p-adic Path Integral Overview

• p-adic Path Integral is a non-archimedean generalization of the Feynman path integral, defined over fields like Qₚ using discretization and ultrametric analytic techniques.
• It underpins key applications in quantum mechanics, field theory, and arithmetic geometry, yielding explicit propagators, renormalization methods, and links to L-functions.
• The framework also extends to combinatorial and automorphic structures, enabling analysis of fractal measures, symbolic paths, and automorphic distributions in p-adic settings.

A $p$-adic path integral is a generalization of the classical path integral formalism to the context of $p$-adic number fields, $\mathbb{Q}_p$ and their completions. It arises in mathematical physics, number theory, and arithmetic geometry as a method for encoding summation or integration over $p$-adic “paths,” states, or field configurations, with deep connections to $p$-adic quantum mechanics, arithmetic L-functions, automorphic forms, and gauge-theoretic analogies. This article synthesizes principal methodologies, analytic tools, and significant applications from foundational and recent research, including analytic quantum mechanics, arithmetic gauge theory, quantum field theory over $p$-adics, and automorphic analysis.

1. Analytic Construction of $p$-adic Path Integrals

The $p$-adic path integral formalism closely mirrors the Feynman path integral in real quantum mechanics but is defined over spaces such as $\mathbb{Q}_p$ or $\mathbb{C}_p$ using non-archimedean analytic techniques. The key construction, exemplified in (Hu et al., 21 Oct 2025), involves:

• Discretization of the time interval $[t_0, t]$ into $N$ subintervals of length $\epsilon$.
• Replacement of a continuous trajectory $x(t)$ by a tuple $(x_0, x_1, \ldots, x_N)$ with $x_0 = x$ and $x_N = y$.
• The path integral kernel (propagator) for a free particle is given by:

$$U(y, t; x, t_0) = \lim_{\epsilon \to 0, N \to \infty} \int \ldots \int_{\mathbb{Z}_p} \chi_a\left( \sum_{i=0}^{N-1} \frac{m}{2} \frac{(x_{i+1} - x_i)^2}{\epsilon} \right) d\mu_1(x_1) \cdots d\mu_{N-1}(x_{N-1}),$$

where $\chi_a$ is a $p$-adic character ($\chi_a(x) = e_p(ax)$), $d\mu_i$ are $p$-adic measures (Haar, Dirac, $\mu_{-1}$), and integration is in the $p$-adic metric.

A key feature is the exact evaluation for free particles, yielding a propagator structurally analogous to the classical Gaussian kernel:

$$U(y, t; x, t_0) = e_p\left( a \frac{m}{2} \frac{(y-x)^2}{t-t_0} \right).$$

This formula is derived by inductively integrating out intermediate positions with Dirac or Haar measures, utilizing $p$-adic quadratic exponential sums, and taking limits with respect to the $p$-adic topology. Notable differences in the analytic structure arise due to the total disconnectedness and ultrametric properties of $p$-adic spaces.

2. $p$-adic Path Integrals in Quantum Mechanics and Field Theory

$P$-adic path integrals are central to $p$-adic quantum mechanics, where the standard spectral operator approach is challenged due to the absence of a canonical Laplacian or Hermitian operator. Instead, path integrals define evolution kernels, expectation values, and correlation functions using $p$-adic multiple integrals, $p$-adic actions, and $p$-adic characters.

For quadratic Lagrangians, analytic evaluation extends to arbitrary finite-dimensional systems (Dragovich, 2010):

$$K_v(x'', t''; x', t') = N_v(t'', t') \cdot \chi_v(-S(x'', t''; x', t')/h),$$

where $v$ is a place of $\mathbb{Q}$ ($\infty$ or $p$), $N_v$ is a normalization factor involving determinants of second derivatives of the action, and $\chi_v$ is the corresponding additive character (real or $p$-adic). These expressions are invariant under field interchange and lay the groundwork for adelic quantum mechanics where

$$K_A(x'', t''; x', t') = K_\infty(x'', t''; x', t') \prod_p K_p(x'', t''; x', t').$$

For $p$-adic quantum field theory, functional integrals are constructed as probability measures on spaces of distributions over $p$-adic manifolds (e.g., $Q_p^3$). The construction involves Gaussian and non-Gaussian measures, Wick-ordered interactions, and rigorous renormalization schemes (Abdesselam et al., 2012), with scaling limits producing anomalous dimensions for composite fields via dynamical renormalization group analysis.

3. $p$-adic Path Integrals and Arithmetic L-functions

Arithmetic path integrals encode deep connections between quantum field theoretic summations and special values of $p$-adic L-functions. In (Carlson et al., 2022), for an odd prime $p$ and odd integer $k \neq 1$, an arithmetic path integral formula expresses the inverse $p$-adic absolute value of the Kubota-Leopoldt $p$-adic L-function at roots of unity:

$$\left| \prod_{j=0}^{p^n-1} z_{1-k}(\exp(2\pi i j/p^n) - 1)^{-1} \right|_p = \lim_{m \to \infty} \sum_{(a, b) \in F_m} \exp(2\pi i \cdot BF(a,b)),$$

where $z_{1-k}(T)$ is a distinguished power series, $BF$ is a functional pairing (analogous to Chern-Simons action), and $F_m$ is a cohomological moduli space. This is a direct arithmetic analogue of summing over fields weighted by the exponential of an action. The approach extends to $p$-adic L-functions of elliptic curves (Park et al., 2023) using Selmer groups, Iwasawa theory, and the Mazur control theorem, where the path integral formula incorporates Tamagawa factors and Néron model arithmetic data.

4. $p$-adic Measures, Integration Theory, and Automorphic Distributions

$P$-adic integration is canonically defined via the Haar measure on locally compact groups such as $(\mathbb{Q}_p, +)$, with scaling property:

$d(ax) = |a|_p dx,$

and normalization $\int_{Z_p} dx = 1$. $P$-adic integrals are foundational for defining $p$-adic Fourier analysis, Mellin transforms, and distributions on cosets. In (Gelbart et al., 2010), explicit $p$-adic measures are constructed from nonconstant Fourier coefficients of Eisenstein series on $SL(2)$, yielding bounded $p$-adic distributions whose Mellin transforms are reciprocals of Dirichlet $L$-functions:

$$\int_{Z_p^*} \chi(x) x^{k-1} d\mu^*(x) = (1 - \chi(p)p^{k-1})^{-1} L(1-k, \chi)^{-1}.$$

This construction is a $p$-adic analog of the Langlands-Shahidi method and suggests a framework for $p$-adic analytic continuation and automorphic $p$-adic path integration.

5. Algebraic and Combinatorial Path Structures

Beyond analytic integration, $p$-adic path structures arise in graph-directed fractal constructions and automaton theory (Abram et al., 2012). Closed sets of $p$-adic integers, $C(\mathbb{Z}_p)$, are characterized by sequences of digits determined by paths in labeled finite automata. Combinatorial "path integration" in this context sums over admissible infinite walks and gives rise to fractals with robust closure properties under $p$-adic addition, multiplication, and Minkowski sums. Hausdorff dimension of such sets is computed as:

$d_H(Y) = \frac{\log \rho(A)}{\log p},$

where $\rho(A)$ is the spectral radius of the automaton's adjacency matrix, relating entropy and scaling in the $p$-adic topology.

6. $p$-adic Path Integrals in Geometry, Quantum Connections, and Hodge Theory

Papers such as (Seidel, 1 Mar 2025) introduce $p$-adic operations—quantum Steenrod operations parametrized by $p$-adic integers—acting on quantum cohomology and consistent with rich geometric structures. The lifting of splittings via $p$-adic power series with logarithmic decay into splittings of the quantum connection provides insight on covariant constant decompositions. While not developing explicit $p$-adic path integrals, these constructions suggest analogies where "integration over discrete symmetries" or summation over curves with arithmetic constraints mimics features of $p$-adic gauge-theoretic summations.

7. Broader Implications and Applications

$P$-adic path integrals unify non-archimedean analysis, quantum field theory, arithmetic geometry, and automorphic representation theory. They provide fundamental objects for adelic quantum mechanics (Dragovich, 2010), rigorous construction of stochastic processes (Abdesselam et al., 2012), probabilistic measures on Skorokhod spaces and diffusion equations (Weisbart, 2020), arithmetic formulas for $L$-functions (Carlson et al., 2022, Park et al., 2023), fractal geometry (Abram et al., 2012), and geometric quantization (Seidel, 1 Mar 2025). The methodology is foundational in applications to $p$-adic string theory, quantum gravity, modular forms, and arithmetic topology, and continues to serve as a powerful lens in exploring number-theoretic phenomena through physical analogies.

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Summary Table: Main $p$-adic Path Integral Constructions

Main Method	Mathematical Formula / Scheme	References
$p$-adic Feynman Path Integral	$U(y, t; x, t_0) = \int \chi_a(S[x(t)]) D\mu(x)$	(Hu et al., 21 Oct 2025, Dragovich, 2010)
Arithmetic Path Integral for $L$-functions	$$\lim_{m \to \infty} \sum_{(a,b)} \exp(2\pi i BF(a,b))$$	(Carlson et al., 2022, Park et al., 2023)
Automorphic Measure Construction	$\int_{Z_p^*} \chi(x) x^{k-1} d\mu^*(x)$	(Gelbart et al., 2010)
Functional Integral for Field Theory	$\int e^{-V(\phi)} d\mu_{C_r}(\phi)$ (Gaussian/non-Gaussian measures)	(Abdesselam et al., 2012, Weisbart, 2020)
Combinatorial Path Sets / Fractals	Symbolic sums over automata, dimension formula $d_H(Y)$	(Abram et al., 2012)

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The $p$-adic path integral formalism comprises both analytic and combinatorial summation structures adapted to ultrametric fields, providing unifying formulas and powerful computational techniques for quantum, geometric, and arithmetic problems.