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Symmetries Near the Horizon

Henry W. Lin, Juan Maldacena, Ying Zhao

arXiv:1904.12820v2hep-thgr-qc

TL;DR

Near-horizon translations in nearly-AdS2 gravity and SYK raise the question of how physical symmetries can move matter relative to dynamical boundaries. The paper constructs gauge-invariant SL(2) generators and relates them to bulk displacements and approximate SYK operators. In the semiclassical limit, they act on boundary time and organize states into primaries and descendants; charge growth saturates beyond the scrambling time as out-of-time-order correlators decay to zero.

  • Problem

    The two near-horizon translation symmetries are less understood than the boost, despite their ability to move matter into a black-hole interior.

    Source: §1 · p. 3

  • Method

    The paper constructs gauge-invariant SL(2) generators relative to boundary positions and develops exact and approximate versions for nearly-AdS2 gravity and SYK.

    Source: §1 · p. 5, §3.1 · p. 12, §7.2 · p. 47

  • Results

    In the semiclassical limit the generators act as conformal symmetries of boundary time and organize states into primaries and descendants; beyond the scrambling time, excitation momentum or energy stops growing as out-of-time-order correlators decay to zero.

    Source: §1 · p. 5, §7.2 · p. 48

  • Takeaways & Limitations

    The generators describe bulk matter displacements across the horizon and connect those displacements to chaos and out-of-time-order correlators.

    Source: §7.2 · p. 47

    Bulk evolution cannot be extended beyond the dotted red lines because the matter boundary conditions there are not guaranteed to match those at a physical boundary.

    Source: §5.3 · p. 35

Abstract

from arXiv · show

We consider a nearly-AdS$_2$ gravity theory on the two-sided wormhole geometry. We construct three gauge-invariant operators in NAdS which move bulk matter relative to the dynamical boundaries. In a two-sided system, these operators satisfy an SL(2) algebra (up to non-perturbative corrections). In a semiclassical limit, these generators act like SL(2) transformations of the boundary time, or conformal symmetries of the two sided boundary theory. These can be used to define an operator-state mapping. A particular large N and low temperature limit of the SYK model has precisely the same structure, and this construction of the exact generators also applies. We also discuss approximate, but simpler, constructions of the generators in the SYK model. These are closely related to the "size" operator and are connected to the maximal chaos behavior captured by out of time order correlators.

1 Introduction and motivation

The paper constructs physical SL(2) symmetries for nearly-AdS2 gravity and SYK, addressing how near-horizon translations can move matter into a black-hole interior. It also examines their semiclassical boundary-time action and the limits of the exact construction.

Motivation

  • Near-horizon translations are less understood than the boost symmetry and matter can be moved into the black-hole interior by these translations.The paper explicitly constructs these symmetries for nearly-AdS2 gravity and the related SYK model.Source: §1 · p. 3

Physical generators

  • Three generators act on the physical Hilbert space, obey an exact SL(2) algebra, and are gauge invariant, though they do not commute with the Hamiltonian.Because the boundary positions evolve, the paper describes a construction for obtaining “conserved” charges by undoing their evolution.Source: §1 · pp. 3–5

Scope

  • The exact discussion applies in a low-temperature scaling limit with fixed Schwarzian coupling, not at finite βJ or finite N.In SYK variables, the limit is N → ∞ and βJ → ∞ with N/(βJ) fixed; the generators need not make sense at finite N.Source: §1 · p. 4

Semiclassical limit

  • In the semiclassical limit, the charges act as conformal symmetries of boundary time and organize the Hilbert space into primaries and descendants with a state-operator correspondence.The correspondence is described as analogous to higher-dimensional versions.Source: §1 · p. 5

SYK and bulk physics

  • The paper relates approximate SYK generators to the “size” operator and to out-of-time-order correlators, while connecting them to bulk matter displacements across the horizon.The approximate global time-translation operator is essentially the coupled Hamiltonian in [16].Source: §1 · pp. 4–5

2 Review

The review sets out the AdS2 isometries, the boundary and gauge structure of nearly-AdS2 gravity, and the corresponding low-energy structure in SYK.

2.1 Review of the symmetries of AdS2

  • AdS2 has an SO(2,1) ≃ SL(2,R) symmetry; near the bifurcation point, its generators act locally as energy, momentum, and boost, with an SL(2) rather than Poincare algebra.The generators are E, P, and B, and [E, P] = iB.Source: §2.1 · p. 6
  • AdS2 particle charges are conserved, and field charges defined from Killing vectors are slice-independent when the fields obey reflecting boundary conditions.A particle’s trajectory obeys Q·Y = q.Source: §2.1 · pp. 7–8

2.2 Review of the nearly-AdS2 gravity theory

  • Nearly-AdS2 gravity reduces to matter and two boundary systems linked by an overall SL(2)g constraint, so only relative positions are physical.The boundary dynamics is treated in the Schwarzian limit, with matter effectively experiencing exact AdS2.Source: §2.2 · pp. 8–9
  • The Schwarzian boundary-position operators satisfy the Poincare algebra with Casimirs r2 = 0 and qx = ±1 in the canonically quantized theory.The left and right boundaries have opposite signs of qx.Source: §2.2 · pp. 9–10

2.3 Review of SYK

  • At low energies, SYK has a Schwarzian soft mode and an SL(2)g redundancy that connects otherwise decoupled action terms through the total gauge constraint.The review identifies the SYK structure with three systems connected by an overall gauge constraint, analogous to nearly-AdS2 gravity.Source: §2.3 · p. 11

3 Exact generators

The paper constructs gauge-invariant SL(2) charges that move bulk matter relative to the boundaries, then relates them to boundary quantities and defines reference-time charges by evolving only the boundary modes.

3.1 Construction of gauge invariant SL(2) generators

  • The three gauge-invariant charges move matter relative to the boundaries: one translates along their connecting geodesic, while the others leave one boundary point fixed and can move matter outside the Wheeler-de-Witt patch.The boost is around the geodesic midpoint; the third generator acts locally as a time translation orthogonal to the geodesic.Source: §3.1 · p. 12
  • The physical Hilbert space is infinite dimensional because it includes matter degrees of freedom and their descendants.Source: §3.1 · p. 13
  • The construction assumes a strip-like topology with large φ0, suppressing topology changes; the boundary-momentum expression also assumes decoupled left and right evolution.Source: §3.1 · p. 13, §3.1.1 · p. 15
  • The generators obey the SL(2) algebra, and their matter Casimir is gauge invariant and commutes with the Hamiltonian.Source: §3.1 · p. 14

3.1.1 Writing the charges purely in terms of boundary quantities

  • The total momentum can be written using boundary gravitational quantities and interpreted as matter momentum in the frame set by the boundary positions.For a coordinate along the connecting geodesic, the matter momentum is minus the sum of the boundary-particle momenta.Source: §3.1.1 · p. 14

3.2 Defining “conserved” charges

  • The charges are not conserved under the left or right Hamiltonian, but reference-time charges are defined by undoing the boundary-mode evolution without undoing matter evolution.The resulting quantum Schwarzian expressions are complex and difficult to express in correlators at the current-time basis.Source: §3.2 · pp. 15–16

4 Approximate expressions for the generators

This section develops approximate, state-dependent generators in semiclassical settings, relating them to boundary-time transformations, Euclidean operator insertions, and exponentially growing correlator corrections. It also examines how matter insertions affect the generators beyond the scrambling time, and considers one-sided generators and momentum, including chaos-related limits.

4.1 The generators in the semiclassical limit

  • In Schwarzian units, β ≪ 1 (s ≫ 1) is semiclassical; small boundary shifts permit a charge expansion whose equations of motion ensure time-independent charges.The expansion assumes the matter-induced changes to the boundary trajectories are small.Source: §4.1 · pp. 16–17

4.2 The semiclassical limit and SL(2) symmetries of the physical boundary time

  • Euclidean evolution over β/2 with operator insertions defines a map between operators and wormhole states, and the charges act on those states as expected from boundary-time reparametrizations.The state is represented by a path integral over one half of the Euclidean circle, with the other half preparing its bra.Source: §4.2 · pp. 18–19
  • Around classical states, the exact generators act as SL(2) transformations of boundary time, an approximate, β-dependent symmetry that is not expected away from the semiclassical limit.The symmetry action is state-dependent, and the semiclassical picture also breaks down for sufficiently early Lorentzian matter insertions.Source: §4.2 · p. 22

4.2.1 Inserting matter at early lorentzian times

  • Earlier Lorentzian insertions produce exponentially growing energy and momentum in the semiclassical formulas, which cease to suffice beyond the scrambling time u_scr = β/(2π) log s.The insertion must be slightly smeared to create a relatively low-energy state in the conformal regime.Source: §4.2.1 · p. 23

4.2.2 Inserting matter beyond the scrambling time and corrections to the semiclassical limit

  • Beyond the scrambling time, exact generators for a one-sided insertion saturate at order s, while with matter sent from both sides, global energy continues to increase exponentially.For one-sided insertion, momentum also saturates and boost approaches zero; the exact generators remain usable beyond the perturbative regime.Source: §4.2.2 · pp. 23–24

4.3 Other semiclassical expressions for the generators

  • The boost generator is approximately the difference of the left and right Hamiltonians, which is also the modular Hamiltonian for the left-right split and an exact symmetry of the thermofield double.Source: §4.3 · p. 25
  • The coupled-system construction [16] identifies its Hamiltonian with global AdS2 time translation and recovers the semiclassical expression when boundary coupling has little effect on bulk matter.The paper states that the final expression agrees with the earlier semiclassical generator.Source: §4.3 · pp. 25–26
  • The hatted generators are well defined in quantum-dual systems such as SYK, but obey only an approximate SL(2) algebra near a temperature- and synchronization-specific thermofield double.A relatively simple approximate expression for ˜E is obtained by coupling the two sides.Source: §4.3 · p. 26

4.4 Order from chaos

  • The difference between a decaying out-of-time-order correlator and the boost-invariant disconnected thermofield-double correlator yields exponentially growing corrections used to define the generators.The correlator’s initial decay appears as an exponentially growing deviation from the disconnected contribution.Source: §4.4 · p. 27
  • The algebraic generator relation implies that perturbed thermofield-double correlators contain growth at the maximal Lyapunov exponent, providing an algebraic derivation of maximal chaos in nearly-AdS2 gravity.The formulas apply in the semiclassical limit s ≫ 1; the construction uses correlator growth to define symmetry, or “order,” from chaos.Source: §4.4 · p. 28

4.5 Generators for the one-sided case

  • One-sided matter charges cannot be determined for general states, and chaos makes candidate one-sided generators fail to commute, with their algebra breaking down after sufficiently long post-insertion evolution.On thermal states the exponentially growing commutator has the maximal Lyapunov exponent; restricted states with matter inserted only from one side may permit detecting the charges.Source: §4.5 · pp. 28–31
  • Using right-boundary operators at two times defines a generator that boosts around their light rays’ intersection, preserves the enclosed causal wedge, and approximates its modular Hamiltonian without bulk gravity.The wedge is formed by a future-directed ray from the earlier time and a past-directed ray from the later time.Source: §4.5 · p. 30
  • For a semiclassical particle inserted from the right, the one-sided momentum approximates the two-sided momentum while the right-boundary time remains below the scrambling time.The geodesic from the right to the left boundary approximately intercepts the left bifurcating surface in this regime.Source: §4.5 · p. 31

5 Exploring the bulk

The charges describe bulk motion relative to dynamical boundaries, enabling exploration within the Wheeler–DeWitt patch and construction of gauge-invariant bulk operators. Their reach is limited at the inner horizon, where the assumed matter boundary conditions cease to be guaranteed.

5.1 Evolving with the charges

  • The gauge-invariant generator ˜E has a β-independent Hamiltonian for which thermofield-double states at any temperature and relative synchronization are ground states; only states with bulk matter have nonzero energy.In a special gauge its expression can appear local, but acting with it requires boundary-position information from left-right correlators, so the full operator is nonlocal.Source: §5.1 · pp. 32–33

5.2 Exploring behind the horizon or moving the horizon

  • The charges move matter within the Wheeler–DeWitt patch, and evolving with ˜E can bring a region previously behind the horizon outside it by moving the horizon.Moving matter and moving the boundaries are equivalent descriptions of this evolution.Source: §5.2 · p. 34

5.3 The inner horizon or Cauchy horizon

  • Bulk evolution cannot be extended past the inner horizon because standard AdS2 matter boundary conditions are not guaranteed beyond the physical boundaries.Irrelevant perturbations or double-trace operators can produce a singularity there; the discussed effects are suppressed in the scaling limit but may matter at finite βJ or finite throat length.Source: §5.3 · pp. 34–36

5.4 Moving operators into the bulk

  • The construction translates bulk operators relative to the boundary and yields mutually commuting operators at distinct geodesic distances, with full gravitational dressing to all orders in GN.The construction assumes a boundary-relative position and is described for the bulk theory, not as an operator construction in the holographic boundary theory.Source: §5.4 · pp. 37–38

6 Connection to SYK and other systems

In the specified large-N, low-temperature SYK scaling limit, the paper constructs exact SL(2) generators despite finite quantum effects of the boundary mode. It relates simpler approximate generators to operator size and out-of-time-order correlators, and compares their structure with Rindler-decomposed quantum field theory.

6.1 Generators in SYK

  • In the scaling limit, low-energy correlator operators can be raised to arbitrary powers or logged to construct the exact generators, while finite-N effects and 1/s corrections remain unaddressed.Topology-changing corrections of order e^-S0 also make the prescription non-perturbatively ill-defined.Source: §6.1 · pp. 38–40
  • In the SYK scaling limit, exact SL(2) generators persist even though boundary-mode quantum effects and thermal-excitation scrambling time remain finite.The paper identifies this limit with the JT-gravity scaling limit in which the boundary is taken far away; finite-N breaking is not analyzed.Source: §6.1 · pp. 38–39

6.2 Relation to “size”

  • The approximate generators connect size growth to corrections in out-of-time-order correlators, while the associated momentum is related to the time derivative of size.The size of an excitation grows as its location moves deeper into the throat, reaching order βJ before it enters the near-Rindler region.Source: §6.2 · pp. 41–42

6.3 Analogy to the Rindler (and AdS-Rindler) decomposition of a higher dimensional field theory

  • The approximate global-energy generator resembles a QFT generator for a system split in two: it combines left-only and right-only terms with operators coupling the two sides.The cross-boundary terms are crucial to the isometry that transfers information between the sides and correspond to the teleportation operator discussed in [15].Source: §6.3 · pp. 43–44
  • Unlike the higher-dimensional CFT comparison, where the corresponding E belongs to an exact conformal algebra, the two-dimensional gravity construction has only an approximate expression for that generator.The analogy is with splitting a QFT into two regions, where the full modular Hamiltonian acts as a boost and global time translation includes cross-region terms.Source: §6.3 · pp. 43–45

7 Discussion

The discussion explains how boundary correlations can serve as a measure of wormhole distance, while emphasizing assumptions and scope limits. It reviews the generators’ action and semiclassical interpretation, and introduces a gauged two-particle SO(3) analogy.

7.1 Measuring distance

  • Boundary correlations can indirectly measure wormhole distance, but the holographic boundary theory has no obvious distance operator, and the proposal’s validity depends on its assumptions.Averaging correlators over many fields protects the distance from changes to one field; the field number must be comparable to S0, and matter interactions must be suppressed as in the stated SYK limit.Source: §7.1 · pp. 45–46
  • The relative time shift between the thermofield-double sides behaves classically and is measurable through correlators, motivating the authors’ proposed “space-time superfluid” description.They relate changes in the time shift to evolution by Hl + Hr and temperature changes to an overall dilation that lengthens the wormhole.Source: §7.1 · p. 46

7.2 Conclusions and open questions

  • The three SL(2) generators move matter relative to fixed boundary positions, organize physical states into representations, and remain well defined beyond the scrambling time in the stated limit.They do not commute with the Hamiltonian; solvable boundary-position evolution permits time-independent charges to be defined.Source: §7.2 · p. 47
  • The discussion’s scaling limit keeps near-extremal entropy fixed as temperature becomes very small and extremal entropy large, while finite-throat and finite-βJ corrections remain open.The authors also identify finite-N topology-changing effects as excluded from the discussion.Source: §7.2 · pp. 47, 49
  • In the semiclassical limit, the generators act approximately as SL(2) transformations of boundary time and yield an operator-state mapping from operators inserted during Euclidean evolution by β/2 to wormhole states.The approximate expressions can be written using products of operators on the two boundaries and depend explicitly on the background temperature.Source: §7.2 · pp. 47–48
  • The approximate generators connect to the size operator, with its time derivative equal to bulk momentum; a more manifest construction from SYK UV operators remains an open goal.The authors also note that size saturates after the scrambling time and call for a better microscopic picture of that behavior.Source: §7.2 · p. 48

A SO(3) analogy

  • The SO(3) analogy models left and right particles constrained to a sphere, matter carrying angular momentum, and a gauge constraint setting total angular momentum to zero.The passage notes that the matter must have integer spin.Source: A SO(3) analogy · p. 49

A.1 Exact SO(3) algebra from two copies

The construction uses two copies and gauge-invariant operators to realize an exact physical SO(3) algebra. Its existence relies on vector operators beyond angular momentum alone.

A.1 Exact SO(3) algebra from two copies

  • The three physical charges form an SO(3) algebra, with the right vector replaced by its orthonormal Gram-Schmidt combination relative to the left vector.The commutators are checked directly in the two-body Hilbert space without imposing the gauge constraint.Source: §A.1 · p. 50
  • Constructing this algebra requires vector operators beyond angular momentum, so the left and right Hilbert spaces represent the Euclidean algebra and are infinite dimensional.The physical charges’ Casimir equals the Casimir of the gauge charges.Source: §A.1 · p. 50
  • The generators move the left and right particles relative to the matter, equivalently describing matter rotation when the particle positions are held fixed.This is the invariant interpretation of the generators’ action.Source: §A.1 · p. 50

A.2 Action on states/operators

Gauge-invariant matter vectors transform under the physical SO(3) charges and generate tensor operators in integer-spin representations. Acting repeatedly on a singlet organizes states into irreducible representations.

A.2 Action on states/operators

  • A gauge-invariant matter vector transforms as a vector under the physical charges and can generate operators in any integer-spin representation.The construction begins with a matter vector operator distinct from the angular momentum operator.Source: §A.2 · p. 51
  • Repeatedly applying the gauge-invariant matter vector to a singlet generates states that can be organized into physical-algebra irreducible representations.The thermofield double is given as an example of a singlet with ψ proportional to e^−βj²/4.Source: §A.2 · p. 51
  • The gauge theory permits only integer-spin matter states, excluding a spin-1/2 qubit in the two-particle Hilbert space.The stated reason is that this Hilbert space contains no half-integer states.Source: §A.2 · p. 51

B Canonical quantization of Schwarzian theory

Canonical quantization of the Schwarzian gives gauge charges that generate the expected transformations and satisfy the SL(2) algebra. The same formalism also shows that the approximate charge construction does not close into an algebra.

B Canonical quantization of Schwarzian theory

  • The canonically quantized Schwarzian charges generate the expected infinitesimal transformations of the boundary times and satisfy the SL(2)g commutation relations.The charges are written in terms of T, T′, and momenta, with momenta ordered to the right of coordinates.Source: B Canonical quantization of Schwarzian… · p. 52
  • The Schwarzian Hamiltonian is the Casimir of the gauge charges, and its equation of motion gives i[H, T] = T′.The canonical variables obey [T, T′] = 0, [T, p1] = i, and [T′, p2] = i.Source: B Canonical quantization of Schwarzian… · p. 52
  • The Schwarzian realizes the postulated Poincare algebra: the embedding-space components commute with one another and transform as a vector under the charges.The Hamiltonian expressions in T, T′, p1, and p2 are agnostic about whether evolution uses the uncoupled or coupled Hamiltonian.Source: B Canonical quantization of Schwarzian… · p. 53
  • The approximate charge construction fails to form a closed algebra even in the Schwarzian limit, partly because the eternal-wormhole spectrum contains a tensor factor that is not SL(2) invariant.The passage says this failure therefore occurs beyond the microscopic fermion description.Source: B Canonical quantization of Schwarzian… · p. 53

C Evaluating commutators perturbatively

The section checks the gauge-charge SL(2) commutators perturbatively using the linearized Schwarzian, then notes why extending the check is cumbersome.

C Evaluating commutators perturbatively

  • The perturbative check rewrites the linearized Schwarzian action using q = −ϵ′′ + ϵ and ghost field r = ϵ′′, following [3].The section focuses on gauge-charge commutators; physical-charge commutators can be checked analogously.Source: C Evaluating commutators perturbatively · p. 53
  • Quadratic-order expansion of the charges reproduces [Q−1, Q±] = ±iQ± and [Q+, Q−] = 2iQ−1.The quadratic terms are needed to reproduce the linear terms in Q± on the right-hand side.Source: C Evaluating commutators perturbatively · pp. 53–54
  • Higher-order checks require expanding the charges and modifying commutation relations for higher-order action terms, which quickly becomes cumbersome.This motivates the preceding section’s alternative construction.Source: C Evaluating commutators perturbatively · p. 54

D Spinor description of the boundary variables

A two-component spinor provides an alternative description of the null boundary vector, with a normalization that simplifies its evolution.

D Spinor description of the boundary variables

  • The boundary vector Xa can be constructed from a two-component spinor λα, with Pauli matrices supplying the vector components.The spinor inner product is antisymmetric, so a single λ cannot form a nonzero SL(2)-invariant and X² = 0 follows automatically.Source: D Spinor description of the boundary… · p. 54
  • Normalizing the spinor by ⟨˙λ, λ⟩ = constant gives ˙X² = −1 and Q·X = 1, yielding a simple evolution expression.The three generators also admit expressions in terms of left and right spinors.Source: D Spinor description of the boundary… · p. 54

D.1 Computation of the charges beyond the scrambling time

The spinor formalism tracks how an early-time perturbation changes a boundary variable, retaining the finite effect in a limit with ωe^−u0 fixed.

D.1 Computation of the charges beyond the scrambling time

  • For a perturbation inserted at zero time, its rescaling effect is negligible as ω → 0, leaving a small infinitesimal rotation as the main effect.The background uses Q̸r = σ3/2 with β = 2π.Source: §D.1 · p. 54
  • For an early insertion, the finite limit keeps ωe^−u0 fixed and gives λr = (1 + 2α̂σ−)λr(0).Here α̂ is proportional to ωe^−u0; the left spinor remains as before, and these values are inserted into the charge computation.Source: §D.1 · p. 55

E OTOC correlators and the expectation values of charges

The section relates OTOC calculations to expectation values of charges by extracting the distance and testing its classical saddle-point limit.

E OTOC correlators and the expectation values of charges

F Stiffer traversable wormholes

The section seeks to make boundary excitations heavy enough to isolate the SL(2) spectrum at low energies, using a modified effective potential. In the stated limit, tuning η/ρ can make the frequency large for 0 < ∆ < 1/2.

F Stiffer traversable wormholes

  • For fixed 0 < ∆ < 1/2, taking η/ρ ≪ 1 with small fixed ρ makes φ_m ≪ 0 and the frequency large, while satisfying the condition to neglect the Schwarzian term.In SYK, ˜∆ = 2/q and ∆ = 1/q.Source: F Stiffer traversable wormholes · p. 57
  • The WKB state count is large in the classical limit but suppressed by η/ρ^1/2, which must be small but not too small for the classical analysis.Source: F Stiffer traversable wormholes · pp. 57–58
  • For ∆ < 1/4, adding N particles of energy ∆ leaves T′ and the frequency nearly unchanged, unlike in the Maldacena-Qi wormhole.In the Maldacena-Qi case, the e^φ term typically dominates the e^(2φ) term and changes both T′ and ω.Source: F Stiffer traversable wormholes · p. 58
  • The first-order matter correction to the ground-state energy is T′E_bulk/N, and the correction is suppressed when ω is large.The second-order correction is obtained by correcting δφ by −T′ * E/(NV′′).Source: F Stiffer traversable wormholes · p. 58
  • Further work could examine the model’s thermodynamics and its behavior at higher temperatures in large-q SYK.Source: F Stiffer traversable wormholes · p. 58

G Measuring distance using a single free field

The section tests whether a single free bulk field can provide a proxy for the distance between the two sides. The operator −log(φ_lφ_r) appears to track distance for small numbers of quanta, with limits as particle number grows.

G Measuring distance using a single free field

  • For a single free bulk field, the proposed distance operator is d_∆ ∼ −log(φ_lφ_r), defined using higher-point correlators continued to m = 0.Source: G Measuring distance using a single… · p. 58
  • With bulk particles present, the correlator includes desired “good” diagrams and “bad” contractions in which φ_l contracts with matter created by φ_b or φ_t.The factor of 2 comes from the choice of connecting φ_l to φ_t or φ_b; the matter operators should be regulated, for instance by point-splitting.Source: G Measuring distance using a single… · pp. 58–59
  • At large distances or large ∆, the leading contribution is the first term, which is precisely the distance.Source: G Measuring distance using a single… · p. 59
  • For small numbers of quanta, −log(φ_lφ_r) seems to be a good distance proxy, but its particle-number dependence may limit its accuracy.With many quanta, including a coherent state, its expectation value may instead be set by classical field values that need not track distance.Source: G Measuring distance using a single… · p. 59
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