I have seen the Hamiltonian for a heat bath written as:
$$ H_B = \hbar \int_0^\infty \omega b(\omega)^\dagger b(\omega) d\omega $$
I was hoping to understand this equation better. This suggests that the heat bath is written as a sum of harmonic oscillators with raising/lowering operators $b^\dagger(\omega)$ and $b(\omega)$. I understand that is that this is just the continuous limit of the quantised electromagnetic field.
Please could someone explain the following:
1) Does the raising operator $b^\dagger(\omega)$ correspond directly to a mode in the heat bath with frequency $\omega$? ie does $b^\dagger(\omega)b(\omega)$ give us the number operator for the number of excitations of frequency $\omega$? or does the argument $\omega$ mean something else?
2) How does one alter this to change the temperature of the heat bath? My current interpretation is that this equation is for the vacuum/zero temperature case, so do we just add a Boltzmann factor inside the integral, to change the temperature or is there more subtlety required?
3) If you have any other intuitions/tips/tricks for thinking about this equation, I would appreciate them!