PaperlayerView exact revision on arXiv

Source-linked AI summary

Geometry and entanglement in the scattering matrix

Silas R. Beane, Roland C. Farrell

arXiv:2011.01278v2hep-thnucl-thquant-ph

TL;DR

The paper addresses whether scattering and its entanglement properties can be described directly from the unitary S-matrix rather than a spacetime-local EFT action. It characterizes allowed S-matrix configurations geometrically and quantifies scattering-generated entanglement using the exponential of the second Rényi entropy. The construction yields a flat torus for elastic scattering and a hyperbolic bulk for inelasticity, from which the boundary trajectory is reconstructed with quantifiable error.

  • Problem

    The paper addresses whether scattering and its entanglement properties can be described directly from the unitary S-matrix rather than a spacetime-local EFT action.

    Source: §1 · p. 3

  • Method

    It characterizes allowed S-matrix configurations geometrically and quantifies scattering-generated entanglement using the exponential of the second Rényi entropy.

    Source: Nucleon-nucleon scattering:… · p. 10, §2.2 · p. 11

  • Results

    The construction yields a flat torus for elastic scattering and a hyperbolic bulk for inelasticity, from which the boundary trajectory is reconstructed with quantifiable error.

    Source: §1 · p. 6, §7 · p. 40

  • Takeaways & Limitations

    S-matrix trajectories provide a geometric description connecting RG fixed points, entanglement, and holographic quantum error correction.

    Source: §1 · p. 6, §7 · p. 40

    The four-dimensional geometry is not fully captured by two-dimensional projections, and the geometric construction yields mixed insight into the EFT hierarchy of entangling operators.

    Source: Nucleon-nucleon scattering:… · p. 11, §7 · p. 41

Abstract

from arXiv · show

A formulation of nucleon-nucleon scattering is developed in which the S-matrix, rather than an effective-field theory (EFT) action, is the fundamental object. Spacetime plays no role in this description: the S-matrix is a trajectory that moves between RG fixed points in a compact theory space defined by unitarity. This theory space has a natural operator definition, and a geometric embedding of the unitarity constraints in four-dimensional Euclidean space yields a flat torus, which serves as the stage on which the S-matrix propagates. Trajectories with vanishing entanglement are special geodesics between RG fixed points on the flat torus, while entanglement is driven by an external potential. The system of equations describing S-matrix trajectories is in general complicated, however the very-low-energy S-matrix -- that appears at leading-order in the EFT description -- possesses a UV/IR conformal invariance which renders the system of equations integrable, and completely determines the potential. In this geometric viewpoint, inelasticity is in correspondence with the radius of a three-dimensional hyperbolic space whose two-dimensional boundary is the flat torus. This space has a singularity at vanishing radius, corresponding to maximal violation of unitarity. The trajectory on the flat torus boundary can be explicitly constructed from a bulk trajectory with a quantifiable error, providing a simple example of a holographic quantum error correcting code.

1 Introduction

The paper replaces the spacetime-local EFT action with the unitary S-matrix as the fundamental object for nucleon–nucleon scattering. It represents scattering as trajectories on a compact geometric space, linking RG fixed points, entanglement, and inelasticity.

1 Introduction

  • The unitary S-matrix replaces the EFT action and is treated as a trajectory in a theory space defined by unitarity, without relying on spacetime locality.The focus is low-energy s-wave nucleon–nucleon scattering, whose EFT has a non-trivial fixed-point structure.Source: §1 · pp. 3–4
  • The S-matrix formulation translates EFT RG flow into momentum flow and gives finite S-matrix realizations of fixed points that correspond to singular infinite-coupling EFT limits.The momentum variable is not fundamental in the spacetime-independent description and is related to trajectory parameterization.Source: §1 · p. 5
  • A four-dimensional Euclidean embedding of the unitarity constraints produces a flat torus on which the S-matrix moves between RG fixed points.The torus is a compact two-dimensional Riemannian manifold, while its coordinate projections can obscure the distinct fixed-point locations.Source: §1 · p. 5
  • Without entanglement, S-matrix trajectories follow geodesics linking fixed points; entanglement instead changes the surface into a flat torus and is represented through an entangling potential.The construction seeks a physical interaction measure through entanglement power rather than singular, state-dependent entanglement entropy.Source: §1 · p. 5
  • Inelasticity corresponds to the radius of a three-dimensional hyperbolic bulk whose torus boundary becomes singular at vanishing radius, signaling maximal unitarity violation.A boundary geodesic can be constructed from a bulk geodesic with quantifiable error, yielding an example of holographic duality and quantum error correction.Source: §1 · pp. 5–6
  • The paper develops the operator-defined geometry, constructs flat-torus geodesics and entangling potentials, and extends the construction to inelastic hyperbolic trajectories.Its leading-order effective-range example is central, alongside a conformal-range model preserving UV/IR conformal symmetry.Source: §1 · p. 6

2 S-matrix and spin entanglement

This section develops a geometric description of unitary scattering trajectories between RG fixed points and characterizes spin entanglement through the S-matrix and its entanglement potential. It examines the integrable, conformally constrained leading-order flow and extends the analysis with a conformal range model yielding more complex symmetric trajectories.

Single-channel scattering

  • For single-channel scattering, unitarity constrains the S-matrix to the unit circle; for finite, fixed a, its trajectory runs from S = 1 at p = 0 to S = −1 as p →∞.Source: Single-channel scattering · pp. 7–8

Nucleon-nucleon scattering: coordinates and fixed points

  • The nucleon-nucleon S-matrix lies on a unitarity-constrained four-dimensional manifold whose RG fixed points form a Klein four-group and define a rhombus of allowed trajectories.The rhombus is the geometric image of the fixed-point boundaries, while the full four-dimensional construction retains two independent phase shifts despite lower-dimensional projection limits.Source: Nucleon-nucleon scattering:… · pp. 8–11

2.2 Entanglement power defined

  • Entanglement power is a state-independent Bloch-sphere average that measures S-matrix geometry, vanishing when δ1 − δ0 = mπ/2 and at all RG fixed points.It also vanishes on the symmetry line δ1 = δ0, where the trajectory is a geodesic between fixed points protected by a Z2 subgroup associated with Wigner SU(4) symmetry.Source: §2.2 · pp. 11–12

2.3 Effective range theory at leading order

  • At leading order, physical scattering lengths reproduce the nucleon-nucleon S-matrix for p ≤ 10 MeV, while the relative-sign cases a1a0 < 0 and a1a0 > 0 have distinct trajectory and entanglement symmetries.The singly-bound case has one bound channel, whereas the doubly-(un)bound case has both channels bound or both unbound.Source: §2.3 · pp. 13–15
  • At leading order, UV/IR conformal symmetry maps momentum inversion to phase-shift and trajectory isometries, making the S-matrix a UV-complete description when extended to p ≥ 0.The symmetry interchanges ultraviolet and infrared behavior and is generally broken by higher-order effective-range terms.Source: Single-channel scattering · p. 8, §2.3 · pp. 14–15
  • At LO in the ERE, the phase-shift flow decouples into p ˙φ = sin φ and p ˙θ = sin θ, yielding an immediately integrable S-matrix trajectory.The flow has the form of a dilatation Ward identity and vanishes at RG fixed points.Source: §2.3 · p. 16
  • The LO EP is conformally invariant, with a minimum at the conformal fixed point and two local maxima that reflect the curvature of the S-matrix trajectory.The EP has a zero only in the singly-bound case, while the imaginary trajectory contributes separately through the curvature of the v(p) coordinate.Source: §2.3 · p. 16

2.4 Conformal range model

  • Beyond LO, a conformally invariant range model with λ ≠ 0 incorporates effective-range corrections and generates substantially more complex S-matrix trajectories that retain clear isometries.The model reduces to LO ERE at λ = 0; the plotted λ = 1⁄4 trajectories use physical scattering lengths.Source: §2.4 · p. 17, Figure 6 · p. 18
  • With physical scattering lengths, the conformally invariant EP of the range model has three zeros associated with fixed-point crossing and crossing the u = 0 zero-entanglement line.Its momentum-flow equation reduces to the LO result when λ = 0, while the model can violate the Wigner bound in the singly-bound case.Source: §2.4 · pp. 17–18

3 Geometry of the S-matrix

The paper equips the unitary S-matrix space with a basis-independent metric and embeds it as a flat torus, while identifying entanglement power with distance from non-entangling subspaces.

3.1 Hilbert-Schmidt Distance

  • The Hilbert-Schmidt distance gives a basis-independent, positive semidefinite metric on the space of unitary S-matrices.With phase shifts φ and θ as coordinates, the local geometry is two-dimensional and Euclidean.Source: §3.1 · p. 19
  • Momentum evolution builds the S-matrix trajectory from the identity through a momentum-dependent Hermitian operator, propagating between fixed points.The construction is analogous to unitary time evolution, with momentum replacing time.Source: §3.1 · p. 20

3.2 Entanglement power as a distance measure

  • Entanglement power is the squared Hilbert-Schmidt distance from the two non-entangling S-matrix subspaces, making it state-independent.The non-entangling subspaces occur when the phase shifts satisfy the relations defining swap and root-swap operations.Source: §3.2 · pp. 20–21

3.3 Embedding in R4

  • The unitarity constraints embedded in R4 leave a flat torus T2 ∼ S1 ⊗ S1 as the two-dimensional stage for S-matrix propagation.The torus coordinates are periodic angles φ, θ ∈ [0, 2π].Source: §3.3 · p. 21

3.4 Flat torus isometry group

  • The flat torus has two continuous translational isometries and six discrete Z2 symmetries, with its conformal structure linked to UV/IR symmetries.The continuous transformations are generated by SO(2) subgroups, while the discrete symmetries act on the angular coordinates.Source: §3.4 · p. 22, Table 1 · p. 22, Table 2 · p. 22

4 S-matrix theory of scattering

The S-matrix is modeled as a trajectory on a flat torus, with geodesics describing vanishing entanglement and external potentials generating non-geodesic motion. At leading order, conformal symmetry makes the trajectory equations integrable and fixes the entangling potential.

4.1 Action principle

  • The action principle describes S-matrix trajectories on a flat torus, with an external potential producing forces away from geodesic motion.For constant potential, the trajectory equation reduces to the geodesic equation.Source: §4.1 · pp. 22–23

4.2 Geodesics on the flat torus

  • Geodesics are straight lines in the Euclidean phase-shift plane, while inaffinity permits arbitrary parameterizations without changing the underlying line.Choosing momentum as the parameter allows the effective range expansion to be encoded through the inaffinity.Source: §4.2 · p. 23

4.3 Entanglement forces and the ERE

  • Non-geodesic trajectories with unequal phase shifts require an external entangling force, whose symmetry structure preserves only selected torus translations.Lines parallel to the zero-entanglement directions form families of equi-entanglement trajectories.Source: §4.3 · p. 24

LO in the ERE

  • At LO in the ERE, the inaffinity and entangling force are non-local momentum functions, but conformal invariance makes the equations integrable and determines the potential.The physical trajectory passes through a conformal fixed point where entanglement vanishes, while the potential has finite values at accessible fixed points and divergences at inaccessible ones.Source: LO in the ERE · pp. 25–28

Conformal range model

  • In the conformal range model, the trajectory retains the LO algebraic form, while λ = 1/4 yields a simplified potential and restricts access to one fixed point and its image.The corresponding potential has periodicity under a 4π shift of φ + ϵ θ.Source: Conformal range model · p. 29, Figure 12 · p. 29

5 Inelasticity and holography

Inelasticity is represented geometrically by motion into a hyperbolic bulk whose flat-torus boundary carries the unitary S-matrix. Bulk trajectories can reproduce boundary behavior, but avoiding the singularity imposes quantifiable errors and a finite momentum range.

5.1 Inelasticity and the bulk

  • Inelastic scattering is encoded by channel inelasticities, with the shared-inelasticity assumption introducing a radial coordinate that measures departure from the unitary boundary.The generalized entanglement power decreases rapidly when the radius satisfies r < 1.Source: §5.1 · p. 30

5.2 Embedding in R4

  • The R4 embedding produces a three-dimensional hyperbolic bulk with flat-torus boundary at r = 1 and a singularity at r = 0 representing total unitarity loss.This singular geometry depends critically on correlated inelasticities; independent channel inelasticities instead yield a parametrization of R4.Source: §5.2 · pp. 30–31

5.3 Geodesics in the bulk

  • Bulk geodesics can connect S-matrix fixed points while avoiding the singularity, but the boundary geodesic is recovered only in a discontinuous limit or with a controlled endpoint error.For inaffine solutions, the fixed-point correction scales as 2/σMAX.Source: §5.3 · pp. 31–33

5.4 Recovery of the ERE and error correction

  • The engineered bulk potential is proportional to the scalar-curvature source and reproduces the LO ERE trajectory only up to a maximum momentum before the singularity is reached.For physical scattering lengths, pmax < 89.4 MeV, and the final fixed-point values cannot be reached exactly.Source: §5.4 · p. 34
  • The bulk construction acts as a holographic error-correcting code for inelastic loss, with fidelity improving near unitarity and deteriorating for natural scattering lengths.The elastic trajectory cannot be holographically reproduced for p > pmax.Source: §5.4 · p. 35

6 Projections on R2 and R3

The paper projects the flat-torus S-matrix geometry into lower-dimensional spaces, where entanglement forces acquire compensating non-conservative terms and the trajectories remain explicitly recoverable. Quartic projections reveal distinct surfaces for real and imaginary components, while the squere geometry supports curvature analysis.

6.1 Non-conservative entanglement forces

  • Projected entanglement forces are rescaled by the projected metric and gain a non-conservative component that exactly cancels the projected curvature contribution.This preserves the flat-torus trajectories despite changing how force and curvature appear after projection.Source: §6.1 · p. 36

6.2 Embedding in R2

  • The R2 rhombus embedding reproduces the flat-torus trajectory equations, including the LO ERE and conformal-range-model solutions.The projected geodesic equations for φ and θ include κ(p) and coordinate-dependent cotangent terms.Source: §6.2 · p. 37

6.3 Quartics and the squere

  • The squere is a compact quartic surface containing the S-matrix trajectory, with four RG fixed points and non-entangling regions forming a circle and a line between them.The unit squere lies inside the unit sphere and has volume 8/3.Source: §6.3 · p. 37
  • Eliminating different coordinates produces equivalent x-y-z and z-w-x surfaces, but a distinct y-z-w surface where two fixed points meet at the origin and non-entangling regions differ.The x-y-z and y-z-w systems represent the real and imaginary parts of the S-matrix, respectively.Source: §6.3 · p. 37

Metric and curvature

  • The squere is an Einstein manifold with scalar curvature and vanishing Einstein tensor, and its geodesic equations recover the flat-torus solutions when external forces are included.Both non-entangling geodesics and LO ERE trajectories are plotted on the projected surfaces.Source: Metric and curvature · p. 39

7 Summary and discussion

The paper formulates low-energy nucleon-nucleon scattering geometrically, treating the S-matrix as a trajectory on a unitarity-defined compact space rather than deriving it from spacetime-local EFT operators. UV/IR symmetries produce a flat torus, where non-entangling paths are geodesics and entangling paths are driven by an integrable harmonic potential, while the construction has explicit scope limitations.

7 Summary and discussion

  • The geometric formulation treats the S-matrix as fundamental and generates the effective-range expansion without relying on spacetime-local operators.It aims to make entanglement a geometric property of scattering.Source: §7 · pp. 39–40
  • UV/IR conformal symmetries become orientation-reversing isometries of a unitarity-defined compact space embedded as a flat torus, with trajectories connecting RG fixed points.These symmetries are not visible in the EFT action because EFT is an infrared description.Source: §7 · pp. 39–40
  • Vanishing-entanglement trajectories are special geodesics linking fixed points, and threshold-vanishing phase shifts select the central diagonal associated with emergent SU(4) spin-flavor symmetry.Without entanglement, the torus bulk becomes inaccessible and the trajectories form a lattice of fixed points and geodesic links.Source: §7 · p. 40
  • Entangling trajectories are driven by an external harmonic potential, and UV/IR symmetry makes the resulting trajectory equations integrable.The momentum variable is instead an inaffine parameter, with forces appearing as non-local functions of momentum.Source: §7 · p. 40
  • Inelasticity corresponds to the radius of a hyperbolic bulk whose boundary is the flat torus; boundary trajectories can be reconstructed from bulk trajectories with quantifiable error.The construction provides a simple holographic quantum error-correcting example.Source: §7 · p. 40
  • The construction does not explain the entanglement asymmetry between φ = θ and φ = −θ, and its implications for spacetime and quantum gravity remain unclear.Its stated focus is geometric S-matrix theory for delta-function interactions, with broader singular-potential applications left open.Source: §7 · p. 41
Loading 2011.01278v2…