Pseudoscalar

2025-10-23T08:11:10Z

86.12.127.232: Magnetic flux is not parity odd: it's the dot product of two pseudovectors and so is a true vector.

{{Short description|Scalar quantity, changing sign in mirrored coordinates}}
{{Use American English|date=March 2019}}{{More citations needed|date=January 2021}}

In [[linear algebra]], a '''pseudoscalar''' is a quantity that behaves like a [[scalar (physics)|scalar]], except that it changes sign under a [[Parity (physics)|parity inversion]]<ref>{{cite book |last=Zee |first=Anthony|author-link=Anthony Zee|title=Quantum field theory in a nutshell|edition=2nd|publisher=Princeton University Press|year=2010 |chapter=II. Dirac and the Spinor II.1 The Dirac Equation § Parity |chapter-url=https://archive.org/details/isbn_9780691140346/page/98 |page=98 |url=https://archive.org/details/isbn_9780691140346|url-access=registration |isbn=978-0-691-14034-6}}</ref><ref>{{cite book|last=Weinberg |first=Steven|author-link=Steven Weinberg|title=The quantum theory of fields|volume=1: Foundations|publisher=Cambridge University Press|year=1995|page=228|isbn=9780521550017 |chapter=5.5 Causal Dirac Fields §5.5.57 |chapter-url=https://books.google.com/books?id=doeDB3_WLvwC&pg=PA228}}</ref> while a true scalar does not.

A pseudoscalar, when multiplied by an ordinary [[Vector (mathematics and physics)|vector]], becomes a ''[[pseudovector]]'' (or ''axial vector''); a similar construction creates the [[pseudotensor]].
A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is the [[scalar triple product]], which can be written as the [[scalar product]] between one of the vectors in the triple product and the [[cross product]] between the two other vectors, where the latter is a pseudovector.

==In physics==
In [[physics]], a pseudoscalar denotes a [[physical quantity]] analogous to a [[scalar (physics)|scalar]]. Both are [[physical quantity|physical quantities]] which assume a single value which is invariant under [[proper rotation]]s. However, under the [[parity transformation]], pseudoscalars flip their signs while scalars do not. As [[Reflection (mathematics)|reflection]]s through a plane are the combination of a rotation with the parity transformation, pseudoscalars also change signs under reflections.

===Motivation===
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, the [[Hodge dual]] of a scalar is equal to a constant times the 3-dimensional [[Levi-Civita symbol|Levi-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 completely [[antisymmetric tensor|antisymmetric]] 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 an ''n''-dimensional space the Hodge dual of an order ''r'' tensor will be an antisymmetric pseudotensor of order {{nowrap|(''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-dimensional [[Levi-Civita symbol|Levi-Civita pseudotensor]].

===Examples===
* The [[stream function]] <math>\psi(x,y)</math> for a two-dimensional, incompressible fluid flow <math>\mathbf{v}(x,y)=\langle \partial_{y}\psi,-\partial_{x}\psi\rangle </math>.
* [[Magnetic charge]] is a pseudoscalar as it is mathematically defined, regardless of whether it exists physically.
* [[Helicity (particle physics)|Helicity]] is the projection (dot product) of a [[Spin (physics)|spin]] pseudovector onto the direction of [[momentum]] (a true vector).
* Pseudoscalar particles, i.e. particles with spin 0 and odd parity, that is, a particle with no intrinsic spin with [[wave function]] that changes sign under [[Parity (physics)|parity inversion]]. Examples are [[pseudoscalar meson]]s.

==In geometric algebra==
{{See also|Pseudoscalar (Clifford algebra)}}

A pseudoscalar in a [[geometric algebra]] is a highest-[[graded vector space|grade]] element of the algebra. For example, in two dimensions there are two [[orthogonal basis]] vectors, <math>e_1</math>, <math>e_2</math> and the associated highest-grade basis element is

:<math>e_1 e_2 = e_{12}.</math>

So a pseudoscalar is a multiple of <math>e_{12} </math>. The element <math>e_{12} </math> squares to −1 and commutes with all even elements – behaving therefore like the imaginary scalar <math>i </math> in the [[complex 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

:<math>(e_1,e_2) \mapsto (u_1, u_2) </math>

is a [[change of basis]] representing an [[orthogonal transformation]], then

:<math>e_1 e_2 \mapsto u_1 u_2 = \pm e_1 e_2, </math>

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

==References==
<references/>

[[Category:Geometric algebra]]
[[Category:Clifford algebras]]
[[Category:Linear algebra]]
[[Category:Scalars]]

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Pseudoscalar