# Explicit quantization of free fermionic field

The canonical quantization of a scalar field $$\phi(x)$$ can explicitly be realized in the space of functionals in fields $$\phi(\vec x)$$ (here $$\vec x$$ is spacial variable) by operators $$\begin{eqnarray} \phi(\vec x,0)\to \mbox{ multiplication by } \phi(\vec x,0),\\ \pi(\vec x,0)\to (-i)\frac{\delta}{\delta \phi(\vec x)}. \end{eqnarray}$$ Thus, as usual, $$[\phi(\vec x,t),\pi(\vec y,t)]=i\delta^{(3)}(\vec x-\vec y)$$. (See e.g. $$\S$$ 9.1 in S. Weinberg's book "The QFT", particularly the discussion after formula (9.1.40).)

I am looking for an analogous realization of canonical quantization of a fermionic field. At the moment the case of a free fermionic field is enough for me. In addition I would be interested to know how the vacuum vector in this space looks like; more concretely I would like to know the explicit form of $$\epsilon$$-terms in (9.5.49) in Weinberg’s book.

The discussion of the fermionic case in Weinberg's book is not as detailed as in the scalar case.

• Peskin does the free fermionic field it is gory detail – InertialObserver Feb 5 '19 at 17:57
• @InertialObserver: In what paragraph? If you mean $\S$ 3.5, it does not answer my question. – MKO Feb 5 '19 at 18:01
• What exactly are you looking for? The way canonical quantization usually works is imposing the (anti)commutation relations on the creation/annihilation operators – InertialObserver Feb 5 '19 at 18:08
• Possible duplicates: physics.stackexchange.com/q/458370/2451 and links therein. – Qmechanic Feb 5 '19 at 18:35