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

1. Problem Statement

The paper addresses a fundamental question in quantum mechanics and quantum field theory (QFT): the structure of the vacuum energy in systems where the standard perturbative expansion (Power Series in the coupling constant $g$) vanishes entirely.

• Context: In standard quantum systems (e.g., double-well potentials, $\sin$-Gordon), physical observables are typically described by trans-series. These combine a divergent perturbative series (PS) with non-perturbative terms of the form $\exp(-\text{const}/g^2)$.
• The Specific Challenge: The author investigates a specific 1D quantum mechanical Hamiltonian proposed by A. Turbiner. The goal is to prove that for this specific potential, the perturbative series coefficients are exactly zero to all orders, yet the vacuum energy remains non-zero. This implies the vacuum is purely non-perturbative, requiring an explanation via semiclassical paths that are not real-valued.

2. Methodology

The paper employs a multi-faceted approach combining analytical perturbation theory, Riccati equation analysis, and numerical semiclassical methods.

A. The Model Hamiltonian

The study focuses on the Hamiltonian:
$$H = (p^2 + x^2 - 1) + (a^2x^4 + 2ax^3 - 2ax)$$
where $a$ is the coupling constant. The potential is constructed such that the $O(a)$ terms involve mixed powers of $x$ ($x^3 - x$), preventing standard coordinate redefinitions from absorbing the coupling.

B. Analytical Proofs of Vanishing Perturbation

The author demonstrates that the perturbative expansion of the ground state energy $E_0(a) = \sum a^n E_n$ vanishes ($E_n = 0$ for all $n$) using three distinct methods:

1. Standard Rayleigh-Schrödinger Perturbation Theory: Explicit calculation of the second-order correction ($O(a^2)$) shows a cancellation between the direct $a^2x^4$ term and the second-order contribution from the $2a(x^3-x)$ term.
2. Matrix Diagonalization: The Hamiltonian is represented as an infinite matrix in the harmonic oscillator basis. Numerical evaluation of the determinant $\det(H)$ for truncated matrices (up to $12 \times 12$ and beyond) reveals that the lowest non-zero terms appear at $O(a^{10})$ or higher, suggesting cancellations persist.
3. Riccati Equation Analysis: By transforming the Schrödinger equation into the Riccati form ($y^2 - y' = V - E$), the author constructs a recursive series for the energy and the wavefunction derivative. It is shown that if $E_1=0$ and $E_2=0$, the recursive relations force all higher-order coefficients $E_n$ and the corresponding wavefunction corrections to vanish identically.

C. Numerical Semiclassical Analysis (Complex Bions)

Since the perturbative series is zero, the vacuum energy must arise from non-perturbative effects. The author searches for classical solutions in the complex plane:

• Holomorphic Newton's Equation (HNE): The author solves the Euclidean equations of motion in the complex plane: $m \frac{d^2z}{dt^2} = \partial V(z)$.
• Complex Bions: Unlike real "bounce" solutions (which describe tunneling between real minima), the relevant solutions here start at the global maximum of the inverted potential, travel into the complex plane, hit a complex turning point, and return. These are termed "complex bions."
• Numerical Implementation: The solutions are found by shooting from the global maximum with various initial angles $\phi$ in the complex plane.

3. Key Results

A. Total Absence of Perturbative Series

The paper rigorously establishes that for the Turbiner potential, all perturbative coefficients $a^n$ are zero. This is a rare example of a system where the perturbative vacuum is exact, yet the physical vacuum is not.

B. Non-Perturbative Vacuum Energy

Numerical diagonalization of the Schrödinger equation confirms that the ground state energy $E_0(a)$ is non-zero.

• At small $a$, $E_0(a)$ is exponentially small, scaling as $\sim \exp(-S_{\text{eff}})$.
• This behavior is distinct from generic potentials where $E_0(a)$ would be dominated by the lowest order perturbative term (e.g., $O(a^2)$).

C. The Complex Bion Solution

The numerical search reveals a family of complex bion solutions parameterized by an initial angle $\phi$.

• Action Invariance: Despite the path shape changing with $\phi$, the total action $S_{\text{tot}}$ remains constant.
• Action Value: For $a=0.2$, the action is calculated as:
  $$S_{\text{tot}} \approx 12.9246 + i\pi$$
• Real Amplitude: Although the action has an imaginary part ($i\pi$), the semiclassical amplitude $\exp(-S)$ remains real because $\exp(-i\pi) = -1$. This ensures the physical contribution to the energy is real.
• Cauchy Theorem: The invariance of the action with respect to the path deformation is attributed to the holomorphic nature of the equation and Cauchy's theorem.

D. Quantitative Agreement

The vacuum energy is modeled as:
$$E_{\text{bions}}(a) = C_{\text{det}} \sqrt{\text{Re}(S)} \exp(-\text{Re}(S))$$
where $C_{\text{det}}$ is an unknown fluctuation determinant. By fitting $C_{\text{det}} \approx 4$, the theoretical prediction based on the complex bion action matches the numerically calculated vacuum energy from the Schrödinger equation with high qualitative accuracy.

4. Significance and Implications

• Purely Non-Perturbative Physics: The paper provides a concrete, solvable example of a quantum system where the physics is entirely non-perturbative. There is no "perturbative background" to expand around; the vacuum is generated solely by complex classical paths.
• Validation of Complex Semiclassics: It validates the "complex bion" concept introduced in previous works (e.g., by Behtash et al.) for a system without supersymmetry or fermions. It demonstrates that complex classical trajectories are not just mathematical artifacts but physically responsible for vacuum energy in specific potentials.
• Resurgence and Trans-Series: The work contributes to the understanding of resurgence relations. In standard trans-series, non-perturbative terms cancel the ambiguities of the divergent perturbative series. Here, since the perturbative series is identically zero, the non-perturbative sector stands alone, offering a clean testbed for resurgence theory.
• QFT Relevance: The author suggests that similar complex classical paths might play a crucial role in 4D Gauge Theories (QFT), potentially explaining phenomena like chiral symmetry breaking or mass generation where perturbative expansions fail or are insufficient.

Conclusion

Edward Shuryak demonstrates that a specific 1D quantum potential possesses a vacuum energy that is strictly non-perturbative. By proving the vanishing of all perturbative coefficients and numerically identifying a family of complex classical paths (complex bions) with a fixed action, the paper shows that these complex trajectories fully account for the observed vacuum energy. This bridges the gap between exact numerical solutions and semiclassical approximations in regimes where standard perturbation theory is completely absent.