104 lines
6.2 KiB
Plaintext
104 lines
6.2 KiB
Plaintext
[S1]
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Today we’re not proving quantum mechanics is discrete.
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We’re running a test.
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One test, named INT5000.
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[S2]
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We already completed some simple tests. We called them the naïve test suite.
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It required us to define what is the Action, how does stuff move around.
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And what is momentum.
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But might be neat, but all it showed is that the CA glider has vaguely particle-like properties.
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But a fundamentally discrete universe is 50% of the option space. The other 50% is a continuous universe. Ruling out 50% of the option space like a decent bang for the buck, so we committed to giving a discrete universe every chance it can have to be true. Even if it sounds unlikely.
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We need to make it pass more tests. Lets introduce the advanced test suite.
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[S3]
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If quantum mechanics is emergent from something discrete, there are many constraints it must satisfy.
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It needs Lorentz invariance, otherwise it contradicts Einstein.
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Most of these tests are beyond my abilities, even with AI assistance.
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But there is one I’m excited about, and it’s this one.
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Can a Cellular Automaton reproduce the spinorial rotation structure - SU(2)?
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[S4]
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In quantum mechanics, certain objects - spinors - behave strangely.
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Rotate them 360 degrees, and they don’t return to their original state.
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They pick up a minus sign.
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Only after 720 degrees do they truly return.
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This behavior is encoded mathematically in a group called SU(2).
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If we can’t reproduce this, we can’t reproduce spin.
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If we can’t do that, QM is out of reach. And that means we’re back to the default scenario.
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[S5]
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And the default scenario is - that the Naga is an neat and weird Cellular Automaton, of no significance to physics at all.
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Just like this channel, with its middle aged amateur creator dressed up in a yellow jumpsuit
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Trying to do stuff I don’t have the necessary training to do.
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This is the most likely outcome. I owning it, not afraid of it. It doesn’t hurt anyone, so lets keep going.
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[S6]
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And as you know, physicists generally don’t understand code-speak, and developers struggle with physics-speak. So we’ll keep translating showing both sides.
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In Code speak - SU(2) is a interface exposed by QM that we must support.
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If we can’t support it, the migration fails.
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In physics speak - we isolate the SU(2) representation structure. Then we evaluate only whether the generators exposed by the proposed structure allows to satisfy the Lie algebra.
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[S7]
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Basically we must derive SU(2) from the Naga in an elegant way. If we make any assumptions doing so, those same assumptions carry over to all other integrations tests.
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That’s the contract.
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[S8]
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Importantly, this test does not require
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Not Lorentz invariance.
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Not isotropy of space.
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Not full quantum field theory.
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Just this single interface Naga - > SU(2)
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[S9]
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It might be unnecessary to say this, but you never know, some people are sloppy when meting out criticism. So passing this test does NOT mean
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1. that QM is discrete
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2. That Bells theorem is not relevant
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3. Or that the theory not falsifiable
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It just mean that given the assumptions and tests, the test either fails or passes.
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If it passes, it earns the right to attempt the next one.
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That’s it. It’s test-driven development.
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[S10]
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A Cellular Automaton is discrete.
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SU(2) is continuous.
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To derive something continuous from something discrete requires that the continuous is an approximation to the discrete.
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And this is very
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[S11]
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So Spinors liven 3D space. Which means it should be possible to rotate them like this cup. I rotate it 360° brings you back.
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But spinors are weird, they don’t work like that.
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It’s kind of like this sock. Now if I rotate this sock 360 degrees, it returns back to where it was. But spinors is kind of like a sock that gets inside out each time it rotates. So when I rotate it 360 degrees it inverts. This inversion is captured by this extra -1. Then if I rotate it again it inverts again. So to return completely to both the same position and the same inside-out, we need two rotations.
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That “extra twist” is not decorative is deep in QM. You’re made from objets which rotate in this way, not the normal way.
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[S12]
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What this means for our CA, is that if it naturally accumulates some kind of twist deficit after 2π that only resolves after 4π,
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we’re in business.
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Mathematically - we need a two-valued memory of rotation that flips sign after 2π.
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[S13]
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So lets have a look at the CA. Let us first identify when it goes back to normal. We look at the head here, and keep iterating until the head is back in the original position. This takes N steps. So N steps is a full rotation. Now what happens if we apply just half of these steps, where do we en up - we end up here.
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Periodicity alone is not enough. We now examine how orientation accumulates during those steps.
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[S14]
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And if we look carefully, we executed half of the X, half of the Y and half the Z. Which means that this midway point corresponds to half a rotation in all three planes. See here xy-plane, half rotated. Xz plane, half rotated. Yz plane half rotated. Meaning that T = N gives us a full rotation in all these three planes.
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A full rotation in one plane is 2pi, so a full rotation in all three plane should be 6pi. That is surprising. But look
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[S15]
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A full rotation in the xy plane is also half a rotation in the xz plane, because the x-components are shared.
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And a full rotation in they yz plane is also a half rotation in the xz plane, because the z components are shared.
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Meaning that a full rotation in two planes gives us the full rotation in the third plane for free. 2/3 out of 6 pi is 4 pi.
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[S16]
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Did that not convince you? Let’s do it differently.
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Suppose you define a rotation as the theta value of the HEAD relative to the center in a chosen plane.
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You will now have three rotation generators.
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You can’t add them together because they are never orthogonal to both other planes at the same time.
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Each theta value is orthogonal to half the theta value in each of the two remaining planes.
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Meaning you have theta = theta_xy + 1/2 theta_xz + 1/2 theta_yz.
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A slightly more general form of this is : 2/3(theta_xz + theta_xy + theta_yz)
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So if after a full rotation in a plane theta_xy = theta_xz = theta_yz = 2\pi, then the full rotation becomes
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2/3 (2 pi + 2pi + 2pi ) = 4 pi
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[S17]
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