Thermal Field Theory Integration over the modes?

[Phymath]

The question is more of a mathematical question then one about physics in the attached file between equations 6 and 7 it says "integration over the modes" i don't know how they go from the integration measure

$$\int D\phi \rightarrow \int D\phi_1 D\phi_2...D\phi_N$$

any advice would be helpful, thanks in advance!

 

[vanhees71]

I don't know, whether it helps you, but have a look at my Green's function manuscript, which contains the equilibrium QFT (mostly in the real-time formalism):

http://theorie.physik.uni-giessen.de/~hees/publ/green.pdf"