Gravity / The doublers

The lattice has four more massless waves.

At κ = 1 the lattice operator vanishes on the light cone at long wavelengths, as gravity requires, and also at four corners of the Brillouin zone, the waves with one coordinate equal to π. These four doublers are massless, each on a cone of its own. At one loop, in the stated diagram model, they make the native graviton’s limit diverge; a weight built from the ledger’s cost of curvature gives all four a gap.

Four doublers at κ = 1: provedGapped by the curvature weight: proved under two identificationsA massless graviton: open

  • Zero modes, graviton sector
  • Zero modes, doubler sector
  • Leading-order cones

Live · solved from Appendix B

Each curve is where the gauge-fixed κ = 1 operator, assembled in your browser from the Hessian printed in Appendix B, has a zero mode on the plane k = a e₀ + b(1,1,1,1), found sector by sector under the permutations of axes 1, 2 and 3. Near the origin the graviton’s branch follows the light cone, which opens around record time (1,1,1,1); near πe₀ the doubler’s branch follows its own cone, which opens around a direction close to e₀. Shading marks each cone’s timelike directions. Away from the two points the branches are lattice modes of the same symmetry.

Record time
−4/3q: timelike for the graviton
190hᵀKh: spacelike for the doubler
Shared
none30q + hᵀKh is positive definite: no direction is timelike or null for both (Theorem 6.15)

The corners

Sixteen corners, four doublers.

The corners of the zone are πδ with δ a string of four bits, so they carry the same labels as the records of a cell. Your browser solves the operator at all sixteen and counts its zero modes.

  1. 1111·
  2. 1110·1101·1011·0111·
  3. 1100·1010·1001·0110·0101·0011·
  4. 1000·0100·0010·0001·
  5. 0000·

The kernel has dimension 10 at the origin, where it is the constant strains; 1 at each axis corner, where it is a doubler; and 0 at the other eleven corners. At the corner πec the kernel is spanned by vc(d) = 2|d|−3 on the seven classes with dc = 0 and 0 on the other eight. The Bloch matrix and the vertex-move columns annihilate it separately, so it is a zero mode for every gauge weight, and it is not a vertex move.

Earlier paper, Proposition 6.1 and Section 6.2

The doubler at πe₀, in position space

Exact · Proposition 6.1

The field is (−1)x₀v₀(d): at every vertex it changes the squared lengths of the seven edges in the cube orthogonal to axis 0 by ¼, ½ and 1 for one, two and three steps (line weight), and the sign flips with each step along axis 0 (solid ink +, dashed red −). Edges that step along axis 0 do not change. As a check, 3·(¼)² + 3·(½)² + 1² = 31/16 = |v₀|². Drawn at one vertex of each layer.

How it works

Why a real-time lattice has doublers.

  1. 1 · The cone runs out

    A Lorentzian symbol vanishes on a cone, and a cone is unbounded.

    On a lattice that cone runs out to the edge of the Brillouin zone, so the operator can vanish far from the origin. A Euclidean lattice operator that is positive away from the origin has no such zeros. Lattice fermions have doublers for a different reason; these belong to real-time discretizations, and at κ = 1 they sit exactly at the axis corners because the postings are null.

    Earlier paper, Section 6.2

  2. 2 · A cone of its own

    Each doubler is massless and propagates on its own cone.

    Near the corner, the doubler’s effective operator is (√3/640) hᵀKh at leading order, where h is the offset from the corner and K has one negative and three positive eigenvalues, with det K = −1013375 = −55²·335. The two kinds of cone point different ways: record time is timelike for the graviton and spacelike for the doubler, and no direction is timelike or null for both.

    Earlier paper, Proposition 6.2 and Theorem 6.15

  3. 3 · No mass from the construction

    Nothing in the construction gives the doublers a mass.

    A flat background that keeps every posting null shifts the four zeros, at first order, by amounts that sum to zero, so it cannot lift all four, and any nonzero shift of that kind makes at least one doubler tachyonic. A strain that moves all four the same way changes the posting lengths: raising κ above 1 gives all four a mass, and it leaves κ = 1. In the whole family κ > ½ the doublers occur exactly at κ = 1.

    Earlier paper, Proposition 6.3 and Theorem 6.10; quantum gravity paper, Theorem 2.1

    Put a constant background on the lattice

One loop

At one loop the doublers make the native limit diverge.

In the stated diagram model, the zero-momentum anisotropy of the graviton’s one-loop self-energy grows like log(1/ε) as the regulator ε is removed after the volume, so the ordered limit does not exist. The logarithm comes from the four doublers.

The doubler term D(ε)

Live · quadrature of Proposition 6.5

D(ε) = (5/(44√335 π²)) ∫₀^∞ e−ετ τ²(1 + τ²)−3/2 dτ, computed here by quadrature; its asymptote, dotted, is (5/(44√335 π²))(log(1/ε) + log 2 − 1 − γ), so the anisotropy is Δ(ε) = (5/(44√335 π²)) log(1/ε) + r₀ + o(1), about 6.3 × 10⁻⁴ log(1/ε). Each axis corner is a pinch: the doubler’s eigenvalue vanishes quadratically there, so no deformation of the integration contour avoids the iε prescription. Every other degenerate zero contributes finitely, so nothing cancels the corners’ logarithm. The logarithm is slow: at every regulator a computation can reach, the anisotropy is dominated by its finite part; the divergence is a property of the limit.

Earlier paper, Propositions 6.5, 6.6 and 6.9, Theorems 6.7 and 6.8

The repair

A weight from the cost of curvature gaps all four.

On flat deformations

0

The weight vanishes on constant strains and vertex moves, the lattice’s flat directions.

Quantum gravity paper, Theorem 5.5

On long waves

|k|⁴

It is of fourth order in the wave number on long waves, so it leaves the classical limit alone.

Quantum gravity paper, Theorem 5.5

On the doubler v₀

0.12255

(10 + 3√3)/124: what the weight gives the doubler at πe₀, per |v₀|². Positive on all four doublers: a gap.

Quantum gravity paper, Section 5.3, by exact computation

Free coefficients

0

None, once the channel count is fixed, given the scaling that makes the phase native.

Quantum gravity paper, Theorem 5.4

What it rests on

Two named identifications: the cost of a hinge record reads its area, as the action does; and the weight of a geometry is the Gibbs weight e−C of its ledger cost. With them, the ledger’s cost of the curvature at each hinge enters the measure at the action’s scale, and at Gaussian order it acts through 𝒟 = ½ Σh (Ah/nh) δ′hδ′h⊤.

Quantum gravity paper, Theorem 5.4

What it gives

With the weighted operator, in the same diagram model with the Feynman regulator, the real part of the anisotropy that diverges for the native path sum has a finite ordered limit, and so do the real parts of each probe’s contribution and of each diagram. A finite limit is not a massless graviton: the zero-momentum remainder stays nonzero for every candidate computed to convergence (numerical), and an action derived from the ledger whose Ward defect vanishes is the precise open target.

Quantum gravity paper, Theorem 7.4 and Target 6.8

The curvature weight along an axis

Sources: the earlier paper is Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice; the quantum gravity paper is Quantum Gravity from the Recognition Ledger, version 1.2.

Words on this page, in plain terms: doubler · Brillouin zone · light cone · κ (kappa) · ordered limit · curvature weight · one-loop diagram model.

The gravity portal · Newton’s law from the lattice · Next: the quantum question · The doublers in the full account · One loop in the full account · The earlier paper (PDF) · The quantum gravity paper (PDF) · Glossary · Search every page

With JavaScript on, this page solves the lattice operator live: where it vanishes on one plane of waves, its zero modes at the sixteen corners of the zone, and the one-loop doubler term.