Planck scales aren't always small

Also: another take on the source of indivisibility

Two “elsewhere” regions (in teal) of size R outside of observer A’s light cone. We use the Carrollian scaling where c is “small” and A’s lightcone closes up. The “elsewhere” region expands after an observation Q in the Planck time tₚ by approximately the Planck length ℓₚ in radius so that the boundary of the elsewhere region (a sphere S² with surface area ~ 4πR² ) changes in the tₚ by much more than a Planck area (see below). I haven’t decided whether looking at the past light cone or future is the correct approach so I drew both.

“Large” Planck scale effects

I put a “wisecrack”¹ in my last post that ℏ is the Planck action in the sense that Planck momentum × Planck length i.e. (ℏ/ℓₚ) × ℓₚ = ℏ. This is of course small, but not like, inaccessible-to-particle-accelerators-small quantities such as ℓₚ or tₚ (Planck length and Planck time). We regularly experience effects due to ℏ such as your hob glowing red instead of, say, in the ultraviolet.

Let’s build up a macro-scale effect from Planck scale changes. If we have some event Q at time t = 0 on the edge of A’s past light cone and we move forward to a time to t = tₚ then the elsewhere region “snuggling” up to A’s light cone will grow by ~ Planck length² in girth. The 2D boundary³ of that 3D region will have its area⁴ change by⁵:

\(\displaystyle \begin{eqnarray} \Delta A & = & 4 \pi (R+\ell_{p})^{2} - 4 \pi R^{2} \\ & = & 8 \pi R \;\ell_{p} + 4 \pi \ell_{p}^{2} \end{eqnarray}\)

In a time Δt = tₚ we have⁶

\(\displaystyle \begin{eqnarray} \frac{\Delta A}{ \Delta t} & = &\frac{8 \pi R \;\ell_{p}}{t_{p}} + \frac{4 \pi \ell_{p}^{2}}{t_{p}}\\ & = &\frac{8 \pi R \;c t_{p}}{t_{p}} + \frac{4 \pi c^{2}t_{p}^{2}}{t_{p}}\\ & = &8 \pi \; R \;c + 4 \pi \; \ell_{p}\; c \end{eqnarray}\)

This has a purely macroscopic first term and a Planck scale second term⁷ despite being constructed from Planck scale changes — the change in surface area of a region along a light sheet emitted from the boundary of that region after a Planck time.

The other interesting piece of this is that (per my previous posts) the units⁸ of ΔA/Δt are L²/T or action per mass, which are the units of Carrollian time. Since this ΔA/Δt represents a minimal change in boundary area (the region only increases by how far light can travel in the Planck time), it is possibly another path to the indivisibility in Barandes’ indivisible stochastic process formulation of quantum mechanics.

Let’s take R to be the Compton wavelength. Then:

\(\displaystyle \begin{eqnarray} \frac{\Delta A}{ \Delta t} & = &8 \pi \; R \;c + 4 \pi \; \ell_{p}\; c \\ & \simeq & 8 \pi \frac{\hbar}{m c} c\\ & =& 8 \pi \frac{\hbar}{m} \end{eqnarray}\)

where I’ve dropped the Planck scale term since it is 24 orders of magnitude smaller. The LHS is some kind of minimal quantity that tracks along with a pre-Carrollian limit time⁹ coordinate s and has the same dimensions while the RHS is … well, um, what is it?

Going back the Rindler wedge “Hamiltonian” K in this post, and looking at Carrollian time evolution operator

\(\displaystyle e^{i \eta K s}\)

where η is a dimensionful constant that makes the exponent dimensionless. Since s has dimensions of L²/T we need to cancel those with η and K. Purely by analogy with quantum mechanics, we could say η = 1/ℏ. That would require the dimensions of K to be mass. We could then say the average value ⟨K⟩ = m in a kind of a “what else could it be?” argument. Our area change formula above becomes:

\(\displaystyle \frac{\Delta A}{\Delta t} \equiv s_{\text{min}} = 8 \pi \frac{\hbar}{\langle K \rangle}\)

Factors of two¹⁰ aside, this is a Carrollian time version of the (Margolus–Levitin) quantum speed limit (see also below). But, via the Bekenstein entropy-area law of black hole thermodynamics, a change in area is related to a change in entropy so this could also be seen as the maximal entropy production rate (see here). Interestingly¹¹:

\(\displaystyle \begin{eqnarray} \frac{\Delta A}{\Delta t} =4 \ell_{p}^2 \frac{\Delta S}{\Delta t} & =& 4 \ell_{p} c\; \Delta S\\ \hookrightarrow \quad 4 \ell_{p} c\; \Delta S& =& 8 \pi \frac{\hbar}{m}\\ \Delta S & = & 2 \pi \frac{\hbar}{mc} \frac{1}{\ell_{p}}\\ & = & 2 \pi \frac{ \lambda}{\ell_{p}} \end{eqnarray}\)

The “maximum” entropy produced in a Planck time for a system of mass m is given by the length of the system’s Compton wavelength measured in Planck lengths¹².

Quantum speed limit and indivisibility

Tangentially related, but I’m going to derive the Mandelstam–Tamm quantum speed limit (QSL) in a manner consistent with Carrollian time s and Barandes’ indivisible stochastic process approach (indivisible in Carrollian time s). Starting from Eqs. 57, 58, 60, and 135:

\(\displaystyle \begin{eqnarray} K & \equiv & i \hbar \frac{\partial U}{\partial s} U^{\dagger}\\ i \hbar \frac{\partial \rho}{\partial s} & = & \left[ K, \rho\right]\\ \frac{\partial M}{\partial s} & = &\frac{i}{\hbar} \left[ K, M\right]\\ \Delta L \; \Delta M & = & \frac{\hbar}{2} \left| \langle i [L, M] \rangle \right|\\ \hookrightarrow \Delta K \; \Delta M & \geq & \frac{\hbar}{2} \left| \langle \partial M /\partial s \rangle \right| \end{eqnarray}\)

where M and L are the arbitrary operators and we used the equation of motion (second equation) on the general uncertainty relation to get the last line (there’s a trace with the density operator ρ leading to the expectation value). Taking M to be the projector onto the initial state at s = 0, then

\(\displaystyle \begin{eqnarray} \Delta M & = & \sqrt{\langle M^{2} \rangle - \langle M\rangle^{2} }\\ & = & \sqrt{\langle M \rangle - \langle M\rangle^{2} } \end{eqnarray}\)

since M² = M as it’s a projector. Plugging the previous equation into the last line of the group of equations before it and integrating from 0 to s we get:

\(\displaystyle \begin{eqnarray} \int_{0}^{s} ds \frac{d}{ds} \text{arcsin} \sqrt{\langle M\rangle } &=& \frac{\pi}{2} - \text{arcsin} \sqrt{\langle M\rangle_{s}}\\ & \leq & \frac{\Delta K}{\hbar} (s - 0)\\ \hookrightarrow \quad s &\geq & \frac{\pi}{2} \frac{\hbar}{\Delta K}\equiv s_{QSL} \end{eqnarray}\)

In going to that last line we have made reference to the physical process we are considering¹³: the final state at (Carrollian) time s is orthogonal to the initial state so the projector M at time s is zero.

Could this be the source of Barandes’ indivisibility? The stochastic process cannot in general be “divided” — i.e. a time s' cannot exist between s and 0 such that:

\(\displaystyle \begin{eqnarray} \Gamma (s\leftarrow 0) & \neq & \Gamma (s\leftarrow s') \; \Gamma (s' \leftarrow 0) \\ & & \\ & & \text{unless} \; s \gg s_{QSL} \end{eqnarray} \)

We can at least say this is a consistent picture. There is a better founded non-Carrollian¹⁴ version of this where we have a time t instead of Carrollian s.

If we add a “velocity scale” v of our quantum system Q to mess with the units (i.e. undo all the Carrollian stuff) the LHS of the quantum speed limit becomes some kind of “length” Λ and the (using K ~ m) RHS becomes:

\(\displaystyle \Lambda \geq \frac{\pi}{2} \frac{\hbar}{m v}\)

Or, another way, 1/4 a de Broglie wavelength¹⁵.

Update 17 May 2025:

I do want to say that this is still “system dependent” in the sense I discuss in the previous post. We haven’t reached a “fundamental” scale of action ℏ — in part because it was assumed¹⁶. But it’s more than that — in particular:

• In the first section, it’s the assumption of R ~ ℏ /mc. This was just assumed to show how you could relate the expansion of a light sheet to the quantum speed limit (Margolus–Levitin version). There is no reason R ~ ℏ /mc should be or has to be the choice.

• In the second section, there is dependence on the system “Hamiltonian” K. We can’t really directly use something like the Bekenstein bound (i.e. get a “fundamental” action scale because if K, with dimensions of mass here, is too large you get a black hole).

Returning to this post, if we turn the dimensionless modular Hamiltonian back into a dimensionful one and use the QSL above, we can rewrite the Bekenstein bound as¹⁷

\(\displaystyle \begin{eqnarray} \Delta S & \leq & \frac{s_{QSL} }{\hbar } \left( \langle K \rangle_{s} - \langle K\rangle_{0} \right)\\ & \leq & \frac{\pi}{2} \; \frac{\langle K \rangle_{s} - \langle K\rangle_{0}}{\Delta K} \end{eqnarray}\)

We can see the ℏ completely dropped out of the equation.

1. I am comparing myself (favorably, obviously) to Edward Witten here who noted in natural units the QED gauge coupling e was ~ 1/3 so the 1/Nc perturbative expansion (e.g. here) for QCD (Nc = 3) was not too different.

2. We can imagine a bunch of laser beams starting at the boundary of the elsewhere region of spacetime and propagating outward (or inward if we’re looking at the top half of the diagram) in the sense of here (pdf).

3. In the sense of Bekenstein’s bound.

4. We are in the very weak gravity limit where the mass-energy of whatever Q does not focus the geodesics causing the area to shrink a bit (i.e. the effect of gravity in general relativity).

5. I should note that the last term is an area ~ Planck area and that a change by a Planck area is directly related to Einstein’s equations in General Relativity. In that paper, there is a deletion of a Planck area in the causal region which leads to Einstein’s equations.

6. It is interesting that this works out in 3+1 dimensions to one term independent of Gℏ (i.e. the Planck area modulo factors of c) and one not. In 1+1 dimensions, (the equivalent of) ΔA is 0. In 2+1 dimensions, we only have the ℓₚ term. In 4+1 dimensions, we get multiple cross terms: 3 R² ℓₚ + 3 R ℓₚ². Since I am hinting the terms independent of R are “quantum gravity” or “thermodynamic gravity” and the terms independent of Gℏ (after dividing by the Planck time for ΔA/Δt) are “quantum mechanics” the lower dimensional universes are very different: 1) neither GR nor QM in 1+1 dimensions, 2) only GR in 2+1 dimensions, and 3) a weird hybrid in 4+1 dimensions. (Q.v. AdS₅, maybe?)

7. The macroscopic term is 0.018 m²/s if R is the Compton wavelength while the Planck scale term is ~10^-26 m²/s.

8. My poor math skills limit me to exclusively discussing dimensional analysis.

9. More like the affine parameter along the light sheet “λ” as discussed here.

10. Ok, 16 = 2^4.

11. In a force of habit, I have consistently used the “reduced” Compton wavelength as “the Compton wavelength” with ℏ instead of h. If I had used the actual Compton wavelength throughout, the factor of 2π would drop out.

12. This isn’t really a new “result” since this is basically the Bekenstein area law for an area ΔA = λ ℓₚ modulo factors of 2 and π.

13. The quantum speed limit is the time it takes for a state to change from one state to an orthogonal (indistinguishable) state. As a side note, the quantum speed limit is all an attempt to make sense of “ΔE Δt ~ ℏ” (or in our notation ΔK Δs ~ ℏ). In the case of Δp Δx ~ ℏ, it follows from a commutator of operators p and x, i.e. [x, p] = 𝒾 ℏ. But there is no “time” operator so you have to use some other operator (our projection operator M above) to create a sensible expression.

14. I am couching a lot of this in terms of the Carrollian limit because it appears to have some relevance to flat space holography (again, the ultimate aim is to prove Bousso’s conjecture / side comment that entropy ~ 1 in Bekenstein’s bound is actually the reason we have quantum mechanics). In the back of my mind, however, is the problem that the Schrodinger group (and the symmetries of the Schrodinger equation) are from the Galilean limit / algebra (Newton-Cartan). The only piece of intuition I cling to is the fact that you can’t just go from non-relativistic quantum mechanics to a quantum field theory that is inherently relativistic. I imagine there is a shadow of QM in both contractions of the Poincare group: Carrollian and Galiliean. The latter does not tell us anything about ℏ, so maybe the former does?

15. Weirdly, this is reminiscent of impedance matching. The even weirder piece is that one way to visualize this is with a Smith chart which is essentially a conformal mapping — which is one of the symmetries making up the Carroll / BMS group. Is conformal symmetry a kind of requirement for “impedance matching” information flows?

16. There is a sense where you can assume the need for a constant with units of action and then appeal to measurement to determine what it is. This is how a lot of physics works. But we’re not doing that — we’re assuming more than just “a constant to be measured later”.

17. There’s a clash of notations between the two posts regarding ΔK so in this post ΔK refers to the square root of the variance and the version from the earlier post has been replaced with the difference between the expectation values at different Carrollian times.