HM08 — Grassmannian Quantisation
| Field | Value |
|---|---|
| Domain | Positive Geometry / Scattering Amplitudes |
| System | Massless scattering in N=4 SYM |
| Group | GL(k) × GL(n−k) acting on Gr⁺(k, n) |
| H^k tier | H³ |
| ISA | Hum (β = it/ℏ) — Grassmannian basis |
| Status | Conjectural (conceptual framework) |
| Opcodes | ORBIT · EMIT |
| Papers | Paper 620 |
Physical system
There are two standard quantisation paradigms:
-
Canonical quantisation (first quantisation): fixed particle number, Hilbert space ψ, operators acting on ℋ. Replace Poisson brackets with commutators. The ISA grading: H¹ (quantum phases, interference, no EMIT).
-
Path-integral quantisation (second quantisation): variable particle number, Fock space ⊕_N ℋ^{⊗N}, field operators ψ̂(x) creating/destroying quanta. The ISA grading: H³ Feynman basis (EMIT + PROPAGATE + RENORM).
The amplituhedron proposes a third:
- Grassmannian quantisation: classical scattering data (n particles, helicities k, external momenta Z) maps to a positive region 𝒜_{n,k,L}(Z) ⊂ Gr⁺(k, k+4; L). The amplitude is the volume of this region. No Fock space, no Lagrangian, no virtual particles.
ISA interpretation
In ISA language, Grassmannian quantisation replaces the Fock-space tower ⊕_N ℋ^{⊗N} with the positive Grassmannian — a real manifold with boundary, not a complex vector space.
First and second quantisation emerge as limits:
- First quantisation = L=0 (external data Z only; no loop variables)
- Second quantisation = L > 0 (loop positivity constraints = virtual effects)
The Hum ISA opcode programme is:
- EMIT: specify the theory (n, k, coupling g, particle content)
- ORBIT(Gr⁺(k, k+4; L)): compute the volume
That is all. PROPAGATE and RENORM do not appear — they are Feynman-basis artefacts of the second-quantisation expansion.
The three quantisation paradigms as ISA programmes
| Paradigm | ISA basis | Key opcodes | Fock space? | RENORM? |
|---|---|---|---|---|
| Canonical (1st) | Origami/Motive | ORBIT, TWIST | No | No |
| Feynman (2nd) | Hum (Feynman) | EMIT, PROPAGATE, RENORM | Yes | Yes |
| Grassmannian (3rd) | Hum (Grassmannian) | EMIT, ORBIT | No | No |
The third paradigm has no standard name in the QFT literature. “Positive geometry quantisation” (Arkani-Hamed) and “amplituhedron quantisation” have been used informally.
The distinction between real and virtual particles
In the Feynman basis: external (real) particles are ORBIT entries; internal (virtual) particles are PROPAGATE steps. The distinction is fundamental — real particles are on-shell (p² = m²), virtual particles are off-shell.
In the Grassmannian basis: both real and virtual particles are columns in the (n+2L) × (k+4) matrix (Z_1, …, Z_n, D^{(1)}, …, D^{(L)}). Real particles = visible columns Z_i; virtual particles = hidden columns D^{(i)} subject to positivity. The real/virtual distinction is not fundamental — it is a labelling of which columns are “visible” (given as input data) vs “hidden” (integrated over).
This dissolves the on-shell/off-shell distinction that underlies the need for PROPAGATE and RENORM.
Connections
- HM07 (Amplituhedron): HM07 is the concrete realisation; HM08 is the conceptual framework. HM07 gives the opcode count (1 vs ≥ 10); HM08 explains why (Fock space vs positive Grassmannian).
- P01 (OPU): the Orbit Processing Unit (Paper 598) runs ORBIT on Gr(k,n) for molecular orbitals. Grassmannian quantisation = QFT running on the OPU. The two applications are: chemistry (orbital geometry) and scattering amplitudes.
- RS01 (Rising Sea ISA, future): the Rising Sea ISA (Paper 621) shows that the β-plane fibration contains Grassmannian quantisation as the β=it/ℏ fibre. The “third paradigm” is not a new foundation but a specific fibre in the categorical fibration.
Open questions
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Gravity at H³: the m=6 generalised amplituhedron (Arkani-Hamed, post-2013) suggests graviton amplitudes live at H³ in the Grassmannian basis. If so, non-renormalisability of perturbative gravity is a Feynman-basis artefact at any loop order.
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Grassmannian Lamb shift: the Lamb shift in the Grassmannian basis would be a boundary-constrained ORBIT. What is the positive geometry? (x620f, open)
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ℤ/4ℤ closure: IMAGINE⁴ = MARK (Paper 621, x621d). Does Grassmannian quantisation extend to ℤ/8ℤ (Bott periodicity)? Is there an 8-periodic positive geometry?