# Can I write a complex field, in some cases, as a real field?

I am learning quantum field theory.

Now I am considering this case:

Suppose a spin-0 particle which obeys the Klein-Gordon field equation and its anti-particle obeying the same equation do not have the same solution to the equation. This means that the field should be a complex scalar field. Hence the field operator should be :

$$\hat{\psi}=\int_{p} \text{e}^{-\text{i}px}\hat{a}_{p}+\text{e}^{\text{i}px}\hat{b}^{+}_{p}$$

$$\hat{a}$$ is annihilation operator for a particle and $$\hat{b}^{+}$$ the creation operator for the corresponding anti-particle.

If the physical process we're interested in, for example, is a low-energy process, where particle numbers are conserved and antiparticles are not involved, can I simply treat the field as a real scalar field? i.e:

$$\hat{\psi}=\int_{p} \text{e}^{-\text{i}px}\hat{a}_{p}+\text{e}^{\text{i}px}\hat{a}^{+}_{p}$$

• Why would you neglect anti-particles, they have the same mass, therefore same energy scale? Commented Aug 11, 2020 at 7:26
• For example, I would like to consider two-particle collision, in which the total energy is small. Thus no extra particles would be created? Commented Aug 11, 2020 at 7:37
• Do you mean 1 or 2 real fields? Commented Aug 11, 2020 at 8:01
• I mean can I just drop the part correspoding to antipaticle and rewrite the complex scalar field as a real scalar field. In my umderstanding, a complex scalar field is equivalent to 2 real scalar fields. Hence dropping the antiparticle part should work. Yet I am not sure. Commented Aug 11, 2020 at 12:26