I Weinberg Lectures on QM (2013 ed.), Equation 6.5.5

[jouvelot]

Hi everyone,

I'm a bit puzzled by the derivation of this formula, in particular since the definition of the "overbar" notation is a bit fuzzy (see Formula 6.4.1). Does anyone have a more formal definition of the correlation function in this setting (I know what a CF is, in general)? In this particular instance, I guess one just needs to consider the average of a product of 4 i.i.d. variables as a product of two averages of 2 i.i.d. variables to get the result. But I would have preferred a clearer definition of the notion involved. Any idea where to find a proper one?.

Thanks,

Pierre

 

[mathman]

You might want to state these formulas explicitly for readers who don't have the boo.

 

[jouvelot]

Sure.

The overline notation is such that (quote from the book) "a line over a quantity indicates an average over fluctuations." Introduce a perturbation Hamiltonian matrix in an electric field $\mathbf{E}$ as (simplified) :
$$H'_{nm}(t) = e \mathbf{x}_{nm}.\mathbf{E}.$$
Now assume that
$$\overline{E_i(t_1)E_j(t_2)} = \delta_{ij}f( t_1 - t_2).$$
Then Equation 6.5.5 states that
$$\overline{H'_{nm}(t_1)H'^*_{nm}(t_2)} = e^2 | \mathbf{x}_{nm} | ^2 f( t_1 - t_2).$$
Even though it's somewhat intuitive, I was looking for a more formal definition of the "overline" notation and of its use here.