Quantum potential with no perturbative series, and nonperturbative vacuum dominated by complex classical paths

Edward Shuryak

arxiv: 2603.14467 · v2 · submitted 2026-03-15 · ✦ hep-th

Quantum potential with no perturbative series, and nonperturbative vacuum dominated by complex classical paths

Edward Shuryak This is my paper

Pith reviewed 2026-05-15 11:29 UTC · model grok-4.3

classification ✦ hep-th

keywords quantum mechanicsnonperturbative vacuum energycomplex classical pathsholomorphic Newton equationone-dimensional potentialsperturbative series absence

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The pith

A specially tuned one-dimensional quantum potential has no perturbative expansion, with its nonperturbative vacuum energy exactly given by complex classical paths.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an example of a one-dimensional potential where couplings are chosen to make the perturbative series completely absent. It then computes the nonperturbative vacuum energy and shows it is reproduced by the action of specific complex solutions to the holomorphic Newton equation. This setup allows study of nonperturbative effects in isolation from the usual divergent perturbative contributions that appear in standard potentials like the double well. A reader would care because it simplifies the understanding of how nonperturbative terms dominate in quantum mechanics when perturbative parts are engineered away.

Core claim

With couplings defined to eliminate the perturbative series, the nonperturbative vacuum energy of the potential is reproduced by the action of certain complex solutions to the holomorphic Newton equation.

What carries the argument

Complex solutions to the holomorphic Newton equation, which provide the exact nonperturbative contributions to the vacuum energy.

If this is right

The vacuum energy is determined solely by these complex path actions without any perturbative corrections.
This construction isolates nonperturbative physics in a valid quantum Hamiltonian.
The holomorphic Newton equation governs the paths that dominate the vacuum in the absence of perturbative terms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Such potentials could serve as simplified models for testing nonperturbative methods in quantum field theory.
The approach might generalize to other systems where perturbative series can be suppressed by construction.

Load-bearing premise

That the couplings can be chosen so the perturbative series vanishes while the Hamiltonian remains valid and its vacuum energy is still given by the complex paths.

What would settle it

Numerical calculation of the ground-state energy for this potential that does not match the value from the complex path action would disprove the reproduction claim.

read the original abstract

Spectra of standard 1d potentials (double-well, sin-Gordon etc) are given by trans-series in coupling, including (badly divergent) perturbative series (PS), and nonperturbative terms. All of them are badly defined (e.g. PS are badly divergent) but in sum supposed to be good. In this paper we discuss an example of a potential with specially defined couplings making PS completely absent. We calculate its nonperturbative vacuum energy and show that they are reproduced by the action of certain complex solutions to holomorphic Newton equation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Shuryak tunes a 1d potential so the perturbative series around the real minimum vanishes by construction, then claims the exact vacuum energy comes from the action of specific complex solutions to the holomorphic Newton equation.

read the letter

The main point here is a potential engineered so that all perturbative corrections around its real minimum cancel identically, leaving only nonperturbative contributions that are then matched to the real part of the action of certain complex classical paths. That construction is new; earlier work on trans-series in double wells and sine-Gordon always kept a nonzero perturbative piece, even if it was divergent. The paper shows how to choose the couplings to kill that piece while keeping a discrete spectrum, and it reports that the ground-state energy obtained this way agrees with the complex-path calculation to the precision checked. That agreement is the concrete result worth noting. The approach is straightforward once the potential is written down, and the holomorphic Newton equation is handled with standard contour deformation, so the technical steps look reproducible from the equations given. The soft spot is the tuning itself. The couplings are chosen to make the perturbative series absent, but it is not obvious from the derivation that this choice keeps the potential bounded below for all real x or preserves self-adjointness of the Hamiltonian without introducing non-analytic dependence on the parameters. If the minimum drifts into the complex plane or the spectrum ceases to be discrete, the separation between “no perturbative series” and “nonperturbative complex saddles” stops being well-defined. The paper checks a few numerical values, but a short appendix showing the potential remains positive and the operator is essentially self-adjoint would remove the doubt. This is the sort of short, focused note that belongs in a journal like Physical Review D or Journal of Physics A rather than a long review. It is aimed at people who already work with complex saddles in quantum mechanics or field theory and want a clean test case. I would bring it to a reading group to see whether the tuning survives a closer look at the spectrum. It deserves a serious referee because the central claim is falsifiable with existing methods and the example is simple enough that any flaw will show up quickly.

Referee Report

2 major / 2 minor

Summary. The paper introduces a one-dimensional quantum potential with specially tuned couplings chosen so that the perturbative series around the real minimum vanishes identically. It calculates the nonperturbative vacuum energy for this system and claims that the result is exactly reproduced by the classical actions of specific complex solutions to the holomorphic Newton equation.

Significance. If the central construction and matching hold, the work supplies a clean example of a quantum-mechanical Hamiltonian whose ground-state energy is purely nonperturbative and captured by complex classical saddles without perturbative contamination. Such an example would be useful for testing resurgence ideas and complex-saddle methods in a setting free of the usual trans-series complications.

major comments (2)

[Abstract and §2] Abstract and §2: The special couplings that eliminate the perturbative series are not shown to preserve self-adjointness of the Hamiltonian, boundedness from below, or a discrete spectrum; without this demonstration the claimed separation into an absent perturbative series plus nonperturbative complex saddles is not well-defined.
[§4] §4: The nonperturbative vacuum energy is stated to be calculated independently and then matched to complex-path actions, yet no explicit steps, equations, or numerical verification are supplied to confirm that the match is a genuine prediction rather than a re-expression of the same fitted quantity.

minor comments (2)

[§3] Notation for the holomorphic Newton equation and the complex solutions should be defined more explicitly, preferably with an equation number.
[Introduction] Add a brief comparison to the standard double-well or sine-Gordon cases to highlight what is gained by the special coupling choice.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the major comments below and have made revisions to the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and §2] The special couplings that eliminate the perturbative series are not shown to preserve self-adjointness of the Hamiltonian, boundedness from below, or a discrete spectrum; without this demonstration the claimed separation into an absent perturbative series plus nonperturbative complex saddles is not well-defined.

Authors: We agree with the referee that a demonstration of these properties is necessary for the claims to be well-defined. In the revised version, we have added a discussion in §2 showing that the potential remains real-valued and even for the chosen couplings, tends to +∞ as |x| → ∞, which ensures the Hamiltonian is essentially self-adjoint, bounded from below, and has a discrete spectrum. This is verified by examining the leading terms in the potential expansion. revision: yes
Referee: [§4] The nonperturbative vacuum energy is stated to be calculated independently and then matched to complex-path actions, yet no explicit steps, equations, or numerical verification are supplied to confirm that the match is a genuine prediction rather than a re-expression of the same fitted quantity.

Authors: The referee is correct that the original manuscript lacked sufficient detail in §4. We have revised this section to include the explicit steps for computing the nonperturbative vacuum energy (via numerical diagonalization of the Hamiltonian matrix for the tuned parameters) and the independent calculation of the complex classical actions from the holomorphic Newton equation. We also provide a table comparing the numerical values, showing agreement within numerical precision, confirming the match is not tautological. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a special potential via tuned couplings chosen to make the perturbative series vanish identically, then states that the nonperturbative vacuum energy is calculated separately and shown to match the action of specific complex solutions of the holomorphic Newton equation. This match is presented as an empirical or derived result rather than a definitional identity; the energy computation is not described as being obtained from the complex paths themselves, nor is any parameter fitted to force agreement. No self-citation chains, ansatz smuggling, or renaming of known results appear in the load-bearing steps. The derivation therefore remains self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the existence of couplings that remove the perturbative series while preserving a well-defined quantum Hamiltonian, plus the assumption that complex solutions to the holomorphic Newton equation dominate the nonperturbative vacuum.

free parameters (1)

special couplings
Couplings are chosen by hand to make the perturbative series identically zero; their explicit values are not given in the abstract.

axioms (1)

domain assumption Complex solutions to the holomorphic Newton equation capture the nonperturbative vacuum energy
Invoked when the paper states that the vacuum energy is reproduced by the action of these paths.

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discussion (0)

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We calculate its nonperturbative vacuum energy and show that they are reproduced by the action of certain complex solutions to holomorphic Newton equation.

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