I. Loopholes and workarounds
A "motivation slide" for a potential path to reconciling GR and QM that doesn't assume the existence of the Planck scale
In my post from about a month ago I argued that you can’t just willy-nilly drop the gravitational coupling constant G into quantum field theory (QFT) even “semi-classically” without it yielding the Planck length — whether it is justified or not, bidden or not. I suggested the “pick two trilemma” in the diagram above where you can select G and c (general relativity, GR) or ℏ and c (QFT) without it immediately sending you down the Planck-length-centric “quantum gravity”1 (QG) path that has, like string theory, not led to any progress2 for 50+ years3. Anyway, after the diagram I wrote:
I am not 100% sure what the theory is where you pick G and ℏ, but selecting G, c, and ℏ simultaneously could result in some serious contradictions.
Re-reading the post later, I was like “duh” — it’s Carrollian4 quantum gravity.
Now the Carrollian limit (c → 0) may be a fad reaching (passing?) the peak of the hype curve. The SYK model seems to be fading after taking off pre-pandemic; however, AdS/CFT and the holographic principle seem to be going strong 30+ years later. Who knows which random ideas will lead to insight into quantum gravity. The background for this post is the Carrollian structure of spacetime (e.g. YouTube lecture here), and Carrollian gravity (e.g. YouTube lecture here, lecture notes here5). I’d like to write down my intuition (*cough* crackpot physics takes *cough*) for why I think it’s at least an interesting direction — in handy listicle format.
Elsewhere
So this is my idée fixe — that the uncertainty in quantum mechanics is directly related to the lack of knowledge of events that occur “elsewhere” on a space-time diagram. Quantum mechanics is essentially just a mechanism to preserve observable information through time (i.e. unitarity6), so when something passes into the elsewhere region (which also includes the interior of black holes BTW), we need a way to propagate the information about a system forward in time from when we lose track of it until we can interact with it again (if ever, in the case of black holes7). We literally cannot know what happens inside that region until a signal reaches us8.
Elsewhere is oddly important in QFT, too. A field for a single particle state exponentially decays at the edge of its light cone — which means it’s non-zero outside the light cone (i.e. “elsewhere”) and so breaks causality. That decay is precisely cancelled by the inclusion of antiparticles via the creation and annihilation operators — the multiparticle Fock states are critical to a consistent, causal, relativistic formulation of quantum mechanics. Fields commuting or anticommuting at spacelike separation preserving causality is also key to the spin-statistics theorem. The fact that we can’t have superluminal propagation, that “elsewhere” exists for observers, is the reason why antimatter has to logically exist along with requiring different statistical behavior for systems that differ by half integer values of angular momentum. I can kind of squint and see the two roots of E² = p² + m² requiring an interpretation of the negative energy states9 but tying the structure of spacetime to intrinsic angular momentum10 is not intuitive11. To me, at least.
Anyway, I first talked about “elsewhere” on this blog in the context of the Carrollian limit a year ago. The Carrollian limit turns everywhere into elsewhere by collapsing every observer’s light cone (the brown triangles in the diagram above turn into a vertical line). However, I’ve been thinking about it for awhile — some back of the envelope arguments are given here, but generally whenever there are quantum effects then a system has passed through the elsewhere region12 of the observer’s space time diagram as depicted above for an EPR-type thought experiment. Most of the “quantum weirdness” requires spacelike separation at some point. I’d like to promote this concept to a general principle that we cannot know anything about what happens in the elsewhere region. Maybe not even the (ordinary, causal) laws of physics? The title of this post is “Loopholes and workarounds” — of the two, I’d consider this a “workaround”.
Carrollian time
In order to make the duality between the Carrollian limit (c → 0) and the Galilean limit (c → ∞) manifest, Carrollian “time” s needs to have units of L²/T (length squared over time) — or another way action per mass. Mass m is the “charge” of gravity and action ℏ is the fundamental scale of QM, so action per mass ~ ℏ/m would have a kind of logic in Carrollian quantum gravity.
Despite being “somenified”, a quantum system in Carrollian spacetime can evolve along its single fiber (noodle) with Carrollian time s. Therefore, it makes sense to talk about the evolution of an internal state, say a spin up state |↑⟩, to an orthogonal (distinguishable) spin down state |↓⟩ for a system of mass m. I worked through the math here, but basically this is just a re-hash of the derivation of the quantum speed limit (QSL) yielding:
This is the analogy of the uncertainty principle but for energy and “time”. There is no time operator in quantum mechanics as it’s just a parameter13, so when you see ΔE Δt ~ ℏ it means something different from Δp Δx ~ ℏ where both are operators with a commutation relation [x, p] = 𝒾 ℏ. The QSL is the robust way to make sense of the “energy-time uncertainty principle”. It means that states in a (Carrollian) time interval ~ ℏ/m are generally indistinguishable. It would also mean dividing a time interval for a process into chunks smaller than ℏ/m isn’t meaningful, which could be a way to understand Jacob Barandes’ indivisible stochastic process interpretation of quantum mechanics where
for s′ between the QSL and 0 that kicked off this growing series of posts. Attempting to create a divisible representation results in interference terms — i.e. QM.
It’s true that this all falls somewhere between vibes and dimensional analysis — but the same could be said for the “motivation” slide of any physics talk. I’m just working through the narrative behind this one publicly on a blog.
Oops, all weird dynamics
One of the drawbacks of the Carrollian limit c → 0 is that it’s kinda boring. In terms of your run-of-the-mill physics, nothing can move — an “ultra-local” limit. You can still have qubits evolving in place (per above), but for the most part there aren’t any “normal” dynamics. However there’s a completely valid “loophole” involving energy conditions that I hinted at in the previous post.
First, being a dimensionful constant, you can’t just take c → 0. You have to compare it to some other velocity scale vₛ (s temporarily just meaning “scale”) so that c/vₛ → 0 or alternatively c ≪ vₛ. What could vₛ be?
Back to that loophole. In many cases in dealing with GR you have to assume an “energy condition” — i.e. a constraint on the components of the stress-energy tensor Tμν. For an eye-opening review on the physics community’s complete lack of a firm theoretical basis for energy conditions, see here. Among other things, energy conditions can sometimes allow negative energy or superluminal propagation. The diagonal components of Tμν are energy density ρ and pressure P, so we could define a speed of sound vₛ² = dP/dρ in whatever mass-energy “fluid” Tμν is representing (the subscript s meaning “sound” now). Ubiquitous energy conditions like the null energy condition14 allow vₛ > c and so would allow us to consider the quasi-Carrollian limit vₛ ≫ c. And as mentioned above, if we’re taking seriously the idea that we can’t know anything about what happens “elsewhere” in the spacetime diagram then maybe we can’t know for sure processes obey the universal speed limit — I mean, tachyons have mathematically valid representations in the Lorentz group15. Besides, “spooky action at a distance” is a real thing16 and attempted measurements of the “speed” is at least ~ 14,000 c so maybe the Carrollian approximation is always valid17?
This one is definitely a “loophole”.
Collecting my thoughts
My main objective here is to collect the various threads of my crackpot intuition into a single unhinged tapestry. Hopefully it will inspire some real work and not just some more dimensional analysis or re-deriving existing results simply in a new context. Whenever you start a physics problem, it’s always best to draw a picture:
That picture in the Carrollian limit when Carrollian time is “short”, i.e. on the order of ℏ/m, looks more like this:
Already this is, as previously noted, suggestive of a quantum circuit diagram18 (see more about quantum computational complexity and gravity in a review here) — some set of qubits is set up at Σⁱⁿ and measured at Σᵒᵘᵗ. In the Carrollian limit, any “signal” moving from one point in space to another is inherently superluminal — but e.g. the collapse of a wavefunction is instantaneous throughout space19 (i.e. on a single time slice like Σᵒᵘᵗ). We don’t necessarily have to take the Copenhagen view. Indeed we might expect to gain some insight as to what the “correct” view actually is — especially given a framework where “spooky action at a distance” is logically permitted. (In fact, it is the only kind of action at a distance.)
I haven’t mentioned gravity much — it’s tangentially related to the energy conditions mentioned above, but energy conditions are mostly about the stress-energy tensor20. However the benefit of a semi-classical treatment of gravity in the Carrollian limit is that it prevents us from implicitly assuming a fundamental length (or time) scale like the Planck scale. It also helps inform the third side of the triangle at the top of this post. The other two are decades-old (GR in 1915, QFT in the late 40s/early 50s), but the third doesn’t yet exist — at least not in any “textbook ready” form (see e.g. here). Firmly establishing all three sides should guide us towards how to put all three of those fundamental constants together.
Here, I am going to use “quantum gravity” in the same way physicists use “dark energy” or “dark matter” — as a placeholder for something we do not understand but there are observable phenomena that behave like a uniform contribution to energy density even in empty space or additional mass that doesn’t show up in telescopes, respectively. Dark energy and dark matter are not theories themselves (YouTube). So in this post “quantum gravity” will have implicit “scare quotes” and will mean whatever eventual ideas that resolve the tensions in the previous post and produce a consistent theory with G and ℏ in it. Maybe I should call it dark gravity?
The “progress” in semi-classical quantum gravity consists mostly of creating problems (the black hole information paradox, “unknown” microstates needed for the black hole entropy, “holography”) and trying to make progress in solving them.
It should be (foot)noted that there is no reason why we should expect to achieve quantum gravity in one or two human lifetimes. Newton happened in 1687 and was basically considered the theory of everything (E&M resides in a Newtonian framework like how QED resides in a QFT framework) until the early 1900s when we got quantum mechanics and relativity (200+ years). IMO the best analogy is Maxwell’s equations from 1861 which had some conceptual issues that didn’t get resolved until Einstein’s special relativity in 1905 (the paper is called in English translation “On the electrodynamics of moving bodies” [pdf] and Maxwell’s equations are there in the first sentence). That was 40+ years even with the possibility of experiments at accessible energies. And it took Einstein. (I totally believe special relativity would have been discovered eventually; just might have been a few more years.)
You could also say it’s non-relativistic (Galilean) “quantum gravity” (NRQG or GQG) with the Galilean limit c → ∞ (which is in fact “a thing” e.g. here), but the Carrollian limit has been tied to the BMS group which is the asymptotic symmetry group of flat Minkowski spacetime which makes me think that the consistent version of the diagram at the top of this post is the Carrollian limit c → 0. Also, there’s a connection to dark energy which is an interesting conundrum between GR and QFT — the naive QFT estimate of the cosmological constant assuming the Planck scale is 120 orders of magnitude off the measured value. Side note: the other two sides of the triangle are in a sense unique — GR is the unique diffeomorphism invariant theory with at most two derivatives, and QFT is the unique combination of SR and QM (though there are multiple equivalent formulations of QM, so you could imagine different formulations of QFT however the content would be the same). There is a duality that relates the Galilean limit with the Carrollian limit, so modulo that duality you could say the third side is also unique.
This is from philosophers of physics but I think philosophy at this point is as useful as any other work — it’s all fumbling about in darkness and bafflement.
The black hole information paradox is essentially a breakdown in unitarity caused by Hawking radiation.
This question is super complicated and falls under the general heading of “the black hole information paradox”.
More later, but the idea that we cannot sensibly divide the time spent in the elsewhere region into smaller segments of time might be described in terms of indivisible stochastic processes.
However, this is not what is happening! There’s actually a cancellation with the time-reversed path.
The representation theory of the Lorentz group is complicated (it is a non-compact group so it has no finite, unitary representation) — and when tied to particles it basically means massive ones to have definite spin, massless particles have definite helicity (“spin dot momentum”), and “tachyons” (negative mass particles) only have definite momentum (“spin, helicity = 0”). It’s another way the causal structure of spacetime connects to angular momentum, so the spin-statistics theorem is not totally out of left field. At least this is the most intuitive way I can picture it.
For times on the order of a nanosecond, the elsewhere region of spacetime — a region that is causally disconnected from you and from which you cannot obtain any information — is 1 foot (30 cm) away.
This is part of the “problem of time” that I discussed in the previous post. This needs a resolution in order to couple GR and QM.
The contraction of Tᵤᵥ with two null vectors kᵘ and kᵛ so that Tᵤᵥkᵘkᵛ ≥ 0.
There are also weird dynamics in the Carrollian limit itself that lead to tachyons (see e.g. here) — or at least their remnant in the limit.
As Tim Maudlin likes to say, we already know quantum mechanics violates special relativity — we have to reinterpret c as a speed limit for a different abstract concept called signals instead of just a speed limit in order to keep it. Per Maudlin:
The confirmed predictions of quantum theory are not merely hard to reconcile with Einstein’s vision of a completely local physics—a physics in which both the ontology and the laws are defined and can be checked locally and nothing propagates faster than light—they are flatly incompatible with Einstein’s vision.
You could almost say this is a conservative take on quantum mechanics. Not only are we considering the weird effects in quantum mechanics bounded by some velocity — as opposed to the Copenhagen interpretation that takes vₛ = ∞ (the subscript s meaning “spooky” now) — but they travel as sound waves in some kind of fluid consistent with GR a component of which has possibly been observed as the cosmological constant. I swear this isn’t as crackpot as it sounds.
BRB; interpreting the links between qubits in a quantum circuit diagram as an exchange of tachyons. One issue though: the Lorentz group representation of tachyons is essentially limited to states of definite momentum |p⟩ i.e. no spin or helicity per the footnote above. I am not sure this is rich enough to fully cover all the quantum circuit logic gates which are all at least 2×2 matrices without introducing e.g. internal quantum numbers. Though there might be another way? (See also here [pdf].) There is also this, with a counterargument here.
There is the “traditional” take that this collapse cannot be used to send a signal, but it does carry information — there is a non-zero KL divergence between the probability distribution |Ψ(Σᵒᵘᵗ, tᵒᵘᵗ)|² and the collapsed wavefunction ~ δ(x−xᵒᵘᵗ).
You can “invert” the constraints on Tμν so that they apply instead to the other side of the Einstein field equations (Gμν = 8πκ Tμν) — the Einstein tensor Gμν. They have a geometric interpretation in that case. However this interpretation is … complicated. It’s especially complicated because there’s no intuitive interpretation of Gμν itself (see footnote 11 here).