Specific examples please
Are you asking why GR and QM in their present form seem incompatible, or are you asking why it's difficult to come up with a quantum theory of gravity?
OK, for starters look at this page.
The page recommended by JesseM seems to say that there is a conflict between special relativity and quantum mechanics: "Even in presumably empty space, this "flatness" gets called into question by the uncertainty principle if you examine space at extremely tiny scales." I don't understand this, since I usually hear that it is GR, not special relativity that conflicts with quantum mechanics.
Here are some comments from Burgess, Quantum Gravity in Everyday Life, Living Reviews in Relativity <http://relativity.livingreviews.org/Articles/lrr-2004-5/>: [Broken] According to this point of view there is no such crisis, because the problems of quantizing gravity within the experimentally accessible situations are similar to those which arise in a host of other non-gravitational applications throughout physics. ... This is not to say that there are no challenging problems remaining in reconciling quantum mechanics with gravity. On the contrary, many of the most interesting issues remain to be solved, including the identification of what the right observables should be, and understanding how space and time might emerge from more microscopic considerations. For the rest of the discussion it is useful to separate these deep, unsolved issues of principle from the more prosaic, technical problem of general relativity’s non-renormalizability.
More from Burgess, Quantum Gravity in Everyday Life, Living Reviews in Relativity <http://relativity.livingreviews.org/Articles/lrr-2004-5/>: [Broken] Once it is understood how to use non-renormalizable theories, the size of quantum effects can be quantified, and it becomes clear where the real problems of quantum gravity are pressing and where they are not. In particular, the low-energy expansion proves to be an extremely good approximation for all of the present experimental tests of gravity, making quantum corrections negligible for these tests. By contrast, the low-energy nature of quantum-gravity predictions implies that quantum effects are important where gravitational fields become very strong, such as inside black holes or near cosmological singularities. This is what makes the study of these situations so interesting: it is through their study that progress on the more fundamental issues of quantum gravity is likely to come.
I think you misunderstand, I interpret the page to be saying that even if there are no massive objects to curve spacetime, at extremely tiny scales vacuum fluctuations will cause non-negligible curvature so it's unrealistic to just take for granted a flat background spacetime. So this is a matter of GR, not SR.
What is renormalizability?
To answer your question simply - There is no simple answer. Laymen wise - whether it be mathematical or theoretical, key principles in each of those fields do not mesh together.
Since you did not state your degree of study in these fields, I will assume you are acquainted with a few of the major theories and maintain a casual curiosity about the universe around you. If this is the case, read Michio Kaku's Hyperspace. It is easy to read yet provides background history of general relativity - how the field theory came about - the attempts to combine the two - and your question as to why it is so difficult. A fundamental example of why these two puzzle pieces do not seem to fit would be the role gravity plays in Classical Physics and Quantum Physics. Classically speaking, the gravitational force is much weaker than the electrical force. It takes the our whole planet to keep a piece of paper on a table, it is also difficult to measure. There is something called quantum corrections. The quantum corrections due to gravity are very large (10^19 billion eV)- this energy will not be achieved in this century or with our current technology. When one quantizes force (ex. light) it is broken up into energy packets called quanta. Theory then says gravity should be broken down into gravitrons. These gravitrons bind objects together. Yet experiments have not shown any indication gravitrons exist. To calculate the quantum corrections to Newton and Einsteins theory of gravity prevail infinite answers - purely un-workable. In short, there is a discrepancy between general relativity's explanation of gravity (which work well for large objects like stars and planets), but when one tries to insert the method used to describe forces that have been sufficiently quantized, infinities arise mathematically. I hope this helps - also Stephen Hawking's A Brief History Of Time - is also useful for background knowledge.
er gravitons* - sorry its late here in NYC
Rovelli, http://www.scholarpedia.org/article/Quantum_gravity: [Broken]
Finally, an approach that must be mentioned is simply the possibility of taking general relativity and merge it with quantum mechanics following the conventional methods of quantum field theory, but circumventing the traditional difficulties by an intelligent retuning of the quantization method itself. Several authors have questioned the conventional assumption that this is manifestly impossible.
Unfortunately, I don't know which, if any, of the references he lists give more details on this possibility.
Being a novice in this field but infinitely interested in this issue, one of the "clashes" I see between SR/GR and QM is the definition of gravity. SR/GR do not specify the mechanical aspects of how gravity works; it only expresses the mathematical notions of how it behaves. QM, also, does not yet express the mechanical mechanism for gravity, either, but QM folks see the answer to encompass the notion of the graviton, which has yet to be detected.
So, in essence, there is a discontinuity between the physical mechanism of gravity, which of necessity must be QM, and the mathematical behavior of gravity expressed by SR/GR. Mathematics that describe QM is different than the math of SR/GR.
I'm sure that are many other aspects that I don't understand, but this is the one item that strikes me most glaringly.
Perhaps nature is giving us a hint. 70 years of assuming that: since GRT is classical, and the quantum world is more fundamental; therefore the manifold is quanitizable. Perhaps hopping on the NOT to such assumption might be appropriate. Perhaps the manifold is fundamental; has an equal weight status of it's own. Was Wolfgang Pauli right about their separate status?
The idea to take general relativity and "merge it with quantum mechanics following the conventional methods of quantum field theory, but circumventing the traditional difficulties by an intelligent retuning of the quantization method itself" that Rovelli mentioned but didn't give references to in his scholarpedia article for may be the "asymptotically safe" field theory of gravity.
From the above-mentioned article by Lauscher and Reuter: The conclusion is that it seems quite possible to construct a quantum field theory of the spacetime metric which is not only an effective, but rather a fundamental one and which is mathematically consistent and predictive on the smallest possible length scales even. If so, it is not necessary to leave the realm of quantum field theory in order to construct a satisfactory quantum gravity. This is at variance with the basic credo of string theory, for instance, which is also claimed to provide a consistent gravity theory.
When you write a solution as a series you find it sums to infinity. Renormalization means there's a magic trick where you arrange the terms of a series so that they sum to something finite. Theories in which you cannot do this magic trick are nonrenormalizable, will sum to infinity, and are perhaps unworkable.
Actually, Kenneth Wilson showed that unrenormalizable theories are workable, but only at low energies. That means that maybe the renormalizability of renormalizable theories is not terribly important after all, ie. we should not conclude that renormalizable theories work at high energies.
for evidence of skepticism.
If i were to respond with a soundbite (which I am) I would say to main problems are:
1) GR is background independent i.e the spacetime structure is created by matter (dynamic). QFT is performed in a static background, usually Minkowski spacetime, though not necessarily.
2) Time is reversible in QFT but GR has a built in direction for time.
I try to understand, why the GR is background free, but I do not succeed.
It is posible visualy to describe? It goes without diffeomorphism?
Let us say that we have 3 dimensional space with c = oo. I imagine that we can put GR in this space. Only what we should do is to stretch and reduce some distances and some time differences?
I imagine special relativity (SR). Here we can have 3 dimensional space with c = oo and we put in it SR space. For observers with v <> 0 streching and reduction of some distances happens. What is new in GR?
In GR, the Energy Momentum Tensor describes all the matter and energy in a region. That Tensor is equal to the Einstein Tensor which describes the geometry of the region. You will have to put in a lot more study I think. You will need to understand more geometry and in particular tensors. I don't think think there is a simple way to describe this other than to quote John Wheeler and say "matter tells space how to curve, and space tells matter how to move".
I know Einsteins's equation and tensors. But I really want to imagine "background free" and to understand more precisely what means "background" free.
For instance, if we imagine" that we send a signal with c = oo. What will be its direction and what it will be different as c = c path of light?
But Einsteins equations and tensors are not the most simple way for understanding of "background free".
Background free means that no prior geometry is assumed, the geometry arises from the solution of the field equations. In GR you usually attempt to solve for the metric. The metric is a tensor, which like it's name suggests, encodes information about distances and angles. The metric defines the curvature of spacetime. In non-background free theories the geometry is pre-defined i.e. SR is calculated in Minkowski spacetime which has a flat metric. QFT is also usually calculated in Minkowski spacetime. QFT can be calculated in curved spacetime but it doesn't generate it.
Basically, GR deals with geometry as the basis. The equation of GR is G=T. Geometry implies precise definitions of position and time. QM tells us that we cannot measure p and x together exactly. When we get to the very small, geometry is not important.
So long as forward difference is a mere approximation of a derivative, how can one determine anything when everything used to measure change approaches zero?
Renormalization is a mathmatical procedure that appears to work [ most of the time], but has not been proven fundamentally sound. I think that is where the problem resides.
This is one of the reasons quantum mechanics tends to deal with momentum rather than velocity. Momentum can be measured e.g. via collisions or via wavelength. But any attempt to calculate [itex]\delta x / \delta t[/itex] as [itex]\delta t \rightarrow 0[/itex] is doomed to failure, because as the uncertainty in x decreases, the uncertainty in momentum increases, and the "expected momentum" becomes infinite, yielding an "expected speed" of c, for any particle whose position is known precisely! This is the maths' way of saying you can't do this.
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