 The similarity of the spectral representations of the imaginary- and real-time Green functions allows us to define the function
which is related to formula_73 and formula_57 by
and
A similar expression obviously holds for formula_58.The relation between formula_78 and formula_79 is referred to as a Hilbert transform.We demonstrate the proof of the spectral representation of the propagator in the case of the thermal Green function, defined as

Due to translational symmetry, it is only necessary to consider formula_81 for formula_82, given by
Inserting a complete set of eigenstates gives

Since formula_85 and formula_86 are eigenstates of formula_87, the Heisenberg operators can be rewritten in terms of Schrödinger operators, giving
Performing the Fourier transform then gives

Momentum conservation allows the final term to be written as (up to possible factors of the volume)
which confirms the expressions for the Green functions in the spectral representation.