# Pseudo-scalar

A quantity that does not change under a translation or rotation of the coordinate axes but changes its sign when the direction of each axis is reversed. As an example of a pseudo-scalar one could take the mixed triple scalar product of three vectors (cf. Mixed product), or the inner product $(\mathbf{a},\mathbf{b})$, where $\mathbf{a}$ is an axial vector and $\mathbf{b}$ is a general vector (based at the origin).

#### Comments

Pseudo-scalars are e.g. used in the context of the Clifford algebra based approach to the foundations of geometry and physics; cf. e.g. various articles in [a1] and [a2]. In the terminology of [a3], a pseudo-scalar as defined above is a $W$-scalar (a $W$-tensor of valency $0$).

#### References

[a1] | J.S.R. Chisholm, A.K. Common, "Clifford algebras and their applications in mathematical physics" , Reidel (1986) |

[a2] | D. Hestenes, "New foundations for classical mechanics" , Reidel (1986) |

[a3] | J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) pp. 11ff (Translated from German) |

**How to Cite This Entry:**

Pseudo-scalar.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pseudo-scalar&oldid=42318