Pseudoscalar - Pseudoscalars in Physics

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 skew-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 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.