The interpretation of Feynman Propagator for Complex scalar field

Question

For real scalar field, we know that the Feynman propagator is defined as $$\langle0|\mathcal{T}(\hat{\phi}(x_A)\hat{\phi}(x_B))|0\rangle=\theta(c\tau)\langle0|\hat{\phi}(x_A)\hat{\phi}(x_B)|0\rangle+\theta(-c\tau)\langle 0|\hat{\phi}(x_B)\hat{\phi}(x_A)|0\rangle$$ where $\tau:=t_A-t_B$. It indicates that a particle is created in a past time and will be annihilated at a future time.

However, I have some trouble to interpret the case for the complex scalar field. For the free complex scalar field, there are three kinds of Feynman propagator:

1.$\langle0|\mathcal{T}(\hat\varphi(x_A)\hat\varphi(x_B))|0\rangle$

2.$\langle0|\mathcal{T}(\hat\varphi^\dagger(x_A)\hat\varphi^\dagger(x_B))|0\rangle$

3.$\langle0|\mathcal{T}(\hat\varphi(x_A)\hat\varphi^\dagger(x_B))|0\rangle$

where this time $\hat\varphi(x_A)=\int d^3k\frac{1}{(2\pi^3)\omega/c}(\hat a(\vec{k})e^{-k\cdot x_A}+\hat b^\dagger(\vec{k})e^{+ik\cdot x_A})=\hat\varphi^{(+)}_a+\hat\varphi^{(-)}_{\bar a}$ which reprerents the annihilation of particle and creation of anti-particle operators respectively, and $\hat\varphi^\dagger(x_A)=\hat\varphi^{(-)}_a+\hat\varphi_{\bar a}^{(+)}$ is the composition of creation of particle and annihilation of antiparticle operators respectively.

Then, through my calculation, 1. can be interpreted as the vacuum cannot create an anti-particle and annihilated as a particle. 2. can be interpreted as creating a particle cannot have an annihilation of anti-particle.

However, I have some trouble to interpret the 3rd case. For 3., according to my calculation, the Feynman propagator is the same as the case for the real scalar field. Nevertheless, the $t_A>t_B$ case being described as creating a particle in the past and annihilate in the future, while $t_B>t_A$ case being described as creating an anti-particle in the past and annihilate in the future. Even though, we know that particle and anti-particle exist as a pair, the above interpretation has some problem that one can only choose $t_A>t_B$ or $t_B>t_A$. That means for some time ordering one can see particle being created and annihilated and for other time ordering, one can see anti-particle being created and annihilated. So, what is wrong with my interpretation? Could you point it out?

Answer 1

With $\hat \phi(x_A)$ a complex scalar field with mode decomposition as you wrote, the correlation functions 1. and 2. in your post vanish identically, as suggestive of the physical interpretation you gave. So, the only independent correlator one can then build is 3. which indeed coincides with the Feynman propagator for a real scalar field.

The two time ordering interpretations you describe are correct. Both orderings are encapsulated in a lorentz covariant manner through the explicit time ordering V.E.V and this is all that we ever work with in perturbation theory.