# Imaginary Hamiltonian

• I
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!