Blame Franklin --

https://xkcd.com/567/The question then would be, why are poselectrons left-handed and not right?

You're welcome.

The reason there's any handedness, is simply the order we assign to the vectors, and the commutativity of the cross product between them. Namely, A x B = -B x A.

As it happens, it's a sign change, but it's worth emphasizing that vectors and matrices (in general, linear algebra; and for fields, its extension, vector calculus) need not have the commutative multiplication we take for granted with scalars.

Which, may seem trivial in grade school or high school algebra classes -- of course numbers are commutative, we've been doing it that way our entire lives, and of course there's no way that 6 x 9 != 9 x 6 = 54. Well, they never get around to providing examples to the contrary, but it turns out there was a reason after all that they made note of these things, because some number systems -- algebras, rather -- indeed do not exhibit commutativity, associativity and such.

But anyway, I don't think the present example is chirality as such, so much as that's the convention, and that we must have

*some* consistent convention to do work in this space. But on that note. . .

That "handedness" is called chirality, it basically means that you can't transform item A into item B (for example: left hand rule into right hand rule) by rotation and translation alone, you have to mirror it (by changing the sign of one vector: let's say 0|0|1 is your perpendicular vector AxB, 0|0|-1 would be its mirror image).

One example of chirality in nature would be enantiomers in chemistry, the interaction of a lot of chiral molecules with your body depends on their handedness (an infamous example would be Thalidomid, one enantiomer was perfectly save while the other had devastating effects).

Indeed, chemistry, and on even smaller scales, quantum mechanics itself, exhibits chirality, in a very fundamental way. In QM, the same vector equations, and linear algebra, that gives us questions like this, also yields a fundamental degree of freedom (spin), and deeper still (QED, Standard Model, etc.), chirality and handedness of fundamental particles. Or going back to EM, there's circular polarization of waves, for instance.

At its most basic, this simply arises because there are certain ways to move through 3-dimensional space (and 4D spacetime when applicable), and every kind of motion, every symmetry, begets a conservation law in our system of equations.

Tim