When should we consider "reverse Heisenberg" evolution of operators?

Question

In Quantum Mechanics, the Heisenberg evolution of an observable $hat{o}$ is defined as

$$ hat{o}(t) = U(t,0)^{dagger} hat{o} U(t,0) $$

where $U(t,0)$ is the unitary time-evolution operator from time $0$ to time $t$. This satisfies the Heisenberg equations of motion

$$ ihbar frac{d}{dt} hat{o}(t) = [hat{o}(t),H(t)],$$

But is there a standard name for the "reverse Heisenberg evolution"

$$ hat{o}_R(t) = U(t,0) hat{o} U(t,0)^{dagger} $$

which satifies the differential equation
$$ ihbar frac{d}{dt} hat{o}_R(t) = [H(t),hat{o}_R(t)],$$

and in which circumstances should one consider it?

It came up because I was thinking about a state $|psirangle$ which is defined to be the (unique, say) eigenstate of some observable $hat{o}$ with eigenvalue $lambda$. Then we see that the time-evolved state $|psi(t)rangle = U(t,0) |psirangle$ can be characterized as the eigenstate of the operator $hat{o}_R(t)$.

doublefelix · Answer

It came up because I was thinking about a state $|psirangle$ which is defined to be the (unique, say) eigenstate of some observable $hat{o}$ with eigenvalue $lambda$. Then we see that the time-evolved state $|psi(t)rangle = U(t,0) |psirangle$ can be characterized as the eigenstate of the operator $hat{o}_R(t)$

This property is necessary for the derivation of the path integral in quantum mechanics and QFT. In the context I saw it, it was used in reverse, but it's the same idea. We had operators which evolve normally
$$X(t) = U^dagger (t)X U(t)$$
$$P(t) = U^dagger (t)P U(t)$$

but states were then defined with backwards evolution:
$$|x,trangle equiv U^dagger (t)|xrangle$$

So that for all time, they remain eigenstates:

$$X(t)|x,trangle=U^dagger (t)X U(t) U^dagger(t)|xrangle$$
$$=U^dagger (t)x |xrangle$$
$$=x |x,trangle$$
(and same with $P$)

If you want to read more, I learned this from Weinberg's The Quantum Theory of Fields, the part he does this is in Chapter 9.1, page 379. His derivation is for Quantum Mechanics but with n degrees of freedom. Taking the $nto infty$ would give the QFT path integral.

You asked if this evolution has a name; I don't know of any special name for it. Reverse evolution seems to get the point right.

When should we consider "reverse Heisenberg" evolution of operators?

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