Pseudovector - Physical Examples

Physical Examples

Physical examples of pseudovectors include the magnetic field, torque, vorticity, and the angular momentum.

Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting the effect of symmetry on the solution to physical systems. For example, consider the case of an electrical current loop in the z = 0 plane, which has a magnetic field at z = 0 that is oriented in the z direction. This system is symmetric (invariant) under mirror reflections through the plane (an improper rotation), so the magnetic field should be unchanged by the reflection. But reflecting the magnetic field through that plane naively appears to change its sign if it is viewed as a vector field—this contradiction is resolved by realizing that the mirror reflection of the field induces an extra sign flip because of its pseudovector nature, so the mirror flip in the end leaves the magnetic field unchanged as expected.

As another example, consider the pseudovector angular momentum L = r × p. Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the "reflection" of this angular momentum "vector" (viewed as an ordinary vector) points to the right, but the actual angular momentum vector of the wheel still points to the left, corresponding to the extra minus sign in the reflection of a pseudovector. This reflects the fact that the wheels are still turning forward. In comparison, the behaviour of a regular vector, such as the position of the car, is quite different.

To the extent that physical laws would be the same if the universe were reflected in a mirror (equivalently, invariant under parity), the sum of a vector and a pseudovector is not meaningful. However, the weak force, which governs beta decay, does depend on the chirality of the universe, and in this case pseudovectors and vectors are added.