Proving Imaginary Hamiltonian is P-Odd, T-Even & Imaginary

[Malamala]

Hello! The Hamiltonian for nuclear spin independent parity violation in atoms is given by: $$H_{PV} = Q_w\frac{G_F}{\sqrt{8}}\gamma_5\rho(r)$$ Here $Q_w$ is the weak charge of the nucleus (which is a scalar), $G_F$ is the Fermi constant and $\rho(r)$ is the nuclear density. From the papers I read, this operator is P-odd, T-even, and purely imaginary. I am having a hard time proving this. I managed to prove that it is P-odd. $\rho(r)$ is P-even, as it depends on the magnitude of $r$, so all I need to prove is that $\gamma_5$ is P-odd. For spinors, the parity operator is $\gamma_0$, and it can be easily shown that $\gamma_0\gamma_5\gamma_0^\dagger = -\gamma_5$. Hence the above Hamiltonian is P-odd. I am not sure how to prove it is T-even and imaginary. The first issue is that I am not sure what is the spinor space operator for the time inversion. Can someone help me with this? Thank you!

 

[eljose79]

Hello! Thank you for your question. I am a scientist and I can help you with your query.

To prove that the Hamiltonian is T-even, we need to show that it remains unchanged under time inversion. In general, the time inversion operator is represented by $\mathcal{T}$ and it is defined as $\mathcal{T} = i\gamma_0\mathcal{K}$, where $\mathcal{K}$ is the complex conjugation operator. For spinors, the time inversion operator acts on the spinor space as $\mathcal{T}\psi(t) = \psi(-t)$.

Now, to prove the T-even property, we need to show that $\mathcal{T}H_{PV}\mathcal{T}^{-1} = H_{PV}$. Let's start by applying $\mathcal{T}$ on the Hamiltonian:

$$\begin{align} \mathcal{T}H_{PV}\mathcal{T}^{-1} &= \mathcal{T}(Q_w\frac{G_F}{\sqrt{8}}\gamma_5\rho(r))\mathcal{T}^{-1} \\ &= Q_w\frac{G_F}{\sqrt{8}}\mathcal{T}(\gamma_5)\mathcal{T}(\rho(r))\mathcal{T}^{-1} \\ &= Q_w\frac{G_F}{\sqrt{8}}(-i\gamma_0\mathcal{K}\gamma_5i\gamma_0\mathcal{K})\mathcal{T}(\rho(r))\mathcal{T}^{-1} \\ &= Q_w\frac{G_F}{\sqrt{8}}(\gamma_5)\mathcal{T}(\rho(r))\mathcal{T}^{-1} \\ &= Q_w\frac{G_F}{\sqrt{8}}(\gamma_5)\mathcal{T}(\rho(r)\mathcal{T}^{-1}) \\ &= Q_w\frac{G_F}{\sqrt{8}}(\gamma_5)\rho(-t) \\ &= Q_w\frac{G_F}{\sqrt{8}}(\gamma_5)\rho(r) \\ &= H_{PV} \end{align}$$

Hence, we have shown that the Hamiltonian is T-even.

To prove that the Hamiltonian is purely imaginary,