Where exactly do the laws of relativity and quantum mechanics disagree?
At very high energies, there are problems between GR (not SR) and quantum mechanics. This shows up in the fact that GR is not "renormalizable".
However, at "low" energies (i.e. energies within human laboratory experience), we have reasonably good "effective field theories" for quantum gravity. What we don't have is a theory of quantum gravity that can be extended to arbitrarily high energies.
so we're talking about charges.. like electromagnetic forces or gravitational forces? what laws don't work together and why not??
I'm not sure how to explain this more simply and still be accurate enough to be useful.
Anyone else care to give it a try?
I think difference people would describe this from different perspectives, depending on your approach, and perhaps there are more than one focus points to the problems.
Here are some personal _philosophical considerations_.
From a foundational and a bit philosophical point of view I would say that:
If we say that quantum mechanics suggests that there exists a set of definite possibilities (hilbert space) on which we can only determined the probability distribution, and the evolution of the probability distribution, the best we can do is really to take our "dice" and play.
Then, general relativity and any extended relational principles suggests that we can not just "pick a dice", we need to define the dice using existing relations, so this dice suddenly becomes alive.
So The full QG should be some kind of probability theory where the dices changes (evolves) between throws. So our job is not only to predict the system by means ot dice playing, we also need to predict the dice! Because outside dices are banned in a proper relational model.
This seems to me at least, impossible to achieve with assuming a given universal hilbert space playground. I think the hilbert space itself will have to be dynamical. Try to imagine how you would define the complete set of possibilities, in terms of existing relations? I'd say you can't, not without arbitration. You can estimate it at best. So we end up knowing our hilbert space only at a certain probability.
That seems to get's hairy.
If you ignore that, and try to merge the hilbert space stuff with a relational model it is not a surprise from the philosophical ponderings that this is bound to fail, except in special cases of course.
My impression is that most unification work has been made in mathematical spirits within the somewhat semiclassical foundation. I think it's a mission with a low probability of success, and I think we need to revise the very foundations.
I would say that part of the problem is how to even compare then properly, since they rest on somewhat different abstractions. For example energy in quantum mechanics isn't really defined in exactly the same way as in classical physics. No to mention time. All models have some axiomatic starting points which serve to attach the abstractions to reality. If this is done differently, it isn't easy to compare them.
So IMO, before technical issues is discussed I think the broken line of reasoning should be rectified. Perhaps then, the technical issues are non-issues.
Quantum physics operates on a flat background, e.g. a three-dimensional reality with a universal measure of time. General relativity has time and space intermingled. It is intermingled in such a way that we cannot (except for trivial cases) unwind it into a flat, three-dimensional reality with a universal measure of time.
The basic problem for the incompatibility between GR and QM is that GR is a classical theory that does not include the uncertainty principle and other postulates of quantum mechanics such as the collapse of the wavefunction.
If space-time is considered to be classical as in general relativity but matter is considered to have quantum behaviour, the instantaneous collapse of the wavefunction of a particle would lead to superluminal propagation of the deformation of space-time. Otherwise, it is needed to incorporate the superposition principle to the definition of space-time states. You can find a detailed explanation of this argumentation in chapter 13 of Wald's book "General Relativity".
You can ask about compatibility of GR and QM from a more wider perspective, namely whether general covariance and quantum principles are incompatible. This means, whether a quantum theory of gravitation can be generally covariant if QM principles are saved, or whether QM principles can be saved mantaining general covariance.
I think the wider perspective is the more healthy one.
More philosophic comments...
QM is linked to probabilistic concepts, but physics aside there have always been the controverses on howto interpret probabilities. The frequency interpretation of an inifinite collection of test systems is clearly something that can never be realized in reality - it's an idealization. Relative frequencies can be used as estimates _in reality_, but there is always an fundamental uncertainty there. It's an abstraction that is obviously useful in many cases, but also obviously not of suffucuent general validity to describe the entire universe. From my philosophical perspective this is unsatisfactory.
This is related to the fixed background issue. The conclusion and I think good spirit to keep from GR is that all knowledge and estimates is fundamentally relative to the observer. And the best we can do is try to find the connections between observers.
But QM adds another crucial point, information must be _measured_. Ie defined in terms of something measureable. This includes probabilities, and as it seems it takes infinite data to measure.
Is it then tempting to think that we can list all the possibilities, and then rank them as per their probability, but there we go again. There is no way this ranking is _definite_ either, because by the same token it would take an infinite number of completely unrealistic trials to establish, and infinite time. By the same token, it's also only an estimate. So at the best stretch, there is a residual fuzzy we have to acknowledge. If we make the mistake to *ignore* the residual fuzz, we are also limiting our own possibility to progress. We should not only, keep all doors open, we need to see that the doors are inferred from relations, and there may well pop up new doors that wasn't there before, and several doors may associate into one.
I think we should to acknowledge this, and build a theory that makes sense from scratch.
My own conclusion is that the seemingly only design that can match this is an fundamentally evolutionary one. Neither QM nor GR does this. Evolutionary also means that we do NOT make an outragoeous list of possibilities on the outse, we let the list grow as well! So as we evolve, things do not only change, the list of possibilities change too.
My personal hunch is that this is the direction to find the answer.
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