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. (1442) that the scalar potential, the vector potential, and the potential 4-vector, are proper tensors.
It follows from Eq. (1459) that under a parity inversion. Tensors such as 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 (1461) and (1462) 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 our use of them. It follows that laws of physics cannot contain mixtures of tensors and pseudo-tensors, otherwise they would depend our choice of handedness convention.2
Let us now consider electric and magnetic fields. We know that