## 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.

reperiendisaid, on 2007 November 29 at 5:10 pmIt has to do with the analogy between statics in n space dimensions and dynamics in (n-1) space and one time dimension. See “A spring in imaginary time.”

Each basis state describes a path in space of a spring parameterized by s; there’s an associated energy to the path. The zeta function tells what the probability is of finding the spring in that state given the spring’s temperature. A spring in equilibrium is in a minimal (really a critical) energy state.

Changing s to it changes the length parameterization to imaginary time and the static problem to a dynamic one. E -> -iS. Then, by summing exp(iS) over all paths, you get the path of least (or critical) action.