The topic of this thread is Wick rotation.
It's used in physics, in relativity. By introducing an imaginary variable in the right place, difficult problems in Euclidean space can be "rotated" into Minkowski space, solved there, and rotated back. (Or vice versa).
But there is a deeper more mysterious and more significant application of Wick rotations, in probability theory. You'll see why it's significant in a moment.
First we will need the concept of moments. You may know these by their colloquial names mean, variance, skewness, kurtosis, and so on. Actually there are an infinite number of such moments, they form an infinite series that describes the probability distribution, one can speak of (and calculate) the n-th order moment, which in the probability theory is defined as E[X^n], and most often we're interested in a moment "around a point" (like, around the mean), and in that case these are "central" moments defined as E[(X-x0)^n].
en.wikipedia.org
The idea is we're sampling the variable n times. In theory if everything is independent and nicely behaved the probability after n outcomes should simply be the product of the n probabilities, hence the outcome raised to the power.
In SOME nicely behaved cases, we can have a function that generates the moments for us, it's called the moment generating function. For example if you have a family of Gaussians with widths varying from infinity to 0, you can parametrize these from the unit interval.
The moment generating function, if it exists, defines the shape of the probability distribution. Which is where this topic gets interesting. There is a deep relationship between the moment generating function and the Fourier transform of the probability distribution. Which occurs in the complex plane and supports "Wick rotations" that link statistical mechanics with quantum mechanics.
en.wikipedia.org
Turns out, understanding Wick rotations is vital for any serious study of chaos and criticality. There is already a bunch of clever methods and a huge literature on phase plane methods in physics, one of them directly represents the Schrodinger equation as a 2n-dimensional probability map and uses it to solve spin glass geometry and things like that. It all leads to Feynman's path integrals, which are directly analogous to moment generating functions.
If you want to know how a particle gets from here to there, you have to know "all possible paths" by which that can happen, and each of the possible paths has a probability assigned to it. You have a "probability distribution" where the outcome is a linear combination of possible paths. There is an "evolution" of the state of the system according to Schrodinger's equation, which in turn maps back to Wiener's original math around Brownian motion. The maths are the same, and it's the same math needed to understand criticality.
One of the best studied chaotic systems in physics is the spin glass. That's what happens to a magnet if you heat it beyond its Curie temperature, the little magnetic dipoles that used to align now start floating around in various directions like a liquid, suddenly the magnetic spins are disordered and chaotic.
And it just so happens that spin glasses are an excellent case study because we have working models for both Wick and non-Wick solutions.
It's used in physics, in relativity. By introducing an imaginary variable in the right place, difficult problems in Euclidean space can be "rotated" into Minkowski space, solved there, and rotated back. (Or vice versa).
But there is a deeper more mysterious and more significant application of Wick rotations, in probability theory. You'll see why it's significant in a moment.
First we will need the concept of moments. You may know these by their colloquial names mean, variance, skewness, kurtosis, and so on. Actually there are an infinite number of such moments, they form an infinite series that describes the probability distribution, one can speak of (and calculate) the n-th order moment, which in the probability theory is defined as E[X^n], and most often we're interested in a moment "around a point" (like, around the mean), and in that case these are "central" moments defined as E[(X-x0)^n].
Moment (mathematics) - Wikipedia
The idea is we're sampling the variable n times. In theory if everything is independent and nicely behaved the probability after n outcomes should simply be the product of the n probabilities, hence the outcome raised to the power.
In SOME nicely behaved cases, we can have a function that generates the moments for us, it's called the moment generating function. For example if you have a family of Gaussians with widths varying from infinity to 0, you can parametrize these from the unit interval.
The moment generating function, if it exists, defines the shape of the probability distribution. Which is where this topic gets interesting. There is a deep relationship between the moment generating function and the Fourier transform of the probability distribution. Which occurs in the complex plane and supports "Wick rotations" that link statistical mechanics with quantum mechanics.
Wick rotation - Wikipedia
Turns out, understanding Wick rotations is vital for any serious study of chaos and criticality. There is already a bunch of clever methods and a huge literature on phase plane methods in physics, one of them directly represents the Schrodinger equation as a 2n-dimensional probability map and uses it to solve spin glass geometry and things like that. It all leads to Feynman's path integrals, which are directly analogous to moment generating functions.
If you want to know how a particle gets from here to there, you have to know "all possible paths" by which that can happen, and each of the possible paths has a probability assigned to it. You have a "probability distribution" where the outcome is a linear combination of possible paths. There is an "evolution" of the state of the system according to Schrodinger's equation, which in turn maps back to Wiener's original math around Brownian motion. The maths are the same, and it's the same math needed to understand criticality.
One of the best studied chaotic systems in physics is the spin glass. That's what happens to a magnet if you heat it beyond its Curie temperature, the little magnetic dipoles that used to align now start floating around in various directions like a liquid, suddenly the magnetic spins are disordered and chaotic.
And it just so happens that spin glasses are an excellent case study because we have working models for both Wick and non-Wick solutions.
