How does the $\beta$ factor appear in eq. (4.33) of Altland, Simons "Condensed Matter Field Theory"?

Question

In Altland and Simons' "Condensed Matter Field Theory" book, on the partition function for the non-interacting gas section, is stated that, after diagonalizing the hamiltonian the action assumes the form $$ S = \sum_a \sum_{\omega_n} \overline \phi_{an} (-i \omega_n + \xi_a) \phi_{an} $$ where $\xi_a \equiv \epsilon_a - \mu$ and $\epsilon_a$ are the single-particle energies. However, immediately after that, a beta factor shows up when computing the Path Integral: $$ \mathcal Z_a = \int D(\overline\phi_a, \phi_a) e^{-\sum_n \overline \phi_{an} (-i \omega_n + \xi_a) \phi_{an}} = \prod_n [\beta(-i \omega_n + \xi_a)]^{-\zeta} $$ where $\zeta =1 (−1)$ for bosonic (fermionic) fields, respectively.

Where is that $\beta$ factor coming from?

Answer 1

Factors of $\beta$ can be somewhat slippery in the Matsubara formualism. The change of variables from $\phi(\tau)$ to $\phi_n$ produces an extra factor of $\beta$ for each $n$ in the Jacobian. See footnote 9 on the preceding page (168).