Cracking the Code of Quantum Mechanics

Scientists have made tremendous progress in understanding the laws that govern the microscopic world of quantum mechanics. However, one enduring mystery has been how to reconcile quantum theory’s perfectly smooth mathematical description with the peculiar discretized, “quantized” energy levels that atoms and molecules can truly possess. A new study published in SciPost Physics finally cracks this code by developing a unified mathematical framework called a “transseries” that captures both the quantized energy levels and the tiny fluctuations—known as instantons—between them.

Quantum mechanics famously dictates that the possible energies an electron can have within an atom are restricted to specific quantized values rather than being able to vary smoothly. This quantization arises from the wave-like nature of matter at the quantum scale. However, within the framework of perturbation theory—which calculates corrections to energies order-by-order based on the strength of quantum fluctuations—the energies appear as a continuous spectrum. Physicists have long suspected that buried within perturbation theory’s results are clues about the true quantized spectrum and instanton effects that cause discrete jumps between energy levels.

Two Dutch physicists, Alexander van Spaendonck and Marcel Vonk, have now sorted out this mysterious connection using powerful mathematical tools originally developed to study complex problems in physics and nonlinear equations. Their approach combines new techniques in “resurgence theory,” which relates divergences in perturbation theory to non-perturbative effects, with “alien calculus,” a language for quantifying subtle discontinuities known as Stokes phenomena that arise when resumming divergent series.

Van Spaendonck and Vonk applied their framework to three archetypal quantum systems: the simple harmonic oscillator subject to a cubic potential, a double-well potential, and a periodic cosine potential. For each, they derived exact expressions capturing the quantized energy levels to all perturbation orders alongside contributions from quantum tunneling between wells. Perhaps their most profound insight is that all this information naturally assembles into a single mathematical object called a “transseries.”

A transseries expands the energy not just as a simple perturbation series but as a combination of series joined together with exponential non-perturbative corrections like e−E/ħ, where E is the energy and ħ is the reduced Planck constant. It represents a minimal structure closed under analytic continuation, ensuring any unphysical discontinuities arising from resummation cancel out. Van Spaendonck and Vonk showed that within a transseries, all information about discrete quantization, tunneling amplitudes between states, and subtle Stokes discontinuities is systematically encoded.

“Using our description Stokes’ phenomenon, we were able to show that certain ambiguities that had plagued the ‘classical’ methods of computing nonperturbative effects — infinitely many, in fact — all dropped out in our method. The underlying structure turned out to be even more beautiful than we originally expected”

Alexander van Spaendonck

To elucidate the transseries structure, they introduced a single parameter σ and formulated a generic expression for the energy in terms of it. All model-specific details are then compressed into coefficients that can be computed explicitly. They extracted the quantization condition and tunneling information by studying how σ transforms across Stokes lines in the complex plane—phenomena dictated by the strange derivative operators of “alien calculus.”

Some universal lessons emerged. In double-well systems, the energy transseries factorizes into a “minimal transseries” plus an auxiliary “median transseries.” And for the periodic cosine potential, they identified topological sectors in the parameter space where quantized energy bands reside. Instanton effects depend on a continuous angular variable θ within each band, linking the study to notions of “resurgence triangles” where topology manifests.

This tour de force establishes quantum mechanical systems as ideal sandboxes for studying resurgence and its role in connecting perturbative and non-perturbative physics. Van Spaendonck and Vonk’s unified transseries framework finally closes the loop between Dirac’s original vision of quantization and subsequent developments in instanton calculus and asymptotic analysis. It provides a Rosetta stone for translating between different mathematical perspectives on quantum phenomena. With further development, their ideas could shed light on everything from condensed matter to quantum field theory and even quantum gravity. By cracking the quantum code, this work brings us closer to a complete solution to the riddle of quantization itself.

Reference(s)

1. Alexander van Spaendonck, Marcel Vonk. Exact instanton transseries for quantum mechanics. SciPost Physics, 2024; 16 (4) DOI: 10.21468/SciPostPhys.16.4.103

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