Pseudovector

In physics and mathematics, a pseudovector is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does not transform like a vector under certain discontinuous rigid transformations such as reflections. For example, the angular velocity of a rotating object is a pseudovector because, when the object is reflected in a mirror, the reflected image rotates in such a way so that its angular velocity "vector" is not the mirror image of the angular velocity "vector" of the original object; for true vectors, the reflection "vector" and the original "vector" must be mirror images.
One example of a pseudovector is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b, that span the plane. The vector is a normal to the plane, and is a pseudovector. This has consequences in computer graphics, where it has to be considered when transforming surface normals.
In three dimensions, the curl of a polar vector field at a point and the cross product of two polar vectors are pseudovectors.
A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and torque. In mathematics, in three dimensions, pseudovectors are equivalent to bivectors, from which the transformation rules of pseudovectors can be derived. More generally, in n-dimensional geometric algebra, pseudovectors are the elements of the algebra with dimension, written ⋀ⁿ⁻¹Rⁿ. 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.

Physical examples

Physical examples of pseudovectors include angular velocity, angular acceleration, angular momentum, torque, magnetic field, and magnetic dipole moment.
Image:Impulsmoment van autowiel onder inversie.svg|thumb|Each wheel of the car on the left driving away from an observer has an angular momentum pseudovector pointing left. The same is true for the mirror image of the car. The fact that the arrows point in the same direction, rather than being mirror images of each other, indicates that they are pseudovectors.
Consider the pseudovector angular momentum. Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the "reflection" of this angular momentum "vector" points to the right, but the actual angular momentum vector of the wheel still points to the left, corresponding to the extra sign flip in the reflection of a pseudovector.
The distinction between polar vectors and pseudovectors becomes important in understanding the effect of symmetry on the solution to physical systems. Consider an electric current loop in the plane that inside the loop generates a magnetic field oriented in the z direction. This system is symmetric under mirror reflections through this plane, with the magnetic field unchanged by the reflection. 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.
In physics, pseudovectors are generally the result of taking the cross product of two polar vectors or the curl of a polar vector field. The cross product and curl are defined, by convention, according to the right hand rule, but could have been just as easily defined in terms of a left-hand rule. The entire body of physics that deals with pseudovectors and the right hand rule could be replaced by using pseudovectors and the left hand rule without issue. The pseudovectors so defined would be opposite in direction to those defined by the right-hand rule.
While vector relationships in physics can be expressed in a coordinate-free manner, a coordinate system is required in order to express vectors and pseudovectors as numerical quantities. Vectors are represented as ordered triplets of numbers: e.g., and pseudovectors are represented in this form too. When transforming between left and right-handed coordinate systems, representations of pseudovectors do not transform as vectors, and treating them as vector representations will cause an incorrect sign change, so that care must be taken to keep track of which ordered triplets represent vectors, and which represent pseudovectors. This problem does not exist if the cross product of two vectors is replaced by the exterior product of the two vectors, which yields a bivector which is a 2nd rank tensor and is represented by a 3×3 matrix. This representation of the 2-tensor transforms correctly between any two coordinate systems, independently of their handedness.

Details

The definition of a "vector" in physics is more specific than the mathematical definition of "vector". Under the physics definition, a "vector" is required to have components that "transform" in a certain way under a proper rotation: In particular, if everything in the universe were rotated, the vector would rotate in exactly the same way. Mathematically, if everything in the universe undergoes a rotation described by a rotation matrix R, so that a displacement vector x is transformed to, then any "vector" v must be similarly transformed to. This important requirement is what distinguishes a vector from any other triplet of physical quantities
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. This is in fact how the bookkeeping was done before the more formal and generalised tensor notation came to be. It still manifests itself in how the basis vectors of general tensor spaces are exhibited for practical manipulation.
The discussion so far only relates to proper rotations, i.e. rotations about an axis. However, one can also consider improper rotations, i.e. a mirror-reflection possibly followed by a proper rotation. Suppose everything in the universe undergoes an improper rotation described by the improper rotation matrix R, so that a position vector x is transformed to. If the vector v is a polar vector, it will be transformed to. If it is a pseudovector, it will be transformed to.
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. The symbol det denotes determinant; this formula works because the determinant of proper and improper rotation matrices are +1 and −1, respectively.

Behavior under addition, subtraction, scalar multiplication

Suppose v₁ and v₂ are known pseudovectors, and v₃ is defined to be their sum,. If the universe is transformed by a rotation matrix R, then v₃ is transformed to
So v₃ is also a pseudovector. Similarly one can show that the difference between two pseudovectors is a pseudovector, that the sum or difference of two polar vectors is a polar vector, that multiplying a polar vector by any real number yields another polar vector, and that multiplying a pseudovector by any real number yields another pseudovector.
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,. If the universe is transformed by an improper rotation matrix R, then v₃ is transformed to
Therefore, v₃ is neither a polar vector nor a pseudovector. For an improper rotation, v₃ does not in general even keep the same magnitude:
If the magnitude of v₃ were to describe a measurable physical quantity, that would mean that the laws of physics would not appear the same if the universe was viewed in a mirror. 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.

Behavior under cross products and curls

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, if v is any vector field, the following equation is always true:
where denotes the curl operation from vector calculus.
Suppose v₁ and v₂ are known polar vectors, and v₃ is defined to be their cross product,. If the universe is transformed by a rotation matrix R, then v₃ is transformed to
So v₃ is a pseudovector. Likewise, one can show that the cross product of two pseudovectors is a pseudovector and the cross product of a polar vector with a pseudovector is a polar vector. In conclusion, we have:

polar vector × polar vector = pseudovector
pseudovector × pseudovector = pseudovector
polar vector × pseudovector = polar vector
pseudovector × polar vector = polar vector

This is isomorphic to addition modulo 2, where "polar" corresponds to 1 and "pseudo" to 0.
Similarly, if v₁ is any known polar vector field and v₂ is defined to be its curl, then if the universe is transformed by the rotation matrix R, v₂ is transformed to
So v₂ is a pseudovector field. Likewise, one can show that the curl of a pseudovector field is a polar vector field. In conclusion, we have:

∇ × polar vector field = pseudovector field
∇ × pseudovector field = polar vector field

This is like the above rule for cross-products if one interprets the del operator ∇ as a polar vector.