Science

Jim__

(14,083 posts) Wed Dec 9, 2015, 04:16 PM Dec 2015

Quantum physics problem proved unsolvable: Godel and Turing enter quantum physics

...

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."

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Quantum physics problem proved unsolvable: Godel and Turing enter quantum physics (Original Post) Jim__ Dec 2015 OP

first major problem in physics for which such a fundamental limitation could be proven? longship Dec 2015 #1

This is different. drm604 Dec 2015 #2

No it isn't. longship Dec 2015 #5

You refer to "uncertainty of the measurement"; Ghost Dog Dec 2015 #7

Well, Heisenberg is not just about measurements. longship Dec 2015 #8

Yes it is. drm604 Dec 2015 #11

Well, I actually studied physics. longship Dec 2015 #12

If so then maybe that article is wrong in comparing it to Turing and undecidability. drm604 Dec 2015 #13

Well, we'll see when the peer review comes in. longship Dec 2015 #15

The article in phys.org agrees with the abstract of the peer reviewed paper in Nature. Jim__ Dec 2015 #17

Not quite Kelvin Mace Dec 2015 #3

No, what Heisenberg said... longship Dec 2015 #14

Plus, Heisenberg's energy/time relationship is important to both quantum foam and dark energy. longship Dec 2015 #16

This message was self-deleted by its author Fortinbras Armstrong Dec 2015 #18

Heisenberg and Schrödinger get pulled over for speeding. central scrutinizer Dec 2015 #4

Well played! nt longship Dec 2015 #6

No. Unfortunately we are limited to copying only 4 paragraphs. Jim__ Dec 2015 #9

I cross-posted to LBN bananas Dec 2015 #10

If the problem is undecidable... DetlefK Dec 2015 #19

It's about a model/description of the material not the material itself. jakeXT Dec 2015 #20

Absolutely fascinating thread, loved reading all the responses.....thank you :) AuntPatsy Dec 2015 #21

longship

(40,416 posts)

1. first major problem in physics for which such a fundamental limitation could be proven?

Reply to Jim__ (Original post)

Wed Dec 9, 2015, 04:20 PM

Dec 2015

Does the author mean other than the Heisenberg uncertainty principle?

drm604

(16,230 posts)

2. This is different.

Reply to longship (Reply #1)

Wed Dec 9, 2015, 04:42 PM

Dec 2015

The uncertainty principle concerns things where there is no answer until the process has played out. Undecidable problems do have an answer, just one that cannot be calculated mathematically.

This isn't about the position of a particle (or wave...), it's about physical law.

longship

(40,416 posts)

5. No it isn't.

Reply to drm604 (Reply #2)

Wed Dec 9, 2015, 04:52 PM

Dec 2015

It is about measurements in which the accuracy of one measurement determines the inaccuracy of the other in certain orthogonal variables.

Delta P * Delta V >= h-bar

Delta E * Delta t >= h-bar

Where P is position, V is velocity, E is energy, t is time, h-bar is Planck constant divided by 2*pi.

Delta is the uncertainty of the measurement.

Ghost Dog

(16,881 posts)

7. You refer to "uncertainty of the measurement";

Reply to longship (Reply #5)

Wed Dec 9, 2015, 05:14 PM

Dec 2015

The OP refers to (theoretical, or posited presumably) "a complete microscopic description".

Not the same?

longship

(40,416 posts)

8. Well, Heisenberg is not just about measurements.

Reply to Ghost Dog (Reply #7)

Wed Dec 9, 2015, 05:29 PM

Dec 2015

As anybody who has studied physics could tell you, the orthogonal variables are actually part of how the universe works.

It fucking ain't just about measurements. It is how nature actually is.

The Delta-E * Delta-t >= h-bar inequality gives us quantum foam, and likely a start of understanding of dark energy.

So there's that.

drm604

(16,230 posts)

11. Yes it is.

Reply to longship (Reply #5)

Wed Dec 9, 2015, 06:09 PM

Dec 2015

This isn't about the uncertainty of a measurement. This is about undecidability, which is a different thing.

While I am far from an expert, I studied this in my computer science courses back in college.

https://en.wikipedia.org/wiki/Undecidable_problem

longship

(40,416 posts)

12. Well, I actually studied physics.

Reply to drm604 (Reply #11)

Wed Dec 9, 2015, 07:24 PM

Dec 2015

So, no it isn't.

See my other responses here. Esp., #8

My best to you.

drm604

(16,230 posts)

13. If so then maybe that article is wrong in comparing it to Turing and undecidability.

Reply to longship (Reply #12)

Wed Dec 9, 2015, 07:29 PM

Dec 2015

I can only go by what's in the article.

longship

(40,416 posts)

15. Well, we'll see when the peer review comes in.

Reply to drm604 (Reply #13)

Wed Dec 9, 2015, 07:36 PM

Dec 2015

That is what science does.

But Heisenberg was on top of something important. And it has been recapitulated by Gödel in number theory and Turing in computability. But only Heisenberg had the cojones to state that nature is actually like that.

So, AFAIK, Heisenberg was first.

Jim__

(14,083 posts)

17. The article in phys.org agrees with the abstract of the peer reviewed paper in Nature.

Reply to drm604 (Reply #13)

Thu Dec 10, 2015, 04:57 AM

Dec 2015

It's published in Nature and the article at phys.org agreees with what is claimed in the paper's abstract:

The spectral gap—the energy difference between the ground state and first excited state of a system—is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang–Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

And yes, this is different from Heisenberg's uncertainty principle. According to an article in Scientific American, when physicists refer to Heisenberg's uncertainty principle, they are referring to Kennard's formulation:

Kennard's formulation is therefore totally different from Heisenberg's. But many physicists, probably including Heisenberg himself, have been under the misapprehension that both formulations describe virtually the same phenomenon. The one that physicists use in everyday research and call Heisenberg's uncertainty principle is in fact Kennard's formulation. It is universally applicable and securely grounded in quantum theory. If it were violated experimentally, the whole of quantum mechanics would break down. Heisenberg's formulation, however, was proposed as conjecture, so quantum mechanics is not shaken by its violation.

And the wikipedia entry on the uncertainty princple has a proof of Kennard's formulation - I can't cut and paste it but you can view it on the cited page - expand the Proof of the Kennard inequality using wave mechanics. So, Heisenberg's uncertainty principle is certainly not a mathematically undecidable problem.

Kelvin Mace

(17,469 posts)

3. Not quite

Reply to longship (Reply #1)

Wed Dec 9, 2015, 04:44 PM

Dec 2015

Heisenberg said, to put it basically, if you know where a particle is, you can't know its momentum. If you know the particle's momentum, you can't know its position. So, you can know both values for a particle, just not at the same time.

longship

(40,416 posts)

14. No, what Heisenberg said...

Reply to Kelvin Mace (Reply #3)

Wed Dec 9, 2015, 07:30 PM

Dec 2015

...is that the universe is actually like that. It is not just measurements. And if you think it is, you do not understand the physics behind quantum theory.

Sorry, my friend.

longship

(40,416 posts)

16. Plus, Heisenberg's energy/time relationship is important to both quantum foam and dark energy.

Reply to Kelvin Mace (Reply #3)

Wed Dec 9, 2015, 11:45 PM

Dec 2015

So Heisenberg is not just about momentum and position.

QED

Response to Kelvin Mace (Reply #3)

Fortinbras Armstrong This message was self-deleted by its author.

central scrutinizer

(11,661 posts)

4. Heisenberg and Schrödinger get pulled over for speeding.

Reply to longship (Reply #1)

Wed Dec 9, 2015, 04:50 PM

Dec 2015

The cop asks Heisenberg "Do you know how fast you were going?"

Heisenberg replies, "No, but we know exactly where we are!"

The officer looks at him confused and says "you were going 108 miles per hour!"

Heisenberg throws his arms up and cries, "Great! Now we're lost!"

The officer looks over the car and asks Schrödinger if the two men have anything in the trunk.

"A cat," Schrödinger replies.

The cop opens the trunk and yells "Hey! This cat is dead."

Schrödinger angrily replies, "Well he is now."

longship

(40,416 posts)

6. Well played! nt

Reply to central scrutinizer (Reply #4)

Wed Dec 9, 2015, 04:56 PM

Dec 2015

Jim__

(14,083 posts)

9. No. Unfortunately we are limited to copying only 4 paragraphs.

Reply to longship (Reply #1)

Wed Dec 9, 2015, 05:44 PM

Dec 2015

The fundamental problem that they are talking about is more fully described in the preceding paragraph:

A mathematical problem underlying fundamental questions in particle and quantum physics is provably unsolvable, ...

The mathematical problem is provably unsolvable. My understanding - based on textual descriptions, not a mathematical understanding of the proofs - is that Heisenberg's Uncertainty Principle has been mathematically proven.

From Scientific American there is an ambiguity attached to the phrase:

And the same article describes Heisenberg's Uncertainty Principle as a physical law:

Ozawa's formulation confirms an emerging trend in probing the foundations of physics: to hew closely to what experimenters directly see in the lab—a so-called operational approach. "The error–disturbance uncertainty relation is much more important than that of fluctuations," says Akio Hosoya, a theoretical physicist at Tokyo Institute of Technology, "because in physics the final say comes from experimental verification." Heisenberg would be pleased that the limitation we can know about the world, which he aimed to expressed, was this time clearly revealed with the new rigorous, experimentally verified formulation. The new uncertainty relation between measurement error and disturbance is no more just conjecture, but physical law.

Wikipedia has a proof of Kennard's version of the inequality - it doesn't really cut and paste due to the notation, but you can read it on the wikipedia page. So, it is not the same thing.

bananas

(27,509 posts)

10. I cross-posted to LBN

Reply to Jim__ (Original post)

Wed Dec 9, 2015, 06:09 PM

Dec 2015

Thanks for posting this, it's a very important result.
LBN thread: http://www.democraticunderground.com/10141284965

DetlefK

(16,423 posts)

19. If the problem is undecidable...

Reply to Jim__ (Original post)

Thu Dec 10, 2015, 08:59 AM

Dec 2015

How does the material know what attributes it should have macroscopically?

jakeXT

(10,575 posts)

20. It's about a model/description of the material not the material itself.

Reply to DetlefK (Reply #19)

Thu Dec 10, 2015, 07:01 PM

Dec 2015

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.

But we don't understand fully how the world works, so how can you model it?

What happened to photonic matter ?

"Most of the properties of light we know about originate from the fact that photons are massless, and that they do not interact with each other," Lukin said. "What we have done is create a special type of medium in which photons interact with each other so strongly that they begin to act as though they have mass, and they bind together to form molecules. This type of photonic bound state has been discussed theoretically for quite a while, but until now it hadn't been observed.

Read more at: http://phys.org/news/2013-09-scientists-never-before-seen.html

AuntPatsy

(9,904 posts)

21. Absolutely fascinating thread, loved reading all the responses.....thank you :)

Reply to Jim__ (Original post)

Thu Dec 10, 2015, 11:30 PM

Dec 2015

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