### Alex Nelson

Shared publicly -**Parity Puzzle.**

In 3+1 dimensions, parity ("reversing handedness symmetry", or spatial inversion) is given by the matrix diag(1,-1,-1,-1).

What is the corresponding symmetry in 2+1 dimensions?

The naive answer diag(1,-1,-1) doesn't seem right because it doesn't perform the desired symmetry (reversing handedness). We can see this because it's a proper Lorentz transformation.

The more sophisticated but still naive answer diag(1,-1,1) privileges one spatial coordinate over another. Which is bad, we want no privileges for any spatial coordinate.

(Hint: think of complex spatial coordinates.)

1

John Baez

+

1

2

1

2

1

Well, if you allow complex spatial coordinates you can do (1, i, i). But if we ignore time (which is not really relevant here, I think) and take R^n to be space, there really is a big difference between n even and n odd. For n odd there's a transformation with determinant -1 that commutes with all rotations, while for n even there's not. So only in the former case can we choose a "parity" map that doesn't break rotational symmetry!

This is one of many ways in which rotations in even dimensions differ from rotations in odd dimensions. For example, a rotation in odd dimensions must leave some nonzero vector fixed.

This is one of many ways in which rotations in even dimensions differ from rotations in odd dimensions. For example, a rotation in odd dimensions must leave some nonzero vector fixed.

Add a comment...