Parity

Definition

x0→x′0=x0
x^i \rightarrow x'^i = -x^i

Group theory

Parity is a group with just one element P with the property \( P^2 = \mathbb{I} \). This group is called \( \mathbb{Z}_2 \).

Parity of system of particles

The parity of a system of particles is given by the product of their intrinsic parities and \( \left( -1 \right) ^l \) , where \( l \) is the orbital angular momentum of the system

Classification of physical observables under Parity

Vectors

Vectors under rotation. Changes sign under parity.

Pseudo vectors

Vectors under rotation. Does not change sign under parity.

Scalars

Scalar under rotation. Does not change sign under parity.

Pseudo scalars

Scalar under rotations. Changes sign under Parity.

Representations

Two irreducible one dimensional representations

\( P = 1 \) and \( P = -1 \)

Representations in the Hilbert space of Quantum mechanics a.k.a intrinsic parity


Representations on the Quantum Hilbert space are projective representations of the original group as we allow an extra phase factor which has no physical significance in the said Hilbert space. Therefore, in this Hilbert space, we only demand (this must be true for all states because phase differences are experimentally measurable):
\[ P^2 = e^{i\theta} \mathbb{I} \]

If the theory has an \( U(1) \) symmetry

We can combine Parity and a \( U(1) \) rotation of \( e^{-i\frac{\theta}{2}} \) to define a new Parity operator
\[ P' = P e^{-i\frac{\theta}{2}} \]
The this new parity operator satisfies
\[ P'^2 = 1 \]
Eigenvalues of such an operator can only take values \( \pm 1 \). Therefore, for fields with an \( U(1) \) symmetry, the associated particle has an intrinsic parity \( \pm 1 \)

If the theory does not have an \( U(1) \) symmetry

Eigenvalues of Parity may be complex phases.
e.g. For Majorana fermions, which do not have an \( U(1) \) symmetry, the intrinsic parity can take values \( \pm i \)

Classical examples

  • momentum
  • 3-position
  • current density

Quantum examples

  • Boost generator

Classical examples

  • angular momentum

Quantum examples

  • spin

Classical examples

  • energy
  • time
  • temparature

Quantum examples

Classical examples

Quantum examples

  • helicity

Intrinsic parity of particles

Why is spin a pseudo vector?

Why is the boost generator a vector?

Why?

Let us express the space part of the wavefunction of a system of particle as \( R(r) Y^l_m (\theta, \phi) \left| r, \theta, \phi \right>. \)
\[ PR(r) Y^l_m (\theta, \phi) \left| r, \theta, \phi \right> = R(r) Y^l_m (\theta, \phi) \left| r, \pi - \theta, \pi + \phi \right> =R(r) Y^l_m (\pi - \theta, \phi - \pi) \left| r, \theta, \phi \right> = (-1)^l R(r) Y^l_m (\theta, \phi) \left| r, \theta, \phi \right> \]
The \( (-1)^l \) coming form the space part simply gets multiplied by the intrinsic parity of each particle to give us the eigenvalue of the overall parity.

Parity conservation

Parity even systems remain even under evolution, parity odd systems remain odd.

Electromagnetic interactions

Conserved!

Strong interactions

Conserved!

Weak interactions

Not conserved! :-O

The \( \theta \tau \) puzzle

  • In the mid-1950’s it was noticed that there were 2 charged particles that had (experimentally) consistent
    masses, lifetimes, and spin = 0, but very different weak decay modes:
    \[ \theta^+ \rightarrow \pi^+ \pi^0 \]
    \[ \tau^+ \rightarrow \pi^+ \pi^- \pi^+ \]
  • Assuming Parity is conserved in these reactions, one can infer that the intrinsic parity of \( \theta^+ = +1 \) and that of \(\tau^+ = -1\) .
  • Most physicists thought \(\tau \) and \(\theta \) were different particles and that Parity was conserved.
  • Young and Lee claimed that they were the same particle with the same intrinsic parity. They explained this by claiming that Parity might not be conserved in reactions involving weak interactions, like the ones above.

Wu experiment (confirmation of parity violation in weak decays)

The Wu experiment had Co-60 atoms nuclei decaying to Ni-60 via beta decay. A magnetic field was applied to align the spin of the Co-60 nuclei. In this state, the nuclei were in a Parity even state (the mirror image and the original image are the same). During the decay, it was found that the electron (beta particle) was emitted preferentially in the direction opposite to the spin of the nuclei. This situation is odd under Parity (in the mirror image, the electrons are emitted in the same direction as the spin). This transition from a Parity even state to a Parity odd state shows that Parity was violated in this interaction.

Fermions

Antiparticles have opposite intrinsic parity compared to their corresponding particles

Bosons

Antiparticles have the same intrinsic parity compared to their corresponding particles

Assignment of parity

We choose the intrinsic parity of the Proton, neutron, and the Electron to be \( +1 \). We can choose the intrinsic parities of three particles simultaneously and arbitrarily as the Standard Model has three \( U(1) \) symmetries: one corresponding to Baryon number, one corresponding to Lepton number and one corresponding to charge.

The parity of every other particle can be inferred from decays that involve electromagnetic and strong interactions (Parity conservation).