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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.)
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    $\begingroup$ 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. $\endgroup$ – knzhou Dec 17 '18 at 15:37
  • $\begingroup$ The second question seems completely independent, and probably should get its own question. $\endgroup$ – knzhou Dec 17 '18 at 15:37
  • $\begingroup$ 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 $\endgroup$ – ohneVal Dec 17 '18 at 15:37
  • $\begingroup$ @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. $\endgroup$ – Buzz Dec 17 '18 at 20:13
  • $\begingroup$ @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. $\endgroup$ – alexchandel Dec 17 '18 at 21:44
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I think you are mixing up several distinct notions in your question. The free bosonic field is an extension of the harmonic oscillator in ordinary QM. You can of course represent states of the harmonic oscillator by a normalized wavefunction, but you can also expand that wavefunction in a basis of energy eigenstates which are labeled by non-negative integers. So the states of the harmonic oscilator can be expressed as a superposition of integers. In QFT this is extended to the Fock states with a distinct integer for each independent momentum.

Similarly the free fermionic field is an extension of a single two level system (a qubit) in ordinary QM. There is nothing mysterious about the states of this system. They are just two complex numbers specifying the superposition of 'occupied' and 'unoccupied.' In QFT this is extended to the Fock states with a distinct two level system for each independent momentum (and spin).

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