[Paper Review] General relativity from $p$-adic strings

An Huang, Bogdan A. Stoica+1 more|arXiv (Cornell University)|Jan 7, 2019

advanced mathematical theories4 citations

TL;DR

This paper proposes a p-adic string action in curved target spacetime, showing that worldsheet scaling symmetry of quantum p-adic strings leads to the vacuum Einstein equations, analogous to the Archimedean case. It identifies spherical vectors of unramified principal series representations of PGL(2, Qp) as plane wave modes and demonstrates that loop diagrams simplify due to unique normalization features. A key result is the connection between the adelic string spectrum and the nontrivial zeros of the Riemann zeta function.

ABSTRACT

For an arbitrary prime number $p$, we propose an action for bosonic $p$-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum $p$-adic strings, similar to the ordinary bosonic strings case. It turns out that certain $p$-adic automorphic forms are the plane wave modes of the bosonic fields on $p$-adic strings, and that the regularized normalization of these modes on the $p$-adic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum field theory, while also exhibiting some new features that make loop diagrams much simpler. Assuming a certain product relation, we also observe that the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann Zeta function.

Motivation & Objective

• To establish that the vacuum Einstein equations in target spacetime emerge from worldsheet scaling symmetry of quantum p-adic strings.
• To generalize the Zabrodin p-adic string action to curved target spacetime using a distance-squared formulation.
• To analyze the role of spherical vectors of unramified principal series representations of PGL(2, Qp) as plane wave modes on the p-adic worldsheet.
• To show that loop diagrams in p-adic string theory simplify due to unique normalization and regularization features.
• To explore the connection between the adelic string spectrum and the nontrivial zeros of the Riemann zeta function under a product relation assumption.

Proposed method

• Propose a new p-adic string action S = Σ⟨ij⟩∈E(Tp) d²(Xi,Xj)/a²⟨ij⟩, generalizing the Zabrodin action to curved target spacetime.
• Identify momentum eigenfunctions on the Bruhat-Tits tree Tp as spherical vectors of unramified principal series representations of PGL(2, Qp).
• Use analytic regularization of geometric sums on the tree, including zeta function regularization for divergent series like ∑1 = ζ(0) = −1/2.
• Apply worldsheet scaling symmetry to derive the one-loop condition that enforces the vacuum Einstein equations.
• Compute the tree-level and one-loop two-point function using p-adic worldsheet path integrals with careful normalization of modes.
• Demonstrate vanishing of real line contributions to the two-point function using regularization in Definition 8, ensuring consistency with p-adic dominance.

Experimental results

Research questions

• RQ1 Does worldsheet scaling symmetry of quantum p-adic strings in curved target spacetime lead to the vacuum Einstein equations?
• RQ2 How do spherical vectors of unramified principal series representations of PGL(2, Qp) serve as plane wave modes on the p-adic worldsheet?
• RQ3 What is the role of regularization in the normalization of p-adic string modes, and how does it simplify loop computations?
• RQ4 Can the adelic spectrum of the bosonic string be related to the nontrivial zeros of the Riemann zeta function under a product relation?
• RQ5 Why do contributions from the real line vanish in the two-point function path integral under the proposed regularization scheme?

Key findings

• The vacuum Einstein equations in the target spacetime are derived from worldsheet scaling symmetry of quantum p-adic strings, analogous to the Archimedean case.
• Spherical vectors of unramified principal series representations of PGL(2, Qp) are identified as the plane wave modes of the p-adic string field theory.
• The regularization of mode normalization on the p-adic worldsheet leads to simplifications in loop diagrams, reducing complexity compared to standard QFT.
• The two-point function’s one-loop correction vanishes under the scaling symmetry condition, implying no metric running and thus the Einstein equations.
• Under a product relation assumption, the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann zeta function.
• Contributions from the real line to the two-point function path integral vanish due to cancellation under the proposed regularization, ensuring p-adic dominance.