Creation and annihilation from basic position/momentum operators I was reading these lecture notes and they show how if you start with two operators $p$ and $q$ such that $[q,p]=i$, you can define $a:=\frac{1}{\sqrt{2}}(q+ip)$ and $a^\dagger:=\frac{1}{\sqrt{2}}(q-ip)$, such that $[a,a^\dagger]=1$, and that this in turn implies that these are bosonic ladder operators. This left me with a few questions.

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*The creation/annihilation commutation relations are different for fermions and bosons. Does that mean that the moment/position commutation relations also differ? And if not, can we then construct bosonic creation/annihiliation ladder operators for all kinds of particles?


*Suppose I have a wavefunction $\vert \psi\rangle$ with position and momentum operators $Q$ and $P$ such that $Q\vert x\rangle=x\vert x\rangle$ for a wavefunction representing a particle precisely at position $x$, and similarly $P\vert p\rangle= p\vert p\rangle$ for a particle with momentum precisely $p$. Then I can define creation and annihilation operators $a=\frac{1}{\sqrt{2}}(Q+iP)$ and $a^\dagger = \frac{1}{\sqrt{2}}(Q - iP)$. These satisfy $[a,a^\dagger]=1$ (up to normalization), so they are ladder operators. Hence, it has a bunch of eigenstates of the form $\vert 0\rangle, \vert 1\rangle, \dots, \vert n\rangle,\dots$. What do these eigenstates represent?


*When I see descriptions of systems like (2), they always seem to use creation/annihilation operators indexed by position or momentum, like $a_p^\dagger$ and $a_p$. This would imply a set of operators $q_p:= \frac{1}{\sqrt{2}}(a_p^\dagger + a_p)$ and $p_p:=\frac{i}{\sqrt{2}}(a_p^\dagger - a_p)$. What do the position/momentum operators $q_p$ and $p_p$ represent?
 A: 
The creation/annihilation commutation relations are different for
fermions and bosons. Does that mean that the moment/position
commutation relations also differ?

No. The position and momentum operator have the usual commutation relations for fermions.
However, to create a spin-1/2 fermion, we don't just need to specify it's position (or momentum), but also whether it has spin component $\pm \hbar/2$ along the $z$ axis. This spin degree of freedom is an extra part of the creation/annhilation operator and leads to the anti-commutation relation. Note that since the vacuum has no spin, to create a fermion state we must apply a creation operator with an associated spin.
If you write down the anticommutation relations carefully, you should get something like
\begin{equation}
\{b_\sigma,b^\dagger_{\sigma'}\} = \delta_{\sigma,\sigma'}
\end{equation}
where $\sigma=\pm 1$ represents the orbital angular momentum quantum number of the fermion.

And if not, can we then construct bosonic creation/annihiliation ladder operators for all kinds of particles?

Mathematically you can construct whatever you want, but bosonic creation/annhilation operators will not help you represent fermionic degrees of freedom.

Suppose I have a wavefunction |⟩ with position and momentum operators  and  such that |⟩=|⟩ for a wavefunction representing a particle precisely at position , and similarly |⟩=|⟩ for a particle with momentum precisely . Then I can define creation and annihilation operators =12√(+) and †=12√(−). These satisfy [,†]=1 (up to normalization), so they are ladder operators. Hence, it has a bunch of eigenstates of the form |0⟩,|1⟩,…,|⟩,…. What do these eigenstates represent?

You can mathematically construct these states, but they don't have any relation to eigenstates of the Hamiltonian and do not carry an obvious physical meaning (at least not that I know of). In particular, none of these states would have spin 1/2.

When I see descriptions of systems like (2), they always seem to use creation/annihilation operators indexed by position or momentum, like † and . This would imply a set of operators :=12√(†+) and :=2√(†−). What do the position/momentum operators  and  represent?

In quantum field theory, we think of each Fourier mode of the field as a harmonic oscillator. So each Fourier mode as an associated creation and annhilation operator. The "position" and "momentum" of the Fourier mode are essentially the (complex-valued) amplitude of the Fourier mode and time derivative of the Fourier mode.
