# Out-of-time-order correlation function in the interaction picture?

Recently there has been interest in understanding the out of time order correlation function (OTOC) $$F$$, which essentially compares the overlap of two operators $$W$$ and $$V$$ acting on a state in two different ways

1. Apply $$V$$ onto a quantum state, wait a time $$t$$, then apply $$W$$
2. Apply $$W$$ at time $$t$$, then reverse time to $$t=0$$ and apply $$V$$

This correlation function then takes the form $$F=\langle W_t^{\dagger} V^{\dagger} W_t V \rangle$$

Roughly speaking, $$F$$ tells you how quickly interactions make the initially commuting $$V$$ and $$W$$ stop commuting.

Preparing the state of (1) above is straightforward, you just apply the operators one after another. However (2) is challenging, because one has to reverse the direction of time.

The most obvious way to implement (2) is to reverse time through changing the sign of the entire Hamiltonian $$H \rightarrow -H$$, which effectively reverses the time evolution. Needless to say, this is very tricky in dissipative/open systems, as you can only reverse the sign of part of the Hamiltonian, not all of it.

My question is the following: is it meaningful to think about OTOCs in the interaction picture? I ask because it is often the case in experiment (e.g. NMR Hahn echo experiments) that you can reverse the sign of the interaction Hamiltonian but not the full Hamiltonian. What does the OTOC tell you with respect to time evolution in the interaction picture?

More concretely, say you have the following spin Hamiltonian

$$H = H_0 + g \sum_i \mathbf{S}_i\cdot \mathbf{B}$$

Where $$H_0$$ is some unknown function of the spin operators.

Assume that you can change the sign of the magnetic field $$\mathbf{B}$$ at will, like in NMR. Is there anything interesting about the resulting OTOC in this case? Or does it not have any of the useful properties of "true" OTOCs?