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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.

MKO
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2 Answers2

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I would suggest that you have a look at grassmann algebra of fermionic fields. I think it will help. Some references include: Quantization of Gauge theories by Hennnaux, Quantum Field Theory Volume 3 by Weinberg, Superfield formalism of supersymmetry (this can be found in most supersymmetry books like Wess Bagger, Buchbinder, etc). I personally haven't seen canonical quantization in these references, but I think these fermionic operators in superfield notation might help create the analogy.

Sreejith
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    These recommendations are far more technical than what the question required, besides (apart from the first) they talk about supersymmetry, which isn't relevant. Moreover, Henneaux's book is about canonical quantization (mostly), so I'm confused by your last claim. – MannyC Feb 08 '19 at 01:03
  • The books are definitely more technical than your question. I think u should stress on Grassmannian (anticommuting coordinates) property of the coordinates used to represent fermionic operator and not on supersymmetry. The fermionic generators of supersymmetry for example. – Sreejith Feb 11 '19 at 07:00
  • I still don't see how this is relevant. In supersymmetry the commutation relation for the supercharge is completely different from the canonical commutation relation. Btw, it's not my question. – MannyC Feb 11 '19 at 13:50
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Quantization for fermionic field works exactly like bosonic field.

You just replace regular functional derivative $\frac{\delta}{\delta \phi(x)}$ as in $$\pi(x)\to (-i)\frac{\delta}{\delta \phi( x)}$$ with Grassmann-odd functional derivative $\frac{\delta}{\delta \psi( x)}$ as in $$\pi(x)\to (-i)\frac{\delta}{\delta \psi( x)}$$

That's all you need. Of course, there are some minor technicality you have to take care of. For example, Dirac field have four components rather than only one component for scalar field. You have to apply the same quantization procedure to each individual component.

MadMax
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