# Parity in Physics

Something that always confused when I was first studying physics was the concept of parity, parity transformations, parity conservation… So I figured that a good way to properly understand the concept of parity and in a way to help others in doubt, is to talk about it in a informal way.

A good analogy for parity would be for you to imagine yourself in a mirror. Your mirror image looks exactly like you, except for the fact it’s inverted. Try raising your right hand, your mirror image raises the left, and so forth. You could that the mirror you is the you who suffered a transformation, where your front is now your back, and your left is your right, so in mathematics, something like

$\begin{pmatrix} x\\y\\z \end{pmatrix} \Rightarrow \begin{pmatrix} -x\\-y\\z \end{pmatrix}$

x and y represent respectively your left-right and front-back direction. Now that is almost like parity. If your mirror image is also upside-down, then it is a true parity transformation. In short, a parity transformation is essentially a transformation that flips the sign of your coordinates. Something like

$\vec{r}\overset{P}{\rightarrow}-\vec{r}$.

So yeah, it’s just that in essence. An inversion operation under all spatial coordinates. And yet it is very important in physics. Let’s say you have a physical quantity dependent on spacial coordinates, and let’s say that quantity suffers a parity transformation. Then if

$f(-\vec{r}) = f(\vec{r})$

the quantity is said to have even parity. Examples of classical quantities with this property are energy, mass, the electric potential, usually scalar quantities. Else if

$f(-\vec{r}) = -f(\vec{r})$

the quantity is said to have odd parity. Examples being those like the position (seems rather obvious), force, or the linear momentum.

Now, those things are not really what we care about in quantum mechanics and nuclear physics. But with this part, you hopefully understand what we mean by even or odd parity.

In nuclear physics specially, the concept of even or odd parity is important to describe gamma decay (or isomeric transition) and nuclear stability. The quantum state of each nucleon has either odd or even parity. I am not gonna prove this, but the parity operator can be represented by its eigenvalue as show below:

$P|\phi\rangle = (-1)^l|\phi\rangle$

That l is a quantum number equivalent to the angular momentum (but not necessarily being an actual angular momentum). So you can pretty much define the parity of a nucleus by the product of the individual nucleons. If both Z and N (number of neutrons) are even, then the parity of the nucleus is even. If (Z+N) is odd, then the parity is determined by the “valence nucleon”, the nucleon at the highest energy level, you could say. If (Z+N) is even, but Z and N respectively are odd, well, there is no way for you to know. The parity of the nucleus determines its stability, so you can see there is importance in this detail.

The parity is also important for isomeric transitions, or gamma decay. But I will come back later to edit it.