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From Random Walks to Random Matrices$
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Jean Zinn-Justin

Print publication date: 2019

Print ISBN-13: 9780198787754

Published to Oxford Scholarship Online: August 2019

DOI: 10.1093/oso/9780198787754.001.0001

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Periodic semi-classical vacuum, instantons and anomalies

Periodic semi-classical vacuum, instantons and anomalies

(p.319) 18 Periodic semi-classical vacuum, instantons and anomalies
From Random Walks to Random Matrices

Jean Zinn-Justin

Oxford University Press

Chapter 18 describes a few systems where the classical action has an infinite number of degenerate minima but, in the quantum theory, this degeneracy is lifted by barrier penetration effects. The simplest example is the cosine periodic potential and leads to the band structure. Technically, this corresponds to the existence of instantons, solutions to classical equations in imaginary time. In all examples, we show that the classical solutions are constrained by Bogomolnyi’s inequalities, which involve topological charges associated to a winding number and defining homotopy classes. In the case of quantum chromodynamics, this leads to the famous strong CP violation problem.

Keywords:   band structure, barrier penetration, cosine potential, Bogomolnyi’s inequality, instanton, homotopy class, topological charge, winding number, strong CP problem

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