(186) |

(187) |

(188) |

(189) |

(190) | |||

(191) |

Up to now we have restricted ourselves to three basic types of coordinate
transformation; namely, translations, rotations, and standard Lorentz
transformations. An arbitrary combination of these three transformations constitutes
a general Lorentz transformation. Let us now extend our investigations to include
a fourth type of transformation known as a parity inversion; *i.e.*,
. A reflection is a combination of a parity
inversion and a rotation. As is easily demonstrated, the Jacobian of a
parity inversion is , unlike a translation, rotation, or standard
Lorentz transformation, which all possess Jacobians of .

The prototype
of all 3-vectors is the difference in coordinates between two points in space,
. Likewise, the prototype of all 4-vectors is the difference in
coordinates between two events in space-time,
. It
is not difficult to appreciate that both of these objects are invariant under
a parity transformation (in the
sense that they correspond to the same geometric object
before and after the transformation).
It follows that any 3- or 4-tensor which is directly
related to and , respectively, is also invariant under a
parity inversion. Such tensors include the distance between
two points in 3-space, the interval between two points in space-time,
3-velocity, 3-acceleration,
4-velocity, 4-acceleration, and the
metric tensor. Tensors which exhibit tensor behaviour under translations, rotations,
special Lorentz transformations, *and* are invariant
under parity inversions, are termed
*proper tensors*, or sometimes *polar tensors*. Since electric
charge is clearly invariant under such transformations (*i.e.*, it
is a proper scalar) it follows that 3-current and 4-current are proper
vectors. It is also clear from Eq. (2.96) that the scalar potential,
the vector potential, and the potential 4-vector, are proper tensors.

It follows from Eq. (2.149) that
under a parity inversion.
Tensors like this, which exhibit tensor behaviour under translations, rotations, and special
Lorentz transformations, but are *not* invariant under parity inversions
(in the sense that they correspond to different geometric objects
before and after the transformation), are
called *pseudo-tensors*, or sometimes *axial tensors*. Equations
(2.150) and (2.151) imply that the cross product of two proper vectors is a
pseudo-vector, and the curl of a proper vector field is a pseudo-vector field.

One particularly simple way of performing a parity transformation is
to exchange positive and negative numbers on the three Cartesian axes.
A proper vector is unaffected by such a procedure
(*i.e.*, its magnitude and direction are the same before and after).
On the other hand, a pseudo-vector ends up pointing in the opposite
direction after the axes are renumbered.

What is the fundamental difference between proper tensors and pseudo-tensors?
The answer is that all pseudo-tensors are defined according to a handedness
convention. For instance, the cross product between two vectors is conventionally
defined according to a right-hand rule. The only reason for this is that the
majority of human beings are right-handed. Presumably, if the opposite were
true then cross products *etc.* would be defined according to a left-hand
rule and would, therefore, take minus their conventional values. The totally
antisymmetric tensor is the prototype pseudo-tensor and is, of course,
conventionally defined
with respect to a right-handed spatial coordinate system. A parity inversion
converts left into right and *vice versa* and, thereby, effectively swaps
left- and right-handed conventions.

The use of conventions
in physics is perfectly acceptable provided that we recognize that they are
conventions and are *consistent*
in their use. It follows that laws of physics
cannot contain mixtures of tensors and pseudo-tensors, otherwise they would
depend our choice of handedness convention.^{6}

Let us now consider electric and magnetic fields. We know that

(192) | |||

(193) |

We have already seen that the scalar and the vector potential are proper scalars and vectors, respectively. It follows that is a proper vector but that is a pseudo-vector (since it is the curl of a proper vector). In order to fully appreciate the difference between electric and magnetic fields let us consider a thought experiment first proposed by Richard Feynman. Suppose that we are in radio contact with a race of aliens and are trying to explain to them our system of physics. Suppose, further, that the aliens live sufficiently far away from us that there are no common objects which we both can see. The question is this: could we unambiguously explain to these aliens our concepts of electric and magnetic fields? We could certainly explain electric and magnetic lines of force. The former are the paths of charged particles (assuming that the particles are subject only to electric fields) and the latter can be mapped out using small test magnets. We could also explain how we put arrows on electric lines of force to convert them into electric field lines: the arrows run from positive charges (