In-In Zonotopes in Flat-Space QFT

• In-In zonotopes are convex polytopes constructed as Minkowski sums that encode the combinatorial and factorization structures of flat-space equal-time scalar correlators.
• They are built from vertex hypercube and graphical zonotopes whose facet inequalities mirror time-ordered and anti-time-ordered subgraphs in the in-in formalism.
• Their canonical form, derived from the polytope's geometry, reproduces the full in-in correlator and links directly to the wavefunction decomposition via dual subdivision.

An in-in zonotope is a convex polytope introduced to organize scalar equal-time correlators computed via the in-in (Schwinger-Keldysh) formalism in flat-space quantum field theory. Given a finite, connected graph $G=(V,E)$ with $|V|=n$ vertices and $|E|=m$ edges, with positive parameters $x_v>0$ assigned to each vertex $v\in V$ and $y_e>0$ to each edge $e\in E$, the in-in zonotope encodes the combinatorial and factorization structure of flat-space correlators in a fully geometric language. This framework discloses a direct mapping between the polyhedral geometry of the in-in zonotope and the physical structure of correlators, their factorization properties, and wavefunction decompositions (Glew, 26 Jan 2026).

1. Definition and Construction

The in-in zonotope $\mathcal{I}(G)$ is constructed as a Minkowski sum of two auxiliary zonotopes in $\mathbb{R}^n$:

• The vertex hypercube zonotope:

$$\mathcal{H}(G) = \bigoplus_{v\in V} [-x_v e_v, +x_v e_v]$$

where $e_v$ is the standard basis vector corresponding to vertex $v$.

• The graphical zonotope:

$$\mathcal{Z}(G) = \bigoplus_{e=\{v,v'\}\in E} [-y_e e_{vv'}, +y_e e_{vv'}]$$

where $e_{vv'} = e_v - e_{v'}$.

The in-in zonotope is then:

$$\mathcal{I}(G) = \mathcal{H}(G) \oplus \mathcal{Z}(G) \subset \mathbb{R}^n$$

After rescaling, this can be written as a zonotope generated by the segment sums $[0, 2x_v e_v]$ for each vertex and $[0, 2y_e e_{vv'}]$ for each edge.

This construction yields a centrally symmetric polytope whose combinatorics are determined by the graph $G$ and the choice of edge and vertex weights.

2. Facet (Inequality) Structure

For each connected, vertex-induced subgraph $\mathfrak g \preceq G$, define:

$$\alpha_{\mathfrak g} = \sum_{v \in V(\mathfrak g)} \alpha_v$$

$$L_{\mathfrak g} = \sum_{v \in V(\mathfrak g)} x_v + \sum_{e: \text{edge with exactly one end in } V(\mathfrak g)} y_e$$

The in-in zonotope $\mathcal{I}(G)$ is described as the intersection of $2 \cdot |\{\mathfrak g\}|$ half-spaces:

$$\mathcal{I}(G) = \left\{ \alpha\in\mathbb{R}^n \,\,\Big|\,\, -L_{\mathfrak g} \leq \alpha_{\mathfrak g} \leq L_{\mathfrak g}\ \ \forall\ \mathfrak g\preceq G\right\}$$

Each facet corresponds to an equality $\alpha_{\mathfrak g} = \pm L_{\mathfrak g}$, and the facet structure mirrors the combinatorial nested structure of time-ordered and anti-time-ordered subgraphs in the in-in rules.

3. Canonical Form and Correlator Evaluation

Convex polytopes containing the origin possess a unique "canonical form" with logarithmic singularities along their facets. For $\mathcal{I}(G)$, the canonical form has the structure:

$$\Omega(\mathcal{I}(G)) = f_{\mathcal{I}(G)}(\alpha)\, d^n\alpha$$

where

$$f_{\mathcal{I}(G)}(\alpha) = \sum_{\{\epsilon_{\mathfrak g}=\pm1\}} \prod_{\mathfrak g\preceq G} \frac{1}{\alpha_{\mathfrak g} + \epsilon_{\mathfrak g} L_{\mathfrak g}}$$

Setting $\alpha=0$ yields a sum over all sign assignments, which exactly reproduces (up to normalization) the in-in scalar correlator:

$$\langle G \rangle \propto f_{\mathcal{I}(G)}(0) = \sum_{\{\epsilon_{\mathfrak g}=\pm1\}} \prod_{\mathfrak g\preceq G} \frac{1}{\epsilon_{\mathfrak g}L_{\mathfrak g}}$$

This is the $2^n$-term sum characteristic of the in-in formalism.

Worked examples include:

• $K_1$ (single vertex): The canonical form gives $f(0) = 2/x_1$, matching the correlator.
• $P_2$ (two-vertex path): $f(0)$ evaluates to the familiar rational function for the path correlator.

4. Factorization Properties of Facets

The boundary structure of the in-in zonotope reflects the factorization of correlators under time-localization:

• For any nonempty, proper connected induced subgraph $\mathfrak g\prec G$, the facet $\alpha_{\mathfrak g} = +L_{\mathfrak g}$ factors as:

$$\mathcal{I}(G)\big|_{\alpha_{\mathfrak g}=L_{\mathfrak g}} \simeq \left[ \mathcal{H}(\mathfrak g) \oplus \mathcal{Z}(\mathfrak g) \right]\oplus \mathcal{I}(\bar{\mathfrak g})$$

More precisely, up to translation,

$$\mathcal{Z}(\mathfrak g) \otimes \bigotimes_{C\in\operatorname{Comp}(\bar{\mathfrak g})} \mathcal{I}(C)$$

where $\operatorname{Comp}(\bar{\mathfrak g})$ denotes the connected components of the complementary graph. This encapsulates the physical residue factorization: when a subset of vertices "go on shell" (corresponding to a propagator pole), the residue splits as a product of the wavefunction for $\mathfrak g$ (graphical zonotope) and the correlators for the remaining connected pieces.

5. Dual Polytope and Wavefunction Subdivision

The polar (dual) polytope $\mathcal{I}(G)^\circ$ is explicitly:

$$\mathcal{I}(G)^\circ = \operatorname{ConvexHull} \left\{ \pm \frac{1}{L_{\mathfrak g}}\sum_{v\in V(\mathfrak g)} e_v \,|\, \mathfrak g\preceq G\right\} \subset (\mathbb{R}^n)^*$$

The volume $\operatorname{Vol}(\mathcal{I}(G)^\circ)$ is proportional to $f_{\mathcal{I}(G)}(0)$ and thus the scalar correlator.

There is a canonical subdivision of the dual zonotope, where subpolytopes correspond to wavefunction terms in the correlator expansion. Each "chamber" of the subdivision localizes to a single term of the wavefunction decomposition, and the sum of the volumes of all chambers matches the total correlator.

Explicit examples:

• For $P_2$, the dual is a hexagon subdivided into three quadrilaterals whose areas match the three 'wavefunction' terms.
• For a single loop-edge, the dual is a segment subdivided at points that correspond to different contraction (shrinking) of the loop, with the respective segment lengths matching the wavefunction contributions.

6. Summary Table: Key Features of In-In Zonotopes

Structural Feature	Mathematical Realization	Physical Interpretation
Definition	Minkowski sum $\mathcal{H}(G) \oplus \mathcal{Z}(G)$	Encodes all time assignments in in-in QFT
Facets	$\alpha_{\mathfrak g} = \pm L_{\mathfrak g}$ for all connected subgraphs	Encodes all possible "cuts"/poles
Canonical form at $0$	$$f_{\mathcal{I}(G)}(0) = \sum \prod \frac{1}{\epsilon_{\mathfrak g}L_{\mathfrak g}}$$	Full in-in correlator
Boundary factorization	$$\mathcal{Z}(\mathfrak g) \otimes \prod \mathcal{I}(C)$$	On-shell residue splits
Dual & subdivision	$\mathcal{I}(G)^\circ$, subdivided into chambers	Wavefunction decomposition

7. Significance and Outlook

In-in zonotopes provide a unifying geometric framework for organizing flat-space, equal-time scalar correlators computed via the in-in formalism. Their combinatorial structure encodes the $2^n$-fold sum over time orderings, their facet structure mirrors pole factorization, and their dual subdivisions achieve a perfect match with the diagrammatic wavefunction decomposition. Explicit evaluation of their canonical form yields the complete correlator, and the geometry of boundaries and duals fully encodes both physical factorization and combinatorial decomposition (Glew, 26 Jan 2026). This construction generalizes the role of graphical zonotopes and introduces new connections between combinatorics, polyhedral geometry, and quantum field-theoretic computations of correlators.