N-point function calculation from a generating function

[guilhermef]

If a have ln Z[J] = ∫ J²f(t)dt+ a∫ J³dt + b∫ J^4dt, where J=J(t), and I would like to get the 3-point and 4-point functions, how do I proceed?

I have tried to use the regular formula for the n-point function, when you derive Z[J] n times in relation to J(t_1)...J(t_n) and after applies J=0, but it doesn't make sense for me, since all the n-point functions will be zero.

Actually, I think this particular Z[J] pretty strange because its integrals are not linear in J(t) as I used to see.

So, could someone give me a hint?

Thanks!

 

[member 553083]

The 3-point and 4-point functions can be found by taking the partial derivatives of Z[J] with respect to J(t1), J(t2), etc. up to J(tn). For example, the 3-point function is given by: F3(t1, t2, t3) = ∂^3Z[J]/∂J(t1)∂J(t2)∂J(t3) Similarly, the 4-point function is given by: F4(t1, t2, t3, t4) = ∂^4Z[J]/∂J(t1)∂J(t2)∂J(t3)∂J(t4) You can then evaluate the derivatives at J = 0 to get the desired 3-point and 4-point functions.