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Inlinear algebra, apseudoscalar is a quantity that behaves like ascalar, except that it changes sign under aparity inversion^[1][2] while a true scalar does not.

A pseudoscalar, when multiplied by an ordinaryvector, becomes apseudovector (oraxial vector); a similar construction creates thepseudotensor.A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is thescalar triple product, which can be written as thescalar product between one of the vectors in the triple product and thecross product between the two other vectors, where the latter is a pseudovector.

Inphysics, a pseudoscalar denotes aphysical quantity analogous to ascalar. Both arephysical quantities which assume a single value which is invariant underproper rotations. However, under theparity transformation, pseudoscalars flip their signs while scalars do not. Asreflections through a plane are the combination of a rotation with the parity transformation, pseudoscalars also change signs under reflections.

One of the most powerful ideas in physics is that physical laws do not change when one changes the coordinate system used to describe these laws. 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, theHodge dual of a scalar is equal to a constant times the 3-dimensionalLevi-Civita pseudotensor (or "permutation" pseudotensor); whereas the Hodge dual of a pseudoscalar is an antisymmetric (pure) tensor of order three. The Levi-Civita pseudotensor is a completelyantisymmetric pseudotensor of order 3. Since the dual of the pseudoscalar is the product of two "pseudo-quantities", the resulting tensor is a true tensor, and does not change sign upon an inversion of axes. The situation is similar to the situation for pseudovectors and antisymmetric tensors of order 2. The dual of a pseudovector is an antisymmetric tensor of order 2 (and vice versa). The tensor is an invariant physical quantity under a coordinate inversion, while the pseudovector is not invariant.

The situation can be extended to any dimension. Generally in ann-dimensional space the Hodge dual of an orderr tensor will be an antisymmetric pseudotensor of order(n −r) and vice versa. In particular, in the four-dimensional spacetime of special relativity, a pseudoscalar is the dual of a fourth-order tensor and is proportional to the four-dimensionalLevi-Civita pseudotensor.

• Thestream function${\displaystyle \psi (x,y)}$ for a two-dimensional, incompressible fluid flow${\displaystyle \mathbf{v}(x,y)=⟨{\mathrm{\partial }}_y\psi ,-{\mathrm{\partial }}_x\psi ⟩}$.
• Magnetic charge is a pseudoscalar as it is mathematically defined, regardless of whether it exists physically.
• Magnetic flux is the result of adot product between a vector (thesurface normal) and pseudovector (themagnetic field).^{[citation needed]}
• Helicity is the projection (dot product) of aspin pseudovector onto the direction ofmomentum (a true vector).
• Pseudoscalar particles, i.e. particles with spin 0 and odd parity, that is, a particle with no intrinsic spin withwave function that changes sign underparity inversion. Examples arepseudoscalar mesons.

A pseudoscalar in ageometric algebra is a highest-grade element of the algebra. For example, in two dimensions there are two orthogonal basis vectors,${\displaystyle e_1}$,${\displaystyle e_2}$ and the associated highest-grade basis element is

${\displaystyle e_1{e}_2=e_{12}.}$

So a pseudoscalar is a multiple of${\displaystyle e_{12}}$. The element${\displaystyle e_{12}}$ squares to −1 and commutes with all even elements – behaving therefore like the imaginary scalar${\displaystyle i}$ in thecomplex numbers. It is these scalar-like properties which give rise to its name.

In this setting, a pseudoscalar changes sign under a parity inversion, since if

${\displaystyle (e_1,e_2)\mapsto (u_1,u_2)}$

is achange of basis representing anorthogonal transformation, then

${\displaystyle e_1{e}_2\mapsto u_1{u}_2=\pm e_1{e}_2,}$

where the sign depends on the determinant of the transformation. Pseudoscalars in geometric algebra thus correspond to the pseudoscalars in physics.

• Geometric algebra
• Clifford algebras
• Linear algebra
• Scalars

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