Definitions

Pseudoscalar

Pseudoscalar

[soo-duh-skey-ler]
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 true scalar does not.

The prototypical example of a pseudoscalar is the scalar triple product. A pseudoscalar, when multiplied by an ordinary vector, becomes a pseudovector or axial vector; a similar construction creates the pseudotensor.

Mathematically, a pseudoscalar is an element of the top exterior power of a vector space. 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.

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, 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 a skew-symmetric (pure) tensor of rank three. The Levi-Civita pseudotensor is a completely skew-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 skew-symmetric tensors of rank 2. The dual of a pseudovector is a skew-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 a skew-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 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 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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