If you’re in a classical-physics simulation that’s guaranteed to have rotational symmetry, then yep, you stay mirrored.
In any messy real world setting (even ignoring quantum mechanics), chaos theory would indeed kick in pretty quickly—the thermal jostling of air molecules, or radiation from outside, would slowly lead to neurons firing at slightly different times.
But maybe the most sciencey way to unmirror yourselves is to suppose you have the ability to prepare and measure an entangled quantum state. E.g. suppose you meet in the middle and put the spin of a pair of atoms into the state |up,down>+|down,up>. This state is rotationally symmetric (for the nitpickers, I think I’ve implied the atoms have integer spin), so you can do this, but when you measure the atoms you’ll get opposite results.
With quantum, you can go further: even if you’re instantiated in a universe that starts out perfectly rotationally symmetric, as long as we’re not modifying physics to guarantee the symmetry is preserved over time, then the vastly overwhelming majority of quantum behavior will not have spurious correlation between rotationally symmetric positions, meaning that the rotational symmetry will almost instantly be lost at the nanoscale, and then it’s just some chaos away from having macroscopic effects. You can think about this under the many worlds interpretation: there’s a vanishingly small fragment of the wavefunction where every time an atom collides with another atom in one position, the quantum random portion of the collision’s outcome is exactly matched in the rotationally symmetric position. From the outside, this means that even picking which classical slice of the wavefunction is still symmetric requires you to specifically encode the concept of symmetry in your description of what slice to take. (I don’t quite know the quantum math necessary to write this down.) From the inside, it means an overwhelmingly high probability of nanoscale asymmetry, and from there it’s only chaos away from divergence.
I expect it would still take at least a few seconds for the chaos in your brains to diverge enough to have any detectable difference in motor behavior. If we were instead running the same trial twice rather than putting two instances in the same room across from each other, I’d expect it would take slightly longer to become a macroscopic difference, because the separate trials don’t get to notice each others’ difference in behavior to speed up the divergence.
Importantly, assuming many worlds, then the wavefunction will stay symmetric—for every classical slice of the wavefunction (“world”) where person on side A has taken on mindstate A and side B has mindstate B, there’s another world in which person on side A has mindstate B and vice versa. This is because if many worlds is correct then it’s a “forall possible outcomes”, and from the outside, it’s not “randomized” until you pick a slice of this forall. It’s unclear whether many worlds is physically true, so this might not be the case.
Can you symmetrically put the atoms into that entangled state? You both agree on the charge of electrons (you aren’t antimatter annihilating), so you can get a pair of atoms into |↑,↑⟩, but can you get the entangled pair to point in opposite directions along the plane of the mirror?
Edit Wait, I did that wrong, didn’t I? You don’t make a spin up atom by putting it next to a particle accelerator sending electrons up. You make a spin up atom by putting it next to electrons you accelerate in circles, moving the electrons in the direction your fingers point when a (real) right thumb is pointing up. So one of you will make a spin-up atom and the other will make a spin-down atom.
Well, I was going by the original post where you’re rotationally symmetric, not mirror-symmetric. But for mirror symmetry, pick the mirror plane to be the z direction, then you just make your spins either both +z or both -z.
If you’re in a classical-physics simulation that’s guaranteed to have rotational symmetry, then yep, you stay mirrored.
In any messy real world setting (even ignoring quantum mechanics), chaos theory would indeed kick in pretty quickly—the thermal jostling of air molecules, or radiation from outside, would slowly lead to neurons firing at slightly different times.
But maybe the most sciencey way to unmirror yourselves is to suppose you have the ability to prepare and measure an entangled quantum state. E.g. suppose you meet in the middle and put the spin of a pair of atoms into the state |up,down>+|down,up>. This state is rotationally symmetric (for the nitpickers, I think I’ve implied the atoms have integer spin), so you can do this, but when you measure the atoms you’ll get opposite results.
With quantum, you can go further: even if you’re instantiated in a universe that starts out perfectly rotationally symmetric, as long as we’re not modifying physics to guarantee the symmetry is preserved over time, then the vastly overwhelming majority of quantum behavior will not have spurious correlation between rotationally symmetric positions, meaning that the rotational symmetry will almost instantly be lost at the nanoscale, and then it’s just some chaos away from having macroscopic effects. You can think about this under the many worlds interpretation: there’s a vanishingly small fragment of the wavefunction where every time an atom collides with another atom in one position, the quantum random portion of the collision’s outcome is exactly matched in the rotationally symmetric position. From the outside, this means that even picking which classical slice of the wavefunction is still symmetric requires you to specifically encode the concept of symmetry in your description of what slice to take. (I don’t quite know the quantum math necessary to write this down.) From the inside, it means an overwhelmingly high probability of nanoscale asymmetry, and from there it’s only chaos away from divergence.
I expect it would still take at least a few seconds for the chaos in your brains to diverge enough to have any detectable difference in motor behavior. If we were instead running the same trial twice rather than putting two instances in the same room across from each other, I’d expect it would take slightly longer to become a macroscopic difference, because the separate trials don’t get to notice each others’ difference in behavior to speed up the divergence.
Importantly, assuming many worlds, then the wavefunction will stay symmetric—for every classical slice of the wavefunction (“world”) where person on side A has taken on mindstate A and side B has mindstate B, there’s another world in which person on side A has mindstate B and vice versa. This is because if many worlds is correct then it’s a “forall possible outcomes”, and from the outside, it’s not “randomized” until you pick a slice of this forall. It’s unclear whether many worlds is physically true, so this might not be the case.
Can you symmetrically put the atoms into that entangled state? You both agree on the charge of electrons (you aren’t antimatter annihilating), so you can get a pair of atoms into |↑,↑⟩, but can you get the entangled pair to point in opposite directions along the plane of the mirror?
Edit Wait, I did that wrong, didn’t I? You don’t make a spin up atom by putting it next to a particle accelerator sending electrons up. You make a spin up atom by putting it next to electrons you accelerate in circles, moving the electrons in the direction your fingers point when a (real) right thumb is pointing up. So one of you will make a spin-up atom and the other will make a spin-down atom.
Well, I was going by the original post where you’re rotationally symmetric, not mirror-symmetric. But for mirror symmetry, pick the mirror plane to be the z direction, then you just make your spins either both +z or both -z.
Mirror symmetry is not rotational symmetry.
The post has been edited since my answer. See two comments below, or five answers below.