Greens function path integral representation

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The discussion focuses on the path integral representation of the Green's function and the transition from the trace formula to this representation. The original poster references their book and an attached image for clarity. They acknowledge that explaining the transition in detail is complex and may not be suitable for a forum post. They suggest that their lecture notes on many-body quantum field theory could provide additional insights. The conversation emphasizes the need for a deeper understanding of the mathematical framework involved.
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In my book the path integral representation of the green's function is given as that on the attached picture. But how do you go from the usual trace formula for the Green's function 2.6 to this equation?
 

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Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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