2026-10-16
Split geometry and positive geometry: correspondence and computational simplicity
One output of an interferometer can become completely dark even though both paths inside the apparatus remain open. To calculate this result, knowing the fraction of the intensity travelling along each path is not enough: their relative phase must be retained. This example lets us compare two geometric descriptions: a split state with complex components and the positive geometry of observable weights. In this article, we construct an explicit mapping between them, express canonical forms in terms of splitting parameters, and show how operator composition simplifies the calculation of a compatible sequence of transitions.
The mathematical foundation is provided by the constructions discussed in “From Euler’s Formula to Split Geometry” and “Origins of the Idempotent Basis”. Only the necessary definitions are used here; physical applications are discussed after the algebraic correspondences have been established.
1. What Is Positive Geometry?
The theory of positive geometries studies an oriented real semialgebraic region in an algebraic space together with its associated canonical differential form [1]. The region is defined by polynomial equalities and inequalities. The canonical form has only logarithmic singularities on its boundaries; its residue on each boundary is the canonical form of that boundary with the induced orientation. In dimension zero, the normalization is determined by the sign assigned to the point.
The simplest example is an interval:
\[\tag{1} \Delta_1=\{x\in\mathbb R:0\leq x\leq1\},\qquad \Omega_{\Delta_1}=\frac{dx}{x(1-x)}=d\ln\frac{x}{1-x}. \] The form is regular in the interior of the interval and has simple poles at its endpoints. For a triangle described by weight coordinates, we obtain:
\[\tag{2} w_1+w_2+w_3=1,\qquad w_r\geq0,\qquad \Omega_{\Delta_2}=\frac{dw_2\wedge dw_3}{w_1w_2w_3}. \] Here, the wedge symbol denotes the exterior product of differentials. The signs of the forms depend on orientation. The edges of the triangle correspond to one weight vanishing, and its vertices to two weights vanishing. These are examples of the positive geometry of a simplex; the general class is much broader. The word “positive” does not mean positive curvature.
The prominence of this field in physics is associated, in particular, with the amplituhedron [2]: a special geometric construction used to describe scattering amplitudes in planar N=4 supersymmetric Yang–Mills theory. This illustrates how a physical calculation can be expressed through a geometric object and its form. Applying the approach to another theory requires a separate correspondence between the geometry and physical quantities.
2. Where the Complexity Arises
An interval and a triangle are simple. In a higher-dimensional problem, one must first choose the space, specify the constraints, identify the admissible boundaries, and construct a form consistent with them. In constructions based on positive Grassmannians, positivity is expressed through conditions on matrix minors. Computing forms requires accounting for the boundary structure and ensuring consistency between different decompositions of the region [1].
It is useful to distinguish three tasks: describing an admissible state, finding a convenient parametrization, and calculating a physical operation. Even explicit nonnegative coordinates do not automatically specify phases or a transition law. We therefore examine whether a more detailed state can serve as the starting point for a space of weights, with the positive geometry obtained as its image.
The comparison is specific: a split state is compared with the positive simplex of its channel weights. We do not assume that all positive geometries admit such a representation or claim a general computational advantage over them.
3. Idempotent Channels and the Phase Operator
Consider the complex extension of the split algebra. The hyperbolic unit and the two idempotents satisfy:
\[\tag{3} \j^2=1,\qquad \ep=\frac{1+\j}{2},\qquad \em=\frac{1-\j}{2}. \] \[\tag{4} \ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0,\qquad \ep+\em=1. \] On the chosen principal branches, the phase state is specified by the operator:
\[\tag{5} J(a,b)=\j^a(-\j)^b=\ep e^{i\pi b}+\em e^{i\pi a}. \] In the mathematical treatment, a and b are real phase parameters. In the Wave Electricity concept, a describes the internal state and b describes external motion. When the two channels below are interpreted as alternative paths in an interferometer, the physical correspondence between these phases and the optical paths must be established separately.
The orthogonality of the idempotents immediately gives the composition law:
\[\tag{6} J(a,b)J(a',b')=J(a+a',b+b'). \] Multiplication by a second operator of the same type adds the phases in the corresponding channels. The moduli of both coefficients remain equal to one. To allow a variable amplitude distribution, we therefore introduce an additional factor.
4. Amplitude Distribution and Choice of Norm
Define:
\[\tag{7} K(\theta)=\ep\cos\theta+\em\sin\theta,\qquad 0\leq\theta\leq\frac\pi2. \] The product contains both amplitudes and phases:
\[\tag{8} \Psi=K(\theta)J(a,b)=\ep\cos\theta\,e^{i\pi b}+\em\sin\theta\,e^{i\pi a}. \] Explicitly define a positive-definite scalar norm of the coefficients. If \(Q=z_+\ep+z_-\em\), then:
\[\tag{9} \|Q\|_2^2=|z_+|^2+|z_-|^2,\qquad \|\Psi\|_2^2=\cos^2\theta+\sin^2\theta=1. \] This norm differs from the algebraic product formed by complex-conjugating the coefficients:
\[\tag{10} J\overline J=1,\qquad \Psi\overline\Psi=\ep\cos^2\theta+\em\sin^2\theta,\qquad \|J\|_2^2=2. \] The different normalizations are thus kept distinct. The operator K specifies a family of normalized states; it is not asserted to be a unitary transformation of an arbitrary input. A law describing the physical preparation or evolution of such a distribution must be supplied separately.
5. The Mapping to Positive Coordinates
For nonzero Q, define the weights of its components:
\[\tag{11} w_+=\frac{|z_+|^2}{|z_+|^2+|z_-|^2},\qquad w_-=\frac{|z_-|^2}{|z_+|^2+|z_-|^2}. \] By construction, they are nonnegative and sum to one. For the state (8):
\[\tag{12} w_+=\cos^2\theta,\qquad w_-=\sin^2\theta,\qquad w_++w_-=1. \] We have thus introduced an explicit mapping from the state space to the positive interval. Its generalization requires no new principle. Let a complete system of idempotents satisfy:
\[\tag{13} P_rP_s=\delta_{rs}P_r,\qquad \sum_{r=1}^{N}P_r=1,\qquad Q=\sum_{r=1}^{N}z_rP_r. \] Declaring these channels orthonormal under the coefficient inner product gives:
\[\tag{14} \boxed{\mathcal W(Q)_r=\frac{|z_r|^2}{\sum_s|z_s|^2},\qquad \mathcal W(Q)\in\Delta_{N-1},\qquad \Delta_{N-1}=\left\{w_r\geq0\um_rw_r=1\right\}.} \] Proposition. The mapping (14) is onto the entire simplex. Indeed, for any point in the simplex, one can choose a normalized state:
\[\tag{15} \Psi=\sum_{r=1}^{N}\sqrt{w_r}\,e^{i\phi_r}P_r,\qquad \|\Psi\|_2^2=1. \] Every choice of phases gives the same weights. Multiplying Q by a common nonzero complex scalar also leaves (14) unchanged. The mapping therefore discards the overall scale and phase information; reconstruction is not unique. When a weight is zero, the phase of its coefficient is undefined.
Mathematically, this mapping involves complex conjugation and is real-algebraic in the real and imaginary parts of the coefficients. By itself, it neither establishes a particular morphism of positive geometries nor proves a rule for transporting canonical forms. Below, the forms are rewritten by direct substitution.
The weights acquire the meaning of probabilities only when the quadratic detection rule is adopted; interpreting them as energy fractions requires an appropriate energy metric. The algebraic construction (14) does not replace these physical assumptions.
| Split description | Positive geometry of weights |
|---|---|
| Two channels | Interval |
| Three channels | Triangle |
| Four channels | Tetrahedron |
| A coefficient vanishes | The image lies on the corresponding face |
| Only the phases change | The weight point remains unchanged in a fixed basis |
6. The Canonical Form in Terms of the Splitting Angle
In the interior of the positive interval, set:
\[\tag{16} x=w_-=\sin^2\theta,\qquad 1-x=w_+=\cos^2\theta,\qquad 0<\theta<\frac\pi2. \] Differentiating and substituting into (1) gives:
\[\tag{17} dx=2\sin\theta\cos\theta\,d\theta,\qquad \Omega_{\Delta_1}=\frac{2\,d\theta}{\sin\theta\cos\theta}=d\ln\tan^2\theta. \] \[\tag{18} \boxed{\Omega_{\Delta_1}=d\ln\frac{w_-}{w_+}.} \] Thus, the canonical form expresses the differential change in the logarithm of the weight ratio. It diverges when either weight vanishes. The form is not a normalized probability density: its integral diverges at the boundaries. A pole does not automatically imply infinite physical energy either.
The form is rational in the coordinate x and trigonometric in the angular coordinate. This is a reparametrization of a known form, rather than a new canonical form derived solely from the existence of an angle.
7. A Hierarchy of Splittings and Factorization of the Form
Suppose the whole is first distributed with weights 1−u and u. The second branch is then split with relative weights 1−v and v. The resulting weights are:
\[\tag{19} w_1=1-u,\qquad w_2=u(1-v),\qquad w_3=uv,\qquad 0\leq u,v\leq1. \] Their sum is identically one. In the interior of the triangle, the inverse correspondence is:
\[\tag{20} u=w_2+w_3,\qquad v=\frac{w_3}{w_2+w_3}. \] At the boundary u=0, both of the last two channels vanish and the parameter v loses its meaning. Thus, the parameter square covers the triangle, but its edge u=0 collapses to a vertex. This is not a globally one-to-one correspondence of all boundary points.
To calculate the form, it suffices to use:
\[\tag{21} dw_2=(1-v)du-u\,dv,\qquad dw_3=v\,du+u\,dv,\qquad dw_2\wedge dw_3=u\,du\wedge dv. \] Substitution into (2) gives:
\[\tag{22} \boxed{\Omega_{\Delta_2}=\frac{du}{u(1-u)}\wedge\frac{dv}{v(1-v)}.} \] Each independent binary distribution in this tree contributes its own logarithmic factor. For a full binary tree with N leaves, the analogous rule in the interior follows by induction: partitioning the leaves into two groups introduces one parameter for their total weights, and the same construction is repeated within each group. Orientation fixes the overall sign.
Hierarchical variables therefore simplify the calculation of the simplex form. If parameters at different nodes are related, for example by self-similarity, the number of independent variables decreases. In that case, one cannot automatically identify the resulting surface with the full simplex or assign the same higher-degree form to it.
8. Interference: Why Retain the Phases?
Consider a linear operation on the column of state coefficients. Multiplication by the original idempotents and by J is represented by the matrices:
\[\tag{23} \rho(\ep)=\begin{pmatrix}1&\&0\end{pmatrix},\qquad \rho(\em)=\begin{pmatrix}0&\&1\end{pmatrix},\qquad \rho(J)=\begin{pmatrix}e^{i\pi b}&\&e^{i\pi a}\end{pmatrix}. \] Denote the coefficient column by a lowercase letter:
\[\tag{24} \psi=\begin{pmatrix}z_\_-\end{pmatrix},\qquad \psi^\dagger\psi=\|\Psi\|_2^2. \] Introduce a norm-preserving mixing operator:
\[\tag{25} H=\frac1{\sqrt2}\begin{pmatrix}1&\&-1\end{pmatrix},\qquad H^\dagger H=I,\qquad H^2=I. \] It transfers amplitudes between channels. Unlike the diagonal operators (23), H is not multiplication by an element of the original commutative algebra. Such coupling between channels is discussed in “The Splitting Principle: Hierarchy, Horizontal Couplings, and Experimental Verification”. Here, H is explicitly specified as an additional admissible operation.
Suppose the first channel is initially occupied. The sequence of splitting, phase evolution, and recombination is:
\[\tag{26} \psi_0=\begin{pmatrix}1\\ 0\end{pmatrix},\qquad \psi_{\mathrm{out}}=H\rho(J(a,b))H\psi_0. \] Introduce the global phase and the phase difference:
\[\tag{27} \chi=\frac{\pi(a+b)}2,\qquad \delta=\pi(b-a). \] Direct multiplication gives:
\[\tag{28} \psi_{\mathrm{out}}=\frac12\begin{pmatrix}e^{i\pi b}+e^{i\pi a\^{i\pi b}-e^{i\pi a}\end{pmatrix}=e^{i\chi}\begin{pmatrix}\cos(\delta/2\\sin(\delta/2)\end{pmatrix}. \] By the quadratic detection rule:
\[\tag{29} \boxed{w_+^{\mathrm{out}}=\cos^2\frac\delta2,\qquad w_-^{\mathrm{out}}=\sin^2\frac\delta2,\qquad \delta=\pi(b-a).} \] This is the standard law of two-channel interference. It can be implemented by a Hadamard transformation–phase operation–Hadamard transformation sequence in a quantum circuit [3]. The optical analogue uses coherent paths, a controlled phase difference, and compatible beam splitters; their matrices depend on the phase convention. The expression above uses one such norm-preserving convention.
Reproducing a known law demonstrates that the chosen representation and operators work. It does not yet distinguish our physical model from other theories that produce the same interference.
9. Demonstrating Information Loss in the Mapping to Weights
Immediately before the final H, the state is:
\[\tag{30} \psi_{\mathrm{mid}}=\frac1{\sqrt2}\begin{pmatrix}e^{i\pi b}\\ e^{i\pi a}\end{pmatrix},\qquad \mathcal W(\psi_{\mathrm{mid}})=\left(\frac12,\frac12\right). \] Its weights are independent of the relative phase. The result of subsequently applying the same H does depend on that phase:
| Phase difference | Weights before recombination | Weights after recombination |
|---|---|---|
| 0 | (1/2, 1/2) | (1, 0) |
| π/2 | (1/2, 1/2) | (1/2, 1/2) |
| π | (1/2, 1/2) | (0, 1) |
Proposition. No single-valued operation F on the simplex of input weights alone can reproduce this mixing for all normalized states:
\[\tag{31} \nexists F:\Delta_1\to\Delta_1\quad\text{such that}\quad \mathcal W(H\psi)=F(\mathcal W(\psi))\quad\text{for all }\psi. \] Proof. All three states in the table have the same image (1/2, 1/2), but different images after H. The function F would have to assign several values to one point, a contradiction.
The full phase state therefore contains information required for prediction that is absent from its weights in a fixed basis. A richer positive construction may retain these data through additional variables. The proof applies specifically to the reduction (14).
For the output interval \(x=\sin^2(\delta/2)\), the canonical form on the branch 0<δ<π is:
\[\tag{32} \Omega_{\Delta_1}=\frac{2\,d\delta}{\sin\delta}=d\ln\tan^2\frac\delta2. \] At δ=0 or π, one output vanishes through interference. The state (28) remains smooth and normalized even though the form has a pole. Thus, a boundary of the output-weight space can represent complete cancellation of one channel while the full state remains regular.
10. Simplifying a Compatible Sequence of Transitions
The global phase does not affect the output weights. After removing it, write:
\[\tag{33} D(\delta)=\begin{pmatrix}e^{i\delta/2}&\&e^{-i\delta/2}\end{pmatrix},\qquad U(\delta)=HD(\delta)H. \] For two cells:
\[\tag{34} U(\delta_2)U(\delta_1)=HD(\delta_2)HHD(\delta_1)H=HD(\delta_1+\delta_2)H. \] By induction, we obtain:
\[\tag{35} \boxed{U(\delta_N)\cdots U(\delta_1)=HD\!\left(\sum_{k=1}^{N}\delta_k\right)H.} \] For an input in the first channel, the entire output calculation reduces to the sum of the phases:
\[\tag{36} w_-^{\mathrm{out}}=\sin^2\!\left(\frac12\sum_{k=1}^{N}\delta_k\right),\qquad w_+^{\mathrm{out}}=1-w_-^{\mathrm{out}}. \] For identical cells:
\[\tag{37} w_-^{\mathrm{out}}=\sin^2\frac{N\delta}{2}. \] Instead of explicitly enumerating intermediate paths, one need only add N phase differences and evaluate a single trigonometric function. The simplification follows from the specific composition structure: identical compatible mixing H, diagonal phase operations, and the absence of intermediate measurement or decoherence. With other interchannel operators, simple phase addition generally no longer works.
This is an advantage of compact operator notation over an explicit sum over paths. The conventional matrix formalism of quantum mechanics permits the same simplification. The result therefore demonstrates the convenience of the split parametrization, but does not establish uniquely superior computational efficiency.
11. An Additional Correspondence: Projections and the Lorentz Factor
The article “The Origin of the Lorentz Factor from Infinite Idempotent Splitting” uses a deep self-similar state. Its orthogonal coefficients give:
\[\tag{38} \boldsymbol\Gamma_\beta=\sum_{n=0}^{\infty}\beta^n\boldsymbol\xi_n,\qquad \boldsymbol\xi_n\cdot\boldsymbol\xi_m=\delta_{nm},\qquad \|\boldsymbol\Gamma_\beta\|^2=\sum_{n=0}^{\infty}\beta^{2n}=\frac1{1-\beta^2}. \] Under the adopted identification β=v/c and for |β|<1, this squared norm equals γ². For a two-component projection description on the branch 0≤β<1, one can independently choose sin θ=β. Then:
\[\tag{39} w_{\mathrm{ext}}=\beta^2,\qquad w_{\mathrm{int}}=1-\beta^2,\qquad \gamma=\frac1{\sqrt{w_{\mathrm{int}}}}. \] Thus, the inverse Lorentz factor corresponds to the amplitude of the internal projection. The two-channel weights in (39) and the infinite series in (38) are distinct constructions linked by their common dependence on β; their spaces and norms are not identified.
Two positive intervals must also be distinguished. The first describes the weights x=β² and discards the sign of the velocity. The second uses the signed velocity β itself, with closure −1≤β≤1. Their forms are:
\[\tag{40} \Omega_{\mathrm{weights}}=\frac{2\,d\beta}{\beta(1-\beta^2)}\quad(0<\beta<1),\qquad \Omega_{\mathrm{velocity}}=\frac{2\,d\beta}{1-\beta^2}=2\,d\eta,\quad\eta=\operatorname{artanh}\beta. \] The boundaries of the first geometry are at β=0 and β=1; those of the second are at β=−1 and β=1. The difference between the forms is therefore expected. The second form is proportional to the differential of rapidity. These correspondences use a specified metric and physical mapping; positivity and idempotency alone do not derive relativistic dynamics.
12. Advantages, Scope, and the Next Step
The correspondence constructed here lets us state precisely where the split description is more useful than the space of positive weights alone.
| Task | Result | Applicability condition |
|---|---|---|
| Positive coordinates | A single formula using normalized squared moduli | Orthonormal channels and a chosen scalar norm |
| Reconstruction | Phases are explicitly added to the weights | Zero components are handled separately |
| Weight hierarchy | Binary parameters automatically preserve total weight | A distribution tree is specified |
| Simplex form | Factorization over independent tree nodes | A complete set of internal parameters and a consistent orientation |
| Interference | Information needed to calculate the outputs is retained | Coherent mixing and a detection rule are specified |
| Sequence of transitions | Compatible cells reduce to a sum of phases | The operators have the structure (33) |
Simplicity comes from separating the tasks: idempotents specify the channels, complex coefficients specify amplitudes and phases, operators specify the operations, and the mapping (14) specifies observable weights. Positive geometry, in turn, supplies canonical forms and a consistent description of boundaries. In the examples considered, the two formalisms complement one another.
For arbitrary positive geometries, more complex interactions, and sequences without compatible mixing, an analogous simplification remains to be established. The proposed representation does not replace the construction of a physical law or prove an advantage over all existing methods of calculation.
The split description retains the information required for interference, while positive geometry arises when this state is mapped to observable weights. A new prediction specific to our theory requires the next step: deriving the physical mixing operators and phase dynamics from its postulates, rather than specifying them as in the example considered.
References
- N. Arkani-Hamed, Y. Bai, T. Lam. Positive Geometries and Canonical Forms. Journal of High Energy Physics, 2017, 039. DOI: 10.1007/JHEP11(2017)039.
- N. Arkani-Hamed, J. Trnka. The Amplituhedron. Journal of High Energy Physics, 2014, 030. DOI: 10.1007/JHEP10(2014)030.
- IBM Quantum Learning. Quantum Coins: superposition and interference. An example of phase interference with Hadamard transformations.