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I could not locate a definition of Fermionic quantum field (like for an electron!) in even Kevin Costello's book, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.4961&rep=rep1&type=pdf

Can anyone help define this thing? And any such definition also has to make sense when the spacetime is a super-manifold.

(Beyond how physics books define it as a plane-wave expansion with anti-commuting operator coefficients - there its very unclear as to how the spinorial structure is getting encoded or what exactly is the algebraic structure of this operator space from which these operators are coming - and how they can be defined to act on the Hilbert space of the QFT when this Hilbert space is defined in the "right way" as : the dual of an implicit vector space whose basis is assumed to be in bijective correspondence with the space of all possible classical configurations/values of all the classical fields - which need not all be Fermionic - in the underlying classical Lagrangian)

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  • $\begingroup$ you want Dirac fermions or Majorana fermions? $\endgroup$ Jun 6, 2018 at 6:19
  • $\begingroup$ Anything to start off with! $\endgroup$ Jun 6, 2018 at 7:07

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Unlike boson fields, the fermion fields do not posses a classical limit. This complicates their construction in a spacetime manifold. The approach developed by DeWitt is to introduce a "superclassical" limit, in which the fermion fields become anticommuting Grassman variables (by taking a Majorana representation the fields can be assumed to be real). This is explained on page 230 of DeWitt's book on Supermanifolds, or more extensively in Chapter 1 of Effective Action Approach to Quantum Field Theory.

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  • $\begingroup$ Thanks for the references. But I still not sure this explains exactly what I am looking for. I guess you are referring to equations 1.8-1.11 in the Avrimidi lectures. Here what they define using this "superclassical fields" are not quantum fields as far as I can see. The so-called "infinite dimensional Grassman algebra" (not clearly defined!) in the RHS of equation 1.11 does not seem to be acting on the Hilbert space of quantum states as it should have if the fields he is talking of were Quantum Fields. What am I missing? $\endgroup$ Jun 6, 2018 at 22:01
  • $\begingroup$ Also I am a bit confused about how to interprete this fact about the anti-commutator of two quantum Fermion fields being non-zero. If that is true then how do we explain equations like ``$Tr[\psi^{even}=0]$" for gauge charged quantum Fermion fields like say whats happening in the footnotes on page 30 of this paper, arxiv.org/pdf/1104.0680.pdf. $\endgroup$ Jun 6, 2018 at 22:22
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One can define a Fermionic quantum field $\psi$ by postulating its algebra of anti-commutation relations $$ \{ \psi(f), \psi^\dagger(g) \} = (f,g), \qquad \{ \psi(f), \psi(g) \} = 0 $$ where $f,g \in \cal H$ are so called test-functions from a (one particle) Hilbert space. (Similarly, but less cleanly, one can choose a singular basis of Dirac delta distributions $f = \delta_x$, $g = \delta_y$ and obtain $\{ \psi(x), \psi^\dagger(y) \} = \delta(x-y)$.)

Then, one looks for representations of that algebra of which the Fock space is one but not the only one of the possibilities. In the common Fock space representation $\psi$ is realized as an operator valued field of a Dirac (bi)spinorial type. Its properties, like spin, can be analyzed within the representation theory assuming Poincare invariance.

(There is no need for using plane-wave expansion, as you find in old physics books, as that approach is very limited.)

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