Pseudoscalar
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In mathematics and physics, a pseudoscalar is a quantity that behaves more or less like a scalar, except that it transforms oddly under the action of a discrete group. Typically, the discrete group is the parity operation on three-dimensional space, and pseudoscalars change sign under a parity inversion. The notation used in geometric algebra provides a mathematically cleaner, less ambiguous notation for the concept, as compared to the traditional physics notation. The prototypical example of a pseudoscalar is the scalar triple product.
A pseudoscalar, when multiplied by an ordinary vector, becomes a pseudovector or axial vector; a similar construction creates the pseudotensor.
[edit] Pseudoscalars in physics
In physics, a pseudoscalar denotes a physical quantity analogous to a scalar. Both are physical quantities which assume a single value which is invariant under proper rotations. However, under the parity transformation, pseudoscalars flip their signs while scalars do not.
[edit] Examples
- the magnetic charge (as it is mathematically defined, regardless of whether it exists physically)
- the pion, the charged particle that mediates nuclear forces. Most mesons are pseudoscalars. Curiously, the pion forms an isospin triplet; the current associated with the pion is an axial vector, known as the axial vector current.
[edit] Pseudoscalars in geometric algebra
A pseudoscalar in a geometric algebra is the highest-grade basis element of the algebra. For example, in two dimensions there are two basis vectors, <math>e_1</math>, <math>e_2</math> and the highest-grade basis element is <math>e_1 e_2 = e_{12}</math>.
This element squares to −1 and commutes with all elements — behaving therefore like the imaginary scalar i in the complex numbers. It is these scalar-like properties which give rise to its name.
Pseudoscalars in geometric algebra correspond to the pseudoscalars in physics. Indeed, the language of the geometric algebra provides for a cleaner notation for the concept of the pseudoscalar than does the traditional physics notation; this is one of the claimed strengths of the geometric algebra notation.de:Pseudoskalar
