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Pseudoscalar

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Title: Pseudoscalar  
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Subject: Blade (geometry), Geometric algebra, Multivector, Spinor, Scalar
Collection: Clifford Algebras, Geometric Algebra, Linear Algebra
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Pseudoscalar

In physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as improper rotations while a true scalar does not.

Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. 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. A pseudoscalar, when multiplied by an ordinary vector, becomes a pseudovector (axial vector); a similar construction creates the pseudotensor.

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.

Pseudoscalars in physics

In physics, a pseudoscalar denotes a physical quantity analogous to a scalar. 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. As reflections 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. 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 2, which are invariant under inversion. The pseudovector is a much simpler representation of that quantity, but suffers from the change of sign under inversion. Similarly, in 3-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 in fact an 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 an 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.

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 an 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 pseudotensor.

Examples

Pseudoscalars in geometric algebra

A pseudoscalar in a geometric algebra is a highest-grade element of the algebra. For example, in two dimensions there are two orthogonal basis vectors, e_1, e_2 and the associated highest-grade basis element is

e_1 e_2 = e_{12}.

So a pseudoscalar is a multiple of e12. The element e12 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.

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

(e1, e2) → (u1, u2)

is a change of basis representing an orthogonal transformation, then

e1e2u1u2 = ±e1e2,

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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