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| In [[physics]], a '''pseudoscalar''' is a quantity that behaves like a [[scalar (physics)|scalar]], except that it changes sign under a [[Parity (physics)|parity inversion]] such as [[improper rotation]]s while a true scalar does not.
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| The prototypical example of a pseudoscalar is the [[scalar triple product]]. A pseudoscalar, when multiplied by an ordinary [[vector space|vector]], becomes a [[pseudovector|pseudovector (axial vector)]]; a similar construction creates the [[pseudotensor]]. | |
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| Mathematically, a pseudoscalar is an element of the top [[exterior power]] of a [[vector space]], or the top power of a [[Clifford algebra]]; see [[pseudoscalar (Clifford algebra)]]. More generally, it is an element of the [[canonical bundle]] of a [[differentiable manifold]].
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| ==Pseudoscalars in physics==
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| 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.
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| 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. The fact 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 3-space, quantities which are described by a pseudovector are in fact anti-symmetric tensors of rank 3, which are invariant under inversion. The pseudovector is a much simpler representation of that quantity, but suffers from the change of sign under inversion. Specifically, in 3-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 in fact a anti-symmetric (pure) tensor of rank three. The Levi-Civita pseudotensor is a completely [[anti-symmetric]] pseudotensor of rank 3. Since the dual of the pseudoscalar is the product of two "pseudo-quantities" it can be seen that 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 anti-symmetric tensors of rank 2. The dual of a pseudovector is a anti-symmetric tensors of rank 2 (and vice versa). It is the tensor and not the pseudovector which is the representation of the physical quantity which is invariant to a coordinate inversion, while the pseudovector is not invariant.
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| The situation can be extended to any dimension. Generally in an ''N''-dimensional space the Hodge dual of a rank ''n'' tensor (where ''n'' is less than or equal to ''N''/2) will be a anti-symmetric pseudotensor of rank ''N''-''n'' and vice versa. In particular, in the four-dimensional spacetime of special relativity, a pseudoscalar is the dual of a fourth-rank tensor which is proportional to the four-dimensional [[Levi-Civita symbol|Levi-Civita pseudotensor]].
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| ===Examples===
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| * the [[magnetic charge]] (as it is mathematically defined, regardless of whether it exists physically),
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| * the [[magnetic flux]] - it is result of a [[dot product]] between a vector (the [[surface normal]]) and pseudovector (the [[magnetic field]]),
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| * the [[Helicity (particle physics)|helicity]] is the projection (dot product) of a [[spin (physics)|spin]] pseudovector onto the direction of [[momentum]] (a true vector).
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| ==Pseudoscalars in geometric algebra==
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| {{See also|Pseudoscalar (Clifford algebra)}}
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| 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
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| :<math>e_1 e_2 = e_{12}.</math> | |
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| So a pseudoscalar is a multiple of ''e''<sub>12</sub>. The element ''e''<sub>12</sub> squares to −1 and commutes with all even elements – behaving therefore like the imaginary scalar ''i'' in the [[complex numbers]]. It is these scalar-like properties which give rise to its name.
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| In this setting, a pseudoscalar changes sign under a parity inversion, since if
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| :(''e''<sub>1</sub>, ''e''<sub>2</sub>) → (''u''<sub>1</sub>, ''u''<sub>2</sub>) | |
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| is a change of basis representing an orthogonal transformation, then
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| :''e''<sub>1</sub>''e''<sub>2</sub> → ''u''<sub>1</sub>''u''<sub>2</sub> = ±''e''<sub>1</sub>''e''<sub>2</sub>,
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| where the sign depends on the determinant of the rotation. Pseudoscalars in geometric algebra thus correspond to the pseudoscalars in physics.
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| [[Category:Geometric algebra]]
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| [[Category:Clifford algebras]]
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| [[Category:Linear algebra]]
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