- Reality, Amplitude Fisher Geometry, and the Ising Configuration Graph
3.1 Reality of the Static Ising Ground State
The spectral representation developed in Section 2 becomes particularly useful in the static Ising problem because the relevant ground state can be chosen real. Consider again the finite-chain Hamiltonian
H(h) = −JΣᵢσᶻᵢσᶻᵢ₊₁ − hₓΣᵢσˣᵢ − hΣᵢσᶻᵢ,
and define
M_z = Σᵢσᶻᵢ,
so that
∂ₕH = −M_z.
In the σᶻ configuration basis, the off-diagonal matrix elements generated by −hₓΣᵢσˣᵢ are non-positive for hₓ>0. For a connected finite chain with a nondegenerate ground state, the Perron–Frobenius structure therefore permits the ground-state coefficients to be chosen strictly real and nonnegative. Write
|Ψ_h⟩ = Σ_s R_h(s)|s⟩,
with
R_h(s) ≥ 0,
Σ_s R_h(s)² = 1.
Here s denotes a complete Ising configuration,
s = (s₁,…,s_N), sᵢ=±1.
The corresponding Born probability distribution is
p_h(s) = R_h(s)².
This reality property has two related geometric consequences. First, along the static h-family there is a gauge in which the Berry connection vanishes locally. Differentiating the normalization condition gives
0 = ½∂ₕ⟨Ψ_h|Ψ_h⟩ = ⟨Ψ_h|∂ₕΨ_h⟩,
because both vectors are real. Thus ∂ₕ|Ψ_h⟩ is already horizontal to the ground-state ray in this gauge.
Second, the quantum metric receives no independent phase contribution along this one-parameter real family. The complete variation occurs through the amplitudes R_h(s), and the Fubini–Study metric becomes
g_hh = Σ_s [∂ₕR_h(s)]².
This is the same g_hh whose spectral representation was obtained in Section 2,
g_hh = Σₙ≠₀ |⟨n|M_z|0⟩|²/(Eₙ−E₀)².
The equality of these two expressions gives an exact bridge between the amplitude description in the Ising configuration basis and the excitation-spectrum description of the interacting ground-state tangent.
This statement does not imply that quantum geometry is generally reducible to amplitude geometry. It holds here because the static Hamiltonian admits a real nondegenerate ground-state family. Once a second physical parameter generates nontrivial phase motion, as in Section 4, the antisymmetric projective structure again becomes active.
3.2 Quantum Fisher Information as Amplitude Fisher Information
The reality condition gives a particularly simple information-geometric identity. Define the h-deformation score of the Born distribution by
q_h(s) ≡ ∂ₕ ln p_h(s).
Since p_h=R_h²,
q_h(s) = 2∂ₕR_h(s)/R_h(s)
wherever R_h(s)≠0. Consequently,
Σ_s p_h(s)q_h(s)² = 4Σ_s[∂ₕR_h(s)]².
The classical Fisher information of the h-indexed probability family is therefore
F_amp(h) ≡ Σ_s p_h(s)[∂ₕlnp_h(s)]² = 4g_hh.
For a pure state,
F_Q^(h) = 4g_hh.
Hence
F_Q^(h) = F_amp(h).
For the real static Ising ground-state family, the pure-state quantum Fisher information for the longitudinal deformation is therefore exactly the classical Fisher information of the Born-amplitude distribution.
This identity is useful because it removes an otherwise ambiguous step between amplitude geometry and conventional many-body quantum geometry. The same physical response can be expressed either as
F_amp(h) = Σ_s p_h(s)q_h(s)²
or spectrally as
F_Q^(h) = 4Σₙ≠₀ |⟨n|M_z|0⟩|²/(Eₙ−E₀)².
Thus the E₈ excitation spectrum and the corresponding magnetization form factors provide the spectral decomposition of a quantity that, in this real representation, is literally amplitude Fisher information.
The type of Fisher information involved must nevertheless remain explicit. The index h labels a family of Hamiltonians and ground states. F_amp(h) measures how the probability distribution p_h(s) changes when the longitudinal coupling changes. It is therefore a deformation Fisher information on the state family.
By contrast, the sPNP tensor Fᴿ,(Q₀)_IJ introduced later has indices I,J belonging to directions inside a physical configuration manifold. Its local precursor is constructed from derivatives such as ∇_Ilnρ(X), not ∂ₕlnp_h. The exact identity F_Q=F_amp therefore supplies an important amplitude benchmark, but it does not turn the external parameter h into a relational configuration coordinate.
This separation will be maintained throughout the paper. Section 4 uses h together with a rotation parameter θ to construct a genuine two-dimensional projective state manifold. Later sections instead construct Ψ(X) directly on a coherent-state configuration manifold, where the indices of the sPNP amplitude geometry are intrinsic configuration directions.
3.3 The Ising Configuration Graph
There is a second, independent sense in which amplitude-gradient structure is already present in the microscopic Ising Hamiltonian.
The σᶻ configurations form the vertices of an N-dimensional hypercube graph. Two configurations s and s′ are connected by an edge when they differ by a single spin flip. Let sⁱ denote the configuration obtained from s by reversing spin i.
The transverse-field operator acts precisely along these graph edges:
σᵢˣ|s⟩ = |sⁱ⟩.
For a normalized real state
|Ψ⟩ = Σ_s R_s|s⟩,
the transverse-field expectation value is
⟨Hₓ⟩ = −hₓΣ_{s,i}R_sR_{sⁱ}.
Every undirected edge is counted twice in the sum over s and i, so
⟨Hₓ⟩ = −2hₓΣ_⟨ss′⟩R_sR_s′.
Using
(R_s−R_s′)² = R_s² + R_s′² − 2R_sR_s′,
and the fact that the N-dimensional hypercube is N-regular,
Σ_⟨ss′⟩(R_s²+R_s′²) = NΣ_sR_s² = N.
Therefore
Σ_⟨ss′⟩(R_s−R_s′)² = N − 2Σ_⟨ss′⟩R_sR_s′,
which gives the exact identity
⟨Hₓ⟩ = hₓΣ_⟨ss′⟩(R_s−R_s′)² − Nhₓ.
With p_s=R_s², define the graph amplitude functional
ℱ_G[p] ≡ 4Σ_⟨ss′⟩(√p_s−√p_s′)².
Then
⟨Hₓ⟩ = (hₓ/4)ℱ_G[p] − Nhₓ.
Writing the diagonal Ising and longitudinal-field contributions as V(s), the complete expectation value becomes
⟨H⟩ = Σ_s p_sV(s) + (hₓ/4)ℱ_G[p] − Nhₓ.
The functional ℱ_G is a graph Dirichlet/Hellinger amplitude energy and is the natural discrete analogue of the continuous identity
F[ρ] = ∫ρ|∇lnρ|² = 4∫|∇√ρ|².
The transverse-field kinetic term can therefore be represented exactly as a Dirichlet energy of the square-root probability amplitude over the Ising configuration graph.
This is a useful microscopic fact. The Hamiltonian does not merely assign energies to abstract Hilbert-space coefficients: it penalizes variations of the amplitude between configurations connected by elementary spin flips.
3.4 Relation to the Later sPNP Geometry
The graph identity shows that amplitude variation is not imposed on the Ising system solely as a later interpretive overlay. The microscopic Hamiltonian itself assigns an energetic cost to changes in R between configurations connected by the graph.
It is nevertheless important not to identify ℱ_G directly with the full sPNP finite-resolution geometry.
First, ℱ_G is a scalar global Dirichlet functional. It sums squared amplitude differences over all graph edges. The sPNP construction instead retains directional tensor information before contraction and integration. Its finite-resolution tensor Fᴿ,(Q₀)_IJ is intended to determine which physical configuration directions remain geometrically resolvable and how their amplitude gradients are organized.
Second, the hypercube graph is a discrete configuration representation in the σᶻ basis, whereas the later construction uses the continuous coherent-state manifold "(CP¹)^N". These representations describe the same finite spin Hilbert space in different ways, but their geometric structures should not be identified without an explicit map.
The two exact results of this section nevertheless complement one another.
The deformation identity
F_Q^(h)=F_amp(h)
shows that variation of the real ground-state amplitude with the longitudinal coupling contains the complete longitudinal fidelity susceptibility.
The graph identity
⟨Hₓ⟩ = (hₓ/4)ℱ_G[p] − Nhₓ
shows that variation of the amplitude across spin configurations contributes directly to the microscopic transverse-field energy.
They concern different derivatives. The first probes how one ground state changes into another as h is varied. The second probes how the amplitude varies among configurations within a fixed state. Keeping those operations separate is essential, but their coexistence is precisely what makes the Ising system useful: it possesses both a well-controlled interacting spectral response and a natural configuration-space amplitude structure.
The static real family alone contains no nontrivial local PhaSe curvature along h. To compare symmetric and antisymmetric projective geometry, a second physical deformation is needed. The rotating-field construction supplies that benchmark.
- Rotating-Field E₈ as a Parameter-Space Kähler Benchmark
4.1 A Two-Parameter E₈ State Family
A useful extension of the static Ising problem is obtained by allowing the transverse field to rotate in the x–y plane. One may write the instantaneous Hamiltonian, with J=1 and ℏ=1, as
H(h,θ) = −Σᵢ[σᶻᵢσᶻᵢ₊₁ + cosθ σˣᵢ − sinθ σʸᵢ + hσᶻᵢ],
where θ may be taken as a slowly varying rotation angle. For small longitudinal field and slow rotation, the instantaneous low-energy theory remains tied to the E₈ magnetic Ising scaling regime, while the motion in θ introduces a nontrivial Berry and quantum-geometric structure.
Define
U(θ) = exp(iθM_z/2),
with
M_z = Σᵢσᶻᵢ.
Up to the orientation convention chosen for θ,
H(h,θ) = U(θ)H(h,0)U(θ)†.
An instantaneous normalized ground state can therefore be chosen as
|0(h,θ)⟩ = U(θ)|0(h,0)⟩.
The pair (h,θ) defines a real two-dimensional surface in projective Hilbert space.
Its two coordinate directions have different physical meanings. Changing h deforms the interacting ground state through the longitudinal perturbation. Changing θ moves the state unitarily through the generator M_z. The resulting two-plane therefore supplies a controlled system in which a response direction and a phase-generating direction can be compared within the ordinary quantum geometric tensor.
At this point the geometry remains parameter-space geometry. Neither h nor θ is a coordinate of the relational configuration manifold used later in sPNP. The calculation instead provides an exact metric–symplectic benchmark whose relation to the intrinsic sPNP construction can subsequently be examined.
4.2 Magnetic and Rotational Tangents
Evaluate the geometry at θ=0. Define
fₙ ≡ ⟨n|M_z|0⟩,
Δₙ ≡ Eₙ−E₀.
For the magnetic direction, first-order perturbation theory gives
|T_h⟩ = P₀⊥∂ₕ|0⟩ = Σₙ≠₀(fₙ/Δₙ)|n⟩.
For the rotational direction,
∂θ|0(h,θ)⟩|{θ=0} = (i/2)M_z|0⟩.
After removing the component parallel to |0⟩,
|T_θ⟩ = (i/2)(M_z−⟨M_z⟩)|0⟩
and therefore
|T_θ⟩ = (i/2)Σₙ≠₀fₙ|n⟩.
The two tangent vectors resolve into the same magnetization-coupled excitation directions but with different spectral weights:
T_h: fₙ/Δₙ,
T_θ: ifₙ/2.
The magnetic tangent emphasizes lower-energy excitations through the inverse gap. The rotational tangent samples the same excitation directions without that denominator.
Define
C₀ ≡ Σₙ≠₀|fₙ|²,
C₁ ≡ Σₙ≠₀|fₙ|²/Δₙ,
C₂ ≡ Σₙ≠₀|fₙ|²/Δₙ².
These are successive moments of the same positive magnetization spectral measure.
C₀ is the connected equal-time magnetization variance,
C₀ = ⟨M_z²⟩−⟨M_z⟩².
C₂ is the magnetic Fubini–Study metric,
C₂ = g_hh.
C₁ is the intermediate inverse-gap moment entering the static response, subject to the conventional factor used in defining the susceptibility.
This common spectral origin is important: the metric and symplectic geometry derived below are not being assembled from unrelated observables. Both tangents sample the same physical excitation channels but weight their energy scales differently.
4.3 The Induced Quantum Geometric Tensor
The tangent norms are
g_hh = ⟨T_h|T_h⟩ = C₂,
and
g_θθ = ⟨T_θ|T_θ⟩ = C₀/4.
Their inner product is
⟨T_h|T_θ⟩ = (i/2)C₁
for the orientation convention chosen above. It is purely imaginary, so
g_hθ = Re⟨T_h|T_θ⟩ = 0.
Using the convention
ω_ab = −2 Im⟨T_a|T_b⟩,
one has
ω_hθ = −C₁.
Its sign reverses if the orientation of θ is reversed and therefore does not affect the compatibility eigenvalue. We may write simply
|ω_hθ| = C₁.
The metric and two-form on the (h,θ) surface are therefore
g = diag(C₂, C₀/4),
and
ω = [[0, ±C₁], [∓C₁, 0]].
Define
A_param = g⁻¹ω.
Direct multiplication gives
−A_param² = [4C₁²/(C₀C₂)]I₂.
The two-dimensional parameter plane therefore possesses the single compatibility scale
κ_param = 2C₁/√(C₀C₂).
This is an exact property of the induced quantum geometric tensor.
With the normalization used here, a real projective two-plane preserved by the ambient complex structure has κ=2. We may consequently introduce its Kähler angle Θ_K through
cosΘ_K = κ_param/2,
so that
κ_param = 2cosΘ_K.
Thus
cosΘ_K = C₁/√(C₀C₂).
The ratio is not merely a convenient spectral diagnostic. It is the metric–symplectic compatibility eigenvalue of the actual two-plane generated by T_h and T_θ in projective Hilbert space.
Its domain must nevertheless remain explicit. A_param acts on the parameter-space plane spanned by T_h and T_θ. It is not the later sPNP operator A=Ĝ_E⁻¹ω_E acting on intrinsic relational configuration directions.
4.4 Spectral Dispersion and the Kähler Angle
Positivity gives
C₁² ≤ C₀C₂
by Cauchy–Schwarz, and therefore
0 ≤ κ_param ≤ 2.
The Cauchy–Schwarz origin of the bound is exactly consistent with its geometric interpretation. On a real two-plane embedded in a Kähler manifold, the magnitude of the restricted symplectic area cannot exceed the corresponding metric area.
The spectral meaning becomes transparent by defining
wₙ ≡ |fₙ|²/C₀,
with
Σₙwₙ=1,
and
Xₙ ≡ 1/Δₙ.
Then
C₁/C₀ = ⟨X⟩_w,
C₂/C₀ = ⟨X²⟩_w,
and hence
κ_param/2 = ⟨X⟩_w/√⟨X²⟩_w.
If
CV_w(X)² = [⟨X²⟩_w−⟨X⟩_w²]/⟨X⟩_w²,
then
κ_param = 2/√[1+CV_w(1/Δ)²].
The Kähler angle therefore measures the spread of inverse excitation scales sampled by the magnetization spectral measure.
If all relevant spectral weight lies at one excitation gap, or within an exactly degenerate set of gaps,
CV_w(1/Δ)=0
and
κ_param=2.
In this case
T_θ ∝ iT_h,
so the two-plane is preserved by the ambient projective complex structure.
When appreciable spectral weight occupies unequal gaps, the inverse-gap factors in T_h change the relative weights of its excitation components. T_h and the complex rotation of T_θ then cease to be exactly aligned, and
κ_param<2.
This result is not unique to E₈. The same construction can be made in other quantum systems with an analogous generator. E₈ is valuable because the relevant interacting spectrum is unusually organized and its form factors are unusually accessible.
4.5 Amplitude Representation of the Same Compatibility Invariant
The reality of the static Ising ground state gives the same parameter-space invariant a direct amplitude representation.
Let
p_h(s)=R_h(s)²
and
q_h(s)=∂ₕlnp_h(s).
From Section 3,
⟨q_h²⟩_p = 4C₂.
Let
M(s)=Σᵢsᵢ
be the magnetization of configuration s. Then
Var_p(M)=C₀.
Normalization gives
⟨q_h⟩_p=0.
Its covariance with M is
Cov_p(M,q_h)
= Σ_s p_h(s)[M(s)−⟨M⟩]q_h(s).
Using q_h=2∂ₕR/R,
Cov_p(M,q_h)=2C₁.
Therefore
Corr_p(M,q_h)
= C₁/√(C₀C₂),
and hence
κ_param = 2|Corr_p(M,q_h)|.
This is an exact alternative expression for the same Kähler-angle invariant.
The amplitude score q_h measures how the configuration probabilities respond when the longitudinal coupling changes. M(s) is the configuration observable that generates the θ rotation. Consequently, κ_param/2 is the ordinary correlation coefficient between the amplitude response and the phase-generating observable.
Perfect parameter-space compatibility,
κ_param=2,
requires
q_h(s) ∝ M(s)−⟨M⟩
almost everywhere with respect to p_h.
In that special case, the amplitude response produced by changing h and the configuration dependence of the θ generator coincide up to scaling and multiplication by the ambient complex structure.
For a genuinely interacting spectrum containing multiple unequal excitation gaps, that proportionality generally fails.
This provides a useful bridge to the later sPNP problem: amplitude response and PhaSe-generating structure can be compared quantitatively. The calculation remains, however, a property of an externally parameterized family of states. The relational construction requires both structures to be defined directly on intrinsic configuration directions X.
4.6 The E₈ Scaling Value as a Separate Spectral Problem
The quantity
κ_param = 2C₁/√(C₀C₂)
is dimensionless. An overall normalization of M_z multiplies C₀, C₁ and C₂ by the same common factor and cancels. A common rescaling of all excitation energies also cancels from the ratio.
The E₈ scaling theory can therefore possess a well-defined continuum value of this parameter-space Kähler invariant.
Determining that value accurately is a separate calculation. It requires a consistent treatment of one-particle poles, multiparticle continua, finite-volume normalization, lattice-to-continuum normalization, and the approach to the magnetic E₈ scaling regime.
The existence and geometric meaning of κ_param do not depend on completing that numerical extrapolation. No specific continuum value is assumed here. A later calculation can determine it using either sufficiently controlled finite-volume methods or a complete spectral/form-factor analysis.
This is preferable to allowing an uncertain numerical estimate to carry conceptual weight that already follows from the exact geometry.
4.7 Boundary Between the Parameter-Space Benchmark and the sPNP Target
The algebraic similarity between κ_param and the compatibility construction of the parent sPNP theory is useful, but the two operators act on different spaces.
Here,
A_param = g_param⁻¹ω_param
acts on
span{T_h,T_θ} ⊂ T_[Ψ]P(ℋ).
The coordinates h and θ label external deformations of the Hamiltonian or instantaneous state.
The relational target instead has
A_rel = Ĝ_E⁻¹ω_E
acting on
E ⊂ T_XQ_rel.
The complete metric Ĝ may contain a relational kinetic sector, the finite-resolution amplitude tensor Fᴿ,(Q₀), the native projective metric and other admissible symmetric contributions. The PhaSe two-form acts on the same intrinsic configuration directions.
The two constructions should therefore be kept in parallel rather than identified:
parameter-space benchmark:
κ_param = 2C₁/√(C₀C₂),
relational target:
−(Ĝ_E⁻¹ω_E)²eₐ = κₐ²eₐ.
The parameter-space calculation establishes that the E₈ Ising system already possesses a meaningful metric–symplectic compatibility problem in ordinary projective quantum geometry. The next task is to construct a configuration arena on which the sPNP amplitude and PhaSe structures inhabit the same tangent bundle.
The spin-coherent lift developed below provides such a candidate. On "(CP¹)^N", the magnetization becomes a moment-map function, the native Berry curvature supplies a genuine configuration-space two-form, and the many-body state becomes a coherent-state wavefunction Ψ(X). The amplitude-generated tensor Fᴿ,(Q₀), the complete metric Ĝ and the native PhaSe form ω can then all be defined intrinsically on the same manifold.
Section 4 is therefore the final parameter-space benchmark rather than the final sPNP result. It establishes an exact spectral and Kähler-geometric reference point while making clear what additional construction is required before the E₈ system can test the actual relational compatibility spectrum.
