The $φ^n$ trajectory bootstrap

• URL: http://arxiv.org/abs/2402.05778v3
• Date: Tue, 29 Oct 2024 01:03:59 GMT
• Title: The $φ^n$ trajectory bootstrap
• Authors: Wenliang Li,
• Abstract summary: We show that the non-integer $n$ results for $langlephinrangle$ or $langle(iphi)nrangle$ are consistent with those from the wave function approach. In the $mathcalPT$ invariant case, the existence of $langle(iphi)nrangle$ with non-integer $n$ allows us to bootstrap the non-Hermitian theories with non-integer powers.
• Score: 1.8855270809505869
• License:
• Abstract: We perform an extensive bootstrap study of Hermitian and non-Hermitian theories based on the novel analytic continuation of $\langle\phi^n\rangle$ or $\langle(i\phi)^n\rangle$ in $n$. We first use the quantum harmonic oscillator to illustrate various aspects of the $\phi^n$ trajectory bootstrap method, such as the large $n$ expansion, matching conditions, exact quantization condition, and high energy asymptotic behavior. Then we derive highly accurate solutions for the anharmonic oscillators with the parity invariant potential $V(\phi)=\phi^2+\phi^{m}$ and the $\mathcal{PT}$ invariant potential $V(\phi)=-(i\phi)^{m}$ for a large range of integral $m$, showing the high efficiency and general applicability of this new bootstrap approach. For the Hermitian quartic and non-Hermitian cubic oscillators, we further verify that the non-integer $n$ results for $\langle\phi^n\rangle$ or $\langle(i\phi)^n\rangle$ are consistent with those from the wave function approach. In the $\mathcal{PT}$ invariant case, the existence of $\langle(i\phi)^n\rangle$ with non-integer $n$ allows us to bootstrap the non-Hermitian theories with non-integer powers, such as fractional and irrational $m$.

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