Pseudovector

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 conceptual 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:

p = a × b.

A simple example of an improper rotation in 3D (but not in 2D) is a coordinate inversion: x goes to −x, y to −y and z to −z. 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 R3 every bivector a ʌ b has a unique dual vector a × b, it is this dual which is more often used.

Physical examples

Physical examples of pseudovectors include the magnetic field, torque, vorticity, 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), so the magnetic field should be unchanged by the reflection. But reflecting the actual magnetic field through that plane changes its sign—this contradiction is resolved by realizing that the mirror reflection of the field induces an extra sign flip because of its pseudovector nature.

As another example, 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 minus sign.

To the extent that physical laws are the same for right-handed and left-handed coordinate systems (i.e. invariant under parity), the sum of a vector and a pseudovector is not meaningful. However, the weak force, which 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 0-12-059815-9)
  • John David Jackson, Classical Electrodynamics (Wiley: New York, 1999). (ISBN 0-471-30932-X)
  • Susan M. Lea, "Mathematics for Physicists" (Thompson: Belmont, 2004) (ISBN 0-534-37997-4)
  • Chris Doran and Anthony Lasenby, "Geometric Algebra for Physicists" (Cambridge University Press: Cambridge, 2007) (ISBN 978-0-521-71595-9)

See also

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