Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

Abstract

We consider specific quantum mechanical model problems for which perturbation theory fails to explain physical properties like the eigenvalue spectrum even qualitatively, even if the asymptotic perturbation series is augmented by resummation prescriptions to "cure" the divergence in large orders of perturbation theory. Generalizations of perturbation theory are necessary which include instanton configurations, characterized by nonanalytic factors exp(−a/g) where a is a constant and g is the coupling. In the case of one-dimensional quantum mechanical potentials with two or more degenerate minima, the energy levels may be represented as an infinite sum of terms each of which involves a certain power of a nonanalytic factor and represents itself an infinite divergent series. We attempt to provide a unified representation of related derivations previously found scattered in the literature. For the considered quantum mechanical problems, we discuss the derivation of the instanton contributions from a semi-classical calculation of the corresponding partition function in the path integral formalism. We also explain the relation with the corresponding WKB expansion of the solutions of the Schrödinger equation, or alternatively of the Fredholm determinant det(H − E) (and some explicit calculations that verify this correspondence). We finally recall how these conjectures naturally emerge from a leading-order summation of multi-instanton contributions to the path integral representation of the partition function. The same strategy could result in new conjectures for problems where our present understanding is more limited.

Exact quantization conditions for the relativistic Toda lattice

“…However, one has to be careful, since quantization conditions often require nonperturbative corrections, beyond the all-orders WKB solution. A well-known example is the double-well potential in Quantum Mechanics, where the all-orders WKB quantization condition requires instanton corrections (see for example [13]). A subtler example is the pure quartic potential, in which one has to consider complex instantons [14,15].…”

Quantum geometry of resurgent perturbative/nonperturbative relations

“…Two distinct types of resurgent behavior have been identified in quantum spectral problems. The first is a generic type of "large order/low order" form of resurgence [6][7][8][9][10][11][12], whereby the large order growth of the perturbative coefficients of fluctuations about a given non-perturbative sector is related to the low-order perturbative coefficients of fluctuations about other non-perturbative sectors. This resurgent structure encodes an intricate network of relations between different non-perturbative sectors, and reflects to a surprising degree the generic resurgent structure of the all-orders steepest descents analysis of ordinary exponential integrals [13,14], and indeed the general resurgent structure of real trans-series [15].…”

Neutral bions in the ℂ $$ \mathbb{C} $$ P N −1 model

“…On the other side, neutral bions (zero topological charge and zero magnetic charge) can be identified as the infrared renormalon [10][11][12][13][14][15][16][17][18][19][41][42][43]. Here imaginary ambiguities arising in bion's amplitude and those arising in non-Borelsummable perturbative series cancel against each other, and it is expected that full semiclassical expansion including perturbative and non-perturbative sectors, which is called "resurgent" expansion [44], leads to unambiguous and self-consistent definition of field theories in the same manner as the Bogomolny-Zinn-Justin (BZJ) prescription in quantum mechanics [45][46][47]. However, it is not straightforward to verify these arguments in gauge theories directly, since it is difficult to find an explicit ansatz of bion configurations.…”