Ryan North of the exceptionally brilliant and somewhat esoteric Dinosaur Comics penned a comic a while back about the possibility of being the perfect detective by using physics. Quoth T-Rex in the strip:
Someone in the room says something to another person, and then they both leave. Assuming you know exactly how they moved when they left (and therefore how they displaced the air!) then any remaining displacement is due to sound waves. By looking at the location of air molecules in the empty room, you can reconstruct what was said in the past!
Of course there's the minor engineering hurdle of knowing the positions and velocities of every molecule in the room and running the necessary calculations. Obviously it's not possible in practice, but in principle? Utahraptor gives his own theoretical objection, but I think it's one that can be overcome. He notes that sound waves are more like oscillations than permanent displacements and so in fact the information might not be recoverable even in theory. In a perfect wave-equation-with-damping world that might be true, but sound waves are composed of individual particles in thermal motion and thus we have in theory individual particles executing classical trajectories that can be extrapolated backwards.
But with that overcome, we're still not out of the woods. Even perfect information about the state of the particles when you examine the room won't be adequate. Being an N-body problem, any imprecision in the computation will rapidly grow until it swamps the actual result. Every extra second in the past you wish to reconstruct will require an exponential increase in your floating-point precision.
That's classically speaking, of course. Quantum mechanically speaking the scheme is toast from the start. You can't have position and velocity of each particle with arbitrary accuracy, it's just not possible. It's not even mathematically meaningful when it comes right down to it. And that's going to severely limit how far back any reconstruction can be carried even with maximal information. The whole indistinguishability thing isn't going to help matters either.
Nice try though, T-Rex!
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I'm not so sure. Obviously it's not going to be a practical possibility, but you're asking whether it would be possible in principle -- and here we might have to ask which principles we're allowed.
It's worth noting, e.g., that the unitary evolution of quantum mechanics is typically seen as ensuring a complete preservation of physical information -- which would in principle allow the retrodiction of an initial state from knowledge of a later state.
You see this point raised, for example, in popular descriptions of the black hole information loss paradox, which then worry that information falling into black holes would be lost (implying nonunitary evolution, which was once advocated by Hawking).
Of course, such discussions typically just assume that we have a fully specified (pure) quantum state and a known Hamiltonian (and no limits on our ability to compute). It's obviously a different matter if we need to experimentally determine the state; generally this is going to require an ensemble of identically prepared systems, which won't work too well for our T-Rex sleuth.
A further, perhaps more relevant, point is that it's not immediately obvious that sound waves in air molecules might not behave classically enough to allow (in principle) a classical calculation to retrodict the utterance. Keep in mind that planets are composed of quantum particles as well, but we can do a pretty good job of calculating their trajectories.
And I suppose if I'm picking nits, I should also comment on
If we are willing to allow a hidden variable interpretation of QM like Bohm's, then in principle it's quite possible to retrodict the initial state from the final one (though obviously not in practice). Perfectly meaningful, mathematically speaking.
Keep in mind that planets are composed of quantum particles as well, but we can do a pretty good job of calculating their trajectories.
That's because we don't actually need to keep track of the individual particles. They are gravitationally bound, so for most purposes we can treat the planet as a point mass. If the shape ever becomes important (e.g., because we want to calculate the orbital evolution of a satellite in low Earth orbit), we can include quadrupole, etc., terms by classical techniques.
Reconstructing speech from sound waves, by contrast, does require us to keep track of what individual particles are doing. A particular section of wave front will decay in amplitude as the square of time (neglecting partial absorption at walls and other objects in the room, and assuming a uniform temperature), to say nothing of its interference with other sections of the same or different wave fronts. Also, in many systems sound can be viewed as quantized (this is where the term "phonon" comes from), and eventually that quantization will kick in. As a practical matter, if you want to reconstruct after the fact what was said in the room, you need to have planted a device which will record the sound in real time.
Stochastic quantum phlegmatics allows the US anti-missile system to Officially cherry pick enemy sub-orbital warheads at will. Proof of concept is that it is terrible in tests when the exact target trajectory is known, empirically validating uncertainty relations.
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