---
title: Weakly-Local Path Integral
url: https://www.emergentmind.com/topics/weakly-local-path-integral
type: topic
---

# Weakly-Local Path Integral

Searching arXiv for papers related to weakly-local path integrals and closely adjacent formulations.
A **weakly-local path integral** is not a single canonical construction but a family of path-integral formalisms in which locality is retained only in a relaxed sense. Across the literature, the weakening can occur at different levels: Bell-factorizable locality may be replaced by local path interference; point-to-point short-time kernels may be replaced by locally composed kernels containing additional \(\hbar\)-dependent structure; exact coincidence of doubled paths may be replaced by finite-width suppression; and point-path transport may be replaced by region-to-region propagation. In each case, the formulation remains local in some operational, geometric, or slice-wise sense while relaxing a stricter locality requirement [1509.02518][2003.12683][1808.04178][2508.17386].

## 1. Terminological scope

The expression has no uniform definition across the cited literature. Instead, several distinct usages recur.

| Setting | Weak-local feature | Representative paper |
|---|---|---|
| Bell/Feynman path integrals | Local dynamics along each side, weaker than Bell-local factorization | [1509.02518] |
| Dirac generalized path integral | Time-sliced locality, but short-time kernel not fixed uniquely by \(L\) | [2003.12683] |
| GRW density-matrix path integral | Finite-width damping penalizes separated histories | [1808.04178] |
| Awkward-action Monte Carlo | Fundamentally local action, quasi-local effective slice kernels | [1611.06685] |
| Relativistic worldline path integrals | Local segment composition modulo hidden local redundancy | [1706.05388], [2012.15242] |
| Wave-optical transport | Region-to-region paths with non-negative local contributions | [2508.17386] |

This suggests that “weakly-local” functions less as a fixed axiom than as a comparative descriptor. What is weakened differs by context, but three motifs recur: locality is preserved in a short-time, local-interference, or bounded-region sense; a stricter global or pointwise notion is relaxed; and the relaxation is used either to reinterpret correlations, regularize the path integral, or recover tractable sampling.

## 2. Weak locality in the Bell–Feynman locality debate

In the path-integral analysis of Bell-type experiments, the relevant contrast is between Bell-locality formulated on a classical probability space and a weaker locality notion tied to local path interference [1509.02518]. The Bell setup is written with local outcomes
\[
A=A(\alpha,\lambda), \qquad B=B(\beta,\lambda),
\]
and correlation
\[
E(\alpha,\beta)=\int A(\alpha,\lambda)\,B(\beta,\lambda)\,\rho(\lambda)\,d\lambda.
\]
The paper argues that this framework already presupposes a Kolmogorov probability space.

In the Feynman path-integral framework, by contrast, the “hidden variables” are identified with the set of all possible paths from source to detector, and amplitudes are obtained by summing over paths:
\[
(vu)=\sum_{\text{paths }x(\omega)\text{ from }u\text{ to }v} e^{iS[x(\omega)]/\hbar}
= k\int_A e^{iS[x(\omega)]/\hbar}\,Dx.
\]
The author’s claim is that path space does not furnish the countably additive measure required by Bell or CHSH. The paper states that there is “no countably-additive measure \(D\)” on the space of paths that weighs all paths equally, and that the path space is noncompact so its measure would be infinite. On that basis, the standard CHSH derivation,
\[
S = A(\alpha,\lambda)B(\beta,\lambda) + A(\alpha',\lambda)B(\beta,\lambda)
+ A(\alpha,\lambda)B(\beta',\lambda) - A(\alpha',\lambda)B(\beta',\lambda),
\]
with \(|S|=2\) and hence
\[
\int S\,\rho(\lambda)\,d\lambda \le 2,
\]
is said not to apply to the FPI setting.

The weaker locality notion is operational rather than Kolmogorov-theoretic. A model is “local” if the outcome on one side does not depend on the remote detector setting or remote outcome, and if the correlations are explained by local dynamics along each particle’s own paths. The paper illustrates this with a phase-like “clock” variable \(e^{iS[\text{path}]/\hbar}\) and with the Rarity–Tapster interferometer, where the two wings are separated by an opaque barrier and the relevant interference is claimed to occur locally on each side. In this usage, weak locality means locality of dynamical evolution along each branch, not Bell-factorization over a classical hidden-variable space.

## 3. Weak locality as local composition with ambiguous short-time kernel

A different weak-locality notion appears in the generalized path-integral reconstruction attributed to Dirac [2003.12683]. The paper identifies a short-time quantum amplitude
\[
\langle q'_{t+\epsilon}\,|\,q'_t\rangle = e^{\frac{i}{\hbar}\tilde S(q'_{t+\epsilon},q'_t)},
\]
or equivalently a full time-sliced integral
\[
\langle q_{t_f}'\,|\,q_{t_i}'\rangle = \int \cdots \int \prod_{i=1}^{N} dq_i' \, e^{\frac{i}{\hbar}\tilde S(\{q_i'\})},
\]
with
\[
\tilde S = S_{cl}+S^*(\hbar).
\]
The additional term \(S^*\) is the mechanism by which the construction admits quantization ambiguities.

Here the path integral remains fully local in its slicing structure. It is built by repeated insertion of completeness relations and by composition of infinitesimal amplitudes, not by a genuinely nonlocal-in-time kernel. The weakening concerns something else: the infinitesimal propagator is not uniquely fixed by the classical Lagrangian alone. The paper contrasts Dirac’s “analogue” with Feynman’s short-time proportionality
\[
\langle q_{t+\epsilon}'|q_t'\rangle = \frac{1}{A(\epsilon)}e^{\frac{i}{\hbar}L\epsilon},
\]
and argues that the latter cannot hold universally because different quantizations with the same classical limit can yield different short-time kernels.

This produces a notion of weak locality that is exact, not approximate. The formulation is **locally time-sliced**, but only **weakly local in the Feynman sense** because the local short-time generator is ambiguous. Each admissible choice of \(S^*\) yields a generically different quantum Hamiltonian while preserving the same classical limit. The paper’s explicit example modifies the short-time coefficient to \(f_0(x)=m+2\lambda\hbar x^2\), leading to
\[
\hat H_\lambda = \frac{1}{4}\left\{ \hat p^2\frac{1}{f_0(\hat x)}+\frac{1}{f_0(\hat x)}\hat p^2 \right\} +V(\hat x),
\]
which reduces to the standard Feynman Hamiltonian only for \(\lambda=0\).

## 4. Regulators, quasi-local kernels, and near-diagonal locality

A third cluster of constructions weakens strict locality by introducing regulators, finite-width penalties, or effective slice kernels [1808.04178][1611.06685][1605.06982].

In the GRW modification of density-matrix evolution, the usual double path integral is multiplied by an additional real damping functional:
\[
\rho(x_T,y_T,T)= \int \mathcal{D}x_t\,\mathcal{D}y_t\;
\exp\!\left[\frac{i}{\hbar}\big(S[x_t]-S[y_t]\big)\right]
\exp\!\left[-\lambda\int_0^T dt\left(1-\exp\frac{-(x_t-y_t)^2}{4r_C^2}\right)\right]
\rho(x_0,y_0,0)\,dx_0dy_0.
\]
The second exponential is interpreted as a GRW-induced regulator. It does not impose strict coincidence \(x_t=y_t\); instead it suppresses histories with \(|x_t-y_t|\gg r_C\) while leaving \(|x_t-y_t|\ll r_C\) nearly unsuppressed. In this sense the path integral is weakly local: locality is enforced softly, through a finite Gaussian width. The same construction yields the von Neumann equation when \(\lambda T\to 0\) and the Liouville equation in the regime \(\lambda T\gg 1\) with \(S\gg\hbar\).

For awkward or very awkward actions, locality is preserved at the level of the original Lagrangian density, but the effective Monte Carlo implementation becomes only “weakly local” or “quasi-local” [1611.06685]. The paper introduces auxiliary derivative variables \(\dot\psi_\ell\) and writes
\[
\begin{split}
Z(\beta) = {}& \int \exp \bigg\{ \int_0^\beta \! \int \Big[ ( i \dot \phi_\ell - \dot \psi_\ell ) \frac{ \partial  L(\phi, \dot \psi) } { \partial \dot \psi_\ell } + L(\phi, \dot \psi) \Big] dt \, d^3x \bigg\}  \\
&\times \left| \det \!\left( \frac{ \partial^2  L(\phi, \dot \psi) } { \partial \dot \psi_k \partial \dot \psi_\ell }  \right) \right| \, D\phi\,D\dot\psi .
\end{split}
\]
The fundamental integrand remains local in slice variables, but the sampled effective distribution is built from precomputed slice kernels such as \(A(q_{j+1},q_j)\) and \(C(q_{j+1},q_j)\). The locality weakening is therefore computational rather than dynamical.

A more rigorous version appears in the time-slicing construction of approximate heat kernels on Riemannian manifolds [1605.06982]. There the short-time kernel is only required to be accurate near the diagonal, with exponentially small leakage away from it. The explicit estimate
\[
|\chi_{>D}[K_1(t_1)*K_2(t_2)]|_{\ker}\le t\,e^{-D^2/(20t)}
\]
is representative. The semigroup law is weakened to an approximate semigroup condition,
\[
|K(t_1)*K(t_2)-K(t)|_{(t)}\le C t^{3/2},
\]
and the fine-partition limit is then proved to converge to the heat kernel. In this usage, weak locality means that only local, near-diagonal control is needed; errors far from the diagonal are exponentially suppressed.

## 5. Relativistic worldlines and hidden local redundancy

For relativistic point particles, weak locality is tied to geometric composition modulo symmetry rather than to a softened interaction kernel [1706.05388][2012.15242]. The Euclidean construction starts from
\[
S=\int_{\lambda_i}^{\lambda_f} d\lambda \; m \sqrt{\left(\frac{d \vec x}{d\lambda}\right)^2},
\]
and emphasizes that a naïve discretized path integral overcounts physically equivalent paths because of local Lorentz symmetry and Weyl/reparametrization invariance. The corrected definition introduces a no-overcounting prescription,
\[
K \equiv \left. \lim_{n\to\infty}\sum_{j=1}^n \mathcal N_{D,j}(t)\prod_{l=1}^j \int dx_l^D \right|_{NOC} \Delta_l e^{\left[-\sum_r t_{r+1,r}\mathcal L_{r+1,r}\right]}.
\]
The paper characterizes this as a “weakly-local” stepwise construction: the path integral is assembled from short segments, but the segments are not independent in the naïve Markovian sense because many finer slicings lie in the same symmetry orbit. After quotienting by symmetry, higher-step kernels collapse to the one-step result.

The Minkowski-space analysis makes this structure more explicit by identifying a hidden local symmetry under local rotations or boosts of the tangent vector \(dx^\mu/d\lambda\) [2012.15242]. The corresponding path integral is written with a Fujikawa determinant \(\Xi_j\),
\[
K(x_i^\mu,x_f^\mu)\equiv \lim_{n\rightarrow \infty}\left. \sum_{j=1}^n K^{(j)}(x_i^\mu,x_f^\mu)\right|_{NOC},
\]
where \(\left.|_{NOC}\right.\) denotes integration without overcounting. Factoring out this hidden local symmetry is what makes the resulting two-point functions satisfy the Chapman–Kolmogorov identity. The paper then classifies paths into orthochronous, time-like non-orthochronous, and space-like sectors, yielding distinct propagators.

In these relativistic constructions, weak locality does not mean a nonlocal action. The worldline action remains local in length elements. What is weakened is the naïve assumption that inserting more local segments automatically produces new physical histories. The appropriate object is the quotient of path space by local equivalence classes.

## 6. Region-to-region weakly-local path integrals in wave optics

The most explicit formalization of a **weakly-local path integral** appears in wave-optical transport [2508.17386]. The paper begins with a bilinear path integral over pairs of point paths,
\[
I = \int_{\Omega\times\Omega} F(\bar x,\bar y)\, d\mu(\bar x)\, d\mu(\bar y),
\]
which models interference exactly but makes sampling intrinsically global because individual path contributions can cancel each other. The weakly-local generalization replaces point paths by sequences of bounded regions:
\[
I = \int_{\Omega} g(R)\, d\mu(R),
\]
where \(R = R_0R_1\ldots R_n\) is a finite sequence of spatial regions. The contribution is written in operator form as
\[
g(R) = \left\langle W,\; T_{R_{n-2}\to R_{n-1}\to R_n} \cdots T_{R_0\to R_1\to R_2} \,L_{R_0\to R_1}(R_0) \right\rangle,
\]
with recursion
\[
\phi_1 = L_{R_0\to R_1}(R_0), \qquad \phi_{j+1} = T_{R_{j-1}\to R_j\to R_{j+1}}\,\phi_j,
\]
and final measurement
\[
g(R)=\langle W,\phi_n\rangle.
\]

The formulation is called weakly-local because transport is confined to small bounded regions rather than to points, yet does not become fully global. Each step carries wavefront shape and finite spatial extent, but each region-path contribution \(g(R)\) is non-negative, so one can still sample paths incrementally. The practical realization uses Gaussian beams enclosed by elliptical cones. Under propagation by distance \(z\),
\[
\mathbf x_0 \to \mathbf x_0 + z\mathbf d, \qquad
a \to a + z\tan \alpha_a, \qquad
b \to b + z\tan \alpha_b.
\]
The cone frustum determines the next interaction region, allowing one beam or one region path to interact with all geometry inside its footprint at once.

This construction differs from the classical path integral in three stated ways: paths become region sequences rather than point sequences; transport becomes region-to-region rather than BSDF point scattering; and wavefront shape becomes part of the propagated state. The paper presents this as a practical compromise between classical ray path tracing and the bilinear wave path integral. It can model diffraction from arbitrary geometry, multi-edge diffractions, interference within a region, and long-wavelength electromagnetic propagation, but it remains approximate because it is built on Gaussian-beam envelopes, region-level transport, and additional approximations in scattering and sampling.

## 7. Conceptual synthesis and limits of the notion

The surveyed literature does not support a single universal definition of weak locality. Instead, it supports several precise but non-equivalent usages. In Bell-type discussions, weak locality means local path interference without Bell-factorization over a classical probability space [1509.02518]. In generalized short-time constructions, it means local composition with an ambiguous local kernel rather than a unique Feynman kernel fixed by the classical Lagrangian [2003.12683]. In collapse models, it means finite-width suppression of separated histories rather than strict coincidence [1808.04178]. In computational constructions, it can mean a fundamentally local action sampled through effective slice kernels obtained by integrating out awkward variables [1611.06685]. In wave optics, it becomes a literal region-to-region path-space integral [2508.17386].

Accordingly, a weakly-local path integral should not be identified automatically with a nonlocal theory. In several of these works the action density, time slicing, or short-step propagation remains local; the weakening concerns probability structure, the uniqueness of the infinitesimal kernel, the support of admissible coupled histories, or the granularity of the transport primitive. Nor is the notion uniformly exact or uniformly heuristic: some versions are interpretive, some are rigorous near-diagonal estimates, some are numerical devices, and some are explicit new path-space formalisms.

This suggests that the enduring significance of the idea is comparative rather than doctrinal. A weakly-local path integral is a path integral that preserves enough locality to support composition, interpretation, or sampling, while relaxing a stronger notion of locality that would otherwise make the formulation either mathematically inapplicable, physically misleading, or computationally unusable.

Source: https://www.emergentmind.com/topics/weakly-local-path-integral