A rigorous Keldysh functional integral for fermions

Abstract We provide a mathematically rigorous Keldysh functional integral for fermionic quantum field theories. We show convergence of a discrete-time Grassmann Gaussian integral representation in the time-continuum limit under very general hypotheses. We also prove analyticity of the effective action and explicit bounds for the truncated (connected) expectation values $$\gamma ^\textrm{c}_{m,\bar{m}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>γ</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mtext>c</mml:mtext> </mml:msubsup> </mml:math> of the non-equilibrium system. These bounds imply clustering with a summable decay in the thermodynamic limit, provided these properties hold at time zero, and provided that the determinant bound $$\delta _C$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>δ</mml:mi> <mml:mi>C</mml:mi> </mml:msub> </mml:math> and decay constant $$\alpha _C$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>C</mml:mi> </mml:msub> </mml:math> of the fermionic Keldysh covariance are bounded uniformly in the volume. We then give bounds for these constants and show that uniformity in the volume indeed holds for a general class of systems. Finally we show that in the setting of dissipative quantum systems, these bounds are not necessarily restricted to short times.