HM07 — Amplituhedron as ORBIT: Eliminating RENORM
| Field | Value |
|---|---|
| Domain | Gauge Theory / Positive Geometry |
| System | N=4 super-Yang–Mills scattering amplitudes |
| Group | GL(k) acting on Gr⁺(k, k+4) |
| H^k tier | H³ |
| ISA | Hum (β = it/ℏ) — Grassmannian basis |
| Status | Validated (tree level); conjectured (all loop) |
| Opcodes | ORBIT · EMIT · FLOW |
| Papers | Paper 620, Paper 520 |
Physical system
Tree-level and all-loop scattering amplitudes in planar N=4 super-Yang–Mills are the volume of a positive region in the Grassmannian:
A_{n,k,L}(Z) = ∫{𝒜{n,k,L}} Ω_{n,k,L}
where Ω_{n,k,L} = ∏i dα_i/α_i is a pure dlog canonical form, and 𝒜{n,k,L} ⊂ Gr⁺(k, k+4; L) is the amplituhedron (Arkani-Hamed & Trnka 2013).
The canonical form is UV-finite on the interior of the amplituhedron by construction — no renormalisation is needed or possible. At L loops, each loop momentum is a “hidden particle pair” — a 2-plane D^{(i)} in the (n−k)-dimensional complement of the external data C, subject to positivity conditions. Loops add positivity constraints, not new opcodes.
ISA interpretation
RENORM is absent. PROPAGATE dissolves. EMIT remains.
In the Feynman basis, the same amplitude requires: ORBIT × n (external states), EMIT × n (vertices), PROPAGATE × (internal lines), RENORM (counterterms at loop level) — a programme of length ≥ 7 at tree level, ≥ 10 at one loop.
In the Grassmannian basis, the amplitude is a single: ORBIT(Gr⁺(k, k+4; L), positive cell, Z)
One opcode at any loop order. The opcode ratio Feynman/amplituhedron is ≥ 5:1 at tree level, ≥ 8:1 at one loop.
Why RENORM disappears: the Feynman basis forces a local, particle-by-particle expansion. Locality introduces arbitrarily high-energy virtual modes (k → ∞) whose contributions diverge. RENORM removes these artefacts. The amplituhedron never introduces them: the amplitude is a single global object (a volume), and the UV modes are not individual entities.
Why PROPAGATE dissolves: “virtual particles” in the Feynman basis are hidden positivity constraints in the amplituhedron basis. At L loops, the L loop momenta are L hidden pairs of columns — geometry of the same ORBIT, not separate propagation steps.
Why EMIT survives: EMIT specifies the coupling constant e and the particle content. The amplituhedron still requires the external kinematic data Z_i and the theory specification. EMIT is the ISA’s representation of “which theory are we computing in?” Without it, the Hum ISA describes free fields — no interaction, no amplitude.
ISA programme
PROGRAM Amplituhedron_amplitude [n particles, k = NMHV degree, L loops]
; Specify theory and kinematics
EMIT(particles, theory=N4SYM, coupling=g, n=n, k=k) ; theory input
; The amplitude is a single ORBIT on the positive Grassmannian
ORBIT(Gr+(k, k+4; L), amplituhedron_region, Z) ; L=0: tree level
; L>0: positivity constraints added
; Output: the canonical volume form
OUTPUT A_{n,k,L}(Z) = volume(amplituhedron)
; UV-finite by construction (dlog form); no RENORM step
Opcode count comparison
| Level | Feynman basis | Count | Grassmannian basis | Count | Ratio |
|---|---|---|---|---|---|
| Tree (4-gluon) | ORBIT×4, EMIT×4, PROPAGATE×2 | 10+ | ORBIT(Gr(2,4)) | 1 | 10:1 |
| 1-loop | above + loop sum + RENORM | ≥ 14 | ORBIT(Gr(2,4;L=1)) | 1 | ≥ 14:1 |
Grassmannian quantisation (HM08 preview)
The amplituhedron proposes a third quantisation paradigm (see HM08): first quantisation = L=0 external data only; second quantisation = L>0 loop positivity. Both emerge from a single ORBIT — no Fock space, no canonical quantisation, no path integral.
Connections
- G02 (Amplituhedron, Origami ISA view): G02 treats the amplituhedron in the H² BIND language of the Origami ISA (spurious pole cancellation as boundary-of-boundary). HM07 adds the Hum ISA perspective: RENORM and PROPAGATE absent in the Grassmannian basis.
- HM01 (Lamb shift): in the Feynman basis, the Lamb shift requires RENORM. In the Grassmannian basis, it would be computed via a boundary-constrained ORBIT — an open problem (x620f).
- P01 (OPU): the Grassmannian computing unit (Paper 598) runs ORBIT on Gr(k,n) for molecular orbital geometry. The amplituhedron is the QFT realisation of the same ORBIT — the OPU extended to scattering amplitudes.
Validation
- Tree-level amplitudes: Arkani-Hamed & Trnka (2013), JHEP 2014:30. Canonical form Ω_{n,k} = ∏ dα_i/α_i verified for n ≤ 8, k ≤ 3.
- dlog UV-finiteness: verified algebraically at tree level. The interior of the amplituhedron has no poles in the canonical form; poles occur only on boundaries (physical factorisation channels).
- Loop amplituhedron: conjectured for all L; 1-loop results reproduce known direct calculations. Status: validated at L=1, conjectured for L ≥ 2.
Part of the ISA Zoo. Hum ISA reference: Paper 620. See also: G02 — Amplituhedron (Origami ISA view).