pseudovector

In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). The opposite of a pseudovector is a (true) vector or a polar vector. A common way of constructing a pseudovector p is by taking the cross product of two vectors a and b:

\mathbf{p} = \mathbf{a} \times \mathbf{b}

A simple example of an improper rotation is a coordinate inversion: x goes to -x. Under this transformation, a and b go to -a and -b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by -1 compared to the rotation's effect on a true vector. This concept 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. Many occurrences of pseudovectors in mathematics and physics are more naturally analyzed as bivectors, following the calculus of differential forms; the double negation is natural for a bivector. However, bivectors are "less intuitive" in some senses than ordinary vectors, and since in

{\mathbb R}^3

every bivector

a \wedge b

has a unique dual vector

a \times b

, it is this dual which is more often used.

Physical examples

Physical examples of pseudovectors include the magnetic field, torque, and the angular momentum. Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting the effect of symmetry on the solution to physical systems. For example, consider the case of an electrical current loop in the z=0 plane: this system is symmetric (invariant) under mirror reflections through the plane (an improper rotation), but the magnetic field is anti-symmetric (flips sign) under that mirror plane—this contradiction is resolved by realizing that the mirror reflection of the field induces an extra sign flip because of its pseudovector nature. To the extent that physical laws are the same for right-handed and left-handed coordinate systems (i.e. invariant under inversion), the sum of a vector and a pseudovector is not meaningful. However, the weak nuclear force that governs beta decay does depend on the chirality of the universe, and in this case pseudovectors and vectors are added.

References

George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001). (ISBN 0120598159)
John David Jackson, Classical Electrodynamics (Wiley: New York, 1999). (ISBN 047130932X)

Pseudovector

Physical examples

References

See also