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?)


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.