Two Major Research Programmes. One Geometric Object.

Plain-language explainer for doi:10.5281/zenodo.21518107 (#680)

The central claim in one sentence

The positive Grassmannian — the geometric space that encodes all scattering amplitudes in the amplituhedron programme — is precisely the PT-unbroken phase of a larger complex space, and its boundary is the exceptional-point locus where PT symmetry breaks.

Background: the amplituhedron

In 2013, Arkani-Hamed and Trnka discovered that the scattering amplitudes of maximally supersymmetric Yang-Mills theory — the quantum field theory that physicists use as a laboratory for understanding the structure of amplitudes in general — could be encoded in a single geometric object called the amplituhedron.

The amplituhedron lives inside a mathematical space called the Grassmannian Gr(k,n): the space of all k-dimensional planes through the origin in n-dimensional space. (For a single particle, think of k=2 and n=4: the space of all 2-planes in 4 dimensions, which has a natural identification with the space of massless momenta in 4D spacetime.)

The physical scattering amplitudes correspond to the interior of a particular region of this Grassmannian called the positive Grassmannian Gr⁺(k,n), defined by requiring all Plücker coordinates to be positive. Plücker coordinates are numbers built from the momenta of the particles; they are the fundamental coordinates on the Grassmannian. When all of them are positive, the kinematics is physical and the amplitude is finite. When one of them vanishes — that is, when the system sits on the boundary ∂Gr⁺(k,n) — the amplitude has a singularity (a collinear or soft divergence).

Background: PT symmetry

In 1998, Bender and Boettcher made a surprising discovery: a quantum-mechanical Hamiltonian can have an entirely real energy spectrum even if it is not Hermitian, provided it obeys a combined PT symmetry — that is, if it is invariant under the simultaneous operations of parity (P: x → −x) and time-reversal (T: t → −t, i → −i).

PT-symmetric Hamiltonians have a phase structure analogous to ferromagnets:

  • PT-unbroken phase: the spectrum is real and the eigenstates are non-degenerate. The system behaves like a standard Hermitian quantum system.
  • Exceptional point (EP): as a parameter is varied, two eigenvalues collide and the two eigenvectors align. This is the phase boundary.
  • PT-broken phase: past the EP, the previously real eigenvalues become a complex conjugate pair. The system acquires gain and loss.

This phase structure has been found in optical waveguides, lasers, gyroscopes, and mass sensors. Near an exceptional point, physical sensors are dramatically more sensitive: a perturbation that shifts a single eigenvalue by ε far from the EP shifts it by √ε at an EP₂, giving a 1/√ε enhancement.

What nobody had noticed

These two research programmes — the amplituhedron (2013–present) and PT symmetry (1998–present) — have developed almost entirely in parallel, with no interaction.

This paper observes that they are the same geometric structure:

  1. The CPT involution of the underlying quantum field theory acts on the Grassmannian as complex conjugation of the momentum spinors. The fixed-point set of this involution — the points left unchanged by CPT — is exactly the positive Grassmannian Gr⁺(k,n), since real positive Plücker coordinates are unchanged by complex conjugation.

  2. Moving off Gr⁺(k,n) into the complex Grassmannian Gr(k,n,ℂ) — by allowing Plücker coordinates to become complex — is exactly the PT-breaking deformation. The interior of Gr⁺ is the PT-unbroken phase. The exterior is PT-broken.

  3. The boundary ∂Gr⁺(k,n), where some Plücker coordinate first vanishes, is the exceptional-point locus: the eigenvalues of the natural non-Hermitian matrix built from the amplitude data coalesce there, with the characteristic EP₂ square-root splitting.

The amplituhedron programme has been, implicitly, working in the PT-unbroken phase all along. The collinear singularities of scattering amplitudes — the places where the amplitude blows up — are the real-kinematic images of the exceptional points.

The construction

For the simplest non-trivial amplitude, A₄,₂ (four particles, two negative helicities), the Grassmannian Gr(2,4) has exactly two BCFW cells: cell (0,2) and cell (1,3). Each cell contributes a weight to the total amplitude.

Near the collinear limit — when particles 0 and 1 become parallel, so their angle bracket ⟨01⟩ = sin ε → 0 — cell (1,3) has ⟨01⟩ in its denominator and diverges as 1/sin ε, while cell (0,2) stays finite. The ratio of the two weights grows without bound.

This asymmetry maps directly to gain-loss imbalance in a PT-symmetric matrix:

\[H(\varepsilon) = \begin{pmatrix} +i\gamma & \kappa \\ \kappa & -i\gamma \end{pmatrix}\]

where γ = (|w₁₃| − |w₀₂|)/(|w₁₃| + |w₀₂|) ∈ [0,1) is the normalised weight imbalance, and κ is the Plücker inner product between the two cells (a coupling that measures how much the two cells “overlap” in wavefunction space).

The eigenvalues of this matrix are λ± = ±√(κ² − γ²):

  • Far from the collinear limit (large ε): γ ≈ 0, weights balanced, eigenvalues real — PT-unbroken.
  • At some ε* where γ = κ: eigenvalues coalesce at zero — exceptional point.
  • Near the collinear limit (small ε): γ > κ, eigenvalues purely imaginary — PT-broken.

Three numerical confirmations

All tests use physical on-shell momenta with exact momentum conservation.

Test 1 — EP₂ splitting (experiment x680a):

The eigenvalue gap |Δλ| was measured as a function of |ε − ε| on a fine grid near the exceptional point at ε ≈ 1.485 radians. A log-log fit gives:

Δλ ~ ε − ε* ^{0.55}, R² = 0.944

The EP₂ prediction is exponent ½. A simple zero crossing would give exponent 1. The measured 0.55 is clearly not 1 and is within 10% of ½.

At the exceptional point, the eigenvector overlap |⟨v₊, v₋⟩| = 0.991 (from 0.38 in the interior of Gr⁺), confirming eigenvector coalescence — the defining property of an EP.

Test 2 — Full Bender-Boettcher phase diagram (experiment x680b):

Scanning ε from 0.01 (near-collinear, PT-broken) to π/2 (orthogonal, PT-unbroken):

  • At ε = π/2: max|Im(λ±)| < 5×10⁻¹⁹ — eigenvalues real to machine precision. This is the deep interior of Gr⁺.
  • At ε = 0.05: max|Re(λ±)| < 10⁻⁶, max|Im(λ±)| = 0.87 — purely imaginary eigenvalues, complex conjugate pair. This is outside Gr⁺.
  • The gap Δλ(ε) forms a perfect V-shape with minimum at ε* ≈ 1.49.

This is the canonical Bender-Boettcher phase diagram, reproduced from the geometry of scattering amplitudes.

Test 3 — Enhanced sensitivity near the boundary (experiment x680c):

Near EPs, physical sensors have enhanced sensitivity. The kinematic analogue is that small changes in the collinear parameter ε produce large changes in the eigenvalue gap when ε is near ε. Measuring |d|Δλ|/dε| as a function of |ε − ε|:

d Δλ /dε ~ ε − ε* ^{−0.68}, R² = 0.989, 28× enhancement near EP

The EP₂ prediction for the sensitivity exponent is −½. The observed −0.68 is steeper, reflecting that the approach to ε* has some curvature (the EP locus is not perfectly flat in kinematic space). The 28× enhancement and power-law scaling are unmistakable EP₂ signatures.

What the snap threshold is

In the Maslov-Gibbs Einsum, the β* snap threshold is the point at which the free energy switches from single-mode behaviour (one BCFW cell dominates) to multi-mode behaviour (several cells contribute comparably). The snap is where the first derivative of the dominance fraction σ₀² is steepest.

This paper identifies the snap threshold as the real-kinematic shadow of the exceptional-point locus: the closest point in physical kinematic space to the complex surface ∂Gr⁺(k,n). The snap is not an arbitrary threshold — it is the projection of the EP locus onto the real slice.

The Raven amplitude (open question)

The results above motivate a conjecture: there exists a natural complexification of the amplituhedron form to the full complex Grassmannian Gr(k,n,ℂ) — a “Raven amplitude” — such that:

  • On Gr⁺(k,n): the Raven amplitude equals the physical scattering amplitude.
  • Near ∂Gr⁺(k,n): the Raven amplitude has EP₂ singularities with √ε splitting.
  • In the PT-broken sector: the Raven amplitude acquires an imaginary part — and this imaginary part is the total inelastic cross-section, i.e. the optical theorem is PT-breaking.

If proved, this would give the optical theorem a geometric explanation for the first time: unitarity of the S-matrix is the statement that physical amplitudes live in the PT-unbroken sector.

The ISA tier picture

The three tiers of the Origami ISA correspond directly to the three PT phases:

ISA tier PT phase Kinematic regime
H⁰ (tropical, single-cell) Deep PT-unbroken Far from any collinear limit
H¹ (multi-cell, near-snap) Near exceptional point Approaching ∂Gr⁺
H² (BIND, boundary) At EP / PT-broken At collinear singularity

The β* snap threshold of the MGE — the transition from H⁰ to H¹ — is the real-kinematic approach to the EP locus. The H² BIND opcode fires at the EP itself.

What this paper does not claim

  • It does not prove the Raven amplitude conjecture — that requires constructing an explicit canonical form on Gr(k,n,ℂ), which is future work.
  • The PT-symmetric Hamiltonian H(ε) is a construction from the BCFW data, not uniquely derived from first principles. The EP location ε* depends on the construction; the qualitative phase structure does not.
  • The results are established numerically for n=4. Extension to general n is expected from the same CPT argument but has not been checked case by case.

Why this matters to each community

For the amplituhedron community: The positive Grassmannian is not just defined by a sign condition — it is the PT-unbroken phase of a richer complex space. Every collinear singularity is an exceptional point. The boundary stratification of the amplituhedron is an EP hierarchy. The symbol alphabet letters are the functions that vanish at each EP stratum.

For the PT-symmetry community: Here is a concrete, well-understood geometric object (the positive Grassmannian with amplituhedron form) that realises the full Bender-Boettcher phase structure. The PT-breaking deformation has a direct physical meaning: complex momenta, loop integrands, Regge limits.

For the ISA community: The snap threshold β* is the real shadow of the EP locus. The H⁰/H¹/H² tier structure is the PT-unbroken/near-EP/PT-broken trichotomy. The SNAP-count invariant counts collinear boundary crossings along a kinematic path.

See also:

For the full technical treatment, see doi:10.5281/zenodo.21518107