A fermion oscillator interacted with a boson oscillator!

It is known to all that the Hamiltonin H=p^2/m+x^2 can describe the boson and fermion particle, but how can embody the fermion properties when a fermion oscillator interacted with a boson oscillator? what is their interaction form?


Science Advisor
Sorry, I do not see how a fermion oscillator should emerge from the boson oscillator. Can you please be a bit more specific?
Thank you for your reply, I just want to build a theoretical model for a fermion interacted with a boson in dissipative system. But I don't know how to build it.


Science Advisor
Insights Author
You should know that the configuration space of the fermion oscillator is [itex]R^{2}_{a}[/itex]. That is, you need two real anticommuting dynamical variables to describe the fermion oscillator. The action for such oscillator is given by
[tex]S = (i/2)\int dt (X_{1}\dot{X}_{1} + X_{2}\dot{X}_{2} + X_{1}X_{2}-X_{2}X_{1})[/tex]
Form this, you find the Hamiltonian;
[tex]H = -iX_{1}X_{2}[/tex]


Science Advisor
That is all a bit vague and allows for a lot of approaches.

One of the simplest models is the Jaynes-Cummings model which is used for the interaction of a single bosonic photon mode (which is of course dissipative e.g. in the presence of a cavity) with a single fermionic two-level system (atom, quantum dot or whatever).

I do not know whether this is what you are looking for, but it might provide a starting point to find similar models.
Thank you for the reply. It's a good information and advice for me. I will take care of it. I still want to know what the exact form about the interaction of boson and fermion or how to build it?
I know that the Hamiltonin have the form:H=1/2(p^2+q^2)-i*x(1)*x(2) when taken into accout the spin of fermion. But I didn't have the ability to build the interaction team. Welcome to give me a advice. x(i) is the odd Grassmann number.


Science Advisor
Insights Author
Couple them by potential;
[tex]H=(1/2)\left(\dot{q}^{2}+[V'(q)]^{2}\right) -iV''(q)X_{1}X_{2}[/tex]
It's a valueable information. I appreciate it. Let me consider more complicated model, just like a boson interacted with its respective boson bath and a fermion interacted with fermion bath. Could I build the coupled team with potential?

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