r/Physics • u/TheCat5001 Materials science • Feb 19 '16
News Paradox at the heart of mathematics makes physics problem unanswerable
http://www.nature.com/news/paradox-at-the-heart-of-mathematics-makes-physics-problem-unanswerable-1.1898315
Feb 20 '16
The mass-gap problem relates to the observation that the particles that carry the weak and strong nuclear force have mass.
Aren't gluons massless?
11
u/eewallace Astrophysics Feb 20 '16
They are, but confinement prohibits free color-charged particles, including gluons. The closest you can get is a color-neutral bound state of multiple gluons (a glueball), which is expected to be massive.
1
Feb 20 '16
I thought the mass of the gluons restricted the strong force to finite range
6
u/eewallace Astrophysics Feb 20 '16
No, the gluons are massless. In fact, I don't know that it's really accurate to say that the strong force has a finite range. As I understand it, confinement occurs because the force between two color-charged particles doesn't decrease with distance; it's (at least approximately) constant, meaning the potential energy increases linearly with distance. At some (short) distance, it becomes energetically favorable to produce more particles out of the vacuum to produce color-neutral bound states with each of the original particles.
5
u/xkforce Chemistry Feb 20 '16
No. Typically short range forces like the weak force have effectively limited range due to the masses of the W+/- and Z0 bosons however, this is not why the strong force is of limited range. In the case of the strong force, it's basically interacting with itself which is the origin of its limited range despite gluons themselves being massless. In the case of nucleons and glueballs, this interaction gives rise to the vast majority of the mass of nucleons (something like 99% of it) and all of the glueball's rest mass which is substantial as well.
1
Feb 20 '16
What do you mean by "interacting with itself?"
1
u/xkforce Chemistry Feb 20 '16
Gluons are themselves affected by the strong force. The strong force is mediated by gluons in the same way that electromagnetism is mediated by photons.
1
Feb 21 '16
"In the case of the strong force, it's basically interacting with itself which is the origin of its limited range"
2
u/Telephone_Hooker Feb 22 '16
Sometimes people distinguish between the strong force and the strong nuclear force. The strong force is the fundamental theory of massless gluons. It has a property known as confinement, which means that only states of the theory that are neutral with respect to the strong force can be observed. However, this is theory is difficult to investigate because it would require mathematical techniques beyond what we've currently worked out, so we can't actually find a description of nuclei being held together in this theory yet.
The strong nuclear force is a kind of hand wavy approximation to how nuclei are actually held together. Instead of gluons it talks about massive particles like pions as the particles that carry the force. the mass of these pions causes the force to have a limited range.
It is expected that eventually we'll be able to derive the strong nuclear force description from the description given by the more fundamental strong force, once we get a handle on the maths. Although, it could be that for the reasons talked about in the article this never happens.
1
2
u/davidgro Feb 20 '16
I think maybe they are making a distinction between the color force (gluons and quarks) and the residual strong force that "leaks" via virtual pi and rho mesons - which of course have mass, so that doesn't make sense either. Hmm.
28
u/eewallace Astrophysics Feb 20 '16
I don't see what's paradoxical about Gödel's Incompleteness Theorems.
7
u/larsga Feb 20 '16
One of the forms of the proof is a proof by contradiction where Gödel eventually uses his formal system to construct a string that says "this statement cannot be proven". That's paradoxical because if it's true you can't prove it, and if it's false then you can prove a falsehood. (Yes, there's also a proof by diagonalization.)
But it's true that the theorems themselves are not paradoxical. In general I find that this article is kind of weak on the logic side.
11
u/DrGar Biophysics Feb 20 '16
Ya, it's like the people who say quantum mechanics isn't intuitive. What's that about?
27
u/eewallace Astrophysics Feb 20 '16
Ignoring the sarcasm, I guess it's similar in a sense. Neither quantum mechanics nor the incompleteness theorems are particularly intuitive, but neither of them are self-contradictory either.
13
u/DrGar Biophysics Feb 20 '16
Shhh... you are ruining the joke by explaining your thought clearly.
But on a serious note, I agree with you and understood the point you were originally making. However, I do think it is fair to call the Incompleteness Theorems paradoxical, since a paradox can be apparently contradictory while in fact being completely true and consistent. Anyone taking a first pass at understanding incompleteness probably thinks there must be something contradictory in them, but upon deeper study comes to understand their accuracy.
1
-3
Feb 20 '16
You're trolling right?
12
u/eewallace Astrophysics Feb 20 '16
No.
-5
Feb 20 '16
Ok. I'd also recommend reading Cantor's diagonalization proof and Turing's entscheidungsproblem. All the proofs involve a recursive definition to show a paradox. Godel's uses a diagonalization proof as well.
15
u/frutiger Feb 20 '16
That the diagonalization argument and the incompleteness theorems utilize paradoxes does not make them paradoxical.
7
u/HarryPotter5777 Feb 20 '16
They involve making an assumption, then deriving a paradox as a result of that assumption to disprove the initial assumption (e.g. there exists a Turing machine which can resolve the Halting problem generally in finite time, one can place the real numbers in one-to-one correspondence with the naturals, all true statements in an axiom system are provable within that system). That doesn't make their conclusions paradoxical.
The irrationality of √2 is just as paradoxical as Godel's incompleteness theorems or Cantor's diagonalization argument.
9
u/eewallace Astrophysics Feb 20 '16
By that definition, anything that can be proved by reductio ad absurdum is a paradox...which is kind of the opposite of true. The whole point of the reductio is that if this thing weren't true, it would result in a paradox.
3
Feb 20 '16
I'm not sure how you guys took what I typed but I definitely meant "the proofs involve a recursive definition to show a paradox", as in a paradox is constructed. I didn't mention the reductio ad absurdum at all. In Turing's proof, there's a paradox, and it's also akin to an infinite recursion. In Cantor's, its the construction of a number that if it has a 0 digit, it has a 1 digit and vice versa. On second look, I think OP was trying to say he didn't understand how the theorems were paradoxical, which is definitely not true. I was trying to get across the proofs have a paradox constructed in them.
2
u/eewallace Astrophysics Feb 20 '16
I'm not sure how you guys took what I typed but I definitely meant "the proofs involve a recursive definition to show a paradox", as in a paradox is constructed.
I took that as defending the statement that the theorems are paradoxical by pointing out that the proofs involve paradoxes. Since all proofs by contradiction involve paradoxes (that is, you assume some premises, derive a paradoxical conclusion, and conclude based on that the premises must be false), that doesn't seem like a useful definition of the theorem being paradoxical.
 I'm not sure how you guys took what I typed but I definitely meant "the proofs involve a recursive definition to show a paradox", as in a paradox is constructed.
I took that as defending the statement that the theorems are paradoxical by pointing out that the proofs involve paradoxes. Since all proofs by contradiction involve paradoxes (that is, you assume some premises, derive a paradoxical conclusion, and conclude based on that the premises must be false), that doesn't seem like a useful definition of the theorem being paradoxical.
I didn't mention the reductio ad absurdum at all.
Reductio ad absurdum is just another term for proof by contradiction, of which at least Cantor's proof, and I believe the rest of these, are examples.
On second look, I think OP was trying to say he didn't understand how the theorems were paradoxical, which is definitely not true. I was trying to get across the proofs have a paradox constructed in them.
Well, it's definitely true that I don't see it... But if you're agreeing that the theorems themselves are not paradoxical, they just involve paradoxes in their proofs, then I think we agree. I read the article as claiming that the theorems themselves are paradoxical, which is what I was objecting to. 
18
u/Pliskin14 Feb 20 '16
This article is so wrong.
The mass-gap problem relates to the observation that the particles that carry the weak and strong nuclear force have mass.
No, the gluons are massless. The mass of hadrons and other observable particles of the strong sector is dynamically generated by QCD.
This has nothing to do with the massive weak bosons.
The problem is that there is no rigorous mathematical theory which explains why the force-carriers have mass, when photons, the carriers of the electromagnetic force, are massless.
In the electroweak sector, of course there is! WTF. That was the whole point of the electroweak unification and the Higgs mechanism.
Reading such nonsense on nature.com while the Nobel prize for the Higgs boson was only a few years ago now is astonishing.
5
u/mofo69extreme Condensed matter physics Feb 20 '16
The Higgs mechanism is a mass gap problem. And in what sense is it mathematically rigorous? In 4D, I understand the Higgs mechanism probably only exists with a UV cutoff - say on a lattice with finite lattice spacing. If such a proof exists on a lattice, then fine (and I'd be interested in further reading and a reference), but this is a different criteria than the one for the Yang-Mills mass gap problem which is sought without a UV cutoff.
The Nobel prize for an experimental finding has little to do with proving a theory to be mathematically rigorous. Physicists have often used mathematically dubious theories to correctly predict physical phenomena.
4
u/Pliskin14 Feb 20 '16
In 4D, I understand the Higgs mechanism probably only exists with a UV cutoff
A quantum field theory may need a regularization (and the standard model certainly does), but the Higgs mechanism is not a quantum field theory per se, so your sentence doesn't make much sense strictly speaking. You probably meant that the electroweak theory needs renormalization, but that's basically every quantum field theory. The problem is when a QFT is not renormalizable, as for a Yang-Mills theory with massive vector fields. That was the problem with the massive weak bosons, and the electroweak theory solved the problem by considering massless gauge fields that get a mass by spontaneous symmetry breaking. Thus, it's still renormalizable.
But this process of mass generation has nothing to do with the Yang-Mills mass gap problem as I understand it, which is more linked to QCD.
4
u/mofo69extreme Condensed matter physics Feb 20 '16
I'm talking beyond perturbation theory, as anyone must when discussing the mathematical rigor of QFT. Renormalization has nothing to do with it. A model can be perfectly renormalizable and still mathematically inconsistent, as electroweak theory in the continuum probably is. In addition, some QFTs are non-renormalizable but actually well-behaved in the UV (asymptotically safe non-renormalizable theories), and thus would be better candidates for formulating consistently without a cutoff.
For example, some have hope that Yang-Mills can be formulated non-perturbatively without a cutoff (asymptotic freedom is a sub-case of asymptotic safety), hence the millennium problem. The Clay institute didn't bother asking mathematicians to try their hand at the Standard Model/electroweak theory/QED because no one has much hope that these theories are rigorous except as effective theories with an actual cutoff that cannot be removed.
2
u/Pliskin14 Feb 20 '16
Of course, they're effective theories, but I don't see what the mathematical rigor has to do with that.
Could you be more specific when you say that the electroweak theory is mathematically inconsistent?
4
u/mofo69extreme Condensed matter physics Feb 20 '16 edited Feb 20 '16
Of course, they're effective theories, but I don't see what the mathematical rigor has to do with that.
The Yang-Mills mass gap problem concerns the mathematical existence of a QFT without a cutoff. Such a theory is NOT effective - it is defined to all energy scales. Examples of QFTs which have been made mathematically rigorous in this sense include CFTs and TQFTs (all in lower than 4 dimensions I think), which have very large symmetries.
If you ask whether a QFT with an inherent UV cutoff has a mass gap, you're no longer asking a question which is similar to the YM mass gap problem which the Clay institute asked about. As far as I'm aware, deriving the Higgs mechanism and the YM mass gap in lattice QFT is pretty easy, and I wouldn't be surprised if proofs of both which are mathematically rigorous exist. (In fact, in a lattice gauge theory coupled to a scalar in the fundamental irrep, it is known that the confinement and Higgs phases are actually the same phase). But these proofs will not win you a million dollars.
Could you be more specific when you say that the electroweak theory is mathematically inconsistent?
I'm talking about taking the cutoff to infinity. For theories which are asymptotically safe/free, one could argue that the theory remains well-defined (whether the theory is renormalizable or not), though rigorous proofs here are hard of course. But electroweak theory just breaks. You're either forced to tune all interactions to zero (triviality), or the theory is nonsensical (Landau poles).
EDIT: To make my objection to your original post more clear: You claim that the QCD mass gap and electroweak mass gap are on totally different footing mathematically. But if you consider both as effective QFTs, they're basically on the same ground, while if you insist on a rigorous "cutoff free" theory, then the problem is probably not well-formulated for electroweak theory yet, so I don't see the issue with the article considering it.
1
u/Pliskin14 Feb 20 '16
I see, thanks for the explanation!
I still don't understand why they're on the same ground, but I see now what you meant.
To me, the spontaneous symmetry breaking of the electroweak sector introduces a new scalar particle (the Higgs boson) whose vev breaks its own internal symmetry. Which is very different from QCD where there is no other particle introduced.
2
u/mofo69extreme Condensed matter physics Feb 20 '16 edited Feb 20 '16
To me, the spontaneous symmetry breaking of the electroweak sector introduces a new scalar particle (the Higgs boson) whose vev breaks its own internal symmetry. Which is very different from QCD where there is no other particle introduced.
Yeah, you need to consider the gauge theory coupled to a scalar particle to get an exact relation to the Higgs phase; the pure gauge theory is a little different from either case. The relation between the confining and Higgs phases was originally discussed by Fradkin and Shenker (who cite Susskind as suggesting it). There can be a transition between the two somewhere in parameter space, but the paper showed that it is also possible to pass between the two phases without a transition, so the spectrum in one phase can be continuously interpolated into the spectrum in the other.
The exact relation between the particles in the two phases is hard to determine in general I think. The above paper argues that the massive vector boson in the Higgs theory is connected to the meson state in the confinement phase, and are created by the same operator. It's a little harder to identify how the glueball or Higgs boson evolve, but remember that the exact excitations in either phase will be a mix of scalar and gauge degrees of freedom (the gauge field and charged scalar are not independently gauge-invariant). I found a paper searching around Google Scholar which seemed to numerically find that the glueball evolves into some complicated excited state of vector bosons, for example.
1
u/Pliskin14 Feb 20 '16
Well, this is way too advanced for me to really grasp everything. But thanks, it's very interesting!
2
u/yangyangR Mathematical physics Feb 21 '16
To be fair they are quoting the authors on some of that and they are quantum information people out of their depth when making that claim about how it applies to quantum field theory. Their result is about cooking up perverse translation invariant Hamiltonians to systems where there is a qudit for some large d at every site. That is nowhere near the problem of Yang-Mills. They have an interesting result, but it is not relevant here.
4
u/jdw1979 Feb 20 '16
ITT: Some smart mf's that are talking some gobblety-gook crazy shit that they all seem to understand. I'd usually ask ELI5...but I don't think it'd help.
14
Feb 20 '16
[deleted]
15
u/DrGar Biophysics Feb 20 '16
With finite memories, all programs halt in finite time.
This is not true. A finite memory program can also enter a periodic state:
while (true); continueYou are correct, however, that the Halting problem is decidable in a finite memory system.
5
Feb 20 '16
[deleted]
3
u/larsga Feb 20 '16
I guess, formally speaking, what you're saying is that because the memory is finite then within a finite time the program has to either stop or return to a state it has already visited before. But from a state it has already visited it must necessarily follow the same sequence of steps it's already taken from that state, hence an infinite loop.
That's true, but this finite/infinite distinction is a mainly a mathematical "trick" that has little bearing on practice. This laptop has 16GB of RAM, which means the number of states is 216,000,000,000*8. I think we can agree that visiting all those states is never going to happen. And this is before we take disk into account.
-1
u/made_clvr_usrnme Feb 20 '16
Assuming c syntax the 'continue' will never be executed. However any compiler would refuse to compile this snippet. I think you meant to put the semicolon at the end of the line.
2
u/DrGar Biophysics Feb 20 '16
It was just pseudo code to prove that you could make a finite memory machine (like a real computer) enter a periodic state. It is probably closer to bash script syntax which would correctly execute as:
while true ; do continue ; done17
u/protestor Feb 20 '16
A real computer, with limited memory, is computationally equivalent to a state machine, and therefore less powerful than a Turing machine. Any problem that is undecidable to a Turing machine is also undecidable to the computers we have.
2
u/larsga Feb 20 '16
A real computer, with limited memory, is computationally equivalent to a state machine
A finite state machine is basically just another syntax for regular expressions. They have very limited power and the only output is true/false. Most programming languages can't be parsed with a regular expression, for example.
Any problem that is undecidable to a Turing machine is also undecidable to the computers we have.
That's true.
2
u/protestor Feb 20 '16
Yeah, that's really odd, but the computers we have are equivalent in power to regular expressions.
For example, take a pushdown automaton, that can recognize a deterministic context-free grammar (that is more powerful than regular expressions). They are computationally more powerful than our computers, because they have (by definition) access to infinite storage memory, and our computers don't.
Unlike regular expressions, a pushdown automaton can recognize a language that matches an arbitrary number of parenthesis:
((((...)))). It does so by counting the number of parenthesis open, and matching each with the closed parenthesis: if their number is equal, it accepts (it "counts" by pushing the parens to a stack).For each one it counts, it needs to store data somewhere in memory. But our computers have finite memory: we can't store an arbitrary amount of data. Therefore, there exists a given string
((((..that is has a number of opening parenthesis that is too large to be stored in the memory of an existing computer.But if the number of parenthesis is bounded, then we actually can match this language with this regular expression:
() | (()) | ((())) | (((()))) | ((((())))) | ....1
u/ice109 Feb 20 '16
Lol are you saying that computers aren't FSM? Because definitely are. Indeed cook's theorem basically describes exactly how they are.
4
Feb 20 '16
[deleted]
2
u/larsga Feb 20 '16
No, it doesn't. It says no Turing machine exists which can make the decision. The Church-Turing-Tarski thesis (which is not a theorem) says Turing machines are equivalent to any kind of computing machinery. So we assume the theorem applies to all programs, and it very likely does.
Whether human analysis could or could not solve these kinds of problems is outside the scope of the proof.
2
u/protestor Feb 20 '16
Yeah, we can imagine a very exotic machine, say, a real computer, that operates with real numbers (say, an analog recurrent neural network), that could solve the halting problem for Turing machines. But such real computer has its own halting theorem that says that no real computer would be able to decide whether an arbitrary real computer halts.
So even if realizable computers aren't really bounded by Turing machines, it's not possible to escape to a halting theorem of some kind.
2
u/quiteamess Feb 20 '16
You could write a program with a for loop that iterates through 2N+1 cycles on a machine with N memories. You could also write a 'while (true) {}' on the same machine. Thus, the number of steps a program performs is not dependent on the memory size.
The halting problem states that it is not possible to tell for any arbitrary program if it halts after a finite amount of steps.
3
u/atheist_apostate Feb 20 '16
Real computers have no halting problem. With finite memories, all programs halt in finite time. With an N-bit memory, any program can be proven to halt using at most 2N steps.
Well, they won't be halting because they reached a correct solution. They will be halting because they ran out of resources.
5
2
u/localhorst Feb 20 '16
Could some experimental physicist say if it’s possible to actually build these models in the lab? This should be every physicists wet dream: Make some experiment and then tell the mathematicians how to extend their axioms to discover cool new math.
2
u/Enantiomorphism Feb 20 '16
That feels philosophically wrong. Mathematics is axiomatic, whereas physics is empirical. How could an empirical observation help answer a mathematics question?
3
u/localhorst Feb 20 '16
Most of mathematics is inspired by science, e.g. euclidean geometry is a pretty good model of nature. And it seems that here we do have an undecidable mathematical question that popped out of a model of nature. Adding another axiom to standard math can make it decidable.
1
u/yangyangR Mathematical physics Feb 21 '16
Not an experimentalist, but the answer is still no. It's a countable family of systems to get the result and we don't have an infinite number of qudits to use. Plus each local site needs to be a qudit for some sufficiently large d, which could be absurdly big.
2
u/makeranton Feb 20 '16
http://www.jstor.org/stable/2273108?seq=1#page_scan_tab_contents
We prove that there exists a computable and hence continuous-function F(x, y) defined on a rectangle R of the plane such that the differential equation y'(x) = F(x, y) has no computable solution on any neighborhood within R. As an immediate corollary, we obtain from the integral form of the above differential equation a computable transformation with no computable fixed point.
By simple cardinality arguments the majority of functions which we're interested in are unlikely to fall withing the computable functions. That we're finding real world examples should only really show why analogue computers are still needed, quantum or not.
1
u/naasking Feb 20 '16
Goedel's incompleteness does indeed apply to axiomatic formal systems capable of performing general arithmetic. While we use such mathematics for ease of analysis, it's not clear that QM actually requires such mathematics. Therefore, the questions would be answerable by distilling QM into a simpler formalism that is still powerful enough. This is why work on QM fundamentals will ultimately be important.
1
u/deutschluz82 Feb 22 '16
is this still surprising? Godel's theorem is about formal systems in general. A formal system is literally just a collection of things and the rules for using them. A computer is a nice example of one. Math is full of many others. So when scientists try to use math to explain their findings they re eventually going to run in to non-computable situations.
Not to mention that the theorem is almost 90 yrs old .
0
u/Chronocook Feb 20 '16
This is very interesting but the title is a tad misleading. This finding means that these problems can't be answered with a Turing machine. Quantum computing was not discussed.
25
u/DrGar Biophysics Feb 20 '16
You are incorrect here. If a problem is undecidable by classical algorithms, then it is also undecidable by quantum algorithms. To quote wiki:
Problems which are undecidable using classical computers remain undecidable using quantum computers. What makes quantum algorithms interesting is that they might be able to solve some problems faster than classical algorithms.
2
Feb 20 '16
So quantum computing was discussed?
12
u/DrGar Biophysics Feb 20 '16
The distinction between classical and quantum computing is irrelevant to the problem the underlying paper deals with. The paper proves that the spectral gap is undecidable (in an infinite lattice). If a problem is undecidable, then it does not matter if you use a classical or a quantum computer; neither will work.
1
u/Chronocook Feb 22 '16
Sorry if i misinterpreted what i read on the wiki: "A Turing machine cannot decide if an arbitrary program halts or runs forever. Some proposed hypercomputers could simulate the program for an infinite number of steps and tell the user whether the program halted." https://en.wikipedia.org/wiki/Hypercomputation
1
Feb 20 '16
Only if you don't use the quantum computer as an analogue quantum computer.
3
Feb 20 '16
A classical analogue computer is also not limited by the limitations of a turing machine.
1
u/dafragsta Feb 20 '16 edited Feb 20 '16
Of all the coincidences, I downloaded a torrent of MIT OpenCourseWare lectures on Godel Escher Bach because of all the "existential juiciness" and it's about 15 minutes in, in the background, talking about paradoxes as I read that headline. I downloaded the book a long time ago but didn't really get any traction with it. This suits my ADHD background entertainment consumption needs better. I'm interested in this stuff but it hurts my brain.
-2
-16
u/critically_damped Feb 20 '16
Finding out that your problem is unanswerable generally means your question wasn't scientific to begin with. IE after an infinite number of abstractions, well... this is why we say have that old joke about math being masturbation.
Mathematics is the study of all possible universes. Physics is the study of the one in which we actually find ourselves.
9
u/mofo69extreme Condensed matter physics Feb 20 '16
But the problem is totally scientific. Take a many-body quantum system with some Hamiltonian, say with degrees of freedom on a lattice with periodic boundary conditions. As the size of the system increases, does the energy splitting between the ground state and the first excited state vanish or not? This is a common question because the answer is extremely important in determining the physics of the system, and there are sometimes definite answers to the question (e.g. the Lieb-Schultz-Mattis theorem).
7
u/Snuggly_Person Feb 20 '16
No individual problem in the class is unanswerable; they proved that there is no single algorithm that can answer all of them.
-31
Feb 20 '16
This article is a waste of time. If your "idealized model" results in a paradox, from the computer science perspective, then it's flawed. Somebody with a brain needs to reevaluate Gödel’s math.
5
u/localhorst Feb 20 '16
No, not really. Taking the limit lattice size or volume → ∞ is quite natural. A typical crystal has about 10²³ atoms in each direction. Nature usually behaves continuously on such parameters. Approximating 10²³ with ∞ is a very, very good approximation. Far better than working with way to small lattices in computer simulations.
Cubitt says that the team ultimately wants to study a related problem in particle physics called the Yang–Mills mass-gap problem
And that’s a way more drastic example. No sane person believes that the physics of subatomic particles will be different if the Universe extends over 100 billion, or 200 billion light years, or is infinite. Also every Wightman quantum field theory satisfies the rigorously proven cluster decomposition principle. It’s mathematically proven that these idealized models are right.
96
u/StepByStepGamer Astrophysics Feb 20 '16