Flat Space Wavefunction in QFT

• Flat Space Wavefunction is defined as the quantum field amplitude computed via a boundary path integral, generating correlation functions and S-matrix elements in asymptotically Minkowski spacetime.
• It integrates both bulk time-ordered representations and polytope-based positive geometry formulations, linking combinatorial graph structures with physical singularities.
• Its analytic structure identifies energy poles and dispersive sum rules, enforcing causality, unitarity, and generalized Steinmann relations within the perturbative framework.

A flat space wavefunction refers to the object in quantum field theory (QFT) that computes, via a boundary path integral, the amplitude for the field to attain a prescribed configuration at a final spacelike slice in asymptotically Minkowski (flat) spacetime. It serves as the “wavefunction of the universe” in the flat limit, or more concretely, as the generating functional for correlation functions or S-matrix elements when all time evolution is integrated out to the boundary. This object is closely intertwined with cosmological correlators, the analytic structure of quantum amplitudes, and the polytope-based “positive geometry” of perturbative QFT.

1. Mathematical Definition and Representations

Given a QFT with a bulk field $\varphi(x,t)$ (scalar for definiteness), the flat space wavefunctional is defined as the overlap between the Bunch–Davies (in-) vacuum and a prescribed field profile $\varphi_\partial(\vec x)$ at the late-time boundary (typically $t=0$): $$\Psi[\varphi_\partial] = \langle \varphi_\partial | \Omega \rangle = \int_{\mathrm{BD}} [\mathcal D \varphi]\, e^{i S[\varphi]}$$ where the path integral is over histories interpolating from the vacuum at $t \to -\infty$ to $\varphi(\vec x, t=0) = \varphi_\partial(\vec x)$. Logarithmic expansion in the boundary profile yields wavefunction coefficients (WFCs): $$\log \Psi[\varphi_\partial] = \sum_{n \geq 2} \int \frac{\prod_{i=1}^n d^3 k_i}{(2\pi)^{3n} (2\pi)^3 \delta^{(3)}(\sum_i k_i)} \, \varphi_{\partial,k_1} \cdots \varphi_{\partial, k_n} \; c_n(k_1, \dots, k_n)$$ The $c_n$ encode the core analytic data of the theory (Chen et al., 26 Dec 2025).

For a graph $G$ corresponding to an $n$-point wavefunction, the canonical “bulk” (time-integral) representation is: $$\Psi_G(\{X_i\}, \{Y_e\}) = \int_{-\infty}^0 \!d\eta_1 \cdots d\eta_n \ \left[ \prod_{i=1}^n e^{X_i \eta_i} \right] \left[ \prod_{e \in E} P_e(Y_e; \eta_i, \eta_j) \right]$$ where $P_e(Y_e; \eta_i, \eta_j)$ is a specific time-ordered propagator kernel (Dunaisky, 24 Jan 2026).

Alternatively, in the “canonical form” (polytope) representation, each graph $G$ is associated to a convex polytope $P_G$, and

$$\Psi_G(\{X_i\}, \{Y_e\}) = \frac{\mathrm{Adj}_{P_G^\vee}(\{X_i\},\{Y_e\})}{\prod_{T \subseteq G} \ell_T(\{X_i\},\{Y_e\})}$$

with $\ell_T$ linear forms corresponding to energy flows in all connected subgraphs (“tubes”), and $\mathrm{Adj}_{P_G^\vee}$ the adjoint of the dual polytope (Dunaisky, 24 Jan 2026).

2. Analytic Structure, Singularities, and Factorization

The flat space wavefunction is a rational function (at tree level) of external (“on-shell”) and internal (“off-shell”) energies. Singularities manifest as simple poles in (i) the total energy $E_T = \sum_i |k_i|$, and (ii) all “partial energies” $E_L^e = \sum_{i \in L} |k_i|$, $E_R^e = \sum_{i \in R} |k_i|$ for every tree-level subgraph/factorization channel (Chen et al., 26 Dec 2025, Bittermann et al., 2022). The flat-space amplitude is recovered as the residue at $E_T \to 0$: $$\lim_{E_T \to 0} c_n(\{k_i\}) = \frac{A_n(\{k_i\})}{E_T}$$ Partial-energy poles similarly encode products of lower-point amplitudes and WFCs,

$$\lim_{E_I \to 0} c_n = \frac{A_{|I|}(\{k_a\}_{a \in I}) \, \widetilde{\psi}_{n-|I|}(\{k_b\}_{b \notin I}, \pm s_I)}{E_I}$$

with $\widetilde{\psi}$ the “discontinuity” of the lower-point WFC (Bittermann et al., 2022).

At loop-level, singularities include branch cuts and are located where an internal energy sum can vanish, governed by Landau analysis and enforcing causality (no support for $E > 0$ on physical sheet) (Salcedo et al., 2022, Lee, 2023).

3. Polytope and Positive Geometry Formulations

The cosmological polytope construction provides an invariant, geometric underpinning for the flat-space wavefunction (Benincasa et al., 2021, Benincasa, 2018, Benincasa et al., 2020). Each perturbative graph $G$ is mapped to a polytope $P_G$ in a projective ambient space. The canonical (volume) form of $P_G$,

$$\omega(Y,P_G) = \Psi_G(\{x_s\},\{y_e\}) \langle Y\, d^{n_s+n_e-1} Y \rangle$$

encodes only physical (non-spurious) poles. “Facets” (codimension-1 faces) of $P_G$ correspond to the physical singularities, and “scattering facet” $E_{\rm tot} = 0$ directly reproduces the S-matrix amplitude. All other facets admit a unique triangulation into products of lower-point scattering facets and WFCs.

Polytope triangulation through the “outside-locus” systematically ensures that only physical poles are present; this structure provides a combinatorial proof of the all-loop “causal representation” of the wavefunction and enforces generalized Steinmann relations (hierarchies of vanishing multiple discontinuities for partially overlapping channels) (Benincasa et al., 2021, Benincasa et al., 2020).

4. Partial Fraction Expansions and Tubings

The wavefunction can be explicitly expanded as a sum over maximal sets of compatible “tubes” (i.e., connected subgraphs), known as tubings. There are multiple inequivalent notions:

• Binary tubing: sums over admissible collections of edge-connected subgraphs (Glew, 17 Mar 2025).
• Unary tubing: sums over vertex-connected subgraphs, leading to “amplitubes.”
• Cut tubings and decorated orientations: organization into acyclic orientations compatible with the combinatorial constraints of the original graph.

This decomposition provides a bridge from the combinatorial structure of Feynman graphs to the algebraic basis of wavefunction coefficients relevant for cosmological correlators (Glew, 17 Mar 2025). In the bulk (time-ordered) form, the wavefunction is expressed in terms of partial fractions, each term associated with a choice of spanning subgraph $H \subseteq G$ and corresponding tubing configuration (Dunaisky, 24 Jan 2026).

5. Causal Structure, Steinmann Relations, and Unitarity

Steinmann relations enforce vanishing double discontinuities for partially overlapping channels—a direct consequence of causality and unitarity. In the polytope formalism, this arises naturally: certain intersections of facets are disallowed, so the numerator of the canonical form ensures the sequential residues at these forbidden intersections vanish (Benincasa et al., 2021, Benincasa et al., 2020). For the flat-space wavefunction, this property generalizes to a hierarchy of “higher codimension” Steinmann-type constraints: $$\operatorname{Disc}_{E_{g_1}} [\operatorname{Disc}_{E_{g_2}} \cdots W_n(\{|\vec p_i|\})] = 0$$ whenever the corresponding subgraphs $g_1, g_2, \ldots$ are “partially overlapping.” This aligns with the absence of certain codimension-$k$ faces in the polytope combinatorics (Benincasa et al., 2021).

Factorization at total and partial energy poles, simultaneously, is mirrored in the residues of the canonical form on the respective facets, with the residue decomposing into products of lower-point amplitudes and phase space measures (Benincasa, 2018, Chen et al., 26 Dec 2025).

6. Analytic and Dispersion Properties

The flat-space wavefunction is analytic in the lower-half complex plane of individual external energies, with singularities constrained to the negative real axis (causality for $t \leq 0$ ensures convergence and thus analyticity) (Salcedo et al., 2022). All-loop Landau analysis confirms that non-analyticities only occur when the total energy flow through a connected subdiagram vanishes.

This analytic structure enables dispersive (UV/IR) sum rules: coefficients in the low-energy expansion of the wavefunction are related to integrals over discontinuities (spectral densities) in the UV completion,

$$\alpha_n = \int_{-\infty}^{-M_L} \frac{d\omega_1}{2\pi i} \frac{\operatorname{Disc}(\omega_T \psi_{\rm UV})}{\omega_1^{n+1}} + C_\infty$$

These sum rules differ from those for amplitudes by constraining total-derivative operators and integrating only along the negative real energy axis. This structure underpins the potential for deriving positivity bounds for effective field theory Wilson coefficients from the analytic wavefunction (Salcedo et al., 2022).

7. Flat-Space Ontology and Foundational Implications

The question of whether the wavefunction is “on space” or “on configuration space” has received rigorous treatment in both nonrelativistic and relativistic quantum frameworks. It is possible to re-express the $N$-particle wavefunction $\psi(x_1,\ldots,x_N)$ as a geometrically natural, multi-layered direct-sum field (or a section of a bundle with an immense internal fiber) over $\mathbb{R}^3$ (Stoica, 2021, Stoica, 2019). These constructions preserve all physical content—unitary evolution is manifestly local in three-space, and transformations under the symmetry group (Euclidean or Poincaré) correspond to expected geometric actions.

This suggests that, from the perspective of both the Erlangen Program and Wigner–Bargmann classification theory, the wavefunction is ontologically equivalent to a geometric object on space(-time), dispelling the foundational worry that quantum mechanics irreducibly requires a $3N$-dimensional configuration ontology. All predictions and correlation functions remain unchanged; only the representation within the Hilbert space is altered (Stoica, 2021, Stoica, 2019).

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References:

(Benincasa et al., 2021, Salcedo et al., 2022, Lee, 2023, Stoica, 2021, Benincasa et al., 2020, Mikovic et al., 2010, Benincasa, 2018, Chen et al., 26 Dec 2025, Stoica, 2019, Dunaisky, 24 Jan 2026, Glew, 17 Mar 2025, Bittermann et al., 2022)