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I've asked the question below on Physics SE but it seems an answer may not be straight forward...

Question: Is there any answers from Epistemology or philosophy-of-science?

Its often said QM is for the very small and GR for the very large. This brings to mind that there should be some limit at which one starts to apply and the other stops. Now I know there are more substantial differences, GR being a continuous description and QM using Planck length, and Planck time, and so on...

But keeping only to the distance scale difference, is there a hard limit where either theory suddenly stops working? Is there an overlap where both theories have similar predictive accuracy? Or is there a bad gap, a distance scale where neither theory is very useful?

Secondly, I imagine QM calculations can quickly become cumbersome when the number of particles increase. Is this image correct, is QM merely computationally impractical for considering planets and satellites etc. Are there other conceptual/practical problems, such as entanglement with something in another galaxy?

Edit: The suggested duplicate question asks whether a theory is quantum. This question asks whether an application of a theory (should) be QM or GR. I can now say from answers that the question is really more about gravity than the dimension of distance.

However it is also clear that although I tried to restrict it to a length scale, there are inextricable considerations. For instance, collision dynamics of very large molecules, large group behavior of particles, gravitational effects on quantum scales inside dense matter structures, etc.

The second part can be clarified: If all relevant particles could be accurately tracked, and all relevant interaction calculations performed, would QM give similar, or better, results over GR? (For instance regarding the orbit of the moon around Earth)

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    The straighforward answer is: mathematical models. None of these theories is true from the pragmatic view. But they are pretty useful. QM reduces to classical one due to central limit theorem. GR reduces to classical mechanics when the "surface" is pretty "flat". – rus9384 Feb 11 at 14:16
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    Overlap ? YES; see Relativistic quantum mechanics. – Mauro ALLEGRANZA Feb 11 at 14:23
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    "If all relevant particles could be accurately tracked, and all relevant interaction calculations performed..." This is simply not possible according to QM. – Mauro ALLEGRANZA Feb 11 at 14:24
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The answers are mostly negative, I am afraid. First, there is no "hard limit" where a theory stops working for almost any theory. Think of classical mechanics vs relativity, the classical accuracy decreases as the ratio v/c increases, but there is no cutoff. More generally, if a subtler theory reduces to a cruder one in the limit it means that the crude one's errors increases gradually, there is no cutoff. Just as there is no particular number of grains at which a handful of them turns into a heap.

The relationship between the quantum and classical mechanics is much more complex than with relativity. There is no single determining parameter, like v/c. The scale dependence is just a crude rule of thumb for "typical" cases, there are plenty of "large" systems that are essentially quantum, e.g. superfluids and superconductors, or the infamous Schrodinger cat. It is said that quantum effects "amplify" to macroscopic scales. The classicality depends on details of decoherence, which only loosely correlate with the size. A good review is Zurek Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time:

"I will begin by showing that the macroscopic size of a system – any system – does not suffice to guarantee its classicality. Thus, quantum theory implies that even the solar system – and every other chaotic system – is in principle indeterministic, and not just “deterministically chaotic”: Classical predictability in the chaotic context would require an ever increasing accuracy of its initial conditions. This is possible in principle in classical physics. But, according to quantum mechanics, simultaneously increasing the accuracy of position and momentum would eventually violate the Heisenberg principle. This time defines the quantum predictability horizon. It is urprisingly short, and – at least for some of its components – definitely less than the age of the solar system. Classicality is restored with the help of environment-induced decoherence, which continuously destroys the purity of the wavepackets. The resulting loss of predictability can be quantified through the rate of entropy production."

Comparing predictions of QM to GR does not make sense, I am afraid, regardless of the number of particles or complexity of computations. If one wishes to describe a quantum system subject to gravity one will have to figure out how to combine QM (or rather QFT, its generalization that incorporates special relativity) and GR to get any results. Or one will have to neglect one or the other. Either way, there are no two distinct things to compare. Some ad hoc heuristics for mixing GR with "quasi-classical" approximations of QFT are known in special cases, e.g. for the Hawking radiation. Alas, they are, strictly speaking, incoherent, because linking QFT observables to the classical quantities of GR, in principle, allows one to track the former to arbitrary precision, and so violates the uncertainty principle. There is a reason why quantum gravity is the greatest unresolved puzzle of modern physics, see Rickles's primer and Burgess's Quantum Gravity in Everyday Life.

  • The big question I'm looking at is in fact: can philosophy help solve this "greatest unresolved puzzle"? Also see comment to Richard's answer. – christo183 Feb 12 at 6:46
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    @christo183 Philosophy is helping. From the linked Rickles' paper, p.2:"Quantum gravity looks like an exercise in pure mathematics or, sometimes, metaphysics... it is one of the few areas of contemporary physics where (some of) the physicists who are central figures in the field actively engage with philosophers, collaborating with them, participating in philosophy conferences, and contributing chapters to philosophical books and journals." For ontological implications see pp. 72 and 94. QFT ontology you mentioned will likely have to be scrapped, it is background-dependent, unlike in GR. – Conifold Feb 12 at 13:01
  • “postmodern physics”, this could be it! ; ) – christo183 Feb 12 at 13:45
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The maths of relativity works. The maths of QM works.. but as yet nobody has found a way of merging them.

Essentially the maths of QM is about particles. And relativity is about fields.

But there things get interesting. Theres no such thing as particles, only part of fields that have high energies. So surely we can describe everything as some sort of field.. job done?

Here is the problem from 50,000 feet.

2+3=3+2 this is a rule called commutation. If a and b are numbers then a+b=b+a.

What other things aside from individual numbers might obey these kinds of arithmetic rules..

Well certain sets, matrices and groups do. On this type of 'thing' we can do 'abstract algebra'. This area of maths is called Group Theory. It was used to solve the Rubiks cube.

But it turns out that there are groups of things which describe the fields and symmetries of sub atomic forces.

These groups are called 'Lie' groups (pronounced Lee). Actually special Lie groups called Gauge groups.

The problem is that the groups that define the strong nuclear force don't work for electrons, much less 'gravitons'. But there is a system of groups in groups that defines all the nuclear forces.

The question is... is there a larger group G(gut) which can contain the Standard Model gauge groups, and some sort of 'relativity group', which obeys all the required axioms.. and accurately predicts observed reality?

Easy? No.

Edit: in fact G(gut) is a subgroup of G(toe) as in 'theory of everything'. A very recent attempt is called the.E8 Lie group.

  • Could one say that the Lie group describing the strong force is a (partial) mathematical expression of an ontology? The same ontology of which gravity is not part, but the strong nuclear force is. The strong nuclear force then being an expression of interaction in that ontology. This is what I'm going at: philosophy.stackexchange.com/q/53408/33787 – christo183 Feb 12 at 6:28
  • @christo183 Yes essentially, that's the current pervading opinion. Ontologicaly the Gauge groups of the standard model are siblings, whereas the mathematics of relativity is a very distant uncle. The idea for the lat 70 odd years has been to bring Relativity into the fold via 'quantum gravity', and create a TOE group which functions correctly. 'Garrett Lisi' recently (~2006) made a stab at slotting everything into a place in the E8 group, which might turn out. The problem is that 1 in 150000 people truly understand relativity. That number drops to 1 in a million for the maths of Lie Groups. – Richard Feb 12 at 10:22
  • "Essentially the maths of QM is about particles. And relativity is about fields:" Nope. It's precisely the opposite. – RAP - Reinstate Monica Cellio Feb 13 at 16:03
  • yes. QM is based on fields.. but it's about the stuff you see in Feynman diagrams. – Richard Feb 13 at 17:21
  • Any math based on nothing practical can work. – Overmind Feb 14 at 12:13
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They're different issues but NOT contradictory. The reason why one is for small scale and one for large is not that they are doing same thing and working at different scales, but that observations are possible on the small or large scale. They're really doing different things though.

Quantum theory says energy is quantized (i.e. digital "steps" of energy levels, the perfect example is emission of certain spectrums of light from electron transitions in atoms). So the electron has different energy levels. So does a baseball. It's just that the levels are SOOOO freaking close together at that scale, that you can't measure different speeds of a baseball and see the steps.

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

Relativity says that energy has mass associated with it. So if I speed up a baseball, it actually gets heavier. (This is why you can't make a rocket go faster than the speed of light...it gets heavier and heavier and you can only approach the speed of light asymptotically.) But of course at the baseball level, this is irrelevant. (It occurs, just the effect is so small it's irrelevant and too small too measure). However for speeds with some reasonable fraction of the speed of light, you can notice it.

This is what Michelson showed with some of his speed of light measurements (it was a constant, rather than "adding" or "subtracting" the speed of the Earth as it moves). He didn't understand the effect, but he was too much of a good scientist to ignore it:

https://en.wikipedia.org/wiki/Albert_A._Michelson

https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment

Now: In general, we "need" quantum mechanics at small scales. (It's both that we need it do do proper math calcs of effects and also that you can really only observe it with instruments at these scales...it's sort of the same issue.) Similarly for relativity at large scales. You don't need EITHER ONE for the baseball. Not only don't you need it (to do practical calcs on the baseball), you really can't observe either effect at the speed and mass involved in a baseball going ~100 mph during a pitch. So the classical Newtonian theory is much more convenient to use. It's like their is zero point in using a telescopic sight on a shotgun. But either theory would still "work" on the baseball if you wanted to bother (the relativistic math is probably a little bit less of a pain than the QM...easier to see it collapse to the classical...QM math is a hassle even at small scales, but you don't have any choice when you need it.)

Note that relativistic effects DO occur in small scales. The perfect example is the electrons of heavy elements. Like the electrons of a heavy element like lead or uranium. Those little buggers are flying! So, it turns out you "need" relativistic QM for doing quantum calculations involving chemistry of very heavy elements. (You may need it for even smaller elements to observe very fine effects, but you totally need it to get even reasonable calculations of orbitals of heavy elements). A classic example of a relativistic QM effect is "why is gold yellow". See here:

https://en.wikipedia.org/wiki/Relativistic_quantum_chemistry#Color_of_gold_and_caesium

Note that the relativistic QM equation dates back to the 20s (Pauli and Dirac). It is a little bit more of a hassle to deal with than the simpler non-relativistic version, so for some problems people don't bother. However, it also has some tangential benefits in dealing with a property called electron "spin", regardless of relativistic speeds. But there is NOT some massive problem of resolving the two effects (energy quantization and relativistic mass). It was done a few years after Schroedinger, all the way back in the 20s.

https://en.wikipedia.org/wiki/Relativistic_quantum_mechanics#History (this article is a little hairier than the others...what can I say, math/physics geeks on wiki sometimes lose the plot in terms of teaching laymen versus using articles to explain things to themselves. But the section I linked "history" gives a bit of the story, I describe above.

Caveat: both QM and relativity are complicated phenomena and I have not discussed all aspects (e.g. uncertainty principle, tunneling, special versus general relativity, etc.) Nor do I have perfect math or physical understanding of them. But I think this answer fundamentally contradicts some of the other discussion here that looks at the two as disparate theories that don't work with each other. They do.

  • You've mentioned that sometimes you "need" QM, does that mean sufficiently advanced QM calculations could account for relativistic effects? I think the situation is somewhat more involved, think of why do particle physics play on a "chess board" and gravity on a continuous "field" . – christo183 Feb 14 at 11:51
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Warning: The following, while approximately correct, is very oversimplified, and size is more of a rule of thumb here.

QM and GR don't really border all that much. As you start increasing the size of a system you're studying through QM, it starts turning into classical mechanics, and as you scale down a system you're studying with GR, it starts turning into classical mechanics. Pretty much everything we see can be modeled by one of those three models. Without a combination of GR and QM, which we don't really have (relativistic quantum mechanics is a combination of QM and Special Relativity), we have problems extrapolating to more extreme conditions.

General Relativity is mostly the study of gravity, which is by far the weakest of the four basic forces, and so it's hard to study in cases where there's much influence by other forces. QM is mostly about the stronger three forces, and generally ignores gravity, since gravitational effects are so much weaker than anything else. In the examples you give, gravity is pretty much undetectable.

QM calculations get REALLY intractable as the number of things considered goes up, but that's not the main reason why it isn't useful for, say, orbital mechanics or black holes. It's not useful because it doesn't really consider gravity, and it's not needed because of the transition into classical mechanics. By the time you need to consider gravity, every other force is either out of effective range (nuclear forces) or balanced out (electromagnetic force). Any body massive enough to have a reasonable gravitational field is not going to show quantum behavior under any circumstances we can manage.

(BTW, quantum entanglement doesn't really change things. If a couple of photons are entangled, it means that we can measure one of them and know something about how the other would be measured. To vastly oversimplify, consider a red card, a black card, and two envelopes. If one card is placed in each envelope, then if we open one envelope we know what card is in the other. There's no causation going on, and having the existence of the other envelope doesn't affect what happens with our envelope.)

  • This is far too simplistic. The dependence of applicability on size is just a rough rule of thumb, in many circumstances quantum effects manifest macroscopically (superfluids, Geiger counters). Gravity being the "weakest" is equally vague, when the distances are small enough, or near black holes, it becomes very "strong", which is why gravitational corrections hopelessly diverge, and we still do not have a settled theory of black hole evaporation. – Conifold Feb 11 at 21:19
  • @Conifold I added an introductory paragraph emphasizing the oversimplification. I hadn't been clear enough about that. Gravity is the weakest force. At any range, the electromagnetic force is stronger. Within their ranges, the strong and weak nuclear forces are stronger. The difference is that gravity always attracts, and works over all distances, whereas electrical charges can attract and repel, and the nuclear forces have limited range. – David Thornley Feb 12 at 14:41

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