinate system. Now suppose we ask how angular momentum,
,
transforms under a general rotation. Angular momentum in the rotated physical system is still defined by
. If we use a right hand coordinate system,
We have to evaluate
We start with Eq.(1) and multiply by
.
Thus
(15)
and
(16)
Cross products of polar vectors do not transform in the same way as polar vectors under active improper
rotations. There is an extra factor of det(R). There is an important difference for physical quantities
resulting from cross products under active rotations that distinguish them from polar vectors. Under a
pure active inversion L does not change signs, but r and p do. Under a passive rotation, proper or
improper, none of them changes sign.
LORENTZ TRANSFORMATIONS
We have been assuming that one observer is simply comparing the components of vectors in different
Cartesian coordinate systems. A Lorentz transformation is different because it explicitly enables two
observers to compare the results of measurements made of the same physical system in different reference
frames. When one gives a coordinate system a boost, the observer is also boosted. When the coordinate
system is inverted, the observer is also inverted. Although Lorentz transformations must be used to
transform between two coordinate systems that are moving relative to each other, it is still relevant to ask
how vectors, quantities with length and direction, transform under rotations between two coordinate
systems at rest with respect to each other and compare the results with the Lorentz transformation.
Consider two observers at rest, but inverted with respect to each other. Assume there is a current loop at
the origin in each system with its plane in the
plane (or
plane). B transforms as components
of a second rank tensor with two spatial indices and it is easy to see that
in one frame is related to
in the other frame by
. Since the z-axes are inverted with respect to each other, the two observers
claim the magnetic fields are in opposite directions. This is exactly what many people claim happens
when one makes a passive coordinate inversion. The observations in the two reference frames appear to
be inconsistent, but they are actually consistent. Each observer uses a right-hand rule in his reference
frame to determine the direction of the magnetic field, but each observer sees the other observer using a
left-hand rule. Taking this into account, they agree.
One possible reason for the confusion surrounding the transformation properties of cross products might
be because of the natural expectation that cross products, represented by second rank tensors in relativity,
should transform the same way under Lorentz transformations as they do under simple rotations as long
as the rotations are limited to coordinate systems at rest with respect to each other. We will see below
that this is not true.
In relativity, electric and magnetic fields transform as components of a second rank tensor. For Lorentz
transformations involving only a rotation, proper or improper, the space and time components do not mix
and the space components transform as (assuming an orthogonal transformation)
.
(17)
It is unnecessary to distinguish between covariant and contravariant indices since there is no mixing of
space and time components. The space components are related to the magnetic field by (up to a possible
sign, depending on which convention is used)
(18)
For example
and, using Eq.(15),
(19)
Now use
so that
(20)
For a Lorentz transformation, there is an extra factor of det(R) in the transformation law compared to
Eq.(8). The Lorentz transformation does not give the same result as an improper rotation of the
coordinate system for coordinate systems at rest.
REFERENCES
1. H. Goldstein, Classical Mechanics, (Addison-Wesley Publ. Co., Reading, Mass., 1980) Second
Edition, p 171-172.
2. L. Landau and E. Lifschitz, Classical Theory of Fields (C, pp. 19-20.
3. P. Morse and H. Feshbach, Methods of Mathematical Physics (McGraw-Hill Book Company Inc.,
New York, 1953), Part I.p.10.
4. G. B. Arfken and H.J. Weber, Mathematical Methods for Physicists (Elsevier Press Academic
Press, Burlington, MA 2005)
5. Richard Feynman, Feynman Lectures on Physics, (Addison-Wesley Publ. Co., Reading, Mass.,
2005) Vol. 1 Chap. 52
6. A. V. Masket, Am. J. Phys. 34 164 (1966)
7. H. Gelman, Am. J. Phys. 38 599 (1970)
8. W. Hauser, Am. J. Phys. 54 168 (1986)
9. See, for example, D. J. Griffiths, Introduction to Electrodynamics, (Prentice-Hall Inc., New
Jersey, 1999) Third Edition, p 12.