← interactive companion · reference index
Notation and conventions
Symbols first, then the three conventions that are easy to get wrong and that cost time when they are got wrong.
Symbols
| symbol | meaning |
|---|---|
L | the circumference of the spatial circle. Throughout the numerics L = π, which makes π/L = 1 |
ε_N | the lattice spacing at scale N, equal to 2^{-N} L. Larger N is a finer lattice |
Λ_N | the lattice at scale N; #Λ_{N+1} is the qubit count in the simulation budget |
Γ_{N,±} | the momentum lattice, + Ramond and − Neveu–Schwarz. With L = π these are ℤ and ℤ + 1/2 |
K | the Daubechies order. The filter has 2K taps and the scaling function support [0, 2K−1] |
h_n | the low-pass filter taps, normalised so that their sum is √2 |
m₀(ξ) | the low-pass symbol, 2^{-1/2} Σ_n h_n e^{-inξ} |
ŝ(ξ) | the transform of the scaling function, the infinite product ∏_{j≥1} m₀(2^{-j}ξ) |
μ(s) | the centre of mass ∫ x s(x) dx, equal to 2^{-1/2} Σ_n n h_n |
σ_K | the Sobolev exponent of the scaling function |
ρ | the pointwise decay exponent of ŝ |
R^N_∞, α^N_∞ | the renormalization maps on the one-particle space and on the algebra |
ℓ_{±,k}, L_{±,k} | the one-particle and second-quantised Virasoro generators, ± for the two chiralities |
δ | the Sobolev index in which rates are stated; the advertised rate is at δ = 2 |
m(S) | the number of eigenvalues of a symbol lying strictly between 0 and 1 |
Three conventions worth stating
Which reflection of the filter. The spectral factorisation used by tools/dbfilters.py and by numerics/os_check4.py produces the mirror image of the filter table hard-coded in numerics/os_check.py. Reversing a real filter replaces m₀(ξ) by e^{-i(2K−1)ξ} conj(m₀(ξ)), so |m₀| and hence |ŝ| are untouched: the zeros, the Sobolev exponents and every aliasing quantity are the same either way. What does change is the centre of mass, which becomes (2K−1) − μ(s), and the centre of mass is exactly what the re-centring results turn on. The manuscript's convention is the constructed one: it gives μ = 5.99 at K = 4 and 16.87 at K = 10, the values Remark 3.5 quotes. The self-test in tools/dbfilters.py pins this.
Two regularity exponents, not one. σ_K controls sums of ŝ against a Sobolev weight; ρ controls a single value of ŝ. They are close, with ρ slightly the larger, and they are not interchangeable: Lemma 4.28 needs both, entering through different factors. Every "ρ-regular" hypothesis in the revision means σ_K > ρ, a strict inequality, which is why 5/2-regularity needs K ≥ 7 and not the K ≥ 6 printed in Theorem 4.25.
Which version's numbering. The revision, the audit and the two Markdown records do not share a numbering. The audit uses the published numbers. The records use the revision's, with a correspondence table in CHANGES-v4-to-v5.md. results.md in this folder is generated from the compiled revision, so it is the one place where the numbers cannot drift.