Path from correspondence between fermionic oscillator and bosonic oscillator to free bosons from interacting fermions
This question is open to edits and corrections. I'm available to make corrections as needed.
I am attempting to understand fermion boson correspondence and bosonization.
What is the correspondence between the spectrum of the fermionic oscillator and the bosonic oscillator? I think it has a definite answer if it exists of the form $E_f = A * E_b$, and perhaps a correspondence exists too between the states.
So $E_f = (n - 1/2)$
= { -1/2,1/2,3/2,5/2 . . . } but should one terminate at 1/2
and $E_b = (n + 1/2)$
= { 1/2,3/2,5/2 . . 346
. . .}
but it seems to be the case that n can only be 0 or 1 for $E_f$ but for $E_b$ n goes from 0 to $\infty$
Also, the "number states" can be found for both cases and compared.
On the level described here, it seems to me that there isn't much happening.
What about at the level of the ladder operators?
If a field can be written in the form $\psi^{\pm} \propto c^{\pm}$ and
a hamiltonian built from $\psi^{\pm}$
then perhaps bosonization can be done.
I feel like these ingredients are enough to put together coherent sentences about how to bosonize a theory. I just have not figured it yet.
I don't know much about the fermionic oscillator except what I think Pauli exclusion means $b^+b^+ | 0\rangle = 0$ and $b | 0 \rangle = 0$
How does one rewrite the fermionic operators as bosonic ones?
I've come across the terms fermion boson correspondence and bosonization. I'm not going to attempt to ask for references on the subject because I'm likely to not truly understand them. What struck me about these sorts of things is that it seems possible to turn a theory of interacting fermions into a theory of free bosons.
If the simplest interaction is then cooked up in 1d for two fermions, derived in some way from whatever scheme I am attempting to prepare here, how can the theory be written as free bosons?
I hope we can do this with basic ladder operators and no path integrals, please. 281