Pseudovector

In physics and mathematics, a pseudovector (or axial vector)[2] is a quantity that behaves like a vector in many situations, but its direction does not conform when the object is rigidly transformed by rotation, translation, reflection, etc.

This has consequences in computer graphics, where it has to be considered when transforming surface normals.

[5] A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity.

The label "pseudo-" can be further generalized to pseudoscalars and pseudotensors, both of which gain an extra sign-flip under improper rotations compared to a true scalar or tensor.

Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left.

The distinction between polar vectors and pseudovectors becomes important in understanding the effect of symmetry on the solution to physical systems.

But reflecting the magnetic field as a vector through that plane would be expected to reverse it; this expectation is corrected by realizing that the magnetic field is a pseudovector, with the extra sign flip leaving it unchanged.

This representation of the 2-tensor transforms correctly between any two coordinate systems, independently of their handedness.

(The coordinate system is fixed in this discussion; in other words this is the perspective of active transformations.)

This important requirement is what distinguishes a vector (which might be composed of, for example, the x-, y-, and z-components of velocity) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular box cannot be considered the three components of a vector, since rotating the box does not appropriately transform these three components.)

(In the language of differential geometry, this requirement is equivalent to defining a vector to be a tensor of contravariant rank one.

In this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks at the same time, denoted by raised and lowered indices within the Einstein summation convention.)

A basic and rather concrete example is that of row and column vectors under the usual matrix multiplication operator: in one order they yield the dot product, which is just a scalar and as such a rank zero tensor, while in the other they yield the dyadic product, which is a matrix representing a rank two mixed tensor, with one contravariant and one covariant index.

As such, the noncommutativity of standard matrix algebra can be used to keep track of the distinction between covariant and contravariant vectors.

It still manifests itself in how the basis vectors of general tensor spaces are exhibited for practical manipulation.

The transformation rules for polar vectors and pseudovectors can be compactly stated as where the symbols are as described above, and the rotation matrix R can be either proper or improper.

On the other hand, suppose v1 is known to be a polar vector, v2 is known to be a pseudovector, and v3 is defined to be their sum, v3 = v1 + v2.

In fact, this is exactly what happens in the weak interaction: Certain radioactive decays treat "left" and "right" differently, a phenomenon which can be traced to the summation of a polar vector with a pseudovector in the underlying theory.

For a rotation matrix R, either proper or improper, the following mathematical equation is always true: where v1 and v2 are any three-dimensional vectors.

Similarly, one can show: This is isomorphic to addition modulo 2, where "polar" corresponds to 1 and "pseudo" to 0.

An alternate approach, more along the lines of passive transformations, is to keep the universe fixed, but switch "right-hand rule" with "left-hand rule" everywhere in math and physics, including in the definition of the cross product and the curl.

Otherwise, the definitions are equivalent, though it should be borne in mind that without additional structure (specifically, either a volume form or an orientation), there is no natural identification of ⋀n−1(V) with V. Another way to formalize them is by considering them as elements of a representation space for

Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated.

For example, a multivector is a summation of k-fold wedge products of various k-values.

In three dimensions, the most general 2-blade or bivector can be expressed as the wedge product of two vectors and is a pseudovector.

The transformation properties of the pseudovector in three dimensions has been compared to that of the vector cross product by Baylis.

[10] He says: "The terms axial vector and pseudovector are often treated as synonymous, but it is quite useful to be able to distinguish a bivector from its dual."

In this context of geometric algebra, this bivector is called a pseudovector, and is the Hodge dual of the cross product.

On the other hand, if the components are fixed and the basis vectors eℓ are inverted, then the pseudovector is invariant, but the cross product changes sign.

This behavior of cross products is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.