Interacting and renormalized completion of Parisi's p-adic functional integrals: real-time path integrals over p-adic trajectories, admissible kinetic operators on p-adic time, evaluation of the amplitude functions a(t) and b(t), and the fate of adelic product formulae

In 1988 Parisi showed that Feynman's functional-integral construction admits exact p-adic analogues. For a free massless scalar over the p-adic numbers $\mathbb{Q}_p$ the functional integral is computable in closed form and its correlation functions obey an adelic product formula (the product over all primes of the p-adic correlators times the real one is constant). For quantum mechanics with real time and p-adic-valued trajectories, a path integral with a regularized nonlocal kinetic weight reduces the amplitude to piecewise-constant functions $a(t)$ and $b(t)$ governed by a nonlocal Schrödinger-type equation. With p-adic time, a quadratic Fourier-space action with kinetic operator $|E|^2$ is exactly solvable. Parisi explicitly left open the evaluation of $a(t)$ and $b(t)$, the extension to interacting theories with simultaneous ultraviolet and infrared control, the renormalization-group analysis of the resulting models (fixed points and critical exponents), the persistence of adelic product relations beyond the free case, and the admissibility of the $|E|^2$ kinetic choice for p-adic time. The later literature resolves substantial parts of the Euclidean sector: p-adic scalar field theory is equivalent to Dyson's hierarchical model and inherits its rigorous renormalization theory; adelic $\phi^4$ theory is renormalizable to third order with cancellation of ultraviolet and infrared poles; and a rigorous non-Gaussian renormalization-group fixed point with Wilson's predicted anomalous dimension exists over three-dimensional p-adic space. The source-faithful core nevertheless remains open: no interacting real-time (p-adic-trajectory) functional integral with a controlled renormalized limit has been constructed, no classification of admissible kinetic operators on p-adic time exists, the functions $a(t)$ and $b(t)$ have never been evaluated, and adelic product relations are known to fail already at five points in the string-amplitude setting, leaving their interacting field-theory status undetermined.

Background

For a prime $p$, the field of p-adic numbers $\mathbb{Q}_p$ consists of formal series $\sum_{k=n}^{\infty} a_k p^k$ with digits $a_k \in \{0,\dots,p-1\}$; it is complete for the p-adic norm $|p^m u|_p = p^{-m}$ (with $u$ a p-adic unit), which satisfies the ultrametric inequality $|x+y|_p \le \max(|x|_p,|y|_p)$. Ultrametricity makes the geometry tree-like: balls are either disjoint or nested, and the 'spheres' $\{y : |y-x|_p = p^m\}$ partition space into distance classes. $\mathbb{Q}_p$ is a locally compact abelian group under addition, so it carries a translation-invariant Haar measure and a Fourier theory: the additive characters $\chi_a(x) = \exp(2\pi i \{ax\}_p)$, where $\{\cdot\}_p$ is the p-adic fractional part, diagonalize the Vladimirov operators $|D|^\alpha$ defined by $\widehat{|D|^\alpha f}(\xi) = |\xi|_p^\alpha \hat f(\xi)$, whose real-space kernels are powers $|x-y|_p^{-(d+\alpha)}$. These nonlocal operators play the role of the Laplacian and generate the kinetic terms below.

The free massless scalar over $\mathbb{Q}_p^d$ with action $S[\phi] = \int |\hat\phi(P)|^2 |P|^2\, d^dP$ has two-point function $\langle\phi(x)\phi(0)\rangle = |x|_p^{-(d-2)}/\Gamma_d(2)$ with $\Gamma_d(2) = (1-p^{\,d-2})/(1-p^{\,2})$; the case $d=2$ degenerates to a logarithmic correlator. Because each prime gives a different completion of the rationals, the rationals embed diagonally in the ring of adeles (the restricted product of all completions $\mathbb{R}$ and $\mathbb{Q}_p$, in which all but finitely many p-adic components are p-adic integers), and one may ask whether the product over all places (all primes plus the real one) of analogous amplitudes simplifies. For string tree amplitudes, Freund and Witten showed that such adelic products collapse; Parisi's free correlators obey $\prod_q G_q(x) \cdot G_\infty(x) = \text{const}$. Marinari and Parisi, however, showed the collapse already fails for the five-point string amplitude.

Parisi's 1988 paper treats three exact constructions. (a) The free p-adic scalar above, where the Gaussian functional integral is exactly computable and vertex-operator insertions (exponential functionals $e^{i\alpha\varphi(x)}$ of the field, whose correlation functions are the string scattering amplitudes) reproduce p-adic string amplitudes. (b) A quantum-mechanical path integral with a real time parameter and p-adic-valued trajectories: an integral over Haar measure on discretized trajectories $(\omega_0,\omega_1,\dots,\omega_n)$, where $\omega_i$ is the p-adic-valued position at the $i$-th time step of size $\epsilon$, with Euclidean-type weight $\exp(-S)$ whose kinetic part is the regularized weight $\sum_i |\omega_i - \omega_{i-1}|^2 \ln(1/\epsilon)$ (a nonlocal p-adic analogue of $m\dot q^2/2$), reduces the amplitude, on each class of trajectories with a given p-adic distance profile, to piecewise-constant functions $a(t)$ and $b(t)$ satisfying a nonlocal Schrödinger-type (heat-like) equation whose kernel is proportional to $|y-u|^{-3}$ with constant $C = p^2/(1+p+p^2)$. (c) A formulation with p-adic time, where the action is quadratic in Fourier space with kinetic term $|E|^2$ and the free propagator is proportional to $|t|_p$. All three are exact only in the free (quadratic) case.

The renormalization-group context: a p-adic scalar theory with ultrametric, scale-invariant propagator is mathematically the same object as Dyson's hierarchical model of statistical mechanics (the Lerner-Missarov equivalence), the ultrametric balls realizing the hierarchical blocking. Hierarchical models are the one class in which renormalization-group (RG) fixed points - scale-invariant interacting theories - and their critical exponents (anomalous powers such as $\nu$ and $\eta$ governing correlators at criticality) can sometimes be controlled rigorously. Parisi's open issues - interacting extension, ultraviolet (short-distance) and infrared (long-distance) divergence control, fixed points and exponents, adelic relations - are thus well-posed within a framework the later literature partially built.

Problem Statement

Complete, in a mathematically controlled way, the program Parisi initiated in 'On p-adic functional integrals' (1988), by resolving any one of the following source-faithful components. Each asks for a definitive result about p-adic functional integrals of exactly the type Parisi defined.

(1) Real-time interacting functional integral. Construct - or prove impossible to construct - an interacting analogue of Parisi's path integral over p-adic-valued trajectories with a real time parameter: actions built from the regularized kinetic weight $|\omega_i - \omega_{i-1}|^2 \ln(1/\epsilon)$ plus non-quadratic potentials, possessing a well-defined continuum limit as $\epsilon \to 0$ with simultaneous ultraviolet and infrared control, and reducing exactly to Parisi's free amplitude in the quadratic case. The construction should identify the renormalized amplitude's governing nonlocal Schrödinger-type equation and clarify what becomes of unitary time evolution in such a construction.

(2) The never-evaluated amplitude functions. Evaluate Parisi's piecewise-constant functions $a(t)$ and $b(t)$ - equivalently, solve the nonlocal Schrödinger-type equation with kernel $\propto |y-u|^{-3}$ and constant $C = p^2/(1+p+p^2)$ on each ultrametric distance class - in closed form or asymptotically for the free kinetic action.

(3) Admissible kinetic operators on p-adic time. Classify the kinetic operators on p-adic time under which a p-adic-time functional integral is well-defined: determine which choices generalizing the Fourier-space kinetic $|E|^2$ yield amplitudes with a well-defined propagator and sensible limiting and consistency behavior - the answer must state its admissibility criterion explicitly - and construct the corresponding propagators. Alternatively, showing that $|E|^2$ is essentially the unique admissible quadratic choice (up to positive scalar multiples) is by itself a resolution of this component: it settles the quadratic subclass and leaves the classification of non-quadratic operators open.

(4) Adelic products beyond the free theory. Determine for which interacting p-adic amplitudes the infinite product over primes $\prod_q G_q \cdot G_\infty$ collapses, as it does for free correlators and four-point string amplitudes, and for which it fails, as it does for the five-point string amplitude (Marinari-Parisi); in particular, decide whether the third-order ultraviolet/infrared pole cancellation in adelic $\phi^4$ theory (Missarov) persists to all orders of perturbation theory. Resolving the adelic $\phi^4$ case alone is by itself a resolution of this component: it settles the $\phi^4$ instance and leaves the determination for other interacting p-adic amplitudes open.

References

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