A conformal metric configuration is a specification of the conformal factor g(x) ∈ℝ at each of L spatial
sites. A two-frame metric pair (g(0),g(1)) is encoded as ψ(x) = g(0)(x) + ig(1)(x), normalised to unit norm.
This is a toy model capturing the qualitative structure of the Riemann decomposition; the full rank-4
tensor treatment is an open problem (Remark 1).
Theorem 2 (Three-way curvature decomposition). For any conformal metric pair ψ = g(0) +
ig(1), the density matrix ρ = |ψ⟩⟨ψ| decomposes exactly as
where
| ρRicci | = diag(ρ), | |
|
| ρtidal | =  (ρW + ρW ⊤), | |
|
| ρgraviton | =  (ρW −ρW ⊤), | | |
and ρW = ρ − diag(ρ). These satisfy: Tr(ρRicci) = 1, Tr(ρtidal) = 0, Tr(ρgraviton) = 0. Furthermore:
- ρgraviton = 0 if and only if g(0) ∝g(1) (static configuration, no wave);
- ρtidal = 0 if and only if ψ is a single-site state (no spatial extent).
Proof. The decomposition is exact by construction. Tr(diag(ρ)) = ∥ψ∥2 = 1. For any matrix M,
Tr(M −M⊤) = 0; hence Tr(ρgraviton) = 0. Similarly Tr(ρW ) = 0 (off-diagonal entries only),
so Tr(ρtidal) = Tr(ρW ) = 0. ρgraviton = 0 iff ρW = ρW ⊤, which holds iff ρxy = ρyx for all
x≠y, i.e. ψxψy∗ = ψyψx∗, i.e. ψ is real up to a global phase, i.e. g(0) ∝ g(1). If ψ = αej, then
ρxy = |α|2δxjδyj is diagonal, giving ρW = 0 and hence ρtidal = 0. □
| Compression part | Riemann analogue | Physical role |
| ρRicci | Ricci tensor Rμν | Local curvature; source term |
| ρtidal | Coulomb Weyl | Tidal forces; static |
| ρgraviton | Radiative Weyl | Gravitational waves; propagating |
The physical identification is an analogy in the toy model, not a proof that ρRicci is the Ricci tensor of
GR. The tracelessness of both Weyl components emerges from the trace-zero property of off-diagonal
density matrices, mirroring the tracelessness of the Weyl tensor in GR, which follows there from the
symmetries of the Riemann tensor.
