I'm currently reading Altland and Simon's Field Theory, and while trying to solve the partition function of the harmonic oscillator I ended up with a question.
Using a Hamiltonian of the form $H=\hbar wa^\dagger a$. Using the formalism of the field integral with chemical potential to 0 and bosonic Matsubara frequencies, I arrived at $$ Z=N\frac{1}{\sinh(\beta\hbar w/2)} $$
Now this constant, $N$, from the derivation of the whole formalism should only have dependencies on temperature from what I understood, $\beta$ and not on the system parameters like $w$. $N\equiv N(\beta)$.
However if I used the standard way of just performing the sum $$ Z=\sum_{n=0}^{\infty}e^{-\beta(\hbar wn)}=\frac{e^{\beta\hbar w/2}}{2\sinh(\beta\hbar w/2)} $$
Even if simillar we see that this constant brings a $w$ dependence, $N\equiv N(\beta,w)$. I'm then confused if the constant $N$ we get from the field integral formalism can bring different physics or do we just ignore it since a partition function up to a multiplicative constant dependent on $\beta$ represents the same physical state.
