A shard of obsidian knapped nine thousand years ago still carries the exact trace of the angle at which force was applied. The curvature of the surface, the position of the Wallner lines, the depth of the bulb of percussion: all of it is legible. The gesture is inscribed in the result. But the result was already inscribed in the structure of the glass, in the distribution of internal stresses, in the properties of the fracture front before it propagated. The final form was not imposed. It was waiting.
Quantum mechanics has a formal version of this situation. In 1964, Aharonov, Bergmann, and Lebowitz show that a quantum system between two measurements is not described by its initial state alone. If the system is prepared in state $|\psi\rangle$ at $t_0$ and one knows it will be found in state $\langle\phi|$ at $t_f$, then the two vectors, the one advancing from preparation and the one receding from measurement, co-determine everything that happens between them.
The two amplitudes enter symmetrically. What comes from the past and what comes from the future carry the same formal weight. The operational asymmetry exists: one prepares the state, one does not prepare the outcome. But in the formula, that asymmetry has vanished.
In 1988, Aharonov, Albert, and Vaidman draw a consequence. If the coupling between the system and the measuring device is sufficiently weak, the conditional mean displacement of the pointer on the post-selected subset is given by the real part of
$A_w$ is not confined to the spectrum of $\hat{A}$. The operator $\hat{\sigma}_z$ has eigenvalues $\pm 1$. Under appropriate pre- and post-selection, $(\sigma_z)_w = 100$. Measured by Ritchie, Story, and Hulet in 1991. Confirmed by Pryde et al. in 2005.
The excess comes from the denominator. When $\langle\phi|\psi\rangle$ is small, when the two states are quasi-orthogonal, when past and future of the system have almost nothing in common, the intermediate values leave the spectrum. They violate no law. They are what happens when a system is constrained by two boundaries instead of one.
What a weak value of 100 physically means for a spin-1/2 is the subject of a disagreement that has lasted thirty-five years. Dressel et al. (2014) provide a state of the art. Vaidman (2017) defends their ontological reality. Sokolovski (2013) reduces them to a statistical artifact of the pointer. The formalism produces correct predictions in all cases. The interpretation does not converge.
Doctrine
A system defined by two boundaries does not behave like a system defined by one. When what one has and what one wants have almost nothing in common, what happens between the two escapes both.
We work in that in-between. The substrate does not resemble the specification. The specification does not resemble the substrate. What happens in the middle resembles nothing that was anticipated.
Vecteur ouvert
If the final form genuinely participated in determining the process that produces it, how would one know? What would distinguish a process constrained by two boundaries from a process constrained by one, observed after the fact?
