They are the imaginary time versions of statistical mechanics partition functions, giving rise to a close connection between these two areas of physics.
Partition functions can rarely be solved for exactly, although free theories do admit such solutions.
Instead, a perturbative approach is usually implemented, this being equivalent to summing over Feynman diagrams.
From this it follows that an equivalent expression for the partition function reminiscent to a power series in source currents is given by[2] In curved spacetimes there is an added subtlety that must be dealt with due to the fact that the initial vacuum state need not be the same as the final vacuum state.
[3] Partition functions can also be constructed for composite operators in the same way as they are for fundamental fields.
[5] The symmetry factors for these types of diagrams differ from those of correlation functions since all external legs have identical
By performing a Wick transformation, the partition function can be expressed in Euclidean spacetime as[6] where
This form is closely connected to the partition function in statistical mechanics, especially since the Euclidean Lagrangian is usually bounded from below in which case it can be interpreted as an energy density.
It also allows for the interpretation of the exponential factor as a statistical weight for the field configurations, with larger fluctuations in the gradient or field values leading to greater suppression.
This connection with statistical mechanics also lends additional intuition for how correlation functions should behave in a quantum field theory.
Most of the same principles of the scalar case hold for more general theories with additional fields.
After differentiation, the currents are set to zero when correlation functions in a vacuum state are desired, but the currents can also be set to take on particular values to yield correlation functions in non-vanishing background fields.
is equivalent in Euclidean formalism to a theory with a compactified temporal direction of length
Partition functions must be modified appropriately by imposing periodicity conditions on the fields and the Euclidean spacetime integrals This partition function can be taken as the definition of the thermal field theory in imaginary time formalism.
is the position space Feynman propagator This partition function fully determines the free field theory.
In the case of a theory with a single free Dirac fermion, completing the square yields a partition function of the form where