physicsand 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 productof 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.
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 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.
* 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)
Wikimedia Foundation. 2010.
Look at other dictionaries:
pseudovector — noun A quantity that transforms like a vector under a proper rotation but gains an additional change of sign under an improper rotation … Wiktionary
pseudovector — … Useful english dictionary
Pseudovector meson — In high energy physics, a pseudovector meson or axial vector meson is a meson with total spin 1 and even parity (usually noted as JP =1+). Compare to a vector meson, which has a total spin 1 and odd parity.The known pseudovector, or axial vector … Wikipedia
pseudovector interaction — pseudovektorinė sąveika statusas T sritis fizika atitikmenys: angl. axial vector interaction; pseudovector interaction vok. Axialvektorwechselwirkung, f; Pseudovektorwechselwirkung, f rus. аксиально векторное взаимодействие, n; псевдовекторное… … Fizikos terminų žodynas
pseudovector field — pseudovektorinis laukas statusas T sritis fizika atitikmenys: angl. pseudovector field vok. Pseudovektorfeld, n rus. псевдовекторное поле, n pranc. champ pseudo vectoriel, m … Fizikos terminų žodynas
pseudovector meson — pseudovektorinis mezonas statusas T sritis fizika atitikmenys: angl. pseudovector meson vok. pseudovektorielles Meson, n rus. псевдовекторный мезон, m pranc. méson pseudo vectoriel, m … Fizikos terminų žodynas
Cross product — This article is about the cross product of two vectors in three dimensional Euclidean space. For other uses, see Cross product (disambiguation). In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on… … Wikipedia
Triple product — This article is about the mathematical product. For the product in nuclear fusion, see Lawson criterion. In mathematics, the triple product is a product of three vectors. The name triple product is used for two different products, the scalar… … Wikipedia
Pseudoscalar — In physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as improper rotations while true scalar does not.The prototypical example of a pseudoscalar is the scalar triple… … Wikipedia
Vector calculus — Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus Derivative Change of variables Implicit differentiation Taylor s theorem Related rates … Wikipedia