Evolution of quantum observables: from non-commutativity to commutativity

Abstract

A fundamental aspect of the quantum-to-classical limit is the transition from a non-commutative algebra of observables to commutative one. However, this transition is not possible if we only consider unitary evolutions. One way to describe this transition is to consider the Gamow vectors, which introduce exponential decays in the evolution. In this paper, we give two mathematical models in which this transition happens in the infinite time limit. In the first one, we consider operators acting on the space of the Gamow vectors, which represent quantum resonances. In the second one, we use an algebraic formalism from scattering theory. We construct a non-commuting algebra which commutes in the infinite time limit.

Appendix

Let us show that formulas (41) and (42) are not equivalent. For simplicity, we shall assume that there is only one resonance so that dim ${\mathcal {H}}^G=2$. The vector $|\psi )$ being arbitrary is a linear combination of $|\psi ^D\rangle$ and $|\psi ^G\rangle$, so that it may be written as a column vector as $\left( \begin{array}{l} a\\ b\\ \end{array} \right)$ with a and b complex. Then,

$$\begin{aligned} (\psi ^D|\psi )= & {} (1,0) B^\dagger \, \left( \begin{array}{l} a \\ b\end{array}\right) \nonumber \\= & {} (1,0) \,i^{1/2} \left( \begin{array}{ll} -i{\sqrt{2}}/2 &{}\quad {\sqrt{2}}/2 \\ {\sqrt{2}}/2 &{}\quad -i{\sqrt{2}}/2\end{array}\right) \left( \begin{array}{l} a \\ b\end{array}\right) \nonumber \\= & {} i^{1/2} \left[ \frac{{\sqrt{2}}}{2}\,b -i\,\frac{{\sqrt{2}}}{2}\,a\right] . \end{aligned}$$

(63)

Similar calculations yield (we have written $\sqrt{-1}=-i$):

$$\begin{aligned} (\psi ^G|\psi )= & {} (\psi ^D|\psi ),\nonumber \\ (\psi |\psi ^D)= & {} (\psi |\psi ^G)=i^{1/2} \left[ \frac{{\sqrt{2}}}{2}\,b^* -i\,\frac{{\sqrt{2}}}{2}\,a^* \right] . \end{aligned}$$

(64)

This gives (41). To obtain (42), we need the following calculation:

$$\begin{aligned} (\psi ^D|T|\psi )= & {} (1,0) \, i^{1/2} \left( \begin{array}{ll} -i{\sqrt{2}}/2 &{}\quad {\sqrt{2}}/2 \\ {\sqrt{2}}/2 &{}\quad -i{\sqrt{2}}/2\end{array}\right) \left( \begin{array}{ll} 0 &{}\quad C \\ C &{}\quad 0 \end{array}\right) \left( \begin{array}{l} a \\ b\end{array}\right) \nonumber \\= & {} i^{1/2}\, \left( -i\,{\sqrt{2}}/2, {\sqrt{2}}/2 \right) \left( \begin{array}{l} b^* \\ a^*\end{array}\right) \nonumber \\= & {} i^{1/2}\, \left[ \frac{{\sqrt{2}}}{2}\,a^* -i\, \frac{{\sqrt{2}}}{2}\,b^* \right] . \end{aligned}$$

(65)

Taken $\sqrt{-1}=-i$, we easily find that $(\psi ^D|T|\psi )=(\psi ^G|T|\psi )$. From the other terms, we write

$$\begin{aligned} (\psi |T|\psi ^D)= & {} (a^*,b^*) \left( \begin{array}{ll} 0 &{}\quad C \\ C &{}\quad 0 \end{array}\right) (-i)^{1/2} \left( \begin{array}{ll} i{\sqrt{2}}/2 &{}\quad {\sqrt{2}}/2 \\ {\sqrt{2}}/2 &{}\quad i{\sqrt{2}}/2\end{array}\right) \left( \begin{array}{l} 1 \\ 0\end{array}\right) \nonumber \\= & {} i^{1/2} (a^*,b^*) \left( \begin{array}{ll} 0 &{}\quad C \\ C &{}\quad 0 \end{array}\right) \left( \begin{array}{l}{\sqrt{2}}/2 \\ -i{\sqrt{2}}/2 \end{array}\right) \nonumber \\= & {} i^{1/2}\, \left[ \frac{{\sqrt{2}}}{2}\,a^* -i\, \frac{{\sqrt{2}}}{2}\,b^* \right] . \end{aligned}$$

(66)

Similarly, we obtain that

$$\begin{aligned} (\psi |T|\psi ^G)=(\psi ^G|T|\psi )=(\psi ^D|T|\psi )=(\psi |T|\psi ^D). \end{aligned}$$

(67)

The obvious conclusion is that (41) and (42) do not coincide. The first and third identities in (67) are not a surprise due to the properties of the time reversal operator T.