A mathematical problem underlying fundamental questions in particle and quantum physics is provably unsolvable, according to scientists at UCL, Universidad Complutense de Madrid - ICMAT and Technical University of Munich.

It is the first major problem in physics for which such a fundamental limitation could be proven. The findings are important because they show that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.

A small spectral gap - the energy needed to transfer an electron from a low-energy state to an excited state - is the central property of semiconductors. In a similar way, the spectral gap plays an important role for many other materials. When this energy becomes very small, i.e. the spectral gap closes, it becomes possible for the material to transition to a completely different state. An example of this is when a material becomes superconducting.

Mathematically extrapolating from a microscopic description of a material to the bulk solid is considered one of the key tools in the search for materials exhibiting superconductivity at ambient temperatures or other desirable properties. A study, published today in *Nature*, however, shows crucial limits to this approach. Using sophisticated mathematics, the authors proved that, even with a complete microscopic description of a quantum material, determining whether it has a spectral gap is, in fact, an undecidable question.

"Alan Turing is famous for his role in cracking the Enigma code," said co-author, Dr Toby Cubitt from UCL Computer Science. "But amongst mathematicians and computer scientists, he is even more famous for proving that certain mathematical questions are `undecidable' - they are neither true nor false, but are beyond the reach of mathematics. What we've shown is that the spectral gap is one of these undecidable problems. This means a general method to determine whether matter described by quantum mechanics has a spectral gap, or not, cannot exist. Which limits the extent to which we can predict the behaviour of quantum materials, and potentially even fundamental particle physics."

**One million dollars to win!**

The most famous problem concerning spectral gaps is whether the theory governing the fundamental particles of matter itself - the standard model of particle physics - has a spectral gap (the `Yang-Mills mass gap' conjecture). Particle physics experiments such as CERN and numerical calculations on supercomputers suggest that there is a spectral gap. Although there is a $1m prize at stake from the Clay Mathematics Institute for whoever can, no one has yet succeeded in proving this mathematically from the equations of the standard model.

Dr Cubitt added, "It's possible for particular cases of a problem to be solvable even when the general problem is undecidable, so someone may yet win the coveted $1m prize. But our results do raise the prospect that some of these big open problems in theoretical physics could be provably unsolvable."

"We knew about the possibility of problems that are undecidable in principle since the works of Turing and Gödel in the 1930s," added Co-author Professor Michael Wolf from Technical University of Munich. "So far, however, this only concerned the very abstract corners of theoretical computer science and mathematical logic. No one had seriously contemplated this as a possibility right in the heart of theoretical physics before. But our results change this picture. From a more philosophical perspective, they also challenge the reductionists' point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description."

**Not all bad news**

Co-author, Professor David Pérez-García from Universidad Complutense de Madrid and ICMAT, said: "It's not all bad news, though. The reason this problem is impossible to solve in general is because models at this level exhibit extremely bizarre behaviour that essentially defeats any attempt to analyse them. But this bizarre behaviour also predicts some new and very weird physics that hasn't been seen before. For example, our results show that adding even a single particle to a lump of matter, however large, could in principle dramatically change its properties. New physics like this is often later exploited in technology."

The researchers are now seeing whether their findings extend beyond the artificial mathematical models produced by their calculations to more realistic quantum materials that could be realised in the laboratory.

**Explore further:**
Physicists explain the unusual behavior of strongly disordered superconductors

**More information:**
Undecidability of the spectral gap, *Nature*, DOI: 10.1038/nature16059

## axemaster

I suspect they're talking about the inability to find an analytic solution or something like that?

## vpoko

## julianpenrod

## julianpenrod

## nwarden

## Hyperfuzzy

## Hyperfuzzy

## Eikka

That is making the metaphysical assumption that all reality is doing is a computation, which is a reasonable assumption but not necessarily true.

For example, how do you compute a random number? You don't - no computer is capable of coming up with a random number by following any rule or law - assuming such things do exist.

## axemaster

That's not really what I meant, and I would say that is NOT a reasonable assumption.

I simply meant, in reality the physical system interacts with itself and the environment, and it goes through whatever motions end up producing the energy gap. Thus "computing" the result, though of course it's not solving an equation or anything.

I agree 100% with this - you CANNOT propose that any physical system (actually any logical system whatsoever) has any form of indeterminacy. This is the major motivation for the various philosophical debates in quantum physics. Quantum mechanics, if you look at the math for calculating observables, uses indeterminate systems, which is simply an unacceptable situation.

## ralph638s

## Protoplasmix

- - -

If you need to know the initial conditions, you can set them, and as Axe mentioned, the physical system has no qualms with the calculation.

But I think the actual question being asked is more like, "does the physical system employ a mass gap to perform its calculations?"

## ralph638s

## elerner

## axemaster

I hardly think I need to point out the logical problems with this statement. Infinite information densities in space and time are not acceptable for a theory that claims to describe actual physical systems. This is also a problem in quantum mechanics that a lot of people seem unaware of - even if you quantize everything you can, you still have a continuous probability wave function - and if you assume that the wave function is a real thing, you now have infinite information required to describe the physical system.

This is why quantization of space and time (or something that accomplishes the same effect) is required for a theory of everything.

## eachus

Hmm. Make that easier to understand. Say there is a limit (in my compiler) of 4000 odd identifiers in one scope. You run into it, and put in a nested block split the identifiers between them and off you go. But giving wrong answers to arithmetic? Very bad.

## Protoplasmix

If the range and domain of a function you're interested in can be accurately modeled with a hydrogen atom, are you saying hydrogen atoms can't be used to perform the calculations of the function? The result imposes subsequent action, and would be the "deciding factor," insofar as hydrogen atoms "decide" things.

## eachus

Come on, this is easy. Can you measure the gap accurately enough to know that it is non-zero, or does quantum uncertainty prevent it? See proof described above. ;-) It may be easy to determine if the gap is greater than (or approximately equal to) one. But that is a different question.

## FredJose

## elerner

Also, band-gap measurements are routine today and are done on macroscopic samples, so yes, they are easy to measure.

## Protoplasmix

Seriously though, I was thinking it should be possible to compile the underlying math of the program, and then use a general algorithm to see if the "compiled, distilled and condensed version" contains the equivalent of a statement about the natural numbers..? Because then you're not running a program, you're evaluating math, and so then you could say if it does or doesn't halt.

Error 42

Undecidable math error

## axemaster

This is only the case for continuous systems, which I am arguing do not exist. Ilya is literally restating my point.

This is not the case. The quantization of space and time will be determined by the underlying logic structure that the universe is operating on. At the very least, we will be able to narrow the possible quantizations to a useful sub-set.

This only applies to volumetric quantization, which I have previously argued makes some very naive assumptions about the underlying structure of the universe. Also, I started arguing against volumetric quantization over 5 years ago, this isn't something I came up with after reading this article.

## antialias_physorg

Not necessarily.

E.g. the "halting on all inputs" problem. Evaluating it mathematically means you could formalize the inputs and solve this (as a single operation or a an operation on finite number of sets of inputs which sum up to all possible inputs).

As with the Gödel theorem you also run into trouble with recursively defined problems. And here is where the issue with 'real' systems lie (I think). They don't just interact but interact recursively (because of time delay of forces acting on each other).

## antialias_physorg

*which you can't because there is no formalism that encapsulates all possible inputs. It would be like writing the "set of all sets which - among other things - include this set...and the set that includes THAT set...and ...".

## panamars

Microcosmos Geometrically Related to Megacosmos:

https://www.youtu...QFXlWvE0

-------------------------------------------------------------------------------------------------------

http://www.stefan...IDES.pdf

Regarfds from Athens,

Panagiotis Stefanides

## Hyperfuzzy

## eltodesukane

(Stern–Gerlach experiment: silver atoms travel through an inhomogeneous magnetic field and are deflected up or down depending on their spin.)

## cgsperling

Good one, nwarden. And I love the name "Dr. Cubitt".

## Hyperfuzzy

Why not simply calculate real conditions. Why do we always try prediction without adequate information of the event at hand?

## Protoplasmix

So what I'm suggesting to solve the halting problem is that the algorithm which works for all programs has to do the math (same as a human): identify the class (or define a new one) for the program in question, and then utilize (or generate) the specific algorithm for that class.

In practical terms, only specific instances of the "grand" algorithm are possible. But no machine would have the time to evaluate an infinite number of programs anyway, and no machine could (or would) write that many programs in the first place.

## Hyperfuzzy

What? This sounds like a poorly defined problem. How can a solution exist?

## Hyperfuzzy

## leDendrite

I know, crazy crackpottery, all good.

also a black hole is not a hole but a massive magnetic object,

https://archive.o...sm1small

https://youtu.be/9yi0WHKtRd4

## Protoplasmix

If H works, then sending it into an infinite loop instead of returning true for program P when P halts (to prevent H from halting when true), just to then make it into a contradictory program P' to prove H can't exist, isn't reductio ad absurdum, it's a priori absurd. Because you can feed P' into a clean version of H, which (if it works) gives you H(P', P) = false, P' doesn't halt with P as input when H(H, P) = true.

I tend to have trouble with, "you can't", even when it's the math saying so ...

## Captain Stumpy

you can't send me 30 million US dollars in small unmarked bills escorted by a dozen scantily clad bikini models who just want to rub bacon grease and blue bell ice cream all over my body!

(sorry - i couldn't resist a test! LMFAO)

## Hyperfuzzy

Best answer, and the prize goes to LMFAO!

## Hyperfuzzy

I've seen code like yours, Proto, when testing a WiMax multiple access, multiple frequency, multiple coded and well defined; but, analysis by an idiot ... probably why you were no longer there, complete garbage. Juz say'n

LMFAO!

## eachus

What makes you think that there is a difference between executing a program and evaluating math? Gödel's proofs only talk about math, and for even a very restricted subset of integer arithmetic proves that there are some undecidable problems. The second proof goes the other direction and shows that there are questions which are undecidable in any consistent math. (Inconsistent math allows you to prove both a theorem and its converse and so is useless. If one both equals and does not equal zero, you can prove just about anything.)

## Protoplasmix

Realizing there are other infinite series which converge faster is evaluating the math. See http://www.geom.u...spi.html

And then evaluating the math by executing programs -- see http://www.codepr...-Sequenc

It's all math, the difference is subtle...

## Protoplasmix

@CS - thanks, it may take a while :)

## Hyperfuzzy

## Hyperfuzzy

Actually it was just a conversation of a tech talking about why he was fired from his last assignment. He tried to tell me how the choice of frequencies followed a Gaussian distribution without using a Monte Carlo technique for creation. I didn't laugh in his face, but I had to sit my coffee down to avoid spitting on him. He actually trapped each transition and counted, so his code was a little bit slow. I can see why his boss asked him to prove whatever he was trying to sell.

## tomb

## Hyperfuzzy

That's because we start with stupid and try to figure stuff from there, instead of a complete dichotomy of the problem.

## GoodElf

## SuperThunder

Then where are the quanta it's simulating and what are they doing? Why do we exist in a simulation of those things? Why not the originals? Why simulate quanta at all? Unless you mean it's a simulator for something not based on anything outside the simulation, in which case it's the thing itself and not a simulation.

When is a simulation of something unsimulated still a simulation? Word salad, that's when.

If you think I'm picking on you, at least I can nudge you with my elbow and we can both laugh at you not being the one who actually said "a complete dichotomy of the problem." Some people have a limited constabulary.

## GoodElf

## anywallsocket

I read GEB and yet fail to see your point. Everyone would prefer a greater resolution description of truth - greater than simply whether or not you've got one. But like you say, we've got to build up to the high resolution from these binary subsets. Or, we can crack downward, like you imply with "de-fuzzification". Either way, we've got to be careful of the logical levels we transcend in doing so, in that we often go "outside the classical logic" - defined by that particular level of resolution, otherwise we run risk of contradiction. So are you saying that this result is just such a case, of "forget[ing]' what statements are allowed in order for a classical proof" ? Or do you suggest that these fundamental probes into the binary subsets are doomed to fail due to our bivalent logic? In other words, is it a mathematical or human error to judge a set from outside that set?

## Hyperfuzzy

Is it linear, self referent, or ... in other words, how is the logic defined to solve the problem? i.e.:

This statement is false.

True or false? and this is easy, try a universal problem that includes all, even you. Is it gross pomposity or just ignorance?

with the excluded middle, it's 1/2 true, how about other mathematics to define truth, get it?