Even minor departures from equilibrium call for a theoretical description entirely different from the ‘Matsubara formalism’ developed in earlier chapters of this book. The applicability of the latter is rigidly tied to the existence of a quantum grand canonical density operator, the hallmark of a many particle equilibrium system. But how then, can an functional theory of nonequilibrium quantum systems be developed? An obvious idea would be to once more go through the different elements of classical nonequilibrium theory developed in the previous chapter, and subject every one of them to an individual quantization scheme. Luckily, there are more efficient ways to achieve our goal: some decades ago, a many particle formalism suitable to describe nonequilibrium systems under the most general conditions was introduced by Keldysh.