Ax_xiom said:
Ok I'll look at that
And another question, is the infinite dimensions thing a view of the entire field of Quantum General Relativity or just a subset of them
Infinite dimensions is a far out of the mainstream hypothesis.
It isn't necessarily a "crackpot" hypothesis. In other words, there may be a line of scientifically logically plausible arguments to support the approach, and its phenomenological implications may be subtle enough that it can't be conclusively ruled out with observational evidence.
But, the big problem with any theory that proposes more than four dimensions of GR and the Standard Model, is that you then have to explain why we observe only three dimensions of space and one dimension of time in high precision observations and experiments.
Mainstream string theory proposes 10-11 dimensions, of which all but four of them cannot be observed. Mostly, these theorist argue that this is because (1) we and all non-gravitational interactions are confined to a four dimensional brane from which only gravity can escape (which is a partial explanation for why gravity isn't closer in strength to the other fundamental forces), or (2) the dimensions in excess of 4 are "curled up" and tiny so that different locations in those dimensions are imperceptible, even in precise scientific observations.
Of course, the lack of a consensus on why we can't observe more than four dimensions in string theory, is itself notable. String theory doesn't have any definitive or final answers about reality for us at this point, and is really just a set of lots of highly similar theories that are intimately related to each other mathematically.
A hypothesis that lacks any observational support to suggest it, and makes theoretical predictions contrary to other theorists in the same subfield (i.e. string theory) do, is what is called an
"ill-motivated" hypothesis.
Now the fact that a hypothesis is ill-motivated doesn't mean it is wrong. New ultra-precise instrumentation to make a novel kind of observations could provide support for this idea at any time. Given the observations available to scientists in the 1700s, both general relativity and the Standard Model of Particle physics would have been ill-motivated. The math to hypothesize them mostly already existed, but in the 1700s there were no instruments at the time that were capable of observing that phenomena that suggested them, and no one has proposed similar theories about potential new physics then.
But an ill-motivated hypothesis also doesn't provide any really compelling reason to take it seriously at this time, because even if there are "new physics" out there waiting to be discovered, there isn't much to recommend this hypothesis relative to the theoretically infinite variety of possible alternatives which are also ill motivated.
What could be wrong with the analysis?
I'll identify a few of the most obvious possibilities, although this list obviously isn't exhaustive.
Possibility one: The infinite series used isn't fundamental
- the paper then points out that one of the formulae used there only has a finite amount of terms, where it should have an infinite amount of terms
But the fact that one calculates something by approximating an infinite series doesn't imply an infinite number of independent degrees of freedom.
For example, one way to calculate π is by summing up terms from an infinite series. But this doesn't imply that π (or the circles that it is a property of) has properties that are inherently tied to its infinite series representation. You could also just define a circle and the radius of a circle and use that to measure π directly without resorting to an infinite series.
Most quantities in the Standard Model which we calculate using truncated infinite series methods (typically only to a handful of loops made up of thousands to hundreds of thousands of individual terms), obviously happen in real life by some method that doesn't require some extra-dimensional computer to calculate an infinite series every time a particle moves or interacts.
There is every reason to think that the infinite series path integral approach to doing those calculations is not truly fundamental. Indeed, the fundamental way that it happens may be closer to Monte Carlo simulations which apply random chance to lots of independent agents in the model, than it is to the analytical path integral calculation that physicists use to make high precision calculations.
If there is a way to make the same calculation that doesn't involve an infinite series, then the mere fact that an infinite series can be used to make a calculation shouldn't imply that this calculation alternative implies that there are an infinite number of fundamental dimensions of space-time. The infinite number of terms in the calculation may simply be a calculation trick rather than saying anything fundamental about the nature of space and time.
As another example, it is possible to think of imaginary numbers as just a calculation trick, since no observables allow you to directly observe some quantity that has an imaginary number value, even though one can infer an imaginary numbered factor in some intermediate step in a calculation that is done to get what you observe.
Possibility two: The dimensions implied aren't truly the kind of independent and fundamental dimensions we mean when we talk about space-time being four dimensional.
Another possible issue could be that the analysis is not properly distinguishing between a loose sense of the word "dimensions" and a more strict sense of the word restricted to fundamental and irreducible dimensions that are fully independent degrees of freedom.
(Or, in the alternative, the distinction is being made properly and the true result isn't actually about the number of fundamental independent dimensions of space-time, but the term "infinite dimensional" is being used in the title as a sort of click bait to get people to read the paper which is reaching a result which has some scientific merit, but is much less awesome and amazing than its title suggests. I haven't dug into this particular paper to confirm that, but it isn't unusual for physics paper titles to imply something more awesome that the paper actually discusses, with the deflating caveats to the paper's conclusion buried away in the body text.)
As an example within the physical sciences, you can define "temperature" in a way that it is a scalar quantity across all of space-time, which looks like a fifth dimension, even though actually, it can be fully derived from the motion of particles within a four dimensional space time.
Similarly, you could model gravity as a tensor field with a sixteen element tensor value at each point in a four dimensional space-time. But because that can be fully determined from the distribution and flux of mass-energy within space-time, the gravitational tensor field isn't a true independent fundamental space-time dimension.
Outside the physical sciences, for example, such as in an economics model, it is routine to design a model in which all sorts of factors (e.g. the prices of goods in all sorts of different locations and times) are modeled as thousands of independent dimensions, even though those dimensions aren't fundamental degrees of freedom in space-time.
Possibility three: The paper is bumping into the concept of emergent dimensionality in space-time from a different perspective.
In some variations of the loop quantum gravity models of quantum gravity, space-time at the most fundamental level is conceptualized as an infinite, or near infinite (e.g. the four dimensional volume of the observable universe divided into Planck distance/Planck time chunks) set of nodes which have a finite number of connections (e.g. three or four) to other nodes. In this conceptualization concepts like the continuity of space-time, and locality, and also the number of dimensions of space-time, are only statistical approximations that emerge from a network of connected nodes that is actually more fundamental.
It would probably be possible to map this concept of emergent space-time dimensions to a model in which there are an infinite number of dimensions, but in which dimensions greater than the number of connections per node are impossible to observe for all practical purposes, because the network arrangements from which more than that number of dimensions emerge are so small and so improbable.
Thus, it could be that the infinite term series from which it is inferring an infinite number of dimensions, is really describing an infinite, or functionally infinite, number of configurations of nodes and connections that correspond to higher numbers of emergent dimensions that are increasingly less important phenomenologically, because the later terms describe systems made up of fewer nodes in less probable configurations.
With respect to the notion of infinite and functionally infinite being difficult to distinguish, Feynman, when developing renormalization, had wondered if the renormalization cutoff was just an arbitrary mathematical trick, or if there was some real physically based cutoff beyond which the infinite number of terms in an idealized path integral formulation did not extend, since the omitted terms below the renormalization cutoff were too small to make an observable difference.
