Arithmetic Path Integral

• Arithmetic path integrals are formulations that reinterpret quantum path integration in discrete arithmetic contexts using finite or profinite spaces.
• They compute arithmetic invariants by summing over combinatorial paths with phase factors derived from number-theoretic and cohomological data.
• Applications span stochastic geometry, arithmetic topology, and function field trace formulas, bridging quantum symmetry with explicit number theory.

An arithmetic path integral is a formalism that reinterprets and generalizes the concept of path integration, familiar from quantum field theory and probability, in intrinsically number-theoretic or “arithmetic” contexts. Arithmetic path integrals appear across several domains: in stochastic geometry (analysis of winding numbers and areas traced by random processes), in the arithmetic topology of number fields, in functional and operator-theoretic frameworks on algebraic or symplectic varieties, and in deep connections between the absolute Galois group, motivic periods, and quantum symmetries. Common to these manifestations is the use of discrete, often combinatorial or profinite spaces as the domain of "paths," with arithmetic invariants or modular data serving as analogues of physical action functionals.

1. Definitions and Foundational Structures

The core principle underpinning the arithmetic path integral is the “summation” over paths or configurations in an arithmetic (or sometimes profinite/combinatorial) space, where each configuration is equipped with a phase (often of exponential or multiplicative character) determined by an arithmetic action or invariant.

In stochastic geometric contexts (e.g., planar Brownian motion), the path integral formalism—analogous to quantum propagators—enables the computation of expectations over trajectories which enclose integer-valued winding sectors. The arithmetic area is computed by integrating or summing contributions from regions labeled by these windings, leading to expressions such as

$$\langle S_n(m) \rangle = \text{(integral over winding angle generator, involving Bessel functions)}$$

In the arithmetic/number-theoretic context, the integral is replaced by finite or profinite summations over discrete sets (e.g., ℓ-torsion points on an abelian variety or Cantorian encodings of automorphism groups), and the "action" functional encodes class field-theoretic, cohomological, or motivic data.

A central formal object in the arithmetic setting is:

$$\mathcal{I}_G = \int_{C(G)} \exp \left( \frac{iS[\gamma]}{\hbar} \right) d\mu_G = \lim_{n \to \infty} \sum_{\gamma \in \{0,1\}^n} \exp \left( \frac{iS[\gamma_n]}{\hbar} \right) \mu_G([\gamma_n])$$

where $G$ may denote $\mathrm{Gal}_{\mathbb{Q}}$, $\mathrm{GT}$, or an arithmetic group, $C(G)$ is the Cantor space (or other discrete configuration space), and $S$ captures arithmetic invariants associated with the path/coding $\gamma$ (Combe, 17 Mar 2025).

2. Arithmetic Path Integrals in Stochastic Geometry and Random Processes

In models of planar Brownian motion and its generalizations, the path integral framework is employed to analyze the so-called "arithmetic area" of winding sectors enclosed by random paths. Each point in the plane is classified by the number of times the path winds around it, populating n-winding sectors whose areas are then summed, possibly with sign or multiplicity.

The key observable—the arithmetic area—admits explicit path integral representation via the propagator of the system, leading to formulas:

• Generating functions are given by expressions such as

$$G_\alpha(x) = e^{-x} \sum_{k=-\infty}^{\infty} I_{|k-\alpha|}(x)$$

with $I_\nu$ the modified Bessel function.

• Averages over winding sectors and areas result from integrating these propagators using cosine (Fourier) transforms and auxiliary variables, with explicit use of combinatorial summation over independent paths (Desbois et al., 2012).

The main asymptotic result shows the total arithmetic area enclosed by $m$ independent closed Brownian paths grows as $(\pi/2)\ln m - (\pi/4)\ln \ln m + O(1)$. Sectors with zero winding, elusive in direct path integral computation, are incorporated using Schramm–Loewner Evolution (SLE) results, introducing crucial empirical ratios (e.g., $q = 1/5$) for sector suppressions.

3. Path Integrals in Arithmetic Topology and Quantum Arithmetic Invariants

The arithmetic path integral formalism has deep analogues in arithmetic topology, notably in the context of Chern–Simons theory over number fields. Here, one replaces the space of classical paths with spaces of arithmetic data—such as ideal classes, Galois cohomology classes, or ℓ-torsion points on Jacobians—and replaces the analytic "action" functional with arithmetic height pairings or cup-products.

For example, in the context of arithmetic linking numbers and Chern–Simons path integrals:

• An arithmetic linking number is defined via Artin–Verdier duality and cup product as

$\operatorname{lkn}(P, Q) := (d^{-1}[P]_n, [Q]_n)$

where $d$ is a differential on cohomological data.

• The arithmetic path integral formula takes the form of a Gaussian sum

$$\sum_{\rho \in H^1(X, \mathbb{Z}/p\mathbb{Z})} \exp\left[2\pi i \big( (\rho,\rho) + \sum_j (\rho, [\xi_j]) \big) \right]$$

with quadratic forms and residues expressing linking and height invariants in terms of $n$th Hilbert symbols (Chung et al., 2017).

• This extensible correspondence between quantum field theory's path integral and arithmetic finite sums yields explicit arithmetic invariants and connects to the quantization of classical arithmetic invariants.

4. Discrete, Finite, and Profinite Arithmetic Path Integrals: Binary Coding and Galois–Grothendieck Theory

Recent developments extend the path integral concept to topological, combinatorial, and profinite arithmetic structures, most notably in the context of the absolute Galois group, $\mathrm{Gal}_{\mathbb{Q}}$, and the Grothendieck–Teichmüller group, $\mathrm{GT}$ (Combe, 17 Mar 2025). These are realized as compact, totally disconnected Cantor sets and can be encoded purely in terms of infinite binary sequences.

• The Cubic Matrioshka Algorithm encodes each element in $\mathrm{Gal}_{\mathbb{Q}}$ or $\mathrm{GT}$ as an infinite binary sequence by recursive partition of the Cantor set into clopen subsets. Each finite truncation corresponds to a vertex of a hypercube, and compatibility with the group structure is maintained across levels.
• The arithmetic path integral in this context sums over all binary-encoded automorphisms:

$$\mathcal{I}_{\mathrm{Gal}} = \lim_{n\to\infty} \sum_{\gamma \in \{0,1\}^n} \exp\left( \frac{i S[\gamma_n]}{\hbar} \right) \mu([\gamma_n])$$

where $S$ is an arithmetic action functional encoding, for example, period rationality or cohomological data. The phase factors track rational structure in periods and coefficients of the Drinfeld associator.

• This formalism gives mechanisms to detect descent of cohomological classes to $\mathbb{Q}$ as well as to compute ranks of arithmetic objects, with the compatibility of binary encodings ensuring correspondence between GT and Galois symmetries.

5. Arithmetic Trace-Path Integral Formulae over Function Fields

In the context of arithmetic geometry over function fields, path integrals are realized as finite sums over torsion points with phases determined by natural pairings (e.g., the Weil pairing). The sum is shown to coincide with the trace of Frobenius acting on the geometric quantization (representation) of the associated Heisenberg group (Cheng, 4 Sep 2025).

Key structures are summarized as follows:

Object	Arithmetic analogue	Quantum field theory analogy
$J[\ell]$ (ℓ-torsion)	Classical phase space (finite vector space)	Phase space ($T^*M$)
$A : J[\ell] \to (1/\ell)\mathbb{Z}/\mathbb{Z}$	Symplectic/arithmetic pairing	Symplectic form/action
$\mathcal{H}$	Quantization space (theta functions)	Hilbert space via geometric quantization
Path integral	$$\sum_{\gamma \in J[\ell](\mathbb{F}_q)} e^{2\pi i A(\gamma)}$$	$\int_{\text{paths}} e^{iS} d\gamma$
Trace of Frobenius	$$\operatorname{tr}(\operatorname{Fr}_q | \mathcal{H})$$	Trace of monodromy

The main theorem is

$$\operatorname{tr}(\operatorname{Fr}_q | \mathcal{H}) = \text{(explicit sign)} \cdot \sqrt{|J[\ell](\mathbb{F}_q)|} = \text{arithmetic path integral}$$

where the sign is given by an explicit Legendre symbol depending on $\det(A)$ and group-theoretic data. This correspondence structurally mirrors trace–path integral dualities in topological quantum field theory, notably Chern–Simons and Rozansky–Witten theories.

6. Universal and Functional-Analytic Generalizations

Arithmetic path integrals are "universally" encompassed within the universal path integral framework, which sums over all possible computable structures (bit strings, programs), with each phase determined by the output of a universal Turing machine—thus including, as sub-integrals, all arithmetic (computable) path integrals (Lloyd et al., 2013).

Furthermore, several mathematical definitions are possible on symplectic and Poisson manifolds (Lackman, 20 Jun 2024), incorporating propagators $\Omega$ with strict algebraic and differential cocycle properties (e.g., integrating prequantum connections, employing the Bergman kernel on Kähler manifolds). In finite or function field contexts, arithmetic path integrals appear as finite analogues of these more general functional-integral constructions.

7. Applications and Implications

Arithmetic path integrals provide a structured framework for:

• Encoding and extracting arithmetic invariants (e.g., rationality of periods, descent of cohomology classes).
• Interpreting explicit sums in arithmetic geometry as quantum traces or invariants.
• Extending quantum field theoretic principles—including quantization, path summation, trace/monodromy correspondences, and dualities—to arithmetic and motivic settings.
• Bridging combinatorial/profinite topology with number-theoretic data via canonical (binary, Cantorian) coding.
• Unifying disparate approaches (stochastic geometry, arithmetic topology, cohomological quantization) under a single formal apparatus, with deep interplay between algebra, analysis, and quantum symmetry.

The arithmetic path integral thus crystallizes as a unifying theme across stochastic analysis, arithmetic topology, quantum field theory analogies, and modern anabelian/quantum motivic geometry, providing new tools and perspectives for both explicit computation and theoretical synthesis in number theory and related fields.