Pseudoscalar

From Wikipedia, the free encyclopedia

In mathematics and physics, a pseudoscalar is a quantity that behaves more or less like a scalar, except that it transforms oddly under the action of a discrete group. Typically, the discrete group is the parity operation on three-dimensional space, and pseudoscalars change sign under a parity inversion. The notation used in geometric algebra provides a mathematically cleaner, less ambiguous notation for the concept, as compared to the traditional physics notation. 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.

[edit] 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 way to describe the physical quantity it is referring to. This is in fact the case. In 3-space, the dual of a pseudoscalar is equal to a constant times the 3-dimensional Levi-Civita pseudotensor (or "permutation" pseudotensor). 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 dual of an 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.

[edit] Examples

• the magnetic charge (as it is mathematically defined, regardless of whether it exists physically)

• the pion, the charged particle that mediates nuclear forces. Most mesons are pseudoscalars. Curiously, the pion forms an isospin triplet; the current associated with the pion is an axial vector, known as the axial vector current.

[edit] Pseudoscalars in geometric algebra

A pseudoscalar in a geometric algebra is the highest-grade basis element of the algebra. For example, in two dimensions there are two basis vectors, e₁, e₂ and the highest-grade basis element is e₁e₂ = e₁₂.

This element 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.

Pseudoscalars in geometric algebra correspond to the pseudoscalars in physics. Indeed, the language of the geometric algebra provides for a cleaner notation for the concept of the pseudoscalar than does the traditional physics notation; this is one of the claimed strengths of the geometric algebra notation.