# What is the content of an occupied QFT fermionic state?

A simple non-interacting quantum field is constructed by analogy to a harmonic oscillator, with $$\hat{x}$$ & $$\hat{p}$$ replaced by operator-valued distributions $$\hat{\phi}(r)$$ & $$\hat{\pi}(r)$$, and with a separate oscillator for each mode $$k$$.

### Question:

What is the mathematical content of an occupied state for a simple, non-interacting, fermionic quantum field $$\hat\psi$$? (e.g. in position space, the non-relativistic wavefunction assigns $$\psi:\mathbb{R}^d\to\mathbb{C}$$, normalized $$\int_V \psi^*\psi\,dV = 1$$.)

Does the state assign, to every $$r$$, the infinite direct sum (over all occupation numbers N) of alternating tensors of N bispinors (neglecting entangled states)?

• If so, does the state assign, to every $$r$$, a separate value in the space of that direct sum for every $$k$$? If not, how are the independent oscillators handled?

(Also, as the Grassmann algebra is the exterior algebra of a complex vector space, does the state equivalently assign a Grassmann element?)

## Related question:

Since the QHO Hamiltonian eigenstates expressed in the eigenbasis of $$\hat{x}$$ are an infinite linear combination centered around $$x=0$$, does this mean a field's occupied state has no determinate $$\phi$$, but could be written as a wavefunction-like probability amplitude distribution over $$\phi$$ values? That is, an infinite linear combination of $$\hat{\phi}$$ eigenvectors centered around $$\phi=0$$? (answer: yes, but the QHO's spectrum only applies to the free bosonic field.)

• If so, would each $$k$$ have a separate set of eigenvectors here? (answer: no, not different field eigenstates, just occupied states.)
• You are confusing path integrals (where Grassmann numbers appear) with canonical quantization. The state of a vector field is just a bunch of harmonic oscillators, labeled by an additional vector index. Similarly, the state of a spinor field has an additional spinor index, and that's all. – knzhou Dec 17 '18 at 15:37
• The second question seems completely independent, and probably should get its own question. – knzhou Dec 17 '18 at 15:37
• Consider defining what you mean by "mathematical content". A quick comment on the last paragraph. Eigenstates are not infinite linear combinations centered around $x=0$, such statement is very confusing, eigenfunctions for the QHO are Hermite polynomials, now a generic wavefunction CAN indeed be an arbitrary linear combination. A definition for a "field's occupied state" would also be much appreciated as to make the last question much clearer. Consider having a look at Fock Space – ohneVal Dec 17 '18 at 15:37
• @knzhou There is actually a non-field theory version of a single-mode "fermionic harmonic oscillator" using a Grassman variable. It is just little used, because it does not have the broad applicability of the bosonuc SHO. I will try to elaborate more in an answer later. – Buzz Dec 17 '18 at 20:13
• @knzhou I didn't confuse path integrals, but I forgot the Grassmann product is limited by the underlying space's dimension, and an algebra wouldn't apply to an isolated $r$. I reworded it. – alexchandel Dec 17 '18 at 21:44