A lattice works flawlessly as a computational tool ── but the moment you say "spacetime itself is a lattice," eighteen orders of magnitude appear
For someone who writes programs, discretizing the universe is an entirely natural idea. Cut space into cells, give each cell a state, update by local rules. A cellular automaton, exactly. And in fact lattice QCD, the front line of particle physics, does precisely that and succeeds.
So one wants to think "maybe spacetime really is a lattice." There is even a natural spacing on offer, the Planck length.
This episode chases how far that road is right, and where it breaks. The conclusion up front ── as a computational tool it is perfectly right, and as the basic structure of spacetime it breaks fatally. And the way it breaks is interesting. What breaks it is not discreteness itself.
Mix this topic together and it always spins its wheels. That is because three problems whose answers are opposite get discussed in the same words.
| the lattice's role | the spacing \(a\) is | Lorentz invariance |
|---|---|---|
| (a) computational tool lattice QCD | an artificial regulator. take \(a\to0\) | comes back (with proof) |
| (b) basic structure of spacetime | physical. the Planck length | breaks (fatally) though it can be saved by adding symmetry ── §08 ① |
| (c) chiral symmetry | — (a different problem from Lorentz) | ← this is where the no-go theorem lives |
(a) and (b) have opposite answers. Argue without distinguishing them and it looks contradictory. And (c) is the place lattice practitioners have genuinely struggled with, and it is not Lorentz invariance. We take them in turn.
Let us get the terminology right. Standard lattice QCD has Euclidean signature. You Wick-rotate, compute, and analytically continue back afterwards. So what is broken on the lattice is not the Lorentz group with its boosts, but ──
The continuous rotation group \(SO(4)\) of 4d Euclidean space drops to the hypercubic group \(H(4)\) (a finite group of 384 elements).
A continuous group becomes a finite group. This is not an approximation; it is an explicit breaking. The axis directions and the diagonal directions of the lattice become plainly different things.
This is the heart of (a), and it is not a hope that "it will probably be fine" — it can be shown constructively.
The idea is Symanzik's effective action. Rewrite the theory with spacing \(a\) as a continuum theory at long distances:
$$\mathcal{L}_{\rm Sym}=\frac{1}{a}\mathcal{L}^{(3)}+\mathcal{L}^{(4)}+a\,\mathcal{L}^{(5)}+a^2\mathcal{L}^{(6)}+\cdots$$\(\mathcal{L}^{(n)}\) is an operator of dimension \(n\). Every term that breaks rotational symmetry comes with a positive power of \(a\). So they die as \(a\to0\).
What is doing the work is the following group-theoretic fact.
Of the operators that are \(H(4)\) invariant but not \(O(4)\) invariant, the lowest dimension is 6.
reason
An \(H(4)\)-invariant rank-2 tensor is automatically proportional to \(\delta_{\mu\nu}\). Which is to say there is no way to write the breaking. Writing it requires the rank-4 invariant tensor \(\sum_\mu \delta_{\mu\alpha}\delta_{\mu\beta}\delta_{\mu\gamma}\delta_{\mu\delta}\), and that corresponds to dimension 6.
conclusion
The breaking starts at \(a^2\mathcal{L}^{(6)}\) ── \(O(a^2)\) suppression, and with no fine-tuning required.
Put differently: \(H(4)\) is a "large enough" finite group that at dimension 5 and below there is no way to break rotational symmetry. We threw away a continuous symmetry, and at low dimension the difference cannot be expressed.
As a contrast, this is worth knowing. What has genuinely tormented people on the lattice is not Lorentz invariance.
For a lattice fermion action, the following four cannot be satisfied simultaneously.
① locality ② chiral symmetry ③ no doublers ④ translation invariance
This is a genuine no-go theorem. Workarounds exist (the Ginsparg–Wilson relation, overlap / domain-wall fermions), but they amount to a rather deep construction that "gives the lattice an exact deformed chiral symmetry."
There is a decisive asymmetry here.
| symmetry | how it fares on the lattice |
|---|---|
| chiral symmetry | there is a no-go theorem. a deep construction is needed |
| Lorentz (rotational) invariance | no no-go theorem. explicitly broken, but comes back by itself |
So the phrasing "a lattice cannot preserve Lorentz invariance" is, for the lattice as a tool, simply wrong. What cannot be preserved is chiral symmetry.
From here on it is (b), and the real subject of this episode.
Naively one thinks: if the fundamental scale is \(a\sim\ell_P\), the breaking can only enter at dimension 6 and above, so at low energy it is suppressed by \((E/M_P)^2\). Even at TeV energies that is \(10^{-32}\). Utterly invisible.
This is wrong. What breaks it is the loop-integral cutoff.
Insert the dimension-6 breaking operator (coefficient \(1/M_P^2\)) into the self-energy. The loop momentum runs up to \(\Lambda\sim M_P\):
$$\delta c\;\sim\;\frac{g^2}{16\pi^2}\cdot\frac{\Lambda^2}{M_P^2}\;\sim\;\frac{g^2}{16\pi^2}\;\sim\;10^{-2}\text{–}10^{-3}$$the Planck mass has vanished
All that is left is the loop factor of the coupling. The suppression is lost entirely.
So even if you put in only dimension-5 and -6 breaking at the starting point, renormalization generates dimension-3 and -4 breaking and brings it down to low energy. Dimension-4 breaking is, for example, the most direct form of all: "the maximum speed of the electron differs from that of the photon" (\(c_e\ne c_\gamma\)).
So how tightly does experiment squeeze this? Synchrotron radiation from the Crab Nebula. From the spectrum radiated by electrons with Lorentz factor \(\gamma\sim3\times10^9\) ──
breaking predicted by radiative corr. ~ 10^-2 (of order α/π) upper limit allowed by the Crab ~ 6 × 10^-20 (SME c coefficient) ────────────── discrepancy 18 orders
To erase this you have no choice but to fine-tune the bare parameters to eighteen digits. This is "the other fine-tuning problem of quantum gravity." It is the same species of disease as the Higgs mass hierarchy problem, and comparably severe or worse.
Why does this happen? In physics there are quantities that "can stay small and stable" and quantities that cannot.
| small quantity | the symmetry protecting it | stable? |
|---|---|---|
| the electron mass | chiral symmetry | stable |
| the photon mass | gauge symmetry | exactly zero |
| the Higgs mass | none | unstable (hierarchy problem) |
| Lorentz violation | none | unstable (18 orders) |
\(m_e\) can stay small because setting it to zero restores chiral symmetry. A symmetry holds it down, saying "do not come back." Lorentz violation has no equivalent restraint.
So the question rewrites itself as follows.
The question to ask is not "is spacetime a lattice."
It is "what is protecting low-energy Lorentz invariance?"
Exact supersymmetry forbids dimension-3 and -4 Lorentz-violating operators (Groot Nibbelink–Pospelov 2005). In the supersymmetric standard model the lowest allowed dimension is 5. There is always suppression by at least one UV scale.
Moreover, what controls the mixing that brings dimension 5 down to dimension 3 is not \(M_P\) but the soft SUSY-breaking scale \(m_{\rm soft}\). The suppression becomes orders of magnitude stronger. As a result, dimension-6 breaking suppressed at the Planck scale is not in conflict with the observational data.
The price: you need SUSY. It has not been found.
"Not in conflict" alone is weak, so let us set down how much current observation allows. Writing the violation of the dispersion relation as \(\delta v\sim(E/E_{\rm LV})^n\), the fate turns on \(n\). Dimension 5 is \(n=1\); dimension 6 is \(n=2\).
| order | strongest lower bound \(E_{\rm LV}\) | ratio to \(M_{\rm Pl}\) | a Planck lattice is |
|---|---|---|---|
| \(n=1\) (dim 5) | \(>10^{5}\,M_{\rm Pl}\) LHAASO PeV photons | 5 orders above | excluded |
| \(n=2\) (dim 6) | \(>10^{-3}\,M_{\rm Pl}\) same (photon decay / shower formation) | 3 orders below | survives (margin \(10^3\)) |
| Time-of-flight (LHAASO, GRB 221009A, 0.2–18 TeV, 95% CL) gives \(E>1.47\times10^{20}\) GeV \(=12\,M_{\rm Pl}\) for \(n=1\) and \(E>1.2\times10^{12}\) GeV \(\simeq10^{-7}M_{\rm Pl}\) for \(n=2\). Time-of-flight is weak for \(n=2\), so the photon-decay family in the row above is the stronger constraint | |||
\(n=1\) is killed with five orders of margin, and \(n=2\) survives with three orders of margin. There are eight orders between them ── so "is dimension 6 the leading player?" is not a vague matter of taste but a question observation is answering.
In Episode 14 we will see that the cutoff-dependent divergence is absorbed entirely into the renormalization of Newton's constant (the entanglement side \(A/48\pi\epsilon^2\) and the \(\delta(1/G)=1/12\pi\epsilon^2\) obtained independently from the heat kernel agree down to the factor of \(A/4\)).
Which is to say the value of the cutoff itself does not show up in observation. The Lorentz violation seen in this episode is the disease of "when the cutoff creates a preferred frame," not the disease of the cutoff's existence ── that is why \(n=2\) in the table above passes.
Conversely, if a physical spacing \(a\) really exists, then the \(N=A/a^2\) of Episode 12 is not an approximation but a genuine count.
The line that "the breaking is at high energy, but the renormalization-group flow pulls it to zero at low energy." An old idea, around since the 1970s.
Weak coupling is not enough. The flow in a gauge theory is logarithmic, so you only get suppression of order \(1/\log\) ── running twenty orders gives a factor of \(1/\log(10^{20})=1/46=0.022\). Far from eighteen orders.
Bednik–Pujolàs–Sibiryakov (JHEP 11 (2013) 064) showed this in a holographic strong-coupling example. The Lorentz-violating correction is suppressed
$$\delta\sim\left(\frac{\Lambda_{\rm IR}}{\Lambda_*}\right)^{\epsilon},\qquad \epsilon=2(1-\nu),\qquad \nu=\sqrt{\frac{d^2}{4}+(\mu L)^2}$$as a power. In the marginal case (\(\nu=1\)) we get \(\epsilon=0\) and are back to logarithms, but in the correlated case \(0\le\nu<1\), \(\epsilon\) can run from 0 to 2.
A power can cover eighteen orders. And computing the required \(\epsilon\) gives a surprisingly small number.
| \(\Lambda_{\rm IR}\) | \(\log_{10}(\Lambda_{\rm IR}/M_P)\) | required \(\epsilon\) | corresponding \(\nu\) |
|---|---|---|---|
| 1 TeV | \(-16.1\) | 1.12 | 0.44 |
| 1 GeV | \(-19.1\) | 0.94 | 0.53 |
| 0.2 GeV | \(-19.8\) | 0.91 | 0.55 |
| The condition for covering eighteen orders. \(\epsilon\simeq1\), i.e. \(\nu\lesssim0.5\) ── exactly half of the allowed window \(0\le\nu<1\) is usable | |||
It amounts to saying you need an anomalous dimension of \(O(1)\). A size that comes out routinely at strong coupling. So this escape route sits in an entirely different place from the despair one sees at weak coupling.
To summarize, the verdict on escape route ② is upgraded from "not enough" to "works for matter; gravity is left over". The price is not SUSY but something strongly coupled at low energy, plus an unresolved gravity sector.
The episode that digs this route to the end: Episode 16, "Matter can be saved. Gravity leaks" ── the usable window is measured as \(10\) TeV \(\le\Lambda_{\rm conf}\le3\times10^9\) GeV. And that violation in gravity leaks into matter via radiative corrections.
A regular lattice fails not because it is discrete, but because it creates a preferred frame.
Boost a lattice and you get a different (contracted) lattice. So "the frame in which the lattice is at rest" becomes special.
A Poisson sprinkling is different. The determinant of a Lorentz transformation is 1, so it preserves the volume measure. Hence a Poisson process is statistically invariant under boosts. There exists no invariant way to extract a preferred direction from the sprinkling (Bombelli–Henson–Sorkin). Discrete, and yet with no preferred frame.
So the problem of §05–07 never arises in the first place. There is nothing for it to break. The price is non-locality.
The spectrum of the area operator is discrete, so naively it looks like "a lattice of Planck areas," and one would think Lorentz contraction makes a mess of it.
Rovelli–Speziale's counter: area is an observer-dependent quantity, and what is discrete is the spectrum of an operator. A boosted observer is measuring a different surface, so the picture of "the minimal area contracting" does not even get off the ground. It is contested, but on the point that "discreteness ≠ a preferred frame" it faces the same direction as causal sets.
Let us put what escape route ③ shows into general form. The reason is geometric, and simple.
In Minkowski spacetime, what is the set of points at constant proper distance from the origin? In Euclidean space it is a sphere, but in Minkowski it is a hyperboloid.
And the volume of a hyperboloid is
infinite. It is not compact. So "nearest neighbour" cannot be a Lorentz-invariant notion ── build a Lorentz-invariant discrete structure and there are infinitely many points at any given proper distance.
discreteness ／ locality ／ Lorentz invariance
── these three cannot hold at once.
What is interesting is how closely the structure resembles Nielsen–Ninomiya. And each approach can be classified by which one it gave up.
| approach | discrete | local | Lorentz invariant |
|---|---|---|---|
| regular lattice / cellular automaton | ○ | ○ | × (preferred frame) |
| causal set (Poisson sprinkling) | ○ | × | ○ |
| ordinary field theory | × | ○ | ○ |
In the figure below, see directly why only the regular lattice creates a preferred frame.
The lattice on the left visibly distorts when you boost. It squashes diagonally and the axes change direction. Which means there exists a speed "at which the lattice looks straight" ── that is a preferred frame.
The sprinkling on the right does not change in appearance under a boost. The individual points move, but the statistical properties are identical. Because a Lorentz transformation preserves area (determinant 1), the density of points does not change. There is no such thing as "the speed at which the sprinkling looks straight."
This three-way choice turns its blade squarely on the position that "the universe can be computed." That is because a cellular automaton is by definition discrete and local. Exactly the first row of the table. 't Hooft's cellular-automaton interpretation, pursued in earnest, is attacked most on precisely this point.
But it is worth thinking one step more carefully here. What does "the universe can be computed" actually require?
What is required is a finite amount of information.
What is not required is a spacing of space.
These two are entirely different claims.
And finiteness can be had without a lattice. The holographic bound.
$$S\le\frac{A}{4G}$$The information that fits in a finite region is finite. But this has not cut space into a lattice. The finiteness lives in the covariant entropy bound, not in a spacing of space. Because it is formulated covariantly, it creates no preferred frame.
Which is to say ── there exists a way to realize a finite amount of information with no lattice and no preferred frame. Being a finite-state machine and having those states laid out on a lattice were different claims. Take only the former and you never touch the \(6\times10^{-20}\) constraint or the eighteen-order problem at all.
The lattice as a tool is perfectly right. The breaking \(SO(4)\to H(4)\) is real, but for group-theoretic reasons it is suppressed to \(O(a^2)\) with no fine-tuning, and it has been measured. What is genuinely hard on the lattice is chiral symmetry, and that is where the no-go theorem is.
The moment you say "spacetime is a lattice," it becomes fatal. Radiative corrections cancel \(M_P\) through the loop cutoff and bring \(10^{-2}\) Lorentz violation down to low energy. The observational limit is \(6\times10^{-20}\). Eighteen digits of fine-tuning.
The essence is "there is no symmetry protecting it." The electron mass is protected by chiral symmetry, the photon mass by gauge symmetry. Lorentz violation has no such restraint. So the question is not "is spacetime a lattice" but "what protects low-energy Lorentz invariance?"
And the three-way choice. Discreteness, locality and Lorentz invariance cannot hold at once. The culprit is not discreteness but regularity ── a Poisson sprinkling is discrete yet Lorentz invariant, and loses locality instead.
Computability does not need a lattice. What is needed is a finite amount of information, not a spacing of space. The holographic bound supplies that with no lattice and no preferred frame.
That said, "lattice + symmetry" does pass, on the numbers. We put figures into §08 ① ── if the breaking is \(n=1\) (dimension 5), the strongest lower bound is \(10^{5}M_{\rm Pl}\) and it is killed with five orders of margin. But at \(n=2\) (dimension 6) the bound is \(10^{-3}M_{\rm Pl}\), and a Planck lattice passes with three orders of margin. Supersymmetry forbids dimensions 3 and 4; CPT forbids dimension 5. A regular lattice with both added is not in conflict with current observation.
And as we will see in Episode 14, the value of the cutoff itself is absorbed into the renormalization of \(1/G\) and does not appear in observation. This episode's disease was not "there is a cutoff" but "the cutoff creates a preferred frame."
This document is Episode 2 of the "Lattice We Build" series, a reading piece for high-school and university students who love physics. Where the sister series "That Clicks" explains known physics, this series shows the work itself.
Among the claims in the text, the \(O(a^2)\) suppression from the Symanzik effective action, the Nielsen–Ninomiya theorem, the fine-tuning problem from radiative corrections due to Collins et al. (PRL 93, 191301, 2004), the \(6\times10^{-20}\) constraint from Crab synchrotron radiation, and the forbidding of dimension-3 and -4 breaking by supersymmetry (PRL 94, 081601, 2005) are all peer-reviewed, established results. On the other hand, the compatibility of the area spectrum with Lorentz invariance in loop quantum gravity (Rovelli–Speziale) is contested and unsettled. And the "three-way choice" is a structural statement that follows from the results above together with the non-compactness of the hyperboloid; there is no single named theorem in exactly this form. All figures are order-of-magnitude arguments; consult the literature for coefficients and signs. The observational numbers attached to escape route ① in §08 were added later. The strongest lower bounds \(>10^{5}M_{\rm Pl}\) for \(n=1\) and \(>10^{-3}M_{\rm Pl}\) for \(n=2\) are as reported in the analysis using LHAASO PeV photons (arXiv:2105.07967, PRD 104, 063012); the time-of-flight values \(n=1>1.47\times10^{20}\) GeV and \(n=2>1.2\times10^{12}\) GeV are from LHAASO's observation of GRB 221009A (arXiv:2312.09079, 95% CL, maximum likelihood, subluminal case). Papers analysing the same burst by other methods report different values, so read these as order-of-magnitude guides. Ratios to \(M_{\rm Pl}\) use \(M_{\rm Pl}=1.22\times10^{19}\) GeV. That dimension 5 violates CPT is an established fact. The strong-coupling results attached to escape route ② were also added later. That Lorentz violation is power-suppressed as \((\Lambda_{\rm IR}/\Lambda_*)^{2(1-\nu)}\), that the marginal case \(\nu=1\) returns to a logarithm, and that gravity is treated as a fixed background and the authors state that extending to dynamical gravity requires an additional degree of freedom defining a preferred frame, are from Bednik–Pujolàs–Sibiryakov, JHEP 11 (2013) 064 (arXiv:1305.0011). The extension to chiral fermions is Kharuk–Sibiryakov, TMP 189 (2016) 1755 (arXiv:1505.04130). The table of required \(\epsilon\) was computed by the author from this suppression form and the eighteen orders of §06. On the other hand, the phrasing "if you want to keep locality, escape route ① is the only choice, and therefore supersymmetry is required" is this series' author's own framing (it has not been shown that no mechanism other than SUSY can forbid dimensions 3 and 4).
Sister series: Cosmology That Clicks／Relativity That Clicks ｜ Episode 1: Let a machine hunt for the ratios nobody has taken yet ｜ Bonus: ① Could physics be written more simply?／② How far does "ct = constant" get you? ── to print, use your browser's "Print" → "Save as PDF."
Print / PDF: ⌘+P (Ctrl+P on Windows). With the slider in the figure you can confirm that only the lattice distorts.