One day while standing at a mirror, something about it looks funny to you. When you reach up to touch it, instead of feeling glass, your hand passes through! An adventurous spirit, you decide to take the plunge and step your whole body into this mirror universe. You can easily distinguish this world from your own–all the letters are reversed, making reading difficult. Clocks run counterclockwise and the sun rises in the west. But you wonder–these are all distinguishable because we humans have created them or labeled them. Is there anything fundamental in the universe that cares if it’s in the mirror or not?
My first exposure to the idea of parity was through math class, with even and odd functions. To test a function f(x), you replace all instances of x with -x and see what you get. When f(-x)=f(x), it’s even; when f(-x)=-f(x), it’s odd, and if it’s neither, it’s neither (but can be written in terms of an odd function plus an even function). Think of the y-axis as a mirror that faces to the right. An even function is one whose reflection in the mirror is the same as its original form (odd functions require another reflection through the x-axis to match).
In physics, parity is a transformation you can apply to vectors. When you think of the vector definition of “magnitude and direction”, parity changes the direction but not the magnitude. In that way, it’s similar to a rotation. For both parity and rotations, you can think of the transformation as a matrix that operates (through matrix multiplication) on the vector (using the “collection of numbers” definition). Here’s an example:
Say I start with the position vector that is 5m long at 30 degrees above the x-axis, which is the same as (4.33, 2.50) m.
If I apply the parity transformation, I’m reflecting the vector through the y-axis. In that case, the result is the x coordinate becomes negative while the y coordinate is unchanged, resulting in (-4.33, 2.50) m.
A matrix that would provide the same effect is
What if, instead of applying parity, I had rotated the vector by 120 degrees counterclockwise? I should end up with the same result, but using a different matrix. The rotation matrix is
Rotating the position vector gives (-4.33, 2.50) m, as we expected. So is there a difference between rotating and doing a parity transformation? Yes–the determinant of a parity matrix is always -1 and of a rotation matrix is always +1. Checking our examples shows for parity Det(P)=(-1*1)-(0*0)=-1 and for rotation Det(R)=(cos(120)*cos(120) – -sin(120)*sin(120))=cos^2(120)+sin^2(120)=1.
In three dimensions, you can either make one coordinate the opposite its previous value (as we did above) OR you can change all three coordinates. Both are legitimate because both have a determinant of -1. If you change all the coordinates in 2D, it has a positive determinant and so isn’t a parity transformation. But in 3D, it works and is called “space inversion”. You turn a right handed coordinate system into a left handed coordinate system.
Doing this sort of parity transformation on a position vector has the effect of r –> –r, which can be classified as “odd”*. Several vectors behave this way, like the momentum vector p = mv. Velocity depends on position, v= dr/dt. Since r –> –r but time doesn’t change, v –> -dr/dt = –v, and p –> -mv = –p (mass also doesn’t change under parity).
Not all vectors, however, end up as the negative of the original vector when you switch all of the coordinates. The angular momentum L is a combination of r and p: L=rxp. Both r and p become negative under parity, so those negative signs cancel and L remains unchanged. For a physical example, think about a clock on a wall. If you curl your right hand’s fingers in the direction the hands move, your thumb points into the wall. If you looked at this clock in a mirror, it would appear to be moving counterclockwise. Now you use your left hand (since it’s a left handed coordinate system) and curl your fingers; again your thumb points into the wall. Since your thumb points along L, you can see it hasn’t changed.
Vectors that change in the way you’d expect–by changing sign, as r does–are called, creatively, “vectors”. Vectors that don’t behave as we’d expect–by not changing sign, as L doesn’t–are called either “pseudovectors” or “axial vectors”.
It turns out elementary particles also have an associated, intrinsic parity, just as they have intrinsic charge and mass and lepton and baryon numbers. We expect these intrinsic properties to be conserved in interactions; that is, we don’t ever create new electric charge or new mass (it’s accounted for by converting into or from energy). However, it turns out that while the total parity of a system isn’t changed by interactions due to gravity, electromagnetism, or the strong force, it is changed by the weak force.
The weak force breaks parity conservation, and that’s the topic of my next post.
*This is the genesis for the blog name: if you consider a person to be a collection of position vectors then the parity transformation is odd.