In my past life, a decade or so ago, I was writing and thinking a lot about topological geon foundations for quantum gravity. The gist of the idea is that general relativity permits closed timelike curves when spacetime is topologically non-trivial (basically, when spacetime has holes in it, like wormholes). I wrote a lot of unpublished thoughts on this idea, it is a general relativity foundation for quantum mechanics. Which if you do not think is super cool, even astounding, I can't help you! It was based on older idea of Wheeler and Misner on geometrodynamics and "charge without charge". Of course, general relativity is already a sort of "mass without mass", since mass-energy can be interpreted as spacetime curvature, if all mass is a sort of singularity. Unfortunately such things are almost impossible to explore experientially due to measurement limitations.
The trouble for Misner and Wheeler was they could not quantize their theory, so it languished. They failed to realise a topological general relativity may not need to be quantized, it is already quantized, you could say "quantization without quantization". I originally thought about this when Kip Thorne came out with a popular science book on black holes and time machines. He had classical GR solutions with causal consistency conditions which allowed non-paradoxical time travel. The toy model is that of a billiard ball which when pocketed springs up from another pocket but earlier in time, then knocks into the present version of itself and sends itself into the pocket from whence it came.
Although a toy model, this immediately suggested to me this is how weird quantum effects could be explained from pure general relativity, with no quantum postulates required. The logic at least seemed tight. The dynamics very difficult, as anyone studying geometrodynamics will know. This was around 1996.
Amazingly a guy at Warwick University in the UK had the exact same idea, completely independently and managed to produce a PhD thesis around this time, "A Gravitational Explanation for Quantum Mechanics". He must have been thinking of it before I was, and I was already doing a different PhD, sucking up all my time. But I never let the idea go. This dude's name is Mark Hadley. He still writes the occasional article on the arXiv about his topological geon foundations for quantum mechanics from general relativity. It's pretty cool stuff, but has no following in mainstream physics, where all the fuss is over string theory or loop quantum gravity.
I've never had the free time to explore a couple of the big obstacles to deriving all of QM from GR. Four of which always bug me are quantized spin-1/2 and charge, and the particle symmetries, and superposition. There are nice approaches to all of these, but it is superposition I will write about here. First a quite glimpse of the other problems:
Spin: spin is not necessarily a property of fundamental particles or 4-geons. In QM quantized spin is a type of consequence of rotational symmetry properties. So, since 4-geons have rotational symmetry properties it is fairly easy to get a theory of quantized spin in topological GR. You have to disabuse yourself of the notion of an actual spinning particle, and instead consider the logic of rotational symmetry. This becomes a little clearer the more you dig into the geon theory. But seasoned QM vets should know the idea. In QM spin is theoretically accessed through matrix operators and their commutation relations. These do not described actual spinning things, they describe the effects of measurements under rotating coordinates and the like. So if a particle was spinning (which we presume most are, since not spinning around is a fairly atypical thing for tiny objects) then the symmetry operator algebra would apply. But the existence of "quantum spin" is not a quantum effect (that's a widespread myth), it exists in general relativity by virtue of rotational symmetries.
Quantized charge and 3. Particle gauge groups: these are lovely problems to explore. I have only just begun. But I think a lot can be learned surprisingly, from string theory. String theory obtains particle symmetry groups from geometry. So there is good reason to believe geometry+topology in pure GR could also yield the standard model symmetry groups. A terrific new lead on this is Cohl Furey, formerly at Waterloo, now at Cambridge. Her work follows a tradition of work of Dixon, Baez, Schmiekel and Trayling & Bayliss & Cabrera and others which have all obtained most of the standard model from Clifford algebras in 6+1 smooth spacetimes. My own suspicion is that 4-spacetime might be sufficient, but not with smooth GR. The problem with smooth 4-spacetime is there are not enough degrees of freedom to obtain the Standard Model symmetry gauge groups. But if you allow 4-spacetime to have topolgical non-trivial structure, then you automatically produce new degrees of freedom locally, just not globally. But the Standard Model gauge groups are based on local gauge invariances, and so, to my mind, this seems like a tantalising, even salivatingly enticing avenue for research. I'm talking Nobel Prize level stuff.
Superposition: this is a relatively boring aspect of QM from GR, but it often perplexes me, because I've never sat down for weeks to work it all out. I had a few thoughts on a lunch break today, so wanted to document them here for priority.
My daily work these days is far more urgent, I have been shocked into a nascent activism by (a) the finding that almost all of mainstream economics is utter crap, and it is literally killing people (I'm talking about neoliberal austerity ideology), and the battle against neoclassical economics and neoliberal politics is therefore, I believe, a moral duty. (b) The climate warming crises demands a realistic economics based on the unassailable foundations of modern fiat money dynamics (also know as MMT) which tells you government deficits are private surplus, and a sovereign government can never run out of money, so can always pay for anything for sale in it's own currency, including all idle labour. This is what should power a Green New Deal and a Job Guarantee, and completely overthrows the abysmal wreck that is mainstream orthodox economics.
In particular, once you realize a monopoly issuer of a fiat currency can never run out of money, you see the Federal government can always buy anything for sale in It's currency, and does not need any tax dollars to "pay for it". In fact, logically, taxes cannot fund a money sovereign. That false idea would be like saying you need to collect all the IOU paper you've handed out before you can give out more IOU's. It makes no sense. government is not tax-payer funded. Then you see how good things like a Green New Deal, debt cancellation, free healthcare, free college and anything tat can be resourced from available labour and materials can always be government funded. Government deficits are good, private household deficits are bad.
You see now why it is a moral imperative, once you have this knowledge, to work to overthrow economic orthodoxy, becasue those old fools are stark raving mad, and their policies are killing people needlessly in the same of budget austerity.
Dollar for dollar the government debt is the private sector surplus. Think about that for lunch. You suddenly realise you want BIG government deficits, don't you?
The government debt is our household + firm sector surplus, our savings. But we have to take it off the bankers and financial rentiers first, who never earned it.
So, as you can tell, this political activism is a moral imperative and has completely destroyed my ability to focus on theoretical physics. But the topological 4-geon bug has refused to let me go.
So here is my little bit on superposition...
How is there interference of single photons in GR?
Suppose particles a topological 4-geons containing CTC's. Then it is natural, even in absence of solutions to GR, that other geons may traverse those CTC's. So, merely as a point of logic, we must include non-causal interactions in topological GR. So clearly that allows a 4-geon to interact with a past trace of itself. So we now only need to show that 4-geons can interact in a wave-like interference fashion.
A natural way interference can occur with topological geons is by the usual classical interference of waves in a physical medium. 4-geons are like soliton waves in spacetime. Is it not plausible they can thus interfere exactly as soliton waves interfere? I think so. The question then is do the wavelengths for quantum states match the wave-like aspects of topological 4-geons? This seems to require the region of significant spacetime curvature of a 4-geon to extend over something like a de~Broglie wavelength. And this is a macroscopic distance for massless photons. Is this a problem for a 4-geon model of photons?
I am not convinced it is. It has been hypothesized, though perhaps not seriously enough, that a photon is a composite particle, like a particle + antiparticle in a kind of bound state. But if that bound state is more like a pair of topological defects in spacetime joined by an ER bridge, or some-such spacetime link, then you can begin at least to see how a photon can be effectively spread out over a large spatial distance.
OK, take all this with a grain of salt. I need a lot mote free time to work on the mathematics. But if you are interested, get in touch with me. I would love to star a research collaboration.
Well, that's all I've got for now. I'm sure form the lack of interest 4-geon theory has received this research will still be waiting for some attention after we've got the MMT political movement going strong.