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When one rewrites the partition function of a grand-canonical ensemble (quantum version) as functional field integral

$$ Z = \operatorname{Tr}_{ \mathscr{F}} \mathrm{e}^{ - \beta \left( H - \mu N \right) } = \ldots = \int \mathscr{D}( \overline{\psi} , \psi) \, \mathrm{e}^{ - S[ \overline{\psi} , \psi]} $$

with action

$$ S[ \overline{\psi} , \psi] = \int_{0}^{ \beta } \mathrm{d} \tau \Big[ \overline{\psi} \cdot \partial_{ \tau } \psi + H( \overline{\psi} , \psi) - \mu N (\overline{\psi} , \psi) \Big] $$

then the action can be brought into a very nice form by using the Matsubara-technique: One writes e.g.

$$ \psi ( \tau) = \frac{1}{ \sqrt{\beta}} \sum_{ \omega_n } \psi_n \mathrm{e}^{ - \mathrm{i} \omega_n \tau } \qquad (\star) $$

and by referring to a concrete single-particle-basis (and assuming the Hamiltonian to be quadratic in the operators) this leads to

$$ S[ \overline{\psi} , \psi] = \sum_{i j n} \overline{\psi}_{ i n } \big[ (- \mathrm{i} \omega_n - \mu ) \delta_{i j} + h_{i j} \big] \psi_{j n} $$

I wanted to use this technique to calculate observables, but I seem to get unphysical results. I think the problem is equation $(\star)$, since this leads to an action with a physical dimension (which is mathematically not well defined).

My idea of solving / circumventing this problem is to change the normalization in the Fourier-transformation to

$$ \psi ( \tau) = \frac{1}{ \sqrt{M}} \sum_{ \omega_n } \psi_n \mathrm{e}^{ - \mathrm{i} \omega_n \tau } $$

(giving a unitary transformation again with trivial functional determinant).

Question: Is this the only way of solving / approaching this problem? It confuses me that it leads to such "ugly" equations which I have not seen in textbooks so far. (And one has to keep track of the factor $M$ which corresponds to the number of slices in the discretization in the construction of the field-integral etc... This seems a little complicated?)

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The thing that was bothering me was the dimension of the action. But agreeing that the fields carry a dimension makes the action dimensionless. My suggested normalization for the Fourier-transformation is possible, but not reasonable. In standard-notation (cf. e.g. Mahan-book) one has to use the convention in $(\star)$ to acquire the Fourier-coefficient for the imaginary-time-Green-function.

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