A Complex Field Meaning

Question

Here are some conceptual questions I have been thinking about but I either am not sure about the answer or do not know if my thoughts are correct. Some might seem silly, but I ask nevertheless. Any feedback, answers, and expansions will be appreciated.

Question 1: Is there more than one "type" of Dirac Field? Loosely, speaking a field creates and destroys particles of a particular type, so it seems to me that a particle field $\phi$ is characterized by the particles it creates and destroys and by what equation $\phi$ satisfies. The Dirac equation is for $Spin-\frac{1}{2}$ fermions. With this logic the quantum field that creates electrons/positrons is different from the field that creates tau/anti-tau? Like, is it safe to label the fields as $\phi_{electron}$ and $\phi_\tau$ and claim that they are different fields? Or is there one quantum field that creates all Spin-1/2 particles?

Question 2: How can we conceptually describe a complex field? A complex field $\Phi$ can be defined as $\Phi = \phi_1 + i\phi_2$. Ideally, $\phi_1$ and $\phi_2$ should be real fields but they do not HAVE to be. How can we conceptually describe a field like $\Phi = \phi_{electron} + i\phi_{\tau}$ and $\Phi = \phi_{electron} + i\phi_{electron}$? Are fields like this just theoretical constructions, or can they be realized in experiments? If they can, how?

Question 3: Are all quantum fields complex or are some purely real fields? What are some examples of purely real fields?

I hope these are not silly questions. My main goal is to find practical and tangible ways to describe these physical objects so that I actually understand physics as physics and not just pure/abstract mathematics.

Answer 1

1. There is a different field for every type of particle. There is a $\psi_\tau$, $\psi_e$, $\psi_{\nu_e}$, etc.

2. You describe a complex field just like you describe a complex number. You can treat it as a complex number $z$, or write it as $z=x+iy$ or as $z=re^{i\theta}$ with $x,y,r,\theta \in {\mathbb R}$. Whatever you find convenient!

3. Real fields are neutral w.r.t. all forces. E.g. the photon is completely uncharged w.r.t. electromagnetic, weak and strong force so it is real. All others are complex.

For the electron field you can write $\Psi_e = \phi_e + i \psi_e$ where $\Psi_e \in {\mathbb C}$ and $\phi_e , \psi_e \in {\mathbb R}$. $\Psi_e$ creates electrons and destroys positrons and $\Psi_e^*$ creates positrons and destroys electrons.

To answer your other question about whether we can write $\phi_e + i \psi_\tau$, we have to go a bit deeper. In QFT, we like to describe fields as objects which have some "nice" properties. For instance, the fields $\Phi_e, \phi_e , \psi_e$ all have the same mass $m_e$ which is to say, they all satisfy the same differential equation $( \Box + m_e^2 ) \Phi_e = 0$ (and similarly $( \Box + m_e^2 ) \phi_e = 0$, etc.). We can therefore combine these 3 fields in whichever way we want.

On the other hand, $\Phi_\tau$, $\phi_\tau$ and $\psi_\tau$ satisfy a different equation, namely $(\Box + m_\tau^2 ) \Phi_\tau = 0$, etc. and we can also combine these three fields however we want.

However, since $\Phi_e$ and $\Phi_\tau$ satisfy different equations ($m_e \neq m_\tau$), any combination of the two will not satisfy a "nice" equation so we do not do so. This is not to say that we are forbidden to do it - we could obviously add and subtract whatever the hell we want, but not all such combinations are useful to us to understand physics.

The only reason one might want to combine fields in this way is if the physical situation requires us to do so. For instance, if our system is in a superposition of the electron and tau state, then combining $\Phi_e$ and $\Phi_\tau$ in some way to describe such a state makes sense.