Open problems in mathematical physics

[fresh_42]

I came across this beautiful pearl
https://arxiv.org/abs/1710.02105
https://arxiv.org/pdf/1710.02105.pdf
which I like to bring to notice.

We present a list of open questions in mathematical physics. After a historical introduction, a number of problems in a variety of different fields are discussed, with the intention of giving an overall impression of the current status of mathematical physics, particularly in the topical fields of classical general relativity, cosmology and the quantum realm. This list is motivated by the recent article proposing 42 fundamental questions (in physics) which must be answered on the road to full enlightenment [1]. But paraphrasing a famous quote by the British football manager Bill Shankly, in response to the question of whether mathematics can answer the Ultimate Question of Life, the Universe, and Everything, mathematics is, of course, much more important than that.

Despite of its title it is heavier on theoretical physics than it is on mathematics, so I placed it in this forum. I think it is equally interesting to those who are interested in history of science, as it is to those who ask "What's left?" from time to time here on PF.

 

[Greg Bernhardt]

ok everyone, let's hunker down and get to work :)

 

[anorlunda]

I agree with @fresh_42 , this list is a pearl. He provided two links. I found a third link to a version that is somewhat more readable in my opinion.

Life, the universe, and everything – 42 fundamental questions, by Roland E. Allen and Suzy Lidstrom
http://people.physics.tamu.edu/allen/42.pdf

But even easier to read, and motivation for PF members to take the trouble to read the whole paper, the following is the author's summary expressed as 42 questions. If you wonder, "What the heck does that mean?", then go read the paper.

1. Why does conventional physics predict a cosmological constant that is vastly too large?
2. What is the dark energy?
3. How can Einstein gravity be reconciled with quantum mechanics?
4. What is the origin of the entropy and temperature of black holes?
5. Is information lost in a black hole?
6. Did the universe pass through a period of inflation, and if so how and why?
7. Why does matter still exist?
8. What is the dark matter?
9. Why are the particles of ordinary matter copied twice at higher energy?
10. What is the origin of particle masses, and what kind of masses do neutrinos have?
11. Does supersymmetry exist, and why are the energies of observed particles so small compared to the most fundamental (Planck) energy scale?
12. What is the fundamental grand unified theory of forces, and why?
13. Are Einstein relativity and standard field theory always valid?
14. Is our universe stable?
15. Are quarks always confined inside the particles that they compose?
16. What are the complete phase diagrams for systems with nontrivial forces, such as the strong nuclear force?
17. What new particles remain to be discovered?
18. What new astrophysical objects are awaiting discovery?
19. What new forms of superconductivity and superfluidity remain to be discovered?
20. What new topological phases remain to be discovered?
21. What further properties remain to be discovered in highly correlated electronic materials?
22. What other new phases and forms of matter remain to be discovered?
23. What is the future of quantum computing, quantum information, and other applications of entanglement?
24. What is the future of quantum optics and photonics?
25. Are there higher dimensions, and if there is an internal space, what is its geometry?
26. Is there a multiverse?
27. Are there exotic features in the geometry of spacetime, perhaps including those which could permit time travel?
28. How did the universe originate, and what is its fate?
29. What is the origin of spacetime, why is spacetime four-dimensional, and why is time different from space?
30. What explains relativity and Einstein gravity?
31. Why do all forces have the form of gauge theories?
32. Why is Nature described by quantum fields?
33. Is physics mathematically consistent?
34. What is the connection between the formalism of physics and the reality of human experience?
35. What are the ultimate limits to theoretical, computational, experimental, and observational techniques?
36. What are the ultimate limits of chemistry, applied physics, and technology?
37. What is life?
38. How did life on Earth begin and how did complex life originate?
39. How abundant is life in the universe, and what is the destiny of life?
40. How does life solve problems of seemingly impossible complexity?
41. Can we understand and cure the diseases that afflict life?
42. What is consciousness?

 

[cosmik debris]

Hmmm, there are a lot of why questions which are not supposed to be answered by physics, so these unanswered philosophy questions?

 

[fresh_42]

I think "why" is not automatically philosophy or unanswerable. Those questions can probably also be formulated without the why, but not similarly short. E.g.

Why does conventional physics predict a cosmological constant that is vastly too large?

Which physical theory can explain the difference between the cosmological constant predicted by conventional physics in comparison to our observation, i.e. a model in which both are included as boundaries, like Planck's law of black-body radiation explains the agreement to conventional physics for long wavelengths and the discrepancy on short wavelengths.

So no "why" needed.

 

[anorlunda]

So no "why" needed.

The questions are the author's wording, not mine.. Here is the author's full statement of #1 leading to the why question #1.

The cosmological constant problem was thoroughly discussed in a 1989 paper by Steven Weinberg [8], shown in Fig. 3: According to standard physics, the vacuum has an enormous energy density ρvac. A typical positive contribution is the zero-point energy of the electromagnetic field, and a typical negative contribution arises from Higgs condensation. All the various contributions are determined independently and there is no reason why they should cancel. Again according to standard physics, ρvac should act as a gravitational source – effectively an enormous cosmological constant Λvac. It should then have an enormous effect on the curvature of spacetime, roughly 120 orders of magnitude larger than is compatible with observation (with the Planck scale providing a natural cutoff).

In the past few decades, Weinberg and others have adopted the point of view that, of all proposed solutions to this problem, the only acceptable one is the anthropic bound that he obtained in 1987 [9]. However, anthropic reasoning – see 5.2 – is not universally accepted by the physics community. And it is likely that essentially all professional physicists would prefer a nonanthropic explanation of this largest of all discrepancies between standard theory and observation. So despite decades of attempts by the best minds in theoretical physics, there is no truly convincing solution to this problem, and it may be signaling the need for a revolutionary conceptual breakthrough.

I'm just an amateur, not on the level of you @fresh_42 or the paper's authors. Are you saying that this is not an unsolved problem? Or do the authors say that "a model in which both are included as boundaries," is anthropic?

 

[fresh_42]

I'm just an amateur, not on the level of you @fresh_42 or the paper's authors. Are you saying that this is not an unsolved problem? Or do the authors say that "a model in which both are included as boundaries," is anthropic?

Neither am I. I'm not even sure to which extend the authors are. I only wanted to point out that the why isn't per se nonphysical. Physical models are always valid in a certain range and might fail in another range. So the question as what happens at the boundary is a very physical and not a philosophical question. The QM forum is full of why questions, especially when quantum phenomena are described in our macroworld language. Nevertheless, the papers are a nice read and probably should be taken with a grain of salt. I think they are a nice appetizer and explain why physics is interesting, and I appreciated your list very much. And although we don't discuss philosophy here for practical reasons, it is not as if physicists wouldn't discuss it. And like us who have experienced, that it leads nowhere and is rarely on a scientific level, physicists usually debate about philosophical questions at a dinner table and not in their lectures.

 

[anorlunda]

I understand that PF has practical reasons for excluding philosophy. Those reasons don't necessarily apply outside PF.

But isn't there an aspirational way to look at it? The state of the art: Scientific theory supported by observational evidence. That's hard-core science and the scientific method. Things beyond the state of the art are not science -- we can harshly call it philosophy... But to expand the state of the art, someone must aspire to transform some increment philosophy into science. Those aspirations should be encouraged and discussed as long as there is hope that they might succeed. Otherwise every research proposal is non-science by definition. We must look beyond the state of the art to advance it.

The kinds of philosophy that waste our time to discuss and deserves harsh treatment are the hopeless cases, or things so far in the future that we can't discuss them intelligently. (I think of FTL drives as an example.)

 

[fresh_42]

I agree on the basic sentiment of what you wrote. The difficulty is where to draw the line? FTL is as impossible as squaring the circle is. There isn't even an idea to overcome this hurdle. On the other hand, many discussions start with topics as perception, consciousness or reality. I always find it disrespectful towards real philosophy to discuss them on a level, which is basically defined by an open internet forum. I haven't met a single person here, who has seriously studied philosophy and knows what he talks about, namely knows what dozens of philosophers have already written. Not even the basic schools are common knowledge. So the rule to avoid philosophical questions, has always been more of a rule of respect to me, than merely a rule to confine discussions and avoid endless argues.

Another aspect is, if we opened the door a little for "fantasies", which I agree completely with you are needed in science, we would instantaneously get many tri-sectionists, cube-doublers and circle-squarers and their physical pendants. It's bad enough what Hawking and Kaku already set into the world in their popular science magazines. Me, too, finds it sad sometimes, that we don't have more brain storming and dinner-table-like discussions and I would like to hear personal views from many of our members, but I certainly won't like to read the flood of nonsense from all who would feel invited to take part. The framework simply doesn't allow it to distinguish - and to establish a judge or a jury would even increase the trouble.

 

[Drakkith]

But isn't there an aspirational way to look at it? The state of the art: Scientific theory supported by observational evidence. That's hard-core science and the scientific method. Things beyond the state of the art are not science -- we can harshly call it philosophy... But to expand the state of the art, someone must aspire to transform some increment philosophy into science. Those aspirations should be encouraged and discussed as long as there is hope that they might succeed. Otherwise every research proposal is non-science by definition. We must look beyond the state of the art to advance it.

I agree on the basic sentiment of what you wrote. The difficulty is where to draw the line? FTL is as impossible as squaring the circle is. There isn't even an idea to overcome this hurdle. On the other hand, many discussions start with topics as perception, consciousness or reality. I always find it disrespectful towards real philosophy to discuss them on a level, which is basically defined by an open internet forum. I haven't met a single person here, who has seriously studied philosophy and knows what he talks about, namely knows what dozens of philosophers have already written. Not even the basic schools are common knowledge. So the rule to avoid philosophical questions, has always been more of a rule of respect to me, than merely a rule to confine discussions and avoid endless argues.

I agree. I think we often use the words "philosophy" or "philosophical" to describe questions or discussions which are either outside the realm of being answered in the near-to-moderate future, or which are too vague and ill-defined to permit a serious discussion within the bounds of the forum guidelines and rules. Technically these are probably still part of philosophy in a very broad way (is there a question which isn't?), but they are very rarely of a quality which professional philosophers would find useful. That's my guess at least, as I'm no philosopher myself.

 

[Dr. Courtney]

I agree with @fresh_42 , this list is a pearl. He provided two links. I found a third link to a version that is somewhat more readable in my opinion.

Life, the universe, and everything – 42 fundamental questions, by Roland E. Allen and Suzy Lidstrom
http://people.physics.tamu.edu/allen/42.pdf

But even easier to read, and motivation for PF members to take the trouble to read the whole paper, the following is the author's summary expressed as 42 questions. If you wonder, "What the heck does that mean?", then go read the paper.

1. Why does conventional physics predict a cosmological constant that is vastly too large?
2. What is the dark energy?
3. How can Einstein gravity be reconciled with quantum mechanics?
4. What is the origin of the entropy and temperature of black holes?
5. Is information lost in a black hole?
6. Did the universe pass through a period of inflation, and if so how and why?
7. Why does matter still exist?
8. What is the dark matter?
9. Why are the particles of ordinary matter copied twice at higher energy?
10. What is the origin of particle masses, and what kind of masses do neutrinos have?
11. Does supersymmetry exist, and why are the energies of observed particles so small compared to the most fundamental (Planck) energy scale?
12. What is the fundamental grand unified theory of forces, and why?
13. Are Einstein relativity and standard field theory always valid?
14. Is our universe stable?
15. Are quarks always confined inside the particles that they compose?
16. What are the complete phase diagrams for systems with nontrivial forces, such as the strong nuclear force?
17. What new particles remain to be discovered?
18. What new astrophysical objects are awaiting discovery?
19. What new forms of superconductivity and superfluidity remain to be discovered?
20. What new topological phases remain to be discovered?
21. What further properties remain to be discovered in highly correlated electronic materials?
22. What other new phases and forms of matter remain to be discovered?
23. What is the future of quantum computing, quantum information, and other applications of entanglement?
24. What is the future of quantum optics and photonics?
25. Are there higher dimensions, and if there is an internal space, what is its geometry?
26. Is there a multiverse?
27. Are there exotic features in the geometry of spacetime, perhaps including those which could permit time travel?
28. How did the universe originate, and what is its fate?
29. What is the origin of spacetime, why is spacetime four-dimensional, and why is time different from space?
30. What explains relativity and Einstein gravity?
31. Why do all forces have the form of gauge theories?
32. Why is Nature described by quantum fields?
33. Is physics mathematically consistent?
34. What is the connection between the formalism of physics and the reality of human experience?
35. What are the ultimate limits to theoretical, computational, experimental, and observational techniques?
36. What are the ultimate limits of chemistry, applied physics, and technology?
37. What is life?
38. How did life on Earth begin and how did complex life originate?
39. How abundant is life in the universe, and what is the destiny of life?
40. How does life solve problems of seemingly impossible complexity?
41. Can we understand and cure the diseases that afflict life?
42. What is consciousness?

This list is actually from different authors (R. E. Allen and S. Lidstrom,“Life, the universe, and everything: 42 fundamental questions”, Phys. Scr. 92 012501 (2017) [Focus Issue on 21st Century Frontiers].) and is acknowledged by Coley to have a lot of questions that are more metaphysics than physics. (p 12 of https://arxiv.org/pdf/1710.02105.pdf ).

One fundamental unsolved problem in mathematical physics I think Coley missed is that of Quantum Chaos: How does the Schrodinger equation give rise to classical chaos? We do not really understand how the Correspondence Principle works in the case of chaotic systems.

This problem seemed to garner a lot more discussion in the 1980s and 1990s than in the 21st century. It seems to have fallen out of favor, not because it is uninteresting or unimportant, but because it is hard.

 

[nikkkom]

Some questions are ill-defined. Some already have answers (IMO).

Is there a multiverse?

Define "multiverse". I know several different definitions.

What explains relativity and Einstein gravity?Why do all forces have the form of gauge theories?
Why is Nature described by quantum fields?

Any sequence of physicals "whys" ends with a set of axioms which by definition have no answer to "why in your theory are these axioms true?". There is no answer; we just assume them to be true and then develop a theory on them.
The above three "whys" are of that sort.

What is the connection between the formalism of physics and the reality of human experience?

Experiments.

What is life?

Self-replicating nanomachines of natural origin.

What is consciousness?

Some species of "self-replicating nanomachines" have a natural "computer" (created by evolution) which improves survival chances for them. We have empirical evidence that more powerful "consciousness" allows species to be more successful (consider sequence of amoeba > fish > cat > human and how improved capabilities to process information make them more able to survive and reproduce in adverse conditions).

 

[anorlunda]

Some questions are ill-defined. Some already have answers (IMO).

I think if you read the actual article you'll see that the questions are better defined than the terse paraphrasing questions. After reading the full definitions, I would be interested to hear if you still think that some of them have been answered.

 

[Auto-Didact]

Some species of "self-replicating nanomachines" have a natural "computer" (created by evolution) which improves survival chances for them. We have empirical evidence that more powerful "consciousness" allows species to be more successful (consider sequence of amoeba > fish > cat > human and how improved capabilities to process information make them more able to survive and reproduce in adverse conditions).

That's a handwaving description or even simply a vague statement/very incomplete definition, not even coming close to an explanation of the mechanism in terms of some mathematical model consistent with physical theory. In other words, no soup for you!

 

[WWGD]

I came across this beautiful pearl
https://arxiv.org/abs/1710.02105
https://arxiv.org/pdf/1710.02105.pdf
which I like to bring to notice.

Despite of its title it is heavier on theoretical physics than it is on mathematics, so I placed it in this forum. I think it is equally interesting to those who are interested in history of science, as it is to those who ask "What's left?" from time to time here on PF.

Sorry to throw off things a bit. Is this where the 42 in your name comes from?

 

[fresh_42]

Sorry to throw off things a bit. Is this where the 42 in your name comes from?

I assume they both are based on the same origin. 5-25 is our common celebration day, I guess.

 

[Svein]

"Life, the universe, and everything: 42 fundamental questions”

Somebody has read "the Hitchhiker's Guide to the Galaxy"!

 

[jerromyjon]

Somebody has read "the Hitchhiker's Guide to the Galaxy"!

Unfortunately the question is NOT what is 6x7, and the answer of "42", as the answer to the ultimate question of the universe, gets us no closer.
On a more serious tone, does proving they are unanswerable or circumvented by a deeper understanding count?

 

[Keith_McClary]

I can't see mention of the "arrow of time" question .

 

[Thuring]

Some of these questions might have Zen- type answers. lol

 

[Mark Harder]

Some species of "self-replicating nanomachines" have a natural "computer" (created by evolution) which improves survival chances for them. We have empirical evidence that more powerful "consciousness" allows species to be more successful (consider sequence of amoeba > fish > cat > human and how improved capabilities to process information make them more able to survive and reproduce in adverse conditions).

Self-reference is such a fundamental component of consciousness that I don't think you can propose a model without it. Pavlovian reflexes may suffice to generate the sorts of protective and generative behaviors that optimize evolutionary utility. We don't really know for certain (or do we?) if other animals and 'higher' life forms are aware of their emotions and can consciously make decisions based on emotional motivation and rational methods. We humans not only feel pleasure and pain; we know, that is we are conscious of, our pleasure and pain. This allows us to employ a more complex set of tools to optimize pleasure and minimize pain, and to plan our behaviors to encompass life factors in the distant future. Perhaps my cat does know why it drops dead mice on my bed (Though I certainly don't.), but I doubt that it is aware that it will have to await the next spring before it becomes as easy to dispatch and deposit little corpses on my bed. (I could be proved wrong somehow, sometime in the future, but that is not part of my model, and that's all it is, at present.

 

[Auto-Didact]

One fundamental unsolved problem in mathematical physics I think Coley missed is that of Quantum Chaos: How does the Schrodinger equation give rise to classical chaos? We do not really understand how the Correspondence Principle works in the case of chaotic systems.

This problem seemed to garner a lot more discussion in the 1980s and 1990s than in the 21st century. It seems to have fallen out of favor, not because it is uninteresting or unimportant, but because it is hard.

This. I would add to this entry another unsolved fundamental mathematical physics problem, namely the creation of a general theory of physical networks, which fully incorporates the fields and methods of nonlinear dynamical systems theory, self-organisation, critical phenomena, complexity theory, fractal geometry and network theory into a single coherent theory.

The creation of such a theory seems to require the construction of a novel branch of mathematics for physics, differing fundamentally in certain aspects from calculus and differential equations, enabling a more accurate description of many natural phenomena. The hope is this new theory would be key in solving the corresponding fundamental theoretical physics problem of describing open system non-equilibrium thermodynamics, paving the way for solving the questions of what life and perhaps consciousness are as well.

 

[nikkkom]

Self-reference is such a fundamental component of consciousness that I don't think you can propose a model without it. Pavlovian reflexes may suffice to generate the sorts of protective and generative behaviors that optimize evolutionary utility. We don't really know for certain (or do we?) if other animals and 'higher' life forms are aware of their emotions and can consciously make decisions based on emotional motivation and rational methods. We humans not only feel pleasure and pain; we know, that is we are conscious of, our pleasure and pain.

I don't see it to be fundamentally different. Cats and dogs seem to have thoughts too, just very, very simplistic ones. Monkeys have somewhat more complex ones. We have very complex ones.

Example: many animals, including domesticated (dogs, for example), make stashes of food. They remember the place where they hid it, and know when to use it (they understand the link between hunger and hidden food). That's rudimentary form of planning.

I think as human evolved from monkeys, the level of awareness was increasing gradually from simplistic chains of proto-thoughts a-la "feeling hungry" -> "need food" -> "have food hidden there" -> "going to take and eat it" to ever more complex constructs (both in terms of more abstract concepts, and in the ability to sustain longer chains of thoughts). I imagine the creation of language and need to use it to communicate ideas to other members of the tribe was an important driver. Even if first "ideas" to communicate were as mundane as "bring me some water".

The alternative hypothesis would be that one day one of our ancestors just "started thinking and be aware" in some weird quantum jump of consciousness. Does not look plausible to me.

 

[Mark Harder]

I can't see mention of the "arrow of time" question .

Yes, that's a favorite of mine. I used to think that it was obvious: Entropy of the universe always increasing implies irreversibility, hence arrow of time. But I'm not so sure it's as simple as that; it could be a circular argument, since dS>dq/T for irreversible processes, and since dq for the universe is 0 (adiabatic system), dS>0. There must be a more subtle answer than that. Books have been written, all above my head.