# Use and understanding of higher-order correlation functions

The correlation function g1 is pretty easy to understand and the relation to young's double slit experiment is also clear to me.

In every quantum optics book I read so far correlation functions $g^n(x_1, ... , x_{2n})$ of an order greater than 1 are defined, but not related to an experiment or explained any further.

What is an example for an experiment that requires a correlation function of a higher order?

In my understanding $g^1(x_1,x_2)$ gives the similarities between two functions for different $\tau$. How is this possible for more than one function, if $g^n(x_1,...,x_{2n})$ is still scalar?

The definitions I used can be found in any book an quantum optics or here http://www.matthiaspospiech.de/files/studium/skripte/QOscript.pdf (p 64)

• could you maybe add a link explaining what g1 means, in the context of the double slit experiment? or define what it means.
– Siva
Apr 8 '13 at 23:49

Okay, so I gather from the link that $G^{(k)}$ in your notation refers to the correlation between field values at $2k$ points, with $\varepsilon^+$ inserted at half of them and $\varepsilon^-$ inserted at the other half.

This concept of an $n$-point correlation function is very similar to the $n$th moment of a random variable or statistical distribution. For simplicity, consider an example with 1 dimension: A field over 1 time dimension (at a single point in space), described by a random variable. Now, we can consider the probability distribution of this random variable and talk of it's moments. The mean value would be called the 1st moment and the variance would be related to the second moment (it is in fact called the second central moment). Similarly, you can generalize to higher order moments which help characerize the distribution. The moments help you characterize the distribution and also give an intuitive feel for the function.

Generalize this concept to random variables which are fields over many-dimensional spacetime. That is what your correlation functions are.

Btw, for a gaussian distribution (non-interacting fields i.e. quadratic action), all odd moments vanish. (That might be the motivation for $G^{(k)}$ to be defined as the correlation between field values at $2k$ points... even though the actual physical theory you're considering will probably be interacting, else all correlation functions are fairly trivial). Also, all even moments beyond the 2nd-moment are completely specified by the 1st and the 2nd moment. Ref1 and Ref2

If you had an interaction term in the hamiltonian/lagrangian involving 4 fields, then the 4-point correlation function would have 2 kinds of contributions:

1. 2 sets of 2-point correlation functions between pairs of points among those 4 points
2. A nontrivial contribution from the interaction term with one of it's field insertions at each of the 4 points.

So you can see that higher order correlations functions give you very important (an unique) information in an interacting physical theory.

Update: The (many) answers to this SE question might also shed some light on the discussion.