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Electricity and magnetism was unified in the 19th century, and unification of electromagnetism with the weak force followed suit, bringing into play the electroweak force.

I've been told that unifying these with the strong force is likely to be far easier than unifying them with gravity, albeit still very hard.

Apparently this is because of the fact that electromagnetic, weak nuclear and weak nuclear force equations are relatively similar, whereas the equations for gravity differ greatly.

What are the fundamental differences between these equations? (If you could write some that would be great, and then explain the technical differences, if necessary.)

What is it about these differences that make them much harder to unify?

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It is a misunderstanding. It is NOT hard. Just non of us could do it just yet. You may be interested that other forces are unified NOT by us (humans - Maxwell) but BY nature - we just observed that we can write them down nicely in a very similar equations. So the question remains -- did the nature unify gravity with other forces or not? If yes one day we will find correct math. –  Asphir Dom Feb 26 '13 at 21:06
    
Related question about the need to quantize gravity: physics.stackexchange.com/q/6980/2451 and physics.stackexchange.com/q/10088/2451, and physics.stackexchange.com/q/52211/2451 –  Qmechanic Feb 26 '13 at 21:06
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Quantum Gravity is non-renormalizable, see e.g. physics.stackexchange.com/q/3901/671 for what that means –  Andre Holzner Feb 26 '13 at 21:12
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In a recent talk, Nima Arkani-Hamed talked about this preconception: we should all take the statement "gravity + other forces -> trouble" with a grain of salt, since every single QM experiment has been done in the presence of gravity. Only at very small distances, our understanding goes out of the window. –  Vibert Feb 26 '13 at 23:17
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I addressed this at physics.stackexchange.com/questions/387/…. –  Matt Reece Mar 6 '13 at 3:31

3 Answers 3

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If you go back to the origins, the difficulty in merging gravity with the other forces mostly stems from general relativity being a purely geometric theory -- again, that's in its original form -- and all the other forces being quantum, by which I mostly mean they are conveyed by well-defined force particles. The photon as the particle that conveys the electromagnetic field is the simplest example, but the idea carries over very well to both the weak and strong forces.

General relativity in contrast works very, very well without even invoking such concepts, or for that matter particles in general. As Einstein formulated it, GR really, truly is all about curved spaces.

By analogy it was subsequently assumed -- I think sometime back in the 1960s or perhaps 1950s? -- that gravity must also have a quantum form, but it's always been an assumption, not an absolute proof, a sort of "it worked here, and here, and here... so surely it also works just as well for the last force, gravity?"

But it's a bit tough to bridge such a huge gap. It's reasonably easy to provide a general description of gravity as a universally attractive force, although even there you quickly get into odd infinity problems not seen with other forces. But if you do that... what happened to all that part about the space being curved? The simplest possible all-attractive quantum force model would simply keep space as a rigid framework and do everything pretty much as with the electromagnetic force, only with a single type of charge (mass).

So, you can do one (curved space), or you can do the other (universal graviton-mediated quantum attraction), but it's not trivial to do both. And no matter how you do it, the other forces don't directly bend space, which keeps gravity pretty unique. Consequently, the details never seem to work out quite right, and many books have been written about why that might be.

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I'm not sure I know what I'm talking about, but I'd say another (other than considering well-defined force particles) relevant aspect of quantum modeling here is that the entities in it don't have a particular position in space. It makes sense that it is hard to apply this idea to space itself. –  Abel Molina Mar 2 '13 at 7:42
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Abel, that's an interesting comment and observation. Early quantum theorists assumed infinitely precise spacetime a backdrop, and for the most part, so do modern quantum field theories. Even without gravity there are problems with that, since because of quantum uncertainty the infinite downward perfection of space presumably fails at some point (the Planck limit). As long as you don't worry about the equivalence of gravitons to curves in said space, it sort of makes sense. But "quanta of curvature" residing fuzzily on the same space? It starts to sound like a Zen koan pretty quickly... –  Terry Bollinger Mar 5 '13 at 0:29
    
I read (another question on here) that there's a geometric theory of electromagnetism as well... if that's the case, would another option be to look at the problem in reverse? i.e, a geometric theory of quanta, for instance? If this is complete rubbish, I apologize. –  Kitchi Mar 5 '13 at 17:51

A detailed account of the subtleties and technicalities that highlight the problem, is a research topic by itself!

An attempt is made here to provide some brief discussion on epistemological arguments, without mathematical jargon or detailed phenomenology, which I am sure some would wish to complement or improve. I would welcome it.

There are a number of reasons which, when combined, can help us understand why we don’t see how to proceed in the endeavour of finding a good theory of quantum gravity and hence unify it with the other forces of nature. The main reason is the physically different structures of the gravitational force and the other forces of nature. When it comes to gravity, even the notion of quntum fluctuations of the fields is already problematic. While for the other forces of nature quantum fluctuations have meaningful interepretation and are relatively "easy" to claculate.

Another possible reason is, the tools we are using and the philosophy we hold on the notion of quantisation of fields, and that we try to push this philosophy to include gravity. This has been realised in the “old” approaches, in which attempts were made to construct perturbatively renormalizable theories of quantum gravity. They have all suffered from one or another shortcoming.

A successful quantum field theory allows us to do calculations, and extract sensible results which then can be tested against experiments. In the two main approaches to quantum gravity, string theory and Loop quantum gravity, such testable calculations have not been possible to access experimentally, due to the large amount of energies required. So the results of these theories remain at best, at the present time, just theoretical speculations. But both these theories are still being developed and by no means they are, at this stage, completed theories of nature.

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In my view, basic thing is that General Relativity is a deterministic theory and Gravity manifests at macroscopic level, whereas Quantum theory is a probabilistic theory and manifests at microscopic level, and other forces operate at this level. Hence it is difficult to unify these theories. There have been serious efforts during past five decades by theoretical physicists / particle physicists / mathematicians to give a satisfactory Quantum Theory of Gravity, and there have been 1. Kuchar - Isham 's Canonical approach 2. B. DeWitt's covariant approach 3. Ashtekar's loop quantum gravity approach 4. Penrose's Twistor theory approach etc. but as yet we don't have a fully satisfactory theory. Perhaps, we need New Mathematical theory like Non-Commutative Geometry to unify these two, who knows !

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