In quantum field theory for condensed matter physics, an observable $\hat{O}$ may be expressed as: $$\langle\hat{O}\rangle=\frac{\int D(\psi,\psi^{\dagger}) \hat{O}(\psi,\psi^{\dagger})e^{-S(\psi,\psi^{\dagger})}}{\int D(\psi,\psi^{\dagger}) e^{-S(\psi,\psi^{\dagger})}}.\tag{1}$$

Because of formula of (1), an arbitary factor can be added to the functional measure, $$D(\psi,\psi^{\dagger})\rightarrow aD(\psi,\psi^{\dagger}),\tag{2}$$ without changing the physical results.

However, I read in most textbooks that thermodynamical potential $$\Omega=-\beta^{-1}\ln{\int D(\psi,\psi^{\dagger}) e^{-S(\psi,\psi^{\dagger})}}\tag{3}$$

Since thermodynamical potential is a function of $T$ (or $\beta$), (2) would result a shift of entropy: $$S\rightarrow S-\frac{\partial \frac{a}{T}}{\partial T}$$

So I think (3) is not complete and should be modified as some form of (1). Yet I couldn't write it down. Could anyone help?


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