# QFT Matter Fields and Anti-Matter Fields

In QFT is it the case that the electron matter field and anti-electron matter field (using the electron as a specific example) are truly distinct physical fields versus different excitation modes of a single electron matter field?

• What is your definition of "distinct physical field" w.r.t. a quantum field? – ACuriousMind Feb 11 '19 at 15:43
• I guess in the case of matter fields, that it is an independent spinor field with its own set of field values extended through spacetime. In the cases of the electron and anti-electron that there are two separate fields required to model reality. I previously thought that there was a single electron matter field and the electron and anti-electron were just conjugate (loose terminology) modes of excitation, but recent reading seemed to indicate otherwise. If separate fields, it seems odd to two totally separate fields so similar in properties and structure. – CSnowden Feb 11 '19 at 20:40

$$\Psi (x)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2E_{p}}}}\sum _{s}\left(a_{\mathbf {p} }^{s}u^{s}(p)e^{-ip\cdot x}+b_{\mathbf {p} }^{s\dagger }v^{s}(p)e^{ip\cdot x}\right).$$
If we act on the vaccum with this, what we get is essentially an electron in a position eigenstate which has spin $$s$$. Similarly for the conjugate field $$\bar{\Psi}$$ but now we have a positron.
So are $$\Psi$$ and $$\bar{\Psi}$$ "independent"? The only way I see fitting to answer this is by simply just looking at the anti commutation relations. For the same field and its conjugate at two different spacetime points, they are are given by $$\{\Psi(x), \bar{\Psi}(y)\}=0$$. So these fields anti commute. Would you call this "independent"? I'm not sure, as it seems by the reasoning above that they at first "appear independent but satisfy a constraint", if you will.