# Time-ordered product

I am studying about QFT and I'm trying to work with a time ordered product of three fields. In particular I have, $$$$T\{\phi(x)\Psi(y)\bar{\Psi}(z)\}$$$$ Where $$T$$ is the time ordered product. I'm writting this as follows $$$$T\{\phi(x)\Psi(y)\bar{\Psi}(z)\}=\theta(x_0-y_0)\theta(y_0-z_0)\phi(x)\Psi(y)\bar{\Psi}(z)-\theta(x_0-z_0)\theta(z_0-y_0)\phi(x)\bar\Psi(z){\Psi}(y)+...$$$$ Where the minus sign is due by the exchange of two fermionic fields and $$\theta$$ is the Heaviside step function. But is this expression correct? I don't know how to work with time ordered product with three operators so I supposed that we will have two step functions in each term but I'm not sure about it.

The two terms you wrote are correct. Assuming the "$$+...$$" you wrote means "no minus sign whenever $$\Psi(y)$$ precedes $$\bar\Psi(z)$$ and include a minus sign whenever $$\bar\Psi(z)$$ precedes $$\Psi(y)$$", then your expression is right.
One way to justify this is Lorentz invariance. If $$y$$ and $$z$$ are space-like separated, there is no Lorentz invariant way to order them. In a frame where $$y^0=z^0$$, the equal time anti-commutation relations require that there is a minus sign if you switch the order of $$\Psi(y)$$ and $$\bar{\Psi}(z)$$. For Lorentz invariance, you need this minus sign to appear in any boosted frame.