# Moments of a Distribution via Laplace Transforms and Wick Rotations

1. Jun 17, 2014

### bolbteppa

On a mathematical level, the statistical mechanical partition function is just a Laplace transform of the microcanonical probability distribution, i.e. it's moment generating function. Understanding the mathematical and physical motivation and meaning behind this is something many have unfortunately been unable to help me with, hence I ask here:

What is the intuitive meaning of the n'th moment of a function or probability distribution?

• I understand that "Broadly speaking a moment can be considered how a sample diverges from the mean value of a signal - the first moment is actually the mean, the second is the variance etc... ", but using this interpretation I can't make sense of what the 5'th moment of $f(x) = x^2$ is, why it's useful, what it says about anything, and how in the world such an interpretation falls out of the mathematical derivation.

Why would one ever say to oneself that they should decompose a function into it's constituent moments via a moment generating function?

• A Fourier series/transform of a function of, say, time is just an equivalent way to characterize the same function in terms of it's amplitude, phase and frequency instead of time (as a sine wave is uniquely characterized by these quantities, and to a physicist a Fourier series is more or less just a big sine wave where these numbers vary at each point). A geometric interpretation of every term in the expansion of the Fourier series of a function can be given. Intuitively, just looking at, say, light or sound there is an immediate physical motivation for using a Fourier transform because it decomposes them into their constituent frequencies, which exist physically. A m.g.f. is a bilateral Laplace transform, and it decomposes a function/distribution into it's moments. Thus the Laplace transform should have an almost similar story behind using it, one that should explain the reason for wanting the n'th moment of $x^2$. (An answer saying that it uniquely characterizes "nice" functions/distributions is to miss the point of my question).

Why is it that the Fourier series of an exponentially damped distribution (the moment generating function or Bilateral Laplace Transform) decomposes a function into it's moments?

• I understand the mathematical derivation, but not the motivation for doing any of this, or why it should work, i.e. why wouldn't the regular Fourier transform just be enough? There has to be meaning behind it, so taking the fact we're using a Fourier series/transform literally, the question arises as to why it is that the constituent waves of an exponentially damped version of a function/distribution allow for an interpretation in terms of moments? (Note "exponentially damped" means using a minus sign in the m.g.f.)

Why does a Wick rotation of the Moment generating function give the Characteristic function?

• The characteristic function is the Fourier transform of a probability distribution, but it is also a wick rotation of the Fourier transform of an exponentially damped distribution. I understand we are analytically continuing the argument to a new axis, but is there any meaning behind this, does it say anything about the relationship between the Laplace transform's moments and the composite waves that a Fourier series decomposes a function into, and does it help explain why there is a mathematical relationship between the Feynman Path integral and the statistical mechanical Partition function?

In other words: What are we doing? Why are we doing it? Why does a particular method for doing it work? Does it explain an interesting connection between two other things? Thanks