Near the end of his life Einstein worked tirelessly to find a way to unite electromagnetism with gravity. He could not, and never did, the notes scattered on his desk scrawled with fruitless probes and useless hypotheticals. Indeed, Einstein passed without even understanding why the two forces could not be united.
Now, with over a century of experience in dealing with quantum problems, we can see the deep roots of the problem. It’s in that infernal probabilistic nature of the theory. There are…many…approaches to solving problems in quantum mechanics, different mathematical paths one can take to arrive at useful predictions. For our story, the story of attempting to unify electromagnetism (and, eventually, the other forces) with gravity, it’s best to describe quantum machinations through a technique developed by Richard Feynman.
Let’s return to tossing a ball. Easy, simple, uncomplicated. I throw the ball, and it follows a single path to you. A reliable path. A predictable path.
Now, in the usual quantum fashion, let’s make it weird. Imagine the path the ball takes – a simple parabolic trajectory. But imagine the ball taking a different path, an improbable path. Imagine the ball, immediately after leaving my hand, jumping one hundred miles into the sky, then falling into your hand.
How about another. This time, the ball spirals around my head a half dozen times, then leaps over to you and spirals around your head before finally settling in your hand.
Try this one: the ball rockets out into space, lazily orbiting the Andromeda galaxy for a few billion years before finally returning to Earth and landing in your very patient hand.
Imagine all the possible paths, however ridiculous and improbable they may be. What’s stopping the ball from taking those alternate paths?
Well, to be perfectly honest, nothing. From a strict physics point of view, all that matters is that energy and momentum are conserved; at first glance, physics has nothing to say about the path traversed, just the starting and ending points. But if we take all the possible paths – and I really do mean all the possible paths – and take the average, then an insane path on one side (say, the one that flies up to the sky) will cancel out another one on the opposite (say, the one that slithers along the ground like a demented snake). Every possible trajectory will cancel out another one, leaving only one path remaining: the one we’re familiar with.
Feynman took this thought experiment and applied it not to the motion of tossed baseballs, but to the movement of electrons and other subatomic particles. In this language, known as the path integral formalism for those of you who love jargon, to properly calculate the behavior of a subatomic particle performing some interaction (or even just wandering through space, minding its own business) you must first trace every possible path and every possible mode of interaction that the particle can take.
It’s rather complicated and takes a furious amount of math, but it works. And the quantum weirdness doesn’t stop there (otherwise you could rightly ask what the point of all this is). When we examine the behavior of a subatomic particle, like an electron or a photon, that infinite branching multitude of alternate paths do matter, at least a little bit. We still play the same averaging game as before, taking into account every possible exhausting path, but this time each path gets a little weight associated with it. Paths closest to the “true” path (as much as that word makes sense in quantum mechanics) get a higher weight, while paths farther from it get a small weight…but not zero, and they still contribute.
Thus in quantum mechanics to make a final determination of a probability of a particular interaction or a particular trajectory, you must account for, with a suitable amount of grim determination, every possible interaction or trajectory.
Oh, did I mention that subatomic particles can spontaneously change identities, becoming and unbecoming other kinds of particles as they travel?
Grim determination indeed.
So if we want to calculate the particulars of, say, an electron bouncing off of a photon, we have not only compute the interaction strength of the bare electron slamming into the photon (as we would in the macroscopic world, getting it over and done with), we also have to sum together every single possible way that the electron could interact with the photon, including, but not limited to: the photon spontaneously transforming itself into an electron and a positron; the electron and photon first interacting via the weak nuclear force, turning into neutrinos, then transforming back; the electron creating a Z boson, which in turn interacts with the original electron, before proceeding, and so on and so on.
There are an infinite number of ways that any two particles can interact. Thankfully for all but the most intense calculations involving interactions in our high-energy particle colliders, only a handful of these options are required, as they are the “closest” to the “true” interaction, and hence they have the greatest weight.
If you were to list all the people you know, you would start with your immediate family, your dearest friends. Only then do you begin listing distant cousins, work colleagues, and half-remembered acquaintances. You could know an infinite number of people and you would still begin your list this way, because those closest to you carry the greatest weight. At some point, you give up, as your list is good enough for whatever purposes suit you.
Like I said, quantum computations are difficult and exhausting, because as a matter of course every interaction involves an infinite multiplicity of possibilities. These infinities bedeviled the early quantum mechanics, who feared that there would be no escape, and hence no useful theory of quantum mechanics. But Feynman provided a mathematical trick, known as renormalization, to, in essence, sweep those infinities under the rug, allowing computations – and the careers of physicists – to proceed.
And this, this, is why we cannot unite gravity with quantum mechanics. When we include gravity in our calculations, we must not only account for all the possible paths and interactions and splitting and combining of the particles involves, but also all the different ways that spacetime itself can twist, fold, and bend throughout the process of the interaction.
Infinities appear in these calculations just as easily as they appear in ones not involving gravity. It is on one hand just an extension of the earlier troubles with the quantum. It is on the other a complete and total nightmare. The reason is that there are simply too many infinities, and the techniques we have grown accustomed to in half a century of quantum fiddling fall far short. No matter how hard we try, the infinities stubbornly refuse to budge, and our calculations become lost in a tangled mass.
I’m not sure the article does itself a service when it first claims equiprobability of paths, when retract that for unstated reasons of what matters (in both cases).
And this:
“When we include gravity in our calculations, we must not only account for all the possible paths and interactions and splitting and combining of the particles involves, but also all the different ways that spacetime itself can twist, fold, and bend throughout the process of the interaction. Infinities appear in these calculations just as easily as they appear in ones not involving gravity.”
That is assuming that you want to include high energy effects in an effective theory. For all purposes, quantizing gravity seems to be pretty forward: “Quantum gravity as a low energy effective field theory”, John F Donoghue, Scholarpedia. It even makes up for general relativity non-linear first order derivatives approximation with including all orders of derivatives in its own attempt of effective description.
But like general relativity effective quantum gravity field theory breaks down as you go for Planck energies. As far as I understand not because you may insist on introducing general relativity’s own effective approximation of curved space but because you are encouraged to do the impossible: introduce an infinite number of parameters from observation.
The question is, seeing how our most basal physics theories are effective, do you need to? Inflation is observed to happen far from Planck energies, and we don’t know if there was any initial conditions to that. And IIRC a review of black hole models had 5 out of 15 constructed with no Planck energy density problems.