# What is a formal definition of a Fermionic quantum field?

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)

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

• 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. Jun 6, 2018 at 22:22
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.