Pseudoscalar

A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector.

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.

That a pseudoscalar reverses its sign when the coordinate axes are inverted suggests that it is not the best object to describe a physical quantity.

In 3D-space, quantities described by a pseudovector are antisymmetric tensors of order 2, which are invariant under inversion.

The pseudovector may be a simpler representation of that quantity, but suffers from the change of sign under inversion.

Similarly, in 3D-space, the Hodge dual of a scalar is equal to a constant times the 3-dimensional Levi-Civita pseudotensor (or "permutation" pseudotensor); whereas the Hodge dual of a pseudoscalar is an antisymmetric (pure) tensor of order three.

The dual of a pseudovector is an antisymmetric tensor of order 2 (and vice versa).

In this setting, a pseudoscalar changes sign under a parity inversion, since if is a change of basis representing an orthogonal transformation, then where the sign depends on the determinant of the transformation.