Worth pointing out that we aren't Turing equivalent machines - infinite storage is not a computability class that is realizable in the universe, so far as we know (and such a claim would require extraordinary evidence).
As well, perhaps, worth noting that because a subset of the observable universe is performing some function, then it is an assumption that there is some finite or digital mathematical function equivalent to that function; a reasonable assumption but still an assumption. Most models of the quantum universe involve continuously variable values, not digital values. Is there a Turing machine that can output all the free parameters of the standard model?
> Is there a Turing machine that can output all the free parameters of the standard model?
Sure, just hard code them.
> As well, perhaps, worth noting that because a subset of the observable universe is performing some function, then it is an assumption that there is some finite or digital mathematical function equivalent to that function; a reasonable assumption but still an assumption. Most models of the quantum universe involve continuously variable values, not digital values.
Things seem to be quantised at a low enough level.
Also: interestingly enough quantum mechanics is both completely deterministic and linear. That means even if it was continuous, you could simulate it to an arbitrary precision without errors building up chaotically.
(Figuring out how chaos, as famously observed in the weather, arises in the real world is left as an exercise to the reader. Also a note: the Copenhagen interpretation introduces non-determinism to _interpret_ quantum mechanics but that's not part of the underlying theory, and there are interpretations that have no need for this crutch.)
> Things seem to be quantised at a low enough level.
Some things. But others are very much not: in particular, space and time are not quantized, and in fact are not even quantizable. A theory in which there is some discrete minimal unit of space (or of spacetime) is trivially incompatible with special relativity, so it is incompatible with quantum mechanics (QFT, specifically).
This is easy to see from the nature of the Lorenz transformation: if two objects are at a distance D = n*min for some observer, they will be at a distance D' = n*min*gamma for some other observer, where gamma is always < 1 for an o server moving at a higher speed in a direction aligned with the two objects. So the distance for that second observer will be a non-integer multiple of the minimum distance, so your theory is no longer quantized.
We do not know if spacetime is quantized or not, and there are theories claiming that it is (LQG etc). And sure, we don't have a coherent "theory of everything" that includes quantized spacetime, and the models that we do have are contradictory if spacetime is quantized. But we already know that those models are deficient.
Our current theories only work if spacetime is not quantized. Any theory that tries to quantize spacetime needs to somehow replace special relativity and its Lorrentz transform with something completely different, while still remaining consistent with the huge number of observations that confirm SR works - especially the extremely precise experiments that confirmed QFT. This is one of the reasons why LQG is almost certainly wrong, by the way.
Note that this is separate from the problem with GR-QFT inconsistencies. All of our current theories are based on and only work if spacetime is continuous. While it's not impossible that a new theory with quantized spacetime could exist and work, it's not at all required.
The one thing about spacetime that we do believe might be quantizable, and would have to be quantized for GR and QFT to be compatible, is the curvature of spacetime. But even if spacetime can only be curved in discrete quanta, that would not mean that position would be quantized.
That would be super lucky if possible - almost all reals are not computable. How would we initialize or specify this Turing machine? Going to use non-constructive methods?
Given how quickly reality unfolds, it's a bit of a stretch to assume that "simulate it to arbitrary precision" means "computable in a digital representation in real time." I mean, if we have Turing machines that did each computation step n in 2^{-n} time.
> That would be super lucky if possible - almost all reals are not computable. How would we initialize or specify this Turing machine? Going to use non-constructive methods?
The standard model doesn't use arbitrary reals. All the parameters are rational numbers with finite precision.
Obviously, your Turing machine can only hard code a finite amount of information about the parameters, eg whatever finite prefix of their decimal expansion is known.
Btw, the speed of light is one of those parameters that's 'known' with absolute precision thanks to some clever definitions. We can do similar tricks with many of the other parameters, too.
I don’t think that’s true, there are parameters that must be measured experimentally: https://spinor.info/weblog/?p=6355 (I am not a physicist, so I welcome education to clarify this).
As well, perhaps, worth noting that because a subset of the observable universe is performing some function, then it is an assumption that there is some finite or digital mathematical function equivalent to that function; a reasonable assumption but still an assumption. Most models of the quantum universe involve continuously variable values, not digital values. Is there a Turing machine that can output all the free parameters of the standard model?