Time ordering on Kelydsh countour

I want to compute a time-ordering product but I have a question concerning this time ordering product. First, we consider A,E,I,L,N,O and V some second quantized operators without specifing what operators they are. Let $$t_{1}>t_{2}>t_{3}>t_{4}>t_{5}>t_{6}>t_{7}$$. I want to compute this time ordering product :

$$$$ Here $$T_{k}$$ is the time ordering on a Keldysh countour. The Keldysh contour goes from −∞ to +∞ (+ time label) and back from +∞ to −∞ (− time label).

We consider also that the operators are bosonic. This time ordering product contains 9 operators and the operators are written in Heisenberg representation. Since I don't know what these operators are, there are many pairing possible, I am planning so to use the wick theorem to do this (although it will be very fastidious) but I don't know how to use the wick theorem with an odd number of operators. Usually, the time ordering product I encounter is always with an even number of operators.

My questions are :

How can I use the wick theorem with an odd number of operators? Is there another method to compute this quantity (since there are 9 operators there are so many pairing possible...)?

• Are these creating/annihilation operators? What is the averaging -over states with a fixed number of particles or is there a BE condensate involved? It also seems that your questions have nothing specific to Keldysh. – Vadim Feb 24 at 11:33