The partition function and Wick rotation
I was trying to understand Wick rotation by applying it in the case of a finite-dimensional Hilbert space, and came up with something strange. The way I’ve worked it out, it seems to map classical observables to quantum states! I’ve never heard anything like that before.
Say we have an -qubit Hilbert space . This has the set of -bit binary strings as a basis. For brevity’s sake, I’ll write these as Let
where the are real.
Now define the operators
and . is the discrete Fourier transform on a 2-dimensional space; is called the Walsh-Hadamard transform. defines a conjugate basis to the qubit basis. The most important property of for my purposes here is the fact that
where the normalizing factor Modulo the normalizing factor, this is a sum over all possible states.
- What’s the probability amplitude that when you start in the state and evolve according to , the system will still be in the state
Except for a factor of in the exponent and some normalization, this is the partition function for . It’s been “Wick rotated.”
- What’s the probability amplitude that when you start in the state and evolve according to H, the system will move to the arbitrary state Well,
where each is an arbitrary complex number and is a normalizing factor. So
If we divide this by the quantity above, we get the expectation value of a classical observable at “temperature”
This mapping from classical to quantum is not quantization. That maps classical observables to Hermetian operators, not to states—although, one might hit the state with the “Currying” isomorphism between states and linear transformations and get something useful.
I’m trying to work out how to connect this to a sum over paths instead of a sum over states; there’s some interesting stuff there, but I haven’t grokked it yet.