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That all of physics was perfectly beautiful and symmetric except for hyperspace, artificial gravity, shields and a few weapon types.

Oh, this is genius. I love this.

Ahhh! Yes, this is very helpful! Thanks for the explanation.

Question: if I'm considering an isolated system (~= "the entire universe"), you say that I can swap between state-vector-format and matrix-format via

. But later, you say...

If  is uncoupled to its environment (e.g. we are studying a carefully vacuum-isolated system), then we still have to replace the old state vector picture  by a (possibly rank ) density matrix ...

But if , how could it ever be rank>1?

(Perhaps more generally: what does it mean when a state is represented as a rank>1 density matrix? Or: given that the space of possible s is much larger than the space of possible s, there are sometimes (always?) multiple s that correspond to some particular ; what's the significance of choosing one versus another to represent your system's state?)

That is... a very interesting and attractive way of looking at it. I'll chew on your longer post and respond there!

I have an Anki deck in which I've half-heartedly accumulated important quantities. Here are mine! (I keep them all as log10(value in kilogram/meter/second/dollar/whatever seems natural), to make multiplication easy.)

QuantityValue
Electron mass-30
Electron charge-18.8
Gravitational constant-10.2
Reduced Planck constant-34
Black body radiation peak wavelength-2.5
Mass of the earth24.8
Moon-Earth distance8.6
Earth-sun distance11.2
log10( 1 )0
log10( 2 )0.3
log10( 3 )0.5
log10( 4 )0.6
log10( 5 )0.7
log10( 6 )0.8
log10( 7 )0.85
log10( 8 )0.9
log10( 9 )0.95
Boltzmann constant-22.9
1 amu-26.8
1 mi3.2
1 in-1.6
Earth radius6.8
1 ft-0.5
1 lb-0.3
world population10
US federal budget 202312.8
SWE wage (per sec)-1.4
Seattle min wage (per sec) 2024-2.3
1 hr3.6
1 work year6.9
1 year7.5
federal min wage (per sec)-2.7
1 acre3.6

I thank you for your effort! I am currently missing a lot of the mathematical background necessary to make that post make sense, but I will revisit it if I find myself with the motivation to learn!

This is a good point! I'll send you $20 if you send me your PayPal/Venmo/ETH/??? handle. (In my flailings, I'd stumbled upon this "fractional step" business, but I don't think I thought about it as hard as it deserved.)

How are you defining "basically equivalent"

Nyeeeh, unfortunately, sort of "I know it when I see it." It's kinda neat being able to take a fractional step of a classical elementary CA, but I'm dissatisfied because... ah, because the long-run behavior of the fractional-step operator is basically indistinguishable from the long-run behavior of the corresponding classical CA.

So, tentative operationalization of "basically equivalent":  is "basically equivalent" to a classical elementary CA if the long-run behavior of  is very close to the long-run behavior of some , i.e., uh, 

...but I can already think of at least one flaw in this operationalization, so, uh, I'm not sure. (Sorry! This being so fuzzy in my head is why I'm asking for help!)

I was imagining the tape wraps around! (And hoping that whatever results fell out would port straightforwardly to infinite tapes.)

I've never been familiar enough with group-theory stuff to memorize the names (which, warning, also might mean that it will take you a lot of time to write a sufficiently-dumbed-down version), but the internet suggests is related to... the Minkowski metric? I would be flabbergasted to learn that something so specific-to-our-universe was relevant to this toy mathematical contraption.

I think compared to the literature you're using an overly restrictive and nonstandard definition of quantum cellular automata.

That makes sense! I'm searching for the simplest cellular-automaton-like thing that's still interesting to study. I may have gone too far in the "simple" direction; but I'd like to understand why this highly-restricted subset of QCAs is too simple.

Specifically, it only makes sense to me to write as a product of operators like you have if all of the terms are on spatially disjoint regions.

Hmm! That's not obvious to me; if there's some general insight like "no operator containing two ~'partially overlapping' terms like  can be unitary," I'd happily pay for that!

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