pseudovectors

StatusX

i dont get the point of pseudovectors. the only example I ever see is the cross product, and their reasoning is that it should still be defined as the determinant of the matrix with the basis vectors on top, the components of the first vector in the second row, and the components of the second vector in the third row. so if you define it this way, it will not be invariant under a coordinate transform that changes the handedness of the system (ie, a reflection). then they say that since in real life the coordinate system obviously doesnt matter, you have to be careful in defining physical quanities in terms of cross products, like the magnetic field and angular velocity. So why not just define cross products to be normal vectors in the first place? The only example ive seen of a pseudotensor is the levi-cevita symbol, which is another way of defining the cross product. thats really circular reasoning, at least in my opinion.

mathman

The sign of a cross product is ambiguous. It is defined to perpendicular to the plane determined by the 2 vectors which are crossed, but the direction is determined by convention only. A similar example of this ambiguity is square root.

Dr Transport

a vector is not inversion symmetric, where as a pseudovector is. examples of each are the position vector and the angular momentum vector.

StatusX

i know, but im saying pick a rule and stick to it. use, say, the right hand rule, no matter what the coordinate system. to me, it seems like were defining the cross product differently in right and left handed systems, and then when it doesnt look the same in both, we label it a pseudovector and label the tensor we used to define it a pseudotensor. this all seems so unnecessary. can anyone explain why we do it this way?

Dr Transport

The right-hand rule always applies, we are not defining the cross-product differently.

Think about it, the magnetic induction field is defined by , although it has vector components, it is a pseudo-vector because it is inversion symmetric, unlike the vector potential .

StatusX

if the right hand rule always applied, there would be no pseudovectors, because you can apply the right hand rule independent of a coordinate system. if your example showed why this is wrong, i didnt see it.

Dr Transport

It is the definition, plain and simple. Pseudo-vectors are invariant with respect to inversion. The magnetic field is always defined by the curl of the vector potential, and it always uses the right hand rule, as does any other quantity using a cross-product.

Jcpaks3

Most people don't believe it, but there is actually no difference between the transformation properties of vectors and cross products under coordinate transformations. This should be obvious since the direction of the cross product is determined by the right hand rule and is independent of using a coordinate system. This means that vectors and cross products of vectors have to transform the same way under proper and improper rotations. It is pretty easy to show that the formalism is consistent with this conclusion. Would it make any sense to have a formalism where the angular momentum of a spinning sphere depended on whether you chose to use a right or left handed coordinate system?

One way to see that this must be true is to consider the way coordinate axes are chosen. The x and y axes are chosen to lie in a plane. The z-axis is perpendicular to this plane with an orientation conventionally chosen by using the right hand rule. According to most people, our set of basis vectors consists of two "polar" vectors and one "axial" vector. This is clearly nonsense unless the transformation properties of vectors and their cross products are identical. It is only under "active" rotations (proper and improper) of the physical system that cross products of vectors and the vectors themselves have different properties.

The claim that "polar" vectors have a natural direction is bogus. The electric field is a polar vector whose direction is defined to be away from positive charges. Even the velocity vector, which seems to have "natural" direction, is actually a definition. The velocity vector is tangent to the trajectory, but we choose it to point in the direction of motion rather than in the opposite direction. We define the direction of "polar" vectors according to our prejudices and consider it to be "natural." We define the direction of "axial" vectors by a right hand rule. Both of these choices are made by convention.