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. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ...

A common way of constructing a pseudovector p is by taking the cross product of two vectors a and b: For the crossed product in algebra and functional analysis, see crossed product. ...

p = a × b

A simple example of an improper rotation in 3D (but not in 2D) 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. In mathematics, a pseudoscalar in a geometric algebra is the highest-grade basis element of the algebra. ... A pseudotensor is a generalization of the pseudovector concept, and changes its sign under inversion by some transformation matrix. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

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 ∧ b has a unique dual vector a × b, it is this dual which is more often used. A bivector is an element of the antisymmetric tensor product of a tangent space with itself. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the Hodge star operator or Hodge dual is a signficant linear map introduced in general by W. V. D. Hodge. ...

Physical examples

Physical examples of pseudovectors include the magnetic field, torque, vorticity, and the angular momentum. This template is misplaced. ... Torque applied via an adjustable end wrench Relationship between force, torque, and momentum vectors in a rotating system In physics, torque (or often called a moment) can informally be thought of as rotational force or angular force which causes a change in rotational motion. ... Vorticity is a mathematical concept used in fluid dynamics. ... This gyroscope remains upright while spinning due to its 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. This article or section does not cite its references or sources. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

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 force, which governs beta decay, does depend on the chirality of the universe, and in this case pseudovectors and vectors are added. The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... In nuclear physics, beta decay (sometimes called neutron decay) is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted. ... A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. ...

References

See also

• Grassmann algebra