Time Reversal

Consider time-reversal invariance, $t\rightarrow t^\prime=-t$. This time we start with the Dirac equation in Hamiltonian form

$$i\frac{\partial\psi(\vec{x},t)}{\partial t} = H\psi = [\vec{... ...-i\vec{\nabla} - e\vec{A}) + \beta m + e\phi]\psi(\vec{x},t).$$ (5.253)

Define the transformation $\mathcal{T}$ such that $t^\prime = -t$ and $\psi^\prime(t^\prime) = \mathcal{T}\psi(t)$, we have

$$\frac{\partial}{\partial t} (\mathcal{T}i\mathcal{T}^{-1})\p... ...ime) = -\mathcal{T} H \mathcal{T}^{-1} \psi^\prime(t^\prime).$$ (5.254)

Since $\vec{A}$ is generated by currents which reverse sign when the sense of time is reversed,

$$\vec{A}^\prime(t^\prime) = -\vec{A}(t) ,$$ (5.255)

$$\phi^\prime(t^\prime) = +\phi(t) .$$ (5.256)

Also $\vec{\nabla}^\prime = +\vec{\nabla}$, since $\vec{x}^\prime = +\vec{x}$. The transformation must cause $i\rightarrow -i$ to get the correct form, therefore $\mathcal{T}$ can be defined as:

1. take complex conjugate,
2. multiply by $4\times 4$ constant matrix $T$.

$$\psi^\prime(t^\prime) = T\psi^*(t).$$ (5.257)

Therefore

$$i\frac{\partial\psi^\prime(t^\prime)}{\partial t^\prime} = [... ...^*T^{-1})m + e\phi^\prime(t^\prime) ] \psi^\prime(t^\prime) .$$ (5.258)

This implies $T$ must commute with $\alpha_2$ and $\beta$ and anticommute with $\alpha_1$ and $\alpha_3$. Therefore we can try

$$\fbox{$\displaystyle T = -i\alpha_1\alpha_3 = -i\alpha_1\gam... ... = i\gamma^0\alpha_1\gamma^0\alpha_3 = i\gamma^1\gamma^3 $}\ .$$ (5.259)

The phase factor is arbitrary.

We apply $\mathcal{T}$ to a plane-wave solution for a free particle of positive energy. Since

$$\mathcal{T}\not{\;\!\!\!p}= T\not{\;\!\!\!p}^* = i\gamma^1\g... ...amma^3p_\mu & \textrm{for}& \mu = 1,2,3 \end{array} \right. ,$$ (5.260)

and

$$\mathcal{T} \left( \frac{\not{p} + m}{2m} \right) \left( \frac{1 + \gamma_5\not{s}}{2} \right) \psi(\vec{x},t) = T \left( \frac{\not{p}^* + m}{2m} \right) T^{-1}T \left( \frac{1 + \gamma_5\not{s}^*}{2} \right) T^{-1} \psi^\prime(\vec{x},t^\prime)$$

$$= \left( \frac{\not{p}^\prime + m}{2m} \right) \left( \frac{1 + \gamma_5\not{s}^\prime}{2} \right) \psi^\prime(\vec{x},t^\prime) ,$$ (5.261)

where $p^\prime = (p_0,-\vec{p})$; $s^\prime = (s_0,-\vec{s})$ Therefore $\mathcal{T}$ projects out a free-particle solution with reversed direction of momentum $\vec{p}$ and spin $\vec{s}$. This is known as ``Wigner time reversal''.