When is it useful to distinguish between vectors and pseudovectors in experimental & theoretical physics?

Question

My understanding of pseudovectors vs vectors is pretty basic. Both transform in the same way under a rotation, but differently upon reflection. I might even be able to summarize that using an equation, but that's about it.

Similarly, I can follow arguments that pseudovectors behave differently in "mirrors" than vectors. But my response to this is always: Okay, so what? When would I ever "do physics" in a mirror?

The usefulness eludes me. I'd like to gain a better understanding of the importance of this difference.

• When is it useful for an experimental physicist to distinguish between the two?
• When is it useful for a theoretical physicist to distinguish between the two?

I believe symmetry is important to at least one of these, but would appreciate a practical rather than abstract argument of when one has to be careful about the distinction.

Answer 1

[Disclaimer: I'm not providing an argument where the distinction would be useful. I am providing an argument that pseudovectors and vectors describe intrinsically different geometrical concepts, and should, for clarify of argument, never be conflated just because they look so similar]

The point is that pseudovectors, by their very nature, are not the same objects as vectors:

A vector, as commonly understood in physics, is an element of the vector space $\mathbb{R}^n$ spanned by the standard basis $e_i$. It points in a direction, and is geometrically connected to a line, i.e. a one-dimensional subspace of $\mathbb{R}^n$.

A pseudovector, as almost no one ever explicitly will tell you, is an element of the sub-top degree of the exterior algebra $\Lambda^{n-1}\mathbb{R}^n$, the space spanned by $e_{i_1} \wedge \dots \wedge e_{i_{n-1}}$. This does not directly point into a direction, but is geometrically the $n-1$-dimensional hyperplane spanned by the vectors $e_{i_1},\dots,e_{i_{n-1}}$, and can then be interpreted as pointing in the direction perpendicular to that hyperplane. Formally, this translation from hyperplanes into normal vectors is the Hodge dual mapping $\Lambda^k\mathbb{R}^n$ to $\Lambda^{n-k}\mathbb{R}^n$.

And there you see why pseudovectors are different from vectors under reflection, geometrically: In $\mathbb{R}^3$, i.e. our ordinary world, the planes are spanned by two vectors - if both change their signs, the pseudovector described by them will not (since the wedge $\wedge$ is linear and anticommutative).

One importance of these considerations is when you want to step from $\mathbb{R}^3$ to higher dimensions. You lose the cross product (which is really just the concatenation of the wedge and the Hodge), and your former pseudovectors are now suddenly no vectors in the ordinary sense at all anymore, since $\Lambda^2 \mathbb{R}^n$ (the "space of planes") does not map to unique normal vectors by the Hodge dual in dimensions that are not three. Now you need to genuinely tell your former pseudovectors and vectors apart, since they now have a different number of independent coordinate entries.

Answer 2

Not only can you do physics "in a mirror", but I've been part of an experiment involving exactly that.

The weak interaction is, well, weak. And that makes it very hard to get access to in any physical process which can also proceed through other interactions. So, you can see the weak interaction at work in beta decay, but to leading order you can't see it at work when a electron scatters off of a proton (because the signal from the electromagnetic interaction is about $10^5$ larger).

But there is a caveat.

You see the electromagnetic interaction respects parity at a conserved quantity, and the weak interaction does not. This is equivalent to saying that the electromenetic interaction is represented by a vector and the weak interaction by the sum of a vector and a pseudo vector (though for historical reasons we call it an "axial vector" which is a synonym). All of this means that if you set up a scattering interaction in which the outcome is different when parity is respected and when it is violated then all of the parity violation that you observe can be attributed to the weak interaction.

Enter $G^0$ which measured the form-factors of the proton as seen by the weak interaction (and which I was a part of) and Q-weak which is a fundamental test of the weak interaction.

Answer 3

Other answers being good, i'll try to give a different perspective.

What is a vector? As Feynman used to say ("Feynman lectures on physics") not every bunch of numbers (i.e $\left( a_1, a_2,..,a_n\right)$) makes a vector simply because it has $n$ components. Why? Because vectors have a specific relation (or more correctly transformational relation) with the underlying basis of the space they are part of. This makes a vector (or polar vector).

Obviously axial vectors (or pseudo-vectors) do not share this property of vectors (as other answers have also noted).

Why is this? What is the relation of a vector to an axial vector? And what is the physical representation of each. Well the physical representation is that vectors represent translational transformations whereas axial vectors represent rotational transformations. It is not correct that axial vectors do not represent direction, they represent the direction of rotation (i.e left-handedness vs right-handedness).

There you have it. This makes clear why the transformation properties of the two are different. Since a rotation co-relates the basis components of the space in a specific way (unlike a translation or scaling), when the basis components change (i.e coordinate transformation), the pseudo-vectors change in such a way to maintain or compensate the representation they have i.e the direction (and magnitude) of rotation (and not the direction and magnitude of translation as vectors do).

The meaning of the above in both theoretical and experimental contexts is the behavior of these entities wrt to transformations of the experimental apparatus and/or transformations of the underlying space they are part of.

Answer 4

Interesting difference for theoretical physics is that $n$-dimensional generalizations of quantities which are vectors have $n$ components, while $n$-dimensional generalizations of quantities which are pseudovectors, such as angular momentum, have $\frac{1}{2}n(n-1)$ components.

This coincides for 3 dimensions, which is why the same vector notation is usually being used for both, instead of using it only for vectors and using $n\times n$ antisymmetric matrices for quantities which are pseudovectors.

Answer 5

All terms in a sum or both sides of an equality should be of the same kind, either vector or pseudo vector. Otherwise the expression will break reflection symmetry. This is a useful check of formulas and possible physical explains of different phenomena.