- #1
mersecske
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Why?
Some measurements confirm this statement?
Or this is a theoretical conclusion?
Some measurements confirm this statement?
Or this is a theoretical conclusion?
It is a dimensional analysis argument.dpackard said:I too would like to know how it is we know quantum effects become important in this regime. Other than knowing we need QG before we get down to a singularity, what in particular makes us think the Planck scale is selected by nature as the transition scale?
Theoretical. I don't know if there are any situations where they both make clear predictions that contradict each other, but see here for a discussion of one of the main problems in figuring out how to reconcile them, having to do with the fact that the uncertainty principle would seem to allow for huge uncertainty in energy at sufficiently small scales, but in GR big energies cause significant curvature of spacetime, and my understanding is that physicists only know how to make predictions in quantum field theory if they have a specific known background spacetime. I guess another more general conflict is that quantum field theories treat the other set of forces using a common set of rules, but if you try to apply these rules to gravity you get infinities which can't be "renormalized" as in the case of the other forces.mersecske said:Why?
Some measurements confirm this statement?
Or this is a theoretical conclusion?
No. You are assuming that there is still a smooth manifold in your statements there, as we need that for GR.mersecske said:"quantum foam" means a difficult topology, but what's the problem. GR can describe any kind of topology. If Wheeler states that spacetime is a "quantum foam", this means that he uses GR at that scale.
JustinLevy said:For example, some attempts to quantize GR directly have found a fractal spacetime that only in length scales much greater than Planck lengths does a smooth spacetime with 4 dimensions emerge ... on smaller length scales the fractal dimension appears closer to 2. So at least in some approaches, Wheeler's intuition is playing out.
JustinLevy said:No. You are assuming that there is still a smooth manifold in your statements there, as we need that for GR.
So "quantum foam" doesn't just mean a difficult topology.
This answer seems incomplete insofar as it doesn't have anything specifically to do with gravity. After all, before physicists had figured out how to quantize classical electrodynamics, no one expected that the characteristic scale of quantum electrodynamics should be the Planck scale, and of course it isn't! I found an interesting paper here which gives a series of basic arguments for the Planck scale should be the scale of quantum gravity, summarized on p. 3:bcrowell said:The answer to #2 is that the Planck scale is the only scale you can construct out of the appropriate physical constants.
For example, argument (4) involves the idea that to probe what's going on in smaller and smaller volumes of space, you need probes of higher and higher frequency and therefore higher energy, and if the volume were as small as the Planck scale the energy density would be so high as to form a Planck-scale black hole.Lacking real experiments we use thought experiments (Gedankenexperiment) in this note. We give plausible heuristic arguments why the Planck length should be a sort of fundamental minimum - either a minimum physically meaningful length, or the length at which spacetime displays inescapable quantum properties i.e. the classical spacetime continuum concept loses validity. Specifically the six thought experiments involve: (1) viewing a particle with a microscope; (2) measuring a spatial distance with a light pulse; (3) squeezing a system into a very small volume; (4) observing the energy in a small volume; (5) measuring the energy density of the gravitational field; (6) determining the energy at which gravitational forces become comparable to electromagnetic forces. The analyses require a very minimal knowledge of quantum theory and some basic ideas of general relativity and black holes, which we will discuss in section II. Of course some background in elementary classical physics, including special relativity, is also assumed.
JesseM said:This answer seems incomplete insofar as it doesn't have anything specifically to do with gravity. After all, before physicists had figured out how to quantize classical electrodynamics, no one expected that the characteristic scale of quantum electrodynamics should be the Planck scale, and of course it isn't!
Still seems like an overly handwavey argument, the fact that you can construct length, distance and energy densities from some fundamental constants which include the gravitational constant doesn't give any clear reason why this should be the characteristic scale of quantum gravity, the arguments in the paper I linked to are more physical. For a different electrodynamics analogy, would any pre-QED physicists have argued that quantum electrodynamics effects don't become significant until we reach the Planck charge?Parlyne said:The constants that Planck scale quantities are derived from are the speed of light, Planck's constant, and Newton's gravitational constant (which also appears in the GR field equations). The presence of Newton's constant is what makes these quantities refer to gravity and why no one should expect quantum electrodynamics to have any relation to the Planck scale (since Newton's constant doesn't show up anywhere in classical electrodynamics).
bcrowell said:The answer to #2 is that the Planck scale is the only scale you can construct out of the appropriate physical constants. This is not an absolutely secure argument. For example, if large extra dimensions http://en.wikipedia.org/wiki/Large_extra_dimensions exist, then we could see quantum gravity at the LHC.
yogi said:You can also get a set of dimensions by combining e, c and G. I believe these were first formulated by Irish Physicist George Johnson Stoney. The dimensions are different than Planck units - one of the reasons I have always considered Planck units as more or less metaphysical numerology - it all started as a dimensional analysis without physics and it still has no physics -
bcrowell said:The Planck scale includes Planck's constant, and that's why it's expected to be the scale at which quantum gravity effects become strong. Stoney's units could be fine for some other purpose, but they don't make use of Planck's constant.
What do you mean by the "scale" of "gravitational interactions"? And does "the quantum length scale" refer simply to combining various constants to get the Planck length, or does it refer to some more physical idea like a statement that some physical quantity becomes significant at that length scale?JustinLevy said:The question is merely: at what scale do quantum corrections become important? Look at the length scale where the gravitational interactions become on the scale of the quantum length scale.
Yes this argument is a dimensional analysis argument.
It sounds like your questions are on the concepts of dimensional analysis itself. Dimensional analysis is quite powerful when used appropriately, and just like statistical mechanics, can cut through all the "un-needed details" to give you very useful broad results.JesseM said:What do you mean by the "scale" of "gravitational interactions"? And does "the quantum length scale" refer simply to combining various constants to get the Planck length, or does it refer to some more physical idea like a statement that some physical quantity becomes significant at that length scale?
But when you say "dimensional analysis", do you mean there aren't even any rough physical arguments, it's purely a matter of shuffling various constants to reach some physical conclusion?JustinLevy said:It sounds like your questions are on the concepts of dimensional analysis itself. Dimensional analysis is quite powerful when used appropriately, and just like statistical mechanics, can cut through all the "un-needed details" to give you very useful broad results.
These techniques are used for all kinds of things. While engineers may not always learn where they come from, they use dimensional analysis in many classes. Looking up pressure curves for non-ideal gases, often use scaled pressures and temperatures, etc. and the gases, despite being non-ideal, are now all described almost identically on a graph. Consider Reynolds number, for fluid dynamics etc.
OK, this seems like a more physical argument (just to be clear, when you talk about the energy needed in QM I assume you mean something like the minimum depth a potential well of width L would need in order for there to be at least one bound state for a photon, i.e. if L is the wavelength then a bound state which fits about one wavelength in the well would have energy approximately equal to E=hf=hc/L according to Planck's equation, so the potential must be at least that deep for there to be a bound state). Perhaps a physicist would make a purely dimensional argument when talking to an audience of physicists who would be assumed to know a way to translate this into a more physical argument, but what I question is whether it's meaningful to use a purely dimensional argument if you don't have a more physical argument in mind. How would you tell "good" dimensional arguments from "bad" ones (like a hypothetical argument saying that quantum gravity effects should apply to any particle with a mass smaller than the Planck mass, or that quantum electrodynamics effects only become important for charges smaller than the Planck charge) if you don't have recourse to more specific physical arguments?JustinLevy said:To help "bridge the gap", I'll attempt to make the dimensional analysis "more physical" sounding to you here.
1] Length scale in classical GR
for black hole of energy E, on order of
[tex] L_{gr} = \frac{GE}{c^4}[/tex]
2] Length scale from quantum mechanics
Due to quantum wavelength
[tex]\Delta x \Delta p \ge \frac{\hbar}{2}[/tex]
to contain in a box of size L requires an energy on order of
[tex]E = \frac{\hbar c}{L_{qm}}[/tex]
these two become comparable at:
Length scale = Planck length = [tex]\sqrt{\hbar G/c^3}[/tex]
Reworded to sound more physical:
If we have a huge black hole, quantum mechanics doesn't have much to say... GR should hold fine. However, for very small length scales, GR could claim there is a black hole, but quantum mechanics says the very wave nature of particles prevents it from being confined in such a small volume. This is the scale at which quantum effects (wave nature of particles) make the assumptions of GR (classical fields and particles) become incorrect enough that we wouldn't be approximating the real result with GR anymore. We'd need a quantum theory of gravity.
Well, if you divide Planck's constant by (c*some mass) you get something with units of length, but in purely dimensional terms how do you decide whether to use the mass of an electron or the mass of a proton, since the two differ by three orders of magnitude? Again it seems like you need some sort of physical argument, even if it's one a physicist would find obvious enough to just leave it implicit...JustinLevy said:Test your knowledge:
Classical electrodynamics says there cannot be any stable arrangement of charges. Quantum mechanics disagrees with the assumptions of classical electrodynamics, and luckily for us allows atoms to form. Without solving Schrodinger's equations, using dimensional analysis, can you find what length scale classical electrodynamics breaks down for an electron interacting with a proton, and therefore what size you would expect atoms to be?
I think it more dificult. Does spacetime have a length scale at all? I think the answer is no, except for static spacetimes. Do you agree?JustinLevy said:It is a dimensional analysis argument.
Quantum mechanics gives us a time or length scale for an object of particular energy or momentum.
General relativity gives us a length scale given a mass (or energy).
Therefore we can look at where these length scales are comparible. The Planck units are units defined using G, hbar, c, etc.
Huh? I'm not sure what you mean here. I'm not talking about the length scale of spacetime all by its lonesome, I'm talking about the length scale of gravitational interactions. Sort of like asking what the length scale of the electric field is without any context, verse asking what the length scale of the electric interaction is in that last post.Passionflower said:I think it more dificult. Does spacetime have a length scale at all? I think the answer is no, except for static spacetimes. Do you agree?
I think key in this discussion should be the background independence issue.
I don't see how it could have been a misunderstanding since I didn't make any positive claims, I was just asking a question about what you meant by "dimensional analysis", since I'm not too familiar with the term (I've only seen 'dimensional analysis' used in simpler contexts like making sure both sides of an equation have the same dimension). If you didn't mean to imply that one could reach physical conclusions just by playing with constants and without any physical arguments (explicit or implicit), that's fine with me, but then that would suggest that bcrowell's argument lacked the needed physical argument to justify the conclusion that the Planck scale should be the scale of quantum gravity. On the other hand, if you did mean to suggest this, then it seems to me the answer to my initial question should just be "yes". Or perhaps you think I am making a false dichotomy between "physical arguments" and "pure shuffling of constants", I don't see what the third alternative would be though.JustinLevy said:JesseM,
Your beginning questions seem to be misunderstanding dimensional analysis. I'm very sorry, but I don't feel I'm well spoken enough to give a compelling statement of dimensional analysis. Maybe someone else here can.
Yes, more along these lines. I have not attempted to explain dimensional analysis outright, as I don't feel I could do it justice. I don't want to accidentally make it sound like hog-wash to you because of my poor choice in wording. I'm still hoping someone else will take a stab at it, and I'd learn too. For if I can't explain it well to another, then I clearly don't fully understand it myself. Right?JesseM said:Or perhaps you think I am making a false dichotomy between "physical arguments" and "pure shuffling of constants"
We teach that to physics students as well. But that is just the lowest level of using dimensional information. Often physicists just refer to that as "checking units". It is a useful tool to find where an error occurred in a calculation.yogi said:In Engineering, dimensional analysis is exactly what Jesse described - by keeping track of the units you can check the vaidity of both sides of an equation, and quickly spot an error - some of the literature regarding Planck units has appropriated the term "dimensional analysis' - but it is actually a case of "dimensional inference"
JesseM said:Thanks for the explanation Justin. I think you may be right that I was posing a false dichotomy, the notion of "dimensional analysis" you're describing does involve some physical intuitions such as the choice of "relevant" parameters, and also general ideas like the notion that quantum gravity should have a characteristic length scale (whereas it doesn't have a characteristic mass scale), but it comes short of more detailed physical arguments.
No, not saying the Planck mass is meaningless, but it would be a mistake to think that quantum gravity is needed anytime you are analyzing a system with less mass than the Planck mass (though you do if the mass is compressed down to around the Planck length), whereas quantum gravity would be needed anytime you're analyzing interactions on the scale of the Planck length or Planck time--that's exactly what I meant when I said you still needed some basic physical intuitions to do dimensional analysis, even if you don't need detailed physical arguments.yogi said:The same 3 constants (G, C and h) also lead to a unit of mass - are you saying its ok to ignor the mass that falls out of the combination as meaningless - but not the length
The Planck scale is the scale at which the effects of gravity become comparable to those of quantum mechanics. It is the smallest scale at which our current understanding of physics can accurately describe the behavior of matter and energy.
General relativity, which is the theory of gravity, breaks down at the Planck scale because it does not take into account the principles of quantum mechanics. At this scale, the fabric of space-time becomes highly curved and chaotic, making it impossible for general relativity to accurately describe the behavior of matter and energy.
One consequence is that our current understanding of the universe breaks down at this scale, making it difficult to accurately predict and understand the behavior of matter and energy. It also means that a new theory, such as quantum gravity, is needed to accurately describe the behavior of matter and energy at the Planck scale.
Currently, we do not have the technology or tools to directly observe the effects of general relativity breaking down at the Planck scale. However, scientists are working on theories and experiments that may help us understand the behavior of matter and energy at this scale in the future.
The Planck scale is an important factor in the search for a theory of everything, which is a single theory that can accurately describe all of the fundamental forces and particles in the universe. As the smallest scale at which our current theories break down, the Planck scale provides important clues and constraints for scientists in their search for a theory of everything.