Parity

Parity $\hat{P}$ is a transformation that inverts a spatial coordinates of a system. It is a discrete symmetry. It operates over a vector as

It is a specular reflection about the origin.

Parity is a unit operator: $\hat{P}^{2}=\hat{\mathbf{1}}$. Its eigenvalues are therefore $P=\pm1$. These eigenvalues are generally called even (+1) and odd (-1).

Quantities#

Quantities can be assigned a parity depending on whether the function that governs is even or odd. Here are a few important ones:

• Linear momentum is odd.
• Angular momentum is even.
• Spin is even.
• Helicity is odd.

As a rule of thumb, scalars are even (e.g. spin) and vectors are odd (e.g. linear momentum). Pseudovectors (e.g. angular momentum) are instead even, which is why they're called pseudovectors in the first place. Pseudoscalars (e.g. helicity) are odd.

In particle physics#

As in > Quantities, each fundamental interaction can assigned a parity by seeing if the laws that govern that force are even or odd:

• Electromagnetism is even (because Maxwell's equations are).
• Weak interaction actually violates parity symmetry and is therefore weird. See > Wu experiment below.

Parity can also be assigned to individual particles by seeing their wavefunction is even or odd¹. This function may be either even ($\psi(\mathbf{r})=\psi(-\mathbf{r})$) or odd ($\psi(\mathbf{r})=-\psi(-\mathbf{r})$). This is the intrinsic parity of the particle. The parity of a system of many particles is the product of all individual parities: $P_\text{tot}=\prod_{i} P_{i}$.

Wu experiment#

In the 1940s, physicists where wracking their head over a weird phenomenon. Two known particles, $\Theta$ and $\tau$, seemed to have the same mass and mean lifetime². This suggested the idea that they're actually the same thing, but they decayed in different ways, specifically ways that were incompatible with each other:

(detected in cloud chambers). These were weak force decays, but the issue is that they end parity of these was different. The $\Theta^{+}$ decayed into $P=(-1)^{2}=1$ and the $\tau^{+}$ decay into $P=(-1)^{3}=-1$. Alright, so different particles then?

Well, there's a huge hole here, and it's called CPT theorem. This theorem, already proven at the time, showed that this sharing of mass and mean lifetime should not be possible. So, same particle then? But that would mean weak decay violates parity conservation. This question was called the $\Theta$-$\tau$ puzzle.

The solution would come in 1956 thanks to Chinese American physicist Wú Jiànxióng. The experiment wanted to show that weak decay does, in fact, violate parity conservation. To do so, the parity of unstable cobalt-60 nuclei would be measured before and after a beta minus decay:

followed up by nickel-60's double gamma decay:

which gives two high-energy photons. Where the electron gets shot to is the key. Its direction of emission depends on the polarization state of the nucleus, but this is affected by the parity of the nucleus. If parity is conserved, the number of electrons "up" the polarization axis should be identical to the one "down" it. If it's violated, this does not need to be true. This difference (or lack thereof) is the point of the measurement.

The photons aren't strictly necessary, but measuring their distribution too helps to determine that cobalt is truly polarized (spatial distribution depends on the polarization) to rule out a cause for error. We know EM interaction conserves parity, so this is true regardless of the experiment's results.

The nuclei were polarized with a uniform magnetic field (temperature was also kept near zero to prevent random thermal breaking of alignment). In this environment, polarization goes like

where $\mu$ is permeability and $k_{B}$ is the Boltzmann constant. There's a couple complications:

1. This polarization must be kept up for the entire duration of the experiment, meaning for as long as decay measurements are taken.
2. Despite very low temperatures ($T\sim 0.01\text{ K}$), magnetic fields need to be intense ($B\sim 10\text{ T}$). This is because we're polarizing the nucleus! Protons are kept still by the strong force, which is harder to overcome than electromagnetism.

The solution to both of these is adiabatic demagnetization. The cobalt is covered in a thin layer of paramagnetic salt. An initial magnetic field is applied and then waned. While it wanes, the cobalt transfers heat to the salt, which favors the polarization by reducing temperature even further. A second magnetic field is then applied to do the actual polarization. This measurement is conducted with the magnetic field alternating directions to check for emission both ways and see any (if any) differences.

Detectors were located on both (almost) parallel and perpendicular to the polarization axis in order to measure emission distribution by counting the number of detections. Their counts revealed the truth: there was an anisotropy in the electron emission! Changing the polarization did change have an effect, which cannot be explained if parity is conserved. Thus, the weak force must violate parity conservation. As further confirmation, photons were correctly found to not care about polarization, as their distribution did not change.

This solves the $\Theta$-$\tau$ puzzle: $\Theta$ and $\tau$ are the same thing. Later, we'd name it $K^{+}$, the positive kaon.